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Thou shalt not sit \n", "With statisticians nor commit\"\n", "\n", "> -- W.H. Auden [^note1]\n", "\n", "## On the psychology of statistics\n", "\n", "To the surprise of many students, statistics is a fairly significant part of a psychological education. To the surprise of no-one, statistics is very rarely the *favourite* part of one's psychological education. After all, if you really loved the idea of doing statistics, you'd probably be enrolled in a statistics class right now, not a psychology class. So, not surprisingly, there's a pretty large proportion of the student base that isn't happy about the fact that psychology has so much statistics in it. In view of this, I thought that the right place to start might be to answer some of the more common questions that people have about stats...\n", "\n", "A big part of this issue at hand relates to the very idea of statistics. What is it? What's it there for? And why are scientists so bloody obsessed with it? These are all good questions, when you think about it. So let's start with the last one. As a group, scientists seem to be bizarrely fixated on running statistical tests on everything. In fact, we use statistics so often that we sometimes forget to explain to people why we do. It's a kind of article of faith among scientists -- and especially social scientists -- that your findings can't be trusted until you've done some stats. Undergraduate students might be forgiven for thinking that we're all completely mad, because no-one takes the time to answer one very simple question:\n", "\n", ">*Why do you do statistics? Why don't scientists just use **common sense?***\n", "\n", "It's a naive question in some ways, but most good questions are. There's a lot of good answers to it, [^note2] but for my money, the best answer is a really simple one: we don't trust ourselves enough. We worry that we're human, and susceptible to all of the biases, temptations and frailties that humans suffer from. Much of statistics is basically a safeguard. Using \"common sense\" to evaluate evidence means trusting gut instincts, relying on verbal arguments and on using the raw power of human reason to come up with the right answer. Most scientists don't think this approach is likely to work.\n", "\n", "In fact, come to think of it, this sounds a lot like a psychological question to me, and since I do work in a psychology department, it seems like a good idea to dig a little deeper here. Is it really plausible to think that this \"common sense\" approach is very trustworthy? Verbal arguments have to be constructed in language, and all languages have biases -- some things are harder to say than others, and not necessarily because they're false (e.g., quantum electrodynamics is a good theory, but hard to explain in words). The instincts of our \"gut\" aren't designed to solve scientific problems, they're designed to handle day to day inferences -- and given that biological evolution is slower than cultural change, we should say that they're designed to solve the day to day problems for a *different world* than the one we live in. Most fundamentally, reasoning sensibly requires people to engage in \"induction\", making wise guesses and going beyond the immediate evidence of the senses to make generalisations about the world. If you think that you can do that without being influenced by various distractors, well, I have a bridge in Brooklyn I'd like to sell you. Heck, as the next section shows, we can't even solve \"deductive\" problems (ones where no guessing is required) without being influenced by our pre-existing biases.\n", "\n", "\n", "### The curse of belief bias\n", "\n", "People are mostly pretty smart. We're certainly smarter than the other species that we share the planet with (though many people might disagree). Our minds are quite amazing things, and we seem to be capable of the most incredible feats of thought and reason. That doesn't make us perfect though. And among the many things that psychologists have shown over the years is that we really do find it hard to be neutral, to evaluate evidence impartially and without being swayed by pre-existing biases. A good example of this is the ***belief bias effect*** in logical reasoning: if you ask people to decide whether a particular argument is logically valid (i.e., conclusion would be true if the premises were true), we tend to be influenced by the believability of the conclusion, even when we shouldn't. For instance, here's a valid argument where the conclusion is believable:\n", "\n", ">No cigarettes are inexpensive (Premise 1) \\\n", ">Some addictive things are inexpensive (Premise 2)\\\n", ">Therefore, some addictive things are not cigarettes (Conclusion)\n", "\n", "And here's a valid argument where the conclusion is not believable:\n", "\n", ">No addictive things are inexpensive (Premise 1)\\\n", ">Some cigarettes are inexpensive (Premise 2)\\\n", ">Therefore, some cigarettes are not addictive (Conclusion)\n", "\n", "The logical *structure* of argument \\#2 is identical to the structure of argument \\#1, and they're both valid. However, in the second argument, there are good reasons to think that premise 1 is incorrect, and as a result it's probably the case that the conclusion is also incorrect. But that's entirely irrelevant to the topic at hand: an argument is deductively valid if the conclusion is a logical consequence of the premises. That is, a valid argument doesn't have to involve true statements.\n", "\n", "On the other hand, here's an invalid argument that has a believable conclusion:\n", "\n", ">No addictive things are inexpensive (Premise 1)\\\n", ">Some cigarettes are inexpensive (Premise 2)\\\n", ">Therefore, some addictive things are not cigarettes (Conclusion)\n", "\n", "And finally, an invalid argument with an unbelievable conclusion:\n", "\n", ">No cigarettes are inexpensive (Premise 1)\\\n", ">Some addictive things are inexpensive (Premise 2)\\\n", ">Therefore, some cigarettes are not addictive (Conclusion)\n", "\n", "Now, suppose that people really are perfectly able to set aside their pre-existing biases about what is true and what isn't, and purely evaluate an argument on its logical merits. We'd expect 100% of people to say that the valid arguments are valid, and 0% of people to say that the invalid arguments are valid. So if you ran an experiment looking at this, you'd expect to see data like this:\n", "\n", "\n", "| | conclusion feels true| conclusion feels false |\n", "|------------------ |:--------------------:|:----------------------:|\n", "|argument is valid |100% say \"valid\" |100% say \"valid\" |\n", "|argument is invalid|0% say \"valid\" |0% say \"valid\" |\n", "\n", "If the psychological data looked like this (or even a good approximation to this), we might feel safe in just trusting our gut instincts. That is, it'd be perfectly okay just to let scientists evaluate data based on their common sense, and not bother with all this murky statistics stuff. However, you guys have taken psych classes, and by now you probably know where this is going...\n", "\n", "In a classic study, {cite:p}`Evans1983` ran an experiment looking at exactly this. What they found is that when pre-existing biases (i.e., beliefs) were in agreement with the structure of the data, everything went the way you'd hope: \n", "\n", "\n", "| | conclusion feels true| conclusion feels false |\n", "|------------------ |:--------------------:|:----------------------:|\n", "|argument is valid |92% say \"valid\" | |\n", "|argument is invalid| |8% say \"valid\" |\n", "\n", "Not perfect, but that's pretty good. But look what happens when our intuitive feelings about the truth of the conclusion run against the logical structure of the argument:\n", "\n", "| | conclusion feels true| conclusion feels false |\n", "|------------------ |:--------------------:|:----------------------:|\n", "|argument is valid |92% say \"valid\" |**46% say \"valid\"** |\n", "|argument is invalid|**92% say \"valid\"** |8% say \"valid\" |\n", "\n", "Oh dear, that's not as good. Apparently, when people are presented with a strong argument that contradicts our pre-existing beliefs, we find it pretty hard to even perceive it to be a strong argument (people only did so 46% of the time). Even worse, when people are presented with a weak argument that agrees with our pre-existing biases, almost no-one can see that the argument is weak (people got that one wrong 92% of the time!)[^note3]\n", "\n", "If you think about it, it's not as if these data are horribly damning. Overall, people did do better than chance at compensating for their prior biases, since about 60% of people's judgements were correct (you'd expect 50% by chance). Even so, if you were a professional \"evaluator of evidence\", and someone came along and offered you a magic tool that improves your chances of making the right decision from 60% to (say) 95%, you'd probably jump at it, right? Of course you would. Thankfully, we actually do have a tool that can do this. But it's not magic, it's statistics. So that's reason \\#1 why scientists love statistics. It's just *too easy* for us to \"believe what we want to believe\"; so if we want to \"believe in the data\" instead, we're going to need a bit of help to keep our personal biases under control. That's what statistics does: it helps keep us honest.\n", "\n", "\n", "\n", "## The cautionary tale of Simpson's paradox\n", "\n", "The following is a true story (I think...). In 1973, the University of California, Berkeley had some worries about the admissions of students into their postgraduate courses. Specifically, the thing that caused the problem was the gender breakdown of their admissions, which looked like this...\n", "\n", "| | Number of applicants | Percent admitted |\n", "|-------|:--------------------:|:----------------:|\n", "|Males |8442 |44% |\n", "|Females|4321 |35% |\n", "\n", "...and they were worried about being sued. [^note4] Given that there were nearly 13,000 applicants, a difference of 9% in admission rates between males and females is just way too big to be a coincidence. Pretty compelling data, right? And if I were to say to you that these data *actually* reflect a weak bias in favour of women (sort of!), you'd probably think that I was either crazy or sexist. \n", "\n", "Oddly, it's actually sort of true ...when people started looking more carefully at the admissions data they told a rather different story {cite:p}`Bickel1975`. Specifically, when they looked at it on a department by department basis, it turned out that most of the departments actually had a slightly *higher* success rate for female applicants than for male applicants. {numref}`fig-berkleytable` shows the admission figures for the six largest departments (with the names of the departments removed for privacy reasons):\n", "\n", "[^note1]: The quote comes from Auden's 1946 poem *Under Which Lyre: A Reactionary Tract for the Times*, delivered as part of a commencement address at Harvard University. The history of the poem is kind of interesting: http://harvardmagazine.com/2007/11/a-poets-warning.html\n", "\n", "[^note2]: Including the suggestion that common sense is in short supply among scientists.\n", "\n", "[^note3]: In my more cynical moments I feel like this fact alone explains 95% of what I read on the internet.\n", "\n", "[^note4]: Earlier versions of these notes incorrectly suggested that they actually were sued -- apparently that's not true. There's a nice commentary on this here: https://www.refsmmat.com/posts/2016-05-08-simpsons-paradox-berkeley.html. A big thank you to Wilfried Van Hirtum for pointing this out to me!" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/text/html": "

\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
DepartmentMale ApplicantsMale Percent AdmittedFemale ApplicantsFemale Percent Admitted
0A82562%10882%
1B56063%2568%
2C32537%59334%
3D41733%37535%
4E19128%39324%
5F2726%3417%
\n
", "application/papermill.record/text/plain": " Department Male Applicants Male Percent Admitted Female Applicants \\\n0 A 825 62% 108 \n1 B 560 63% 25 \n2 C 325 37% 593 \n3 D 417 33% 375 \n4 E 191 28% 393 \n5 F 272 6% 341 \n\n Female Percent Admitted \n0 82% \n1 68% \n2 34% \n3 35% \n4 24% \n5 7% " }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "berkley-table" } }, "output_type": "display_data" } ], "source": [ "from myst_nb import glue\n", "import pandas as pd\n", "\n", "data = {'Department': ['A', 'B', 'C', 'D', 'E', 'F'],\n", " 'Male Applicants': [825,560,325,417,191,272],\n", " 'Male Percent Admitted': ['62%','63%','37%','33%','28%','6%'],\n", " 'Female Applicants': [108,25,593,375,393,341],\n", " 'Female Percent Admitted': ['82%','68%','34%','35%','24%','7%']}\n", "\n", "df = pd.DataFrame(data)\n", "\n", "glue(\"berkley-table\", df, display=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} berkley-table\n", ":figwidth: 600px\n", ":name: fig-berkleytable\n", "\n", "Admission figures for the six largest departments by gender\n", "```" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Remarkably, most departments had a *higher* rate of admissions for females than for males! Yet the overall rate of admission across the university for females was *lower* than for males. How can this be? How can both of these statements be true at the same time?\n", "\n", "Here's what's going on. Firstly, notice that the departments are *not* equal to one another in terms of their admission percentages: some departments (e.g., engineering, chemistry) tended to admit a high percentage of the qualified applicants, whereas others (e.g., English) tended to reject most of the candidates, even if they were high quality. So, among the six departments shown above, notice that department A is the most generous, followed by B, C, D, E and F in that order. Next, notice that males and females tended to apply to different departments. If we rank the departments in terms of the total number of male applicants, we get **A**>**B**>D>C>F>E (the \"easy\" departments are in bold). On the whole, males tended to apply to the departments that had high admission rates. Now compare this to how the female applicants distributed themselves. Ranking the departments in terms of the total number of female applicants produces a quite different ordering C>E>D>F>**A**>**B**. In other words, what these data seem to be suggesting is that the female applicants tended to apply to \"harder\" departments. And in fact, if we look at the figure below, we see that this trend is systematic, and quite striking. This effect is known as Simpson's paradox. It's not common, but it does happen in real life, and most people are very surprised by it when they first encounter it, and many people refuse to even believe that it's real. It is very real. And while there are lots of very subtle statistical lessons buried in there, I want to use it to make a much more important point ...doing research is hard, and there are *lots* of subtle, counterintuitive traps lying in wait for the unwary. That's reason #2 why scientists love statistics, and why we teach research methods. Because science is hard, and the truth is sometimes cunningly hidden in the nooks and crannies of complicated data." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/text/plain": "" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "berkely-fig" } }, "output_type": "display_data" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "from myst_nb import glue\n", "import os\n", "import pandas as pd\n", "from pathlib import Path\n", "\n", "cwd = os.getcwd()\n", "os.chdir(str(Path(cwd).parents[0]) + '/Data')\n", "\n", "df_berkely = pd.read_csv('berkeley2.csv')\n", "\n", "\n", "import seaborn as sns\n", "\n", "# Plot Berkely data\n", "berkelyplot = sns.lmplot(\n", " data = df_berkely,\n", " x = \"women.apply\", y = \"total.admit\"\n", " #hue = \"depart.size\"\n", ")\n", "\n", "\n", "berkelyplot.set(xlabel =\"Pecentage of Female Applicants\", \n", " ylabel = \"Admission rate (both genders)\")\n", "\n", "glue(\"berkely-fig\", berkelyplot, display=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} berkely_fig\n", ":figwidth: 600px\n", ":name: fig-berkely\n", "\n", "The Berkeley 1973 college admissions data. This figure plots the admission rate for the 85 departments that had at least one female applicant, as a function of the percentage of applicants that were female. Based on data from {cite}`Bickel1975`.\n", "```" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Before leaving this topic entirely, I want to point out something else really critical that is often overlooked in a research methods class. Statistics only solves *part* of the problem. Remember that we started all this with the concern that Berkeley's admissions processes might be unfairly biased against female applicants. When we looked at the \"aggregated\" data, it did seem like the university was discriminating against women, but when we \"disaggregate\" and looked at the individual behaviour of all the departments, it turned out that the actual departments were, if anything, slightly biased in favour of women. The gender bias in total admissions was caused by the fact that women tended to self-select for harder departments. From a legal perspective, that would probably put the university in the clear. Postgraduate admissions are determined at the level of the individual department (and there are good reasons to do that), and at the level of individual departments, the decisions are more or less unbiased (the weak bias in favour of females at that level is small, and not consistent across departments). Since the university can't dictate which departments people choose to apply to, and the decision making takes place at the level of the department, it can hardly be held accountable for any biases that those choices produce. \n", "\n", "That was the basis for my somewhat glib remarks earlier, but that's not exactly the whole story, is it? After all, if we're interested in this from a more sociological and psychological perspective, we might want to ask *why* there are such strong gender differences in applications. Why do males tend to apply to engineering more often than females, and why is this reversed for the English department? And why is it it the case that the departments that tend to have a female-application bias tend to have lower overall admission rates than those departments that have a male-application bias? Might this not still reflect a gender bias, even though every single department is itself unbiased? It might. Suppose, hypothetically, that males preferred to apply to \"hard sciences\" and females prefer \"humanities\". And suppose further that the reason the humanities departments have low admission rates is because the government doesn't want to fund the humanities (Ph.D. places, for instance, are often tied to government funded research projects). Does that constitute a gender bias? Or just an unenlightened view of the value of the humanities? What if someone at a high level in the government cut the humanities funds because they felt that the humanities are \"useless chick stuff\". That seems pretty *blatantly* gender biased. None of this falls within the purview of statistics, but it matters to the research project. If you're interested in the overall structural effects of subtle gender biases, then you probably want to look at *both* the aggregated and disaggregated data. If you're interested in the decision making process at Berkeley itself then you're probably only interested in the disaggregated data. \n", "\n", "In short there are a lot of critical questions that you can't answer with statistics, but the answers to those questions will have a huge impact on how you analyse and interpret data. And this is the reason why you should always think of statistics as a *tool* to help you learn about your data, no more and no less. It's a powerful tool to that end, but there's no substitute for careful thought.\n", "\n", "\n", "## Statistics in psychology\n", "\n", "I hope that the discussion above helped explain why science in general is so focused on statistics. But I'm guessing that you have a lot more questions about what role statistics plays in psychology, and specifically why psychology classes always devote so many lectures to stats. So here's my attempt to answer a few of them...\n", "\n", "\n", "**Why does psychology have so much statistics?**\n", "\n", " To be perfectly honest, there's a few different reasons, some of which are better than others. The most important reason is that psychology is a statistical science. What I mean by that is that the \"things\" that we study are *people*. Real, complicated, gloriously messy, infuriatingly perverse people. The \"things\" of physics include objects like electrons, and while there are all sorts of complexities that arise in physics, electrons don't have minds of their own. They don't have opinions, they don't differ from each other in weird and arbitrary ways, they don't get bored in the middle of an experiment, and they don't get angry at the experimenter and then deliberately try to sabotage the data set (not that I've ever done that...). At a fundamental level, psychology is harder than physics. [^note5]\n", "\n", " Basically, we teach statistics to you as psychologists because you need to be better at stats than physicists. There's actually a saying used sometimes in physics, to the effect that \"if your experiment needs statistics, you should have done a better experiment\". They have the luxury of being able to say that because their objects of study are pathetically simple in comparison to the vast mess that confronts social scientists. It's not just psychology, really: most social sciences are desperately reliant on statistics. Not because we're bad experimenters, but because we've picked a harder problem to solve. We teach you stats because you really, really need it.\n", "\n", "**Can't someone else do the statistics?**\n", "\n", "To some extent, but not completely. It's true that you don't need to become a fully trained statistician just to do psychology, but you do need to reach a certain level of statistical competence. In my view, there's three reasons that every psychological researcher ought to be able to do basic statistics:\n", "\n", "\n", " - Firstly, there's the fundamental reason: statistics is deeply intertwined with research design. If you want to be good at designing psychological studies, you need to at least understand the basics of stats. \n", " - Secondly, if you want to be good at the psychological side of the research, then you need to be able to understand the psychological literature, right? But almost every paper in the psychological literature reports the results of statistical analyses. So if you really want to understand the psychology, you need to be able to understand what other people did with their data. And that means understanding a certain amount of statistics.\n", " - Thirdly, there's a big practical problem with being dependent on other people to do all your statistics: statistical analysis is *expensive*. If you ever get bored and want to look up how much the Australian government charges for university fees, you'll notice something interesting: statistics is designated as a \"national priority\" category, and so the fees are much, much lower than for any other area of study. This is because there's a massive shortage of statisticians out there. So, from your perspective as a psychological researcher, the laws of supply and demand aren't exactly on your side here! As a result, in almost any real life situation where you want to do psychological research, the cruel facts will be that you don't have enough money to afford a statistician. So the economics of the situation mean that you have to be pretty self-sufficient. \n", "\n", "Note that a lot of these reasons generalise beyond researchers. If you want to be a practicing psychologist and stay on top of the field, it helps to be able to read the scientific literature, which relies pretty heavily on statistics. \n", "\n", "**I don't care about jobs, research, or clinical work. Do I need statistics?**\n", "\n", "Okay, now you're just messing with me. Still, I think it should matter to you too. Statistics should matter to you in the same way that statistics should matter to *everyone*: we live in the 21st century, and data are *everywhere*. Frankly, given the world in which we live these days, a basic knowledge of statistics is pretty damn close to a survival tool! Which is the topic of the next section...\n", "\n", "\n", "\n", "## Statistics in everyday life\n", "\n", ">\"We are drowning in information, \n", "> but we are starved for knowledge\"\n", ">\n", ">-Various authors, original probably John Naisbitt\n", "\n", "\n", "When I started writing up my lecture notes I took the 20 most recent news articles posted to the ABC news website. Of those 20 articles, it turned out that 8 of them involved a discussion of something that I would call a statistical topic; 6 of those made a mistake. The most common error, if you're curious, was failing to report baseline data (e.g., the article mentions that 5% of people in situation X have some characteristic Y, but doesn't say how common the characteristic is for everyone else!) The point I'm trying to make here isn't that journalists are bad at statistics (though they almost always are), it's that a basic knowledge of statistics is very helpful for trying to figure out when someone else is either making a mistake or even lying to you. In fact, one of the biggest things that a knowledge of statistics does to you is cause you to get angry at the newspaper or the internet on a far more frequent basis: you can find a good example of this in [this story concerning housing prices in Australia](housingpriceexample). In later versions of this book I'll try to include more anecdotes along those lines. \n", "\n", "\n", "## There's more to research methods than statistics\n", "\n", "So far, most of what I've talked about is statistics, and so you'd be forgiven for thinking that statistics is all I care about in life. To be fair, you wouldn't be far wrong, but research methodology is a broader concept than statistics. So most research methods courses will cover a lot of topics that relate much more to the pragmatics of research design, and in particular the issues that you encounter when trying to do research with humans. However, about 99% of student *fears* relate to the statistics part of the course, so I've focused on the stats in this discussion, and hopefully I've convinced you that statistics matters, and more importantly, that it's not to be feared. That being said, it's pretty typical for introductory research methods classes to be very stats-heavy. This is not (usually) because the lecturers are evil people. Quite the contrary, in fact. Introductory classes focus a lot on the statistics because you almost always find yourself needing statistics before you need the other research methods training. Why? Because almost all of your assignments in other classes will rely on statistical training, to a much greater extent than they rely on other methodological tools. It's not common for undergraduate assignments to require you to design your own study from the ground up (in which case you would need to know a lot about research design), but it *is* common for assignments to ask you to analyse and interpret data that were collected in a study that someone else designed (in which case you need statistics). In that sense, from the perspective of allowing you to do well in all your other classes, the statistics is more urgent. \n", "\n", "But note that \"urgent\" is different from \"important\" -- they both matter. I really do want to stress that research design is just as important as data analysis, and this book does spend a fair amount of time on it. However, while statistics has a kind of universality, and provides a set of core tools that are useful for most types of psychological research, the research methods side isn't quite so universal. There are some general principles that everyone should think about, but a lot of research design is very idiosyncratic, and is specific to the area of research that you want to engage in. To the extent that it's the details that matter, those details don't usually show up in an introductory stats and research methods class.\n", "\n", "[^note5]: Which might explain why physics is just a teensy bit further advanced as a science than we are." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "celltoolbar": "Tags", "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.5" } }, "nbformat": 4, "nbformat_minor": 4 } { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "(studydesign)=\n", "# A brief introduction to research design\n", "\n", ">*To consult the statistician after an experiment is finished is often merely to ask him to conduct a post mortem examination. He can perhaps say what the experiment died of.*\n", ">\n", ">-- Sir Ronald Fisher [^note1]\n", "\n", "\n", "In this chapter, we're going to start thinking about the basic ideas that go into designing a study, collecting data, checking whether your data collection works, and so on. It won't give you enough information to allow you to design studies of your own, but it will give you a lot of the basic tools that you need to assess the studies done by other people. However, since the focus of this book is much more on data analysis than on data collection, I'm only giving a very brief overview. Note that this chapter is \"special\" in two ways. Firstly, it's much more psychology-specific than the later chapters. Secondly, it focuses much more heavily on the scientific problem of research methodology, and much less on the statistical problem of data analysis. Nevertheless, the two problems are related to one another, so it's traditional for stats textbooks to discuss the problem in a little detail. This chapter relies heavily on {cite}`Campbell1963` for the discussion of study design, and {cite}`Stevens1946` for the discussion of scales of measurement. Later versions will attempt to be more precise in the citations. \n", "\n", "(measurement)=\n", "## Introduction to psychological measurement\n", "\n", "The first thing to understand is data collection can be thought of as a kind of ***measurement***. That is, what we're trying to do here is measure something about human behaviour or the human mind. What do I mean by \"measurement\"? \n", "\n", "### Some thoughts about psychological measurement\n", "\n", "Measurement itself is a subtle concept, but basically it comes down to finding some way of assigning numbers, or labels, or some other kind of well-defined descriptions to \"stuff\". So, any of the following would count as a psychological measurement:\n", "\n", "\n", "- My **age** is *33 years*.\n", "- I *do not* **like anchovies**.\n", "- My **chromosomal gender** is *male*. \n", "- My **self-identified gender** is *male*. [^note2]\n", "\n", "\n", "In the short list above, the **bolded part** is \"the thing to be measured\", and the *italicised part* is \"the measurement itself\". In fact, we can expand on this a little bit, by thinking about the set of possible measurements that could have arisen in each case:\n", "\n", "\n", "- My **age** (in years) could have been *0, 1, 2, 3 ...*, etc. The upper bound on what my age could possibly be is a bit fuzzy, but in practice you'd be safe in saying that the largest possible age is *150*, since no human has ever lived that long.\n", "- When asked if I **like anchovies**, I might have said that *I do*, or *I do not*, or *I have no opinion*, or *I sometimes do*. \n", "- My **chromosomal gender** is almost certainly going to be *male (XY)* or *female (XX)*, but there are a few other possibilities. I could also have *Klinfelter's syndrome (XXY)*, which is more similar to male than to female. And I imagine there are other possibilities too.\n", "- My **self-identified gender** is also very likely to be *male* or *female*, but it doesn't have to agree with my chromosomal gender. I may also choose to identify with *neither*, or to explicitly call myself *transgender*.\n", "\n", "\n", "As you can see, for some things (like age) it seems fairly obvious what the set of possible measurements should be, whereas for other things it gets a bit tricky. But I want to point out that even in the case of someone's age, it's much more subtle than this. For instance, in the example above, I assumed that it was okay to measure age in years. But if you're a developmental psychologist, that's way too crude, and so you often measure age in *years and months* (if a child is 2 years and 11 months, this is usually written as \"2;11\"). If you're interested in newborns, you might want to measure age in *days since birth*, maybe even *hours since birth*. In other words, the way in which you specify the allowable measurement values is important. \n", "\n", "Looking at this a bit more closely, you might also realise that the concept of \"age\" isn't actually all that precise. In general, when we say \"age\" we implicitly mean \"the length of time since birth\". But that's not always the right way to do it. Suppose you're interested in how newborn babies control their eye movements. If you're interested in kids that young, you might also start to worry that \"birth\" is not the only meaningful point in time to care about. If Baby Alice is born 3 weeks premature and Baby Bianca is born 1 week late, would it really make sense to say that they are the \"same age\" if we encountered them \"2 hours after birth\"? In one sense, yes: by social convention, we use birth as our reference point for talking about age in everyday life, since it defines the amount of time the person has been operating as an independent entity in the world, but from a scientific perspective that's not the only thing we care about. When we think about the biology of human beings, it's often useful to think of ourselves as organisms that have been growing and maturing since conception, and from that perspective Alice and Bianca aren't the same age at all. So you might want to define the concept of \"age\" in two different ways: the length of time since conception, and the length of time since birth. When dealing with adults, it won't make much difference, but when dealing with newborns it might.\n", " \n", "Moving beyond these issues, there's the question of methodology. What specific \"measurement method\" are you going to use to find out someone's age? As before, there are lots of different possibilities:\n", "\n", "\n", "- You could just ask people \"how old are you?\" The method of self-report is fast, cheap and easy, but it only works with people old enough to understand the question, and some people lie about their age.\n", "- You could ask an authority (e.g., a parent) \"how old is your child?\" This method is fast, and when dealing with kids it's not all that hard since the parent is almost always around. It doesn't work as well if you want to know \"age since conception\", since a lot of parents can't say for sure when conception took place. For that, you might need a different authority (e.g., an obstetrician). \n", "- You could look up official records, like birth certificates. This is time consuming and annoying, but it has its uses (e.g., if the person is now dead). \n", "\n", "\n", "### Operationalisation: defining your measurement\n", "\n", "All of the ideas discussed in the previous section all relate to the concept of ***operationalisation***. To be a bit more precise about the idea, operationalisation is the process by which we take a meaningful but somewhat vague concept, and turn it into a precise measurement. The process of operationalisation can involve several different things:\n", "\n", "\n", "- Being precise about what you are trying to measure. For instance, does \"age\" mean \"time since birth\" or \"time since conception\" in the context of your research?\n", "- Determining what method you will use to measure it. Will you use self-report to measure age, ask a parent, or look up an official record? If you're using self-report, how will you phrase the question? \n", "- Defining the set of the allowable values that the measurement can take. Note that these values don't always have to be numerical, though they often are. When measuring age, the values are numerical, but we still need to think carefully about what numbers are allowed. Do we want age in years, years and months, days, hours? Etc. For other types of measurements (e.g., gender), the values aren't numerical. But, just as before, we need to think about what values are allowed. If we're asking people to self-report their gender, what options to we allow them to choose between? Is it enough to allow only \"male\" or \"female\"? Do you need an \"other\" option? Or should we not give people any specific options, and let them answer in their own words? And if you open up the set of possible values to include all verbal response, how will you interpret their answers?\n", "\n", " \n", "Operationalisation is a tricky business, and there's no \"one, true way\" to do it. The way in which you choose to operationalise the informal concept of \"age\" or \"gender\" into a formal measurement depends on what you need to use the measurement for. Often you'll find that the community of scientists who work in your area have some fairly well-established ideas for how to go about it. In other words, operationalisation needs to be thought through on a case by case basis. Nevertheless, while there a lot of issues that are specific to each individual research project, there are some aspects to it that are pretty general. \n", "\n", "Before moving on, I want to take a moment to clear up our terminology, and in the process introduce one more term. Here are four different things that are closely related to each other:\n", "\n", "\n", "- ***A theoretical construct***. This is the thing that you're trying to take a measurement of, like \"age\", \"gender\" or an \"opinion\". A theoretical construct can't be directly observed, and often they're actually a bit vague. \n", "- ***A measure***. The measure refers to the method or the tool that you use to make your observations. A question in a survey, a behavioural observation or a brain scan could all count as a measure. \n", "- ***An operationalisation***. The term \"operationalisation\" refers to the logical connection between the measure and the theoretical construct, or to the process by which we try to derive a measure from a theoretical construct.\n", "- ***A variable***. Finally, a new term. A variable is what we end up with when we apply our measure to something in the world. That is, variables are the actual \"data\" that we end up with in our data sets.\n", "\n", "\n", "\n", "In practice, even scientists tend to blur the distinction between these things, but it's very helpful to try to understand the differences.\n", "\n", "(scales)=\n", "## Scales of measurement\n", "\n", "As the previous section indicates, the outcome of a psychological measurement is called a variable. But not all variables are of the same qualitative type, and it's very useful to understand what types there are. A very useful concept for distinguishing between different types of variables is what's known as ***scales of measurement***. \n", "\n", "\n", "### Nominal scale\n", "\n", "A **_nominal scale_** variable (also referred to as a ***categorical*** variable) is one in which there is no particular relationship between the different possibilities: for these kinds of variables it doesn't make any sense to say that one of them is \"bigger' or \"better\" than any other one, and it absolutely doesn't make any sense to average them. The classic example for this is \"eye colour\". Eyes can be blue, green and brown, among other possibilities, but none of them is any \"better\" than any other one. As a result, it would feel really weird to talk about an \"average eye colour\". Similarly, gender is nominal too: male isn't better or worse than female, neither does it make sense to try to talk about an \"average gender\". In short, nominal scale variables are those for which the only thing you can say about the different possibilities is that they are different. That's it.\n", "\n", "Let's take a slightly closer look at this. Suppose I was doing research on how people commute to and from work. One variable I would have to measure would be what kind of transportation people use to get to work. This \"transport type\" variable could have quite a few possible values, including: \"train\", \"bus\", \"car\", \"bicycle\", etc. For now, let's suppose that these four are the only possibilities, and suppose that when I ask 100 people how they got to work today, and I get this:\n", " \n", "|Transportation|Number of people|\n", "|:-:|:-:|\n", "| (1) Train | 12|\n", "| (2) Bus | 30|\n", "| (3) Car | 48|\n", "| (4) Bicycle | 10|\n", " \n", "So, what's the average transportation type? Obviously, the answer here is that there isn't one. It's a silly question to ask. You can say that travel by car is the most popular method, and travel by bike is the least popular method, but that's about all. Similarly, notice that the order in which I list the options isn't very interesting. I could have chosen to display the data like this\n", " \n", "\n", " |Transportation|Number of people|\n", "|:-:|:-:|\n", "| (3) Car | 48|\n", "| (1) Train | 12|\n", "| (4) Bicycle | 10|\n", "| (2) Bus | 30|\n", "\n", "and nothing really changes.\n", "\n", "### Ordinal scale\n", "\n", "**_Ordinal scale_** variables have a bit more structure than nominal scale variables, but not by a lot. An ordinal scale variable is one in which there is a natural, meaningful way to order the different possibilities, but you can't do anything else. The usual example given of an ordinal variable is \"finishing position in a race\". You *can* say that the person who finished first was faster than the person who finished second, but you *don't* know how much faster. As a consequence we know that 1st > 2nd, and we know that 2nd > 3rd, but the difference between 1st and 2nd might be much larger than the difference between 2nd and 3rd.\n", "\n", "Here's an more psychologically interesting example. Suppose I'm interested in people's attitudes to climate change, and I ask them to pick one of these four statements that most closely matches their beliefs:\n", "\n", ">(1) Temperatures are rising, because of human activity\n", ">(2) Temperatures are rising, but we don't know why\n", ">(3) Temperatures are rising, but not because of humans\n", ">(4) Temperatures are not rising\n", "\n", "Notice that these four statements actually do have a natural ordering, in terms of \"the extent to which they agree with the current science\". Statement 1 is a close match, statement 2 is a reasonable match, statement 3 isn't a very good match, and statement 4 is in strong opposition to the science. So, in terms of the thing I'm interested in (the extent to which people endorse the science), I can order the items as 1 > 2 > 3 > 4. Since this ordering exists, it would be very weird to list the options like this...\n", "\n", ">(3) Temperatures are rising, but not because of humans\n", ">(1) Temperatures are rising, because of human activity\n", ">(4) Temperatures are not rising\n", ">(2) Temperatures are rising, but we don't know why \n", "\n", "... because it seems to violate the natural \"structure\" to the question. \n", "\n", "So, let's suppose I asked 100 people these questions, and got the following answers:\n", "\n", "|Response | Number|\n", "|-------- |:-----:|\n", "|(1) Temperatures are rising, because of human activity | 51 |\n", "|(2) Temperatures are rising, but we don't know why | 20 |\n", "|(3) Temperatures are rising, but not because of humans | 10 |\n", "|(4) Temperatures are not rising | 19 |\n", "\n", "When analysing these data, it seems quite reasonable to try to group (1), (2) and (3) together, and say that 81 of 100 people were willing to *at least partially* endorse the science. And it's *also* quite reasonable to group (2), (3) and (4) together and say that 49 of 100 people registered *at least some disagreement* with the dominant scientific view. However, it would be entirely bizarre to try to group (1), (2) and (4) together and say that 90 of 100 people said... what? There's nothing sensible that allows you to group those responses together at all.\n", "\n", "That said, notice that while we *can* use the natural ordering of these items to construct sensible groupings, what we *can't* do is average them. For instance, in my simple example here, the \"average\" response to the question is 1.97. If you can tell me what that means, I'd love to know. Because that sounds like gibberish to me!\n", "\n", "### Interval scale\n", "\n", "In contrast to nominal and ordinal scale variables, **_interval scale_** and ratio scale variables are variables for which the numerical value is genuinely meaningful. In the case of interval scale variables, the *differences* between the numbers are interpretable, but the variable doesn't have a \"natural\" zero value. A good example of an interval scale variable is measuring temperature in degrees celsius. For instance, if it was 15$^\\circ$ yesterday and 18$^\\circ$ today, then the 3$^\\circ$ difference between the two is genuinely meaningful. Moreover, that 3$^\\circ$ difference is *exactly the same* as the 3$^\\circ$ difference between 7$^\\circ$ and 10$^\\circ$. In short, addition and subtraction are meaningful for interval scale variables. [^note3]\n", "\n", "However, notice that the 0$^\\circ$ does not mean \"no temperature at all\": it actually means \"the temperature at which water freezes\", which is pretty arbitrary. As a consequence, it becomes pointless to try to multiply and divide temperatures. It is wrong to say that $20^\\circ$ is *twice as hot* as 10$^\\circ$, just as it is weird and meaningless to try to claim that 20$^\\circ$ is negative two times as hot as -10$^\\circ$. \n", "\n", "Again, lets look at a more psychological example. Suppose I'm interested in looking at how the attitudes of first-year university students have changed over time. Obviously, I'm going to want to record the year in which each student started. This is an interval scale variable. A student who started in 2003 did arrive 5 years before a student who started in 2008. However, it would be completely insane for me to divide 2008 by 2003 and say that the second student started \"1.0024 times later\" than the first one. That doesn't make any sense at all.\n", "\n", "### Ratio scale\n", "\n", "The fourth and final type of variable to consider is a ***ratio scale*** variable, in which zero really means zero, and it's okay to multiply and divide. A good psychological example of a ratio scale variable is response time (RT). In a lot of tasks it's very common to record the amount of time somebody takes to solve a problem or answer a question, because it's an indicator of how difficult the task is. Suppose that Alan takes 2.3 seconds to respond to a question, whereas Ben takes 3.1 seconds. As with an interval scale variable, addition and subtraction are both meaningful here. Ben really did take 3.1 - 2.3 = 0.8 seconds longer than Alan did. However, notice that multiplication and division also make sense here too: Ben took 3.1 / 2.3 = 1.35 times as long as Alan did to answer the question. And the reason why you can do this is that, for a ratio scale variable such as RT, \"zero seconds\" really does mean \"no time at all\".\n", "\n", "### Continuous versus discrete variables\n", "\n", "There's a second kind of distinction that you need to be aware of, regarding what types of variables you can run into. This is the distinction between continuous variables and discrete variables. The difference between these is as follows:\n", "\n", "\n", "- A ***continuous variable*** is one in which, for any two values that you can think of, it's always logically possible to have another value in between. \n", "- A ***discrete variable*** is, in effect, a variable that isn't continuous. For a discrete variable, it's sometimes the case that there's nothing in the middle.\n", "\n", "\n", "These definitions probably seem a bit abstract, but they're pretty simple once you see some examples. For instance, response time is continuous. If Alan takes 3.1 seconds and Ben takes 2.3 seconds to respond to a question, then it's possible for Cameron's response time to lie in between, by taking 3.0 seconds. And of course it would also be possible for David to take 3.031 seconds to respond, meaning that his RT would lie in between Cameron's and Alan's. And while in practice it might be impossible to measure RT that precisely, it's certainly possible in principle. Because we can always find a new value for RT in between any two other ones, we say that RT is continuous. \n", "\n", "Discrete variables occur when this rule is violated. For example, nominal scale variables are always discrete: there isn't a type of transportation that falls \"in between\" trains and bicycles, not in the strict mathematical way that 2.3 falls in between 2 and 3. So transportation type is discrete. Similarly, ordinal scale variables are always discrete: although \"2nd place\" does fall between \"1st place\" and \"3rd place\", there's nothing that can logically fall in between \"1st place\" and \"2nd place\". Interval scale and ratio scale variables can go either way. As we saw above, response time (a ratio scale variable) is continuous. Temperature in degrees celsius (an interval scale variable) is also continuous. However, the year you went to school (an interval scale variable) is discrete. There's no year in between 2002 and 2003. The number of questions you get right on a true-or-false test (a ratio scale variable) is also discrete: since a true-or-false question doesn't allow you to be \"partially correct\", there's nothing in between 5/10 and 6/10. The table below summarises the relationship between the scales of measurement and the discrete/continuity distinction. Cells with a tick mark correspond to things that are possible. I'm trying to hammer this point home, because (a) some textbooks get this wrong, and (b) people very often say things like \"discrete variable\" when they mean \"nominal scale variable\". It's very unfortunate.\n", "\n", "[^note1]: Presidential Address to the First Indian Statistical Congress, 1938. Source: http://en.wikiquote.org/wiki/Ronald_Fisher\n", "\n", "[^note2]: Well... now this is awkward, isn't it? This section is one of the oldest parts of the book, and it's outdated in a rather embarrassing way. I wrote this in 2010, at which point all of those facts *were* true. Revisiting this in 2018... well I'm not 33 any more, but that's not surprising I suppose. I can't imagine my chromosomes have changed, so I'm going to guess my karyotype was then and is now XY. The self-identified gender, on the other hand... ah. I suppose the fact that the title page now refers to me as Danielle rather than Daniel might possibly be a giveaway, but I don't typically identify as \"male\" on a gender questionnaire these days, and I prefer *\"she/her\"* pronouns as a default (it's a long story)! I did think a little about how I was going to handle this in the book, actually. The book has a somewhat distinct authorial voice to it, and I feel like it would be a rather different work if I went back and wrote everything as Danielle and updated all the pronouns in the work. Besides, it would be a lot of work, so I've left my name as \"Dan\" throughout the book, and in any case \"Dan\" is a perfectly good nickname for \"Danielle\", don't you think? In any case, it's not a big deal. I only wanted to mention it to make life a little easier for readers who aren't sure how to refer to me. I still don't like anchovies though :-)\n", "\n", "[^note3]: Actually, I've been informed by readers with greater physics knowledge than I that temperature isn't strictly an interval scale, in the sense that the amount of energy required to heat something up by 3$^\\circ$ depends on it's current temperature. So in the sense that physicists care about, temperature isn't actually interval scale. But it still makes a cute example, so I'm going to ignore this little inconvenient truth.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "| | | |\n", "|:--------|:------------:|:------------:|\n", "| | continuous | discrete |\n", "|nominal | | $\\checkmark$ |\n", "|ordinal | | $\\checkmark$ |\n", "|interval | $\\checkmark$ | $\\checkmark$ |\n", "|ratio | $\\checkmark$ | $\\checkmark$ |\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Some complexities\n", "\n", "Okay, I know you're going to be shocked to hear this, but ... the real world is much messier than this little classification scheme suggests. Very few variables in real life actually fall into these nice neat categories, so you need to be kind of careful not to treat the scales of measurement as if they were hard and fast rules. It doesn't work like that: they're guidelines, intended to help you think about the situations in which you should treat different variables differently. Nothing more. \n", "\n", "So let's take a classic example, maybe *the* classic example, of a psychological measurement tool: the ***Likert scale***. The humble Likert scale is the bread and butter tool of all survey design. You yourself have filled out hundreds, maybe thousands of them, and odds are you've even used one yourself. Suppose we have a survey question that looks like this:\n", "\n", ">Which of the following best describes your opinion of the statement that \"all pirates are freaking awesome\" ... \n", "\n", "and then the options presented to the participant are these:\n", "\n", ">(1) Strongly disagree\n", ">(2) Disagree\n", ">(3) Neither agree nor disagree\n", ">(4) Agree\n", ">(5) Strongly agree\n", "\n", "This set of items is an example of a 5-point Likert scale: people are asked to choose among one of several (in this case 5) clearly ordered possibilities, generally with a verbal descriptor given in each case. However, it's not necessary that all items be explicitly described. This is a perfectly good example of a 5-point Likert scale too: \n", "\n", ">(1) Strongly disagree\n", ">(2) \n", ">(3) \n", ">(4) \n", ">(5) Strongly agree\n", "\n", "\n", "Likert scales are very handy, if somewhat limited, tools. The question is, what kind of variable are they? They're obviously discrete, since you can't give a response of 2.5. They're obviously not nominal scale, since the items are ordered; and they're not ratio scale either, since there's no natural zero. \n", "\n", "But are they ordinal scale or interval scale? One argument says that we can't really prove that the difference between \"strongly agree\" and \"agree\" is of the same size as the difference between \"agree\" and \"neither agree nor disagree\". In fact, in everyday life it's pretty obvious that they're not the same at all. So this suggests that we ought to treat Likert scales as ordinal variables. On the other hand, in practice most participants do seem to take the whole \"on a scale from 1 to 5\" part fairly seriously, and they tend to act as if the differences between the five response options were fairly similar to one another. As a consequence, a lot of researchers treat Likert scale data as if it were interval scale. It's not interval scale, but in practice it's close enough that we usually think of it as being ***quasi-interval scale***. \n", "\n", "(reliability)=\n", "## Assessing the reliability of a measurement\n", "\n", "At this point we've thought a little bit about how to operationalise a theoretical construct and thereby create a psychological measure; and we've seen that by applying psychological measures we end up with variables, which can come in many different types. At this point, we should start discussing the obvious question: is the measurement any good? We'll do this in terms of two related ideas: *reliability* and *validity*. Put simply, the **_reliability_** of a measure tells you how *precisely* you are measuring something, whereas the **_validity_** of a measure tells you how *accurate* the measure is. In this section I'll talk about reliability; we'll talk about validity in the next chapter. \n", "\n", "Reliability is actually a very simple concept: it refers to the repeatability or consistency of your measurement. The measurement of my weight by means of a \"bathroom scale\" is very reliable: if I step on and off the scales over and over again, it'll keep giving me the same answer. Measuring my intelligence by means of \"asking my mum\" is very unreliable: some days she tells me I'm a bit thick, and other days she tells me I'm a complete moron. Notice that this concept of reliability is different to the question of whether the measurements are correct (the correctness of a measurement relates to it's validity). If I'm holding a sack of potatos when I step on and off of the bathroom scales, the measurement will still be reliable: it will always give me the same answer. However, this highly reliable answer doesn't match up to my true weight at all, therefore it's wrong. In technical terms, this is a *reliable but invalid* measurement. Similarly, while my mum's estimate of my intelligence is a bit unreliable, she might be right. Maybe I'm just not too bright, and so while her estimate of my intelligence fluctuates pretty wildly from day to day, it's basically right. So that would be an *unreliable but valid* measure. Of course, to some extent, notice that if my mum's estimates are too unreliable, it's going to be very hard to figure out which one of her many claims about my intelligence is actually the right one. To some extent, then, a very unreliable measure tends to end up being invalid for practical purposes; so much so that many people would say that reliability is necessary (but not sufficient) to ensure validity. \n", "\n", "\n", "Okay, now that we're clear on the distinction between reliability and validity, let's have a think about the different ways in which we might measure reliability:\n", "\n", "\n", "- ***Test-retest reliability***. This relates to consistency over time: if we repeat the measurement at a later date, do we get a the same answer?\n", "- ***Inter-rater reliability***. This relates to consistency across people: if someone else repeats the measurement (e.g., someone else rates my intelligence) will they produce the same answer?\n", "- ***Parallel forms reliability***. This relates to consistency across theoretically-equivalent measurements: if I use a different set of bathroom scales to measure my weight, does it give the same answer?\n", "- ***Internal consistency reliability***. If a measurement is constructed from lots of different parts that perform similar functions (e.g., a personality questionnaire result is added up across several questions) do the individual parts tend to give similar answers. \n", "\n", "\n", " \n", "Not all measurements need to possess all forms of reliability. For instance, educational assessment can be thought of as a form of measurement. One of the subjects that I teach, *Computational Cognitive Science*, has an assessment structure that has a research component and an exam component (plus other things). The exam component is *intended* to measure something different from the research component, so the assessment as a whole has low internal consistency. However, within the exam there are several questions that are intended to (approximately) measure the same things, and those tend to produce similar outcomes; so the exam on its own has a fairly high internal consistency. Which is as it should be. You should only demand reliability in those situations where you want to measure the same thing!\n", "\n", "(ivdv)=\n", "## The \"role\" of variables: predictors and outcomes\n", "\n", "Okay, I've got one last piece of terminology that I need to explain to you before moving away from variables. Normally, when we do some research we end up with lots of different variables. Then, when we analyse our data we usually try to explain some of the variables in terms of some of the other variables. It's important to keep the two roles \"thing doing the explaining\" and \"thing being explained\" distinct. So let's be clear about this now. Firstly, we might as well get used to the idea of using mathematical symbols to describe variables, since it's going to happen over and over again. Let's denote the \"to be explained\" variable $Y$, and denote the variables \"doing the explaining\" as $X_1$, $X_2$, etc. \n", "\n", "Now, when we doing an analysis, we have different names for $X$ and $Y$, since they play different roles in the analysis. The classical names for these roles are **_independent variable_** (IV) and **_dependent variable_** (DV). The IV is the variable that you use to do the explaining (i.e., $X$) and the DV is the variable being explained (i.e., $Y$). The logic behind these names goes like this: if there really is a relationship between $X$ and $Y$ then we can say that $Y$ depends on $X$, and if we have designed our study \"properly\" then $X$ isn't dependent on anything else. However, I personally find those names horrible: they're hard to remember and they're highly misleading, because (a) the IV is never actually \"independent of everything else\" and (b) if there's no relationship, then the DV doesn't actually depend on the IV. And in fact, because I'm not the only person who thinks that IV and DV are just awful names, there are a number of alternatives that I find more appealing. The terms that I'll use in these notes are ***predictors*** and ***outcomes***. The idea here is that what you're trying to do is use $X$ (the predictors) to make guesses about $Y$ (the outcomes). [^note4]\n", "\n", "[^note4]: Annoyingly, though, there's a lot of different names used out there. I won't list all of them -- there would be no point in doing that -- other than to note that you may sometimes see \"response variable\" where I've used \"outcome\", and a traditionalist would use \"dependent variable\". Sigh. This sort of terminological confusion is very common, I'm afraid.] This is summarised in the table below:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "|role of the variable |classical name |modern name |\n", "|:--------------------|:-------------------------|:-----------|\n", "|to be explained |dependent variable (DV) |outcome (response) |\n", "|to do the explaining |independent variable (IV) |predictor |" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "(researchdesigns)=\n", "## Experimental and non-experimental research\n", "\n", "One of the big distinctions that you should be aware of is the distinction between \"experimental research\" and \"non-experimental research\". When we make this distinction, what we're really talking about is the degree of control that the researcher exercises over the people and events in the study.\n", "\n", "### Experimental research\n", "\n", "The key features of **_experimental research_** is that the researcher controls all aspects of the study, especially what participants experience during the study. In particular, the researcher manipulates or varies the predictor variables (IVs), and then allows the outcome variable (DV) to vary naturally. The idea here is to deliberately vary the predictors (IVs) to see if they have any causal effects on the outcomes. Moreover, in order to ensure that there's no chance that something other than the predictor variables is causing the outcomes, everything else is kept constant or is in some other way \"balanced\" to ensure that they have no effect on the results. In practice, it's almost impossible to *think* of everything else that might have an influence on the outcome of an experiment, much less keep it constant. The standard solution to this is ***randomisation***: that is, we randomly assign people to different groups, and then give each group a different treatment (i.e., assign them different values of the predictor variables). We'll talk more about randomisation later in this course, but for now, it's enough to say that what randomisation does is minimise (but not eliminate) the chances that there are any systematic difference between groups. \n", "\n", "Let's consider a very simple, completely unrealistic and grossly unethical example. Suppose you wanted to find out if smoking causes lung cancer. One way to do this would be to find people who smoke and people who don't smoke, and look to see if smokers have a higher rate of lung cancer. This is *not* a proper experiment, since the researcher doesn't have a lot of control over who is and isn't a smoker. And this really matters: for instance, it might be that people who choose to smoke cigarettes also tend to have poor diets, or maybe they tend to work in asbestos mines, or whatever. The point here is that the groups (smokers and non-smokers) actually differ on lots of things, not *just* smoking. So it might be that the higher incidence of lung cancer among smokers is caused by something else, not by smoking per se. In technical terms, these other things (e.g. diet) are called \"confounds\", and we'll talk about those in just a moment. \n", "\n", "In the meantime, let's now consider what a proper experiment might look like. Recall that our concern was that smokers and non-smokers might differ in lots of ways. The solution, as long as you have no ethics, is to *control* who smokes and who doesn't. Specifically, if we randomly divide participants into two groups, and force half of them to become smokers, then it's very unlikely that the groups will differ in any respect other than the fact that half of them smoke. That way, if our smoking group gets cancer at a higher rate than the non-smoking group, then we can feel pretty confident that (a) smoking does cause cancer and (b) we're murderers. \n", "\n", "### Non-experimental research\n", "\n", "**_Non-experimental research_** is a broad term that covers \"any study in which the researcher doesn't have quite as much control as they do in an experiment\". Obviously, control is something that scientists like to have, but as the previous example illustrates, there are lots of situations in which you can't or shouldn't try to obtain that control. Since it's grossly unethical (and almost certainly criminal) to force people to smoke in order to find out if they get cancer, this is a good example of a situation in which you really shouldn't try to obtain experimental control. But there are other reasons too. Even leaving aside the ethical issues, our \"smoking experiment\" does have a few other issues. For instance, when I suggested that we \"force\" half of the people to become smokers, I must have been talking about *starting* with a sample of non-smokers, and then forcing them to become smokers. While this sounds like the kind of solid, evil experimental design that a mad scientist would love, it might not be a very sound way of investigating the effect in the real world. For instance, suppose that smoking only causes lung cancer when people have poor diets, and suppose also that people who normally smoke do tend to have poor diets. However, since the \"smokers\" in our experiment aren't \"natural\" smokers (i.e., we forced non-smokers to become smokers; they didn't take on all of the other normal, real life characteristics that smokers might tend to possess) they probably have better diets. As such, in this silly example they wouldn't get lung cancer, and our experiment will fail, because it violates the structure of the \"natural\" world (the technical name for this is an \"artifactual\" result; see later).\n", "\n", "One distinction worth making between two types of non-experimental research is the difference between ***quasi-experimental research*** and ***case studies***. The example I discussed earlier -- in which we wanted to examine incidence of lung cancer among smokers and non-smokers, without trying to control who smokes and who doesn't -- is a quasi-experimental design. That is, it's the same as an experiment, but we don't control the predictors (IVs). We can still use statistics to analyse the results, it's just that we have to be a lot more careful.\n", "\n", "The alternative approach, case studies, aims to provide a very detailed description of one or a few instances. In general, you can't use statistics to analyse the results of case studies, and it's usually very hard to draw any general conclusions about \"people in general\" from a few isolated examples. However, case studies are very useful in some situations. Firstly, there are situations where you don't have any alternative: neuropsychology has this issue a lot. Sometimes, you just can't find a lot of people with brain damage in a specific area, so the only thing you can do is describe those cases that you do have in as much detail and with as much care as you can. However, there's also some genuine advantages to case studies: because you don't have as many people to study, you have the ability to invest lots of time and effort trying to understand the specific factors at play in each case. This is a very valuable thing to do. As a consequence, case studies can complement the more statistically-oriented approaches that you see in experimental and quasi-experimental designs. We won't talk much about case studies in these lectures, but they are nevertheless very valuable tools!\n", "\n", "(validity)=\n", "## Assessing the validity of a study\n", "\n", "More than any other thing, a scientist wants their research to be \"valid\". The conceptual idea behind ***validity*** is very simple: can you trust the results of your study? If not, the study is invalid. However, while it's easy to state, in practice it's much harder to check validity than it is to check reliability. And in all honesty, there's no precise, clearly agreed upon notion of what validity actually is. In fact, there's lots of different kinds of validity, each of which raises it's own issues, and not all forms of validity are relevant to all studies. I'm going to talk about five different types:\n", "\n", "- Internal validity\n", "- External validity\n", "- Construct validity\n", "- Face validity\n", "- Ecological validity\n", "\n", "To give you a quick guide as to what matters here... (1) Internal and external validity are the most important, since they tie directly to the fundamental question of whether your study really works. (2) Construct validity asks whether you're measuring what you think you are. (3) Face validity isn't terribly important except insofar as you care about \"appearances\". (4) Ecological validity is a special case of face validity that corresponds to a kind of appearance that you might care about a lot.\n", "\n", "### Internal validity\n", "\n", "**_Internal validity_** refers to the extent to which you are able draw the correct conclusions about the causal relationships between variables. It's called \"internal\" because it refers to the relationships between things \"inside\" the study. Let's illustrate the concept with a simple example. Suppose you're interested in finding out whether a university education makes you write better. To do so, you get a group of first year students, ask them to write a 1000 word essay, and count the number of spelling and grammatical errors they make. Then you find some third-year students, who obviously have had more of a university education than the first-years, and repeat the exercise. And let's suppose it turns out that the third-year students produce fewer errors. And so you conclude that a university education improves writing skills. Right? Except... the big problem that you have with this experiment is that the third-year students are older, and they've had more experience with writing things. So it's hard to know for sure what the causal relationship is: Do older people write better? Or people who have had more writing experience? Or people who have had more education? Which of the above is the true *cause* of the superior performance of the third-years? Age? Experience? Education? You can't tell. This is an example of a failure of internal validity, because your study doesn't properly tease apart the *causal* relationships between the different variables. \n", "\n", "\n", "\n", "### External validity\n", "\n", "**_External validity_** relates to the **_generalisability_** of your findings. That is, to what extent do you expect to see the same pattern of results in \"real life\" as you saw in your study. To put it a bit more precisely, any study that you do in psychology will involve a fairly specific set of questions or tasks, will occur in a specific environment, and will involve participants that are drawn from a particular subgroup. So, if it turns out that the results don't actually generalise to people and situations beyond the ones that you studied, then what you've got is a lack of external validity.\n", "\n", "The classic example of this issue is the fact that a very large proportion of studies in psychology will use undergraduate psychology students as the participants. Obviously, however, the researchers don't care *only* about psychology students; they care about people in general. Given that, a study that uses only psych students as participants always carries a risk of lacking external validity. That is, if there's something \"special\" about psychology students that makes them different to the general populace in some *relevant* respect, then we may start worrying about a lack of external validity.\n", "\n", "That said, it is absolutely critical to realise that a study that uses only psychology students does not necessarily have a problem with external validity. I'll talk about this again later, but it's such a common mistake that I'm going to mention it here. The external validity is threatened by the choice of population if (a) the population from which you sample your participants is very narrow (e.g., psych students), and (b) the narrow population that you sampled from is systematically different from the general population, *in some respect that is relevant to the psychological phenomenon that you intend to study*. The italicised part is the bit that lots of people forget: it is true that psychology undergraduates differ from the general population in lots of ways, and so a study that uses only psych students *may* have problems with external validity. However, if those differences aren't very relevant to the phenomenon that you're studying, then there's nothing to worry about. To make this a bit more concrete, here's two extreme examples:\n", "\n", "- You want to measure \"attitudes of the general public towards psychotherapy\", but all of your participants are psychology students. This study would almost certainly have a problem with external validity.\n", "- You want to measure the effectiveness of a visual illusion, and your participants are all psychology students. This study is very unlikely to have a problem with external validity\n", "\n", "\n", "Having just spent the last couple of paragraphs focusing on the choice of participants (since that's the big issue that everyone tends to worry most about), it's worth remembering that external validity is a broader concept. The following are also examples of things that might pose a threat to external validity, depending on what kind of study you're doing:\n", "\n", "- People might answer a \"psychology questionnaire\" in a manner that doesn't reflect what they would do in real life.\n", "- Your lab experiment on (say) \"human learning\" has a different structure to the learning problems people face in real life.\n", "\n", "\n", "\n", "### Construct validity\n", "\n", "**_Construct validity_** is basically a question of whether you're measuring what you want to be measuring. A measurement has good construct validity if it is actually measuring the correct theoretical construct, and bad construct validity if it doesn't. To give very simple (if ridiculous) example, suppose I'm trying to investigate the rates with which university students cheat on their exams. And the way I attempt to measure it is by asking the cheating students to stand up in the lecture theatre so that I can count them. When I do this with a class of 300 students, 0 people claim to be cheaters. So I therefore conclude that the proportion of cheaters in my class is 0%. Clearly this is a bit ridiculous. But the point here is not that this is a very deep methodological example, but rather to explain what construct validity is. The problem with my measure is that while I'm *trying* to measure \"the proportion of people who cheat\" what I'm actually measuring is \"the proportion of people stupid enough to own up to cheating, or bloody minded enough to pretend that they do\". Obviously, these aren't the same thing! So my study has gone wrong, because my measurement has very poor construct validity.\n", "\n", "\n", "\n", "### Face validity\n", "\n", "**_Face validity_** simply refers to whether or not a measure \"looks like\" it's doing what it's supposed to, nothing more. If I design a test of intelligence, and people look at it and they say \"no, that test doesn't measure intelligence\", then the measure lacks face validity. It's as simple as that. Obviously, face validity isn't very important from a pure scientific perspective. After all, what we care about is whether or not the measure *actually* does what it's supposed to do, not whether it *looks like* it does what it's supposed to do. As a consequence, we generally don't care very much about face validity. That said, the concept of face validity serves three useful pragmatic purposes:\n", "\n", "- Sometimes, an experienced scientist will have a \"hunch\" that a particular measure won't work. While these sorts of hunches have no strict evidentiary value, it's often worth paying attention to them. Because often times people have knowledge that they can't quite verbalise, so there might be something to worry about even if you can't quite say why. In other words, when someone you trust criticises the face validity of your study, it's worth taking the time to think more carefully about your design to see if you can think of reasons why it might go awry. Mind you, if you don't find any reason for concern, then you should probably not worry: after all, face validity really doesn't matter much.\n", "- Often (very often), completely uninformed people will also have a \"hunch\" that your research is crap. And they'll criticise it on the internet or something. On close inspection, you'll often notice that these criticisms are actually focused entirely on how the study \"looks\", but not on anything deeper. The concept of face validity is useful for gently explaining to people that they need to substantiate their arguments further. \n", "- Expanding on the last point, if the beliefs of untrained people are critical (e.g., this is often the case for applied research where you actually want to convince policy makers of something or other) then you *have* to care about face validity. Simply because -- whether you like it or not -- a lot of people will use face validity as a proxy for real validity. If you want the government to change a law on scientific, psychological grounds, then it won't matter how good your studies \"really\" are. If they lack face validity, you'll find that politicians ignore you. Of course, it's somewhat unfair that policy often depends more on appearance than fact, but that's how things go.\n", " \n", "\n", "\n", "\n", "### Ecological validity\n", "\n", "**_Ecological validity_** is a different notion of validity, which is similar to external validity, but less important. The idea is that, in order to be ecologically valid, the entire set up of the study should closely approximate the real world scenario that is being investigated. In a sense, ecological validity is a kind of face validity -- it relates mostly to whether the study \"looks\" right, but with a bit more rigour to it. To be ecologically valid, the study has to look right in a fairly specific way. The idea behind it is the intuition that a study that is ecologically valid is more likely to be externally valid. It's no guarantee, of course. But the nice thing about ecological validity is that it's much easier to check whether a study is ecologically valid than it is to check whether a study is externally valid. An simple example would be eyewitness identification studies. Most of these studies tend to be done in a university setting, often with fairly simple array of faces to look at rather than a line up. The length of time between seeing the \"criminal\" and being asked to identify the suspect in the \"line up\" is usually shorter. The \"crime\" isn't real, so there's no chance that the witness being scared, and there's no police officers present, so there's not as much chance of feeling pressured. These things all mean that the study *definitely* lacks ecological validity. They might (but might not) mean that it also lacks external validity.\n", "\n", "\n", "\n", "## Confounds, artifacts and other threats to validity\n", "\n", "If we look at the issue of validity in the most general fashion, the two biggest worries that we have are *confounds* and *artifact*. These two terms are defined in the following way:\n", "\n", "\n", "- **_Confound_**: A confound is an additional, often unmeasured variable [^note5] that turns out to be related to both the predictors and the outcomes. The existence of confounds threatens the internal validity of the study because you can't tell whether the predictor causes the outcome, or if the confounding variable causes it, etc.\n", "- ***Artifact***: A result is said to be \"artifactual\" if it only holds in the special situation that you happened to test in your study. The possibility that your result is an artifact describes a threat to your external validity, because it raises the possibility that you can't generalise your results to the actual population that you care about.\n", "\n", "\n", "\n", "As a general rule confounds are a bigger concern for non-experimental studies, precisely because they're not proper experiments: by definition, you're leaving lots of things uncontrolled, so there's a lot of scope for confounds working their way into your study. Experimental research tends to be much less vulnerable to confounds: the more control you have over what happens during the study, the more you can prevent confounds from appearing.\n", "\n", "However, there's always swings and roundabouts, and when we start thinking about artifacts rather than confounds, the shoe is very firmly on the other foot. For the most part, artifactual results tend to be a concern for experimental studies than for non-experimental studies. To see this, it helps to realise that the reason that a lot of studies are non-experimental is precisely because what the researcher is trying to do is examine human behaviour in a more naturalistic context. By working in a more real-world context, you lose experimental control (making yourself vulnerable to confounds) but because you tend to be studying human psychology \"in the wild\" you reduce the chances of getting an artifactual result. Or, to put it another way, when you take psychology out of the wild and bring it into the lab (which we usually have to do to gain our experimental control), you always run the risk of accidentally studying something different than you wanted to study: which is more or less the definition of an artifact.\n", "\n", "Be warned though: the above is a rough guide only. It's absolutely possible to have confounds in an experiment, and to get artifactual results with non-experimental studies. This can happen for all sorts of reasons, not least of which is researcher error. In practice, it's really hard to think everything through ahead of time, and even very good researchers make mistakes. But other times it's unavoidable, simply because the researcher has ethics (e.g., see \\@ref(differentialattrition)). \n", "\n", "Okay. There's a sense in which almost any threat to validity can be characterised as a confound or an artifact: they're pretty vague concepts. So let's have a look at some of the most common examples...\n", "\n", "\n", "### History effects\n", "\n", "***History effects*** refer to the possibility that specific events may occur during the study itself that might influence the outcomes. For instance, something might happen in between a pre-test and a post-test. Or, in between testing participant 23 and participant 24. Alternatively, it might be that you're looking at an older study, which was perfectly valid for its time, but the world has changed enough since then that the conclusions are no longer trustworthy. Examples of things that would count as history effects:\n", "\n", "\n", "- You're interested in how people think about risk and uncertainty. You started your data collection in December 2010. But finding participants and collecting data takes time, so you're still finding new people in February 2011. Unfortunately for you (and even more unfortunately for others), the Queensland floods occurred in January 2011, causing billions of dollars of damage and killing many people. Not surprisingly, the people tested in February 2011 express quite different beliefs about handling risk than the people tested in December 2010. Which (if any) of these reflects the \"true\" beliefs of participants? I think the answer is probably both: the Queensland floods genuinely changed the beliefs of the Australian public, though possibly only temporarily. The key thing here is that the \"history\" of the people tested in February is quite different to people tested in December. \n", "- You're testing the psychological effects of a new anti-anxiety drug. So what you do is measure anxiety before administering the drug (e.g., by self-report, and taking physiological measures, let's say), then you administer the drug, and then you take the same measures afterwards. In the middle, however, because your labs are in Los Angeles, there's an earthquake, which increases the anxiety of the participants. \n", "\n", "\n", "### Maturation effects\n", "\n", "As with history effects, ***maturational effects*** are fundamentally about change over time. However, maturation effects aren't in response to specific events. Rather, they relate to how people change on their own over time: we get older, we get tired, we get bored, etc. Some examples of maturation effects:\n", "\n", "\n", "- When doing developmental psychology research, you need to be aware that children grow up quite rapidly. So, suppose that you want to find out whether some educational trick helps with vocabulary size among 3 year olds. One thing that you need to be aware of is that the vocabulary size of children that age is growing at an incredible rate (multiple words per day), all on its own. If you design your study without taking this maturational effect into account, then you won't be able to tell if your educational trick works.\n", "- When running a very long experiment in the lab (say, something that goes for 3 hours), it's very likely that people will begin to get bored and tired, and that this maturational effect will cause performance to decline, regardless of anything else going on in the experiment\n", "\n", "\n", "\n", " \n", " \n", "### Repeated testing effects\n", "\n", "An important type of history effect is the effect of ***repeated testing***. Suppose I want to take two measurements of some psychological construct (e.g., anxiety). One thing I might be worried about is if the first measurement has an effect on the second measurement. In other words, this is a history effect in which the \"event\" that influences the second measurement is the first measurement itself! This is not at all uncommon. Examples of this include:\n", "\n", "\n", "- *Learning and practice*: e.g., \"intelligence\" at time 2 might appear to go up relative to time 1 because participants learned the general rules of how to solve \"intelligence-test-style\" questions during the first testing session. \n", "- *Familiarity with the testing situation*: e.g., if people are nervous at time 1, this might make performance go down; after sitting through the first testing situation, they might calm down a lot precisely because they've seen what the testing looks like. \n", "- *Auxiliary changes caused by testing*: e.g., if a questionnaire assessing mood is boring, then mood at measurement at time 2 is more likely to become \"bored\", precisely because of the boring measurement made at time 1. \n", "\n", "\n", "\n", "### Selection bias \n", "\n", "***Selection bias*** is a pretty broad term. Suppose that you're running an experiment with two groups of participants, where each group gets a different \"treatment\", and you want to see if the different treatments lead to different outcomes. However, suppose that, despite your best efforts, you've ended up with a gender imbalance across groups (say, group A has 80% females and group B has 50% females). It might sound like this could never happen, but trust me, it can. This is an example of a selection bias, in which the people \"selected into\" the two groups have different characteristics. If any of those characteristics turns out to be relevant (say, your treatment works better on females than males) then you're in a lot of trouble. \n", "\n", "### Differential attrition\n", "\n", "One quite subtle danger to be aware of is called ***differential attrition***, which is a kind of selection bias that is caused by the study itself. Suppose that, for the first time ever in the history of psychology, I manage to find the perfectly balanced and representative sample of people. I start running \"Dan's incredibly long and tedious experiment\" on my perfect sample, but then, because my study is incredibly long and tedious, lots of people start dropping out. I can't stop this: as we'll discuss later in the chapter on research ethics, participants absolutely have the right to stop doing any experiment, any time, for whatever reason they feel like, and as researchers we are morally (and professionally) obliged to remind people that they do have this right. So, suppose that \"Dan's incredibly long and tedious experiment\" has a very high drop out rate. What do you suppose the odds are that this drop out is random? Answer: zero. Almost certainly, the people who remain are more conscientious, more tolerant of boredom etc than those that leave. To the extent that (say) conscientiousness is relevant to the psychological phenomenon that I care about, this attrition can decrease the validity of my results.\n", "\n", "When thinking about the effects of differential attrition, it is sometimes helpful to distinguish between two different types. The first is ***homogeneous attrition***, in which the attrition effect is the same for all groups, treatments or conditions. In the example I gave above, the differential attrition would be homogeneous if (and only if) the easily bored participants are dropping out of all of the conditions in my experiment at about the same rate. In general, the main effect of homogeneous attrition is likely to be that it makes your sample unrepresentative. As such, the biggest worry that you'll have is that the generalisability of the results decreases: in other words, you lose external validity.\n", "\n", "The second type of differential attrition is **_heterogeneous attrition_**, in which the attrition effect is different for different groups. This is a much bigger problem: not only do you have to worry about your external validity, you also have to worry about your internal validity too. To see why this is the case, let's consider a very dumb study in which I want to see if insulting people makes them act in a more obedient way. Why anyone would actually want to study that I don't know, but let's suppose I really, deeply cared about this. So, I design my experiment with two conditions. In the \"treatment\" condition, the experimenter insults the participant and then gives them a questionnaire designed to measure obedience. In the \"control\" condition, the experimenter engages in a bit of pointless chitchat and then gives them the questionnaire. Leaving aside the questionable scientific merits and dubious ethics of such a study, let's have a think about what might go wrong here. As a general rule, when someone insults me to my face, I tend to get much less co-operative. So, there's a pretty good chance that a lot more people are going to drop out of the treatment condition than the control condition. And this drop out isn't going to be random. The people most likely to drop out would probably be the people who don't care all that much about the importance of obediently sitting through the experiment. Since the most bloody minded and disobedient people all left the treatment group but not the control group, we've introduced a confound: the people who actually took the questionnaire in the treatment group were *already* more likely to be dutiful and obedient than the people in the control group. In short, in this study insulting people doesn't make them more obedient: it makes the more disobedient people leave the experiment! The internal validity of this experiment is completely shot. \n", "\n", "\n", "\n", "### Non-response bias\n", "\n", "**_Non-response bias_** is closely related to selection bias, and to differential attrition. The simplest version of the problem goes like this. You mail out a survey to 1000 people, and only 300 of them reply. The 300 people who replied are almost certainly not a random subsample. People who respond to surveys are systematically different to people who don't. This introduces a problem when trying to generalise from those 300 people who replied, to the population at large; since you now have a very non-random sample. The issue of non-response bias is more general than this, though. Among the (say) 300 people that did respond to the survey, you might find that not everyone answers every question. If (say) 80 people chose not to answer one of your questions, does this introduce problems? As always, the answer is maybe. If the question that wasn't answered was on the last page of the questionnaire, and those 80 surveys were returned with the last page missing, there's a good chance that the missing data isn't a big deal: probably the pages just fell off. However, if the question that 80 people didn't answer was the most confrontational or invasive personal question in the questionnaire, then almost certainly you've got a problem. In essence, what you're dealing with here is what's called the problem of ***missing data***. If the data that is missing was \"lost\" randomly, then it's not a big problem. If it's missing systematically, then it can be a big problem.\n", "\n", "\n", "\n", "### Regression to the mean\n", "\n", "***Regression to the mean*** is a curious variation on selection bias. It refers to any situation where you select data based on an extreme value on some measure. Because the measure has natural variation, it almost certainly means that when you take a subsequent measurement, that later measurement will be less extreme than the first one, purely by chance. \n", "\n", "\n", "Here's an example. Suppose I'm interested in whether a psychology education has an adverse effect on very smart kids. To do this, I find the 20 psych I students with the best high school grades and look at how well they're doing at university. It turns out that they're doing a lot better than average, but they're not topping the class at university, even though they did top their classes at high school. What's going on? The natural first thought is that this must mean that the psychology classes must be having an adverse effect on those students. However, while that might very well be the explanation, it's more likely that what you're seeing is an example of \"regression to the mean\". To see how it works, let's take a moment to think about what is required to get the best mark in a class, regardless of whether that class be at high school or at university. When you've got a big class, there are going to be *lots* of very smart people enrolled. To get the best mark you have to be very smart, work very hard, and be a bit lucky. The exam has to ask just the right questions for your idiosyncratic skills, and you have to not make any dumb mistakes (we all do that sometimes) when answering them. And that's the thing: intelligence and hard work are transferrable from one class to the next. Luck isn't. The people who got lucky in high school won't be the same as the people who get lucky at university. That's the very definition of \"luck\". The consequence of this is that, when you select people at the very extreme values of one measurement (the top 20 students), you're selecting for hard work, skill and luck. But because the luck doesn't transfer to the second measurement (only the skill and work), these people will all be expected to drop a little bit when you measure them a second time (at university). So their scores fall back a little bit, back towards everyone else. This is regression to the mean.\n", "\n", "Regression to the mean is surprisingly common. For instance, if two very tall people have kids, their children will tend to be taller than average, but not as tall as the parents. The reverse happens with very short parents: two very short parents will tend to have short children, but nevertheless those kids will tend to be taller than the parents. It can also be extremely subtle. For instance, there have been studies done that suggested that people learn better from negative feedback than from positive feedback. However, the way that people tried to show this was to give people positive reinforcement whenever they did good, and negative reinforcement when they did bad. And what you see is that after the positive reinforcement, people tended to do worse; but after the negative reinforcement they tended to do better. But! Notice that there's a selection bias here: when people do very well, you're selecting for \"high\" values, and so you should *expect* (because of regression to the mean) that performance on the next trial should be worse, regardless of whether reinforcement is given. Similarly, after a bad trial, people will tend to improve all on their own. The apparent superiority of negative feedback is an artifact caused by regression to the mean (see {cite}`Kahneman1973` for discussion).\n", "\n", "\n", "\n", "### Experimenter bias\n", "\n", "***Experimenter bias*** can come in multiple forms. The basic idea is that the experimenter, despite the best of intentions, can accidentally end up influencing the results of the experiment by subtly communicating the \"right answer\" or the \"desired behaviour\" to the participants. Typically, this occurs because the experimenter has special knowledge that the participant does not -- either the right answer to the questions being asked, or knowledge of the expected pattern of performance for the condition that the participant is in, and so on. The classic example of this happening is the case study of \"Clever Hans\", which dates back to 1907 {cite}`Pfungst1911` {cite}`Hothersall2004`. Clever Hans was a horse that apparently was able to read and count, and perform other human like feats of intelligence. After Clever Hans became famous, psychologists started examining his behaviour more closely. It turned out that -- not surprisingly -- Hans didn't know how to do maths. Rather, Hans was responding to the human observers around him. Because they did know how to count, and the horse had learned to change its behaviour when people changed theirs. \n", "\n", "\n", "\n", "The general solution to the problem of experimenter bias is to engage in double blind studies, where neither the experimenter nor the participant knows which condition the participant is in, or knows what the desired behaviour is. This provides a very good solution to the problem, but it's important to recognise that it's not quite ideal, and hard to pull off perfectly. For instance, the obvious way that I could try to construct a double blind study is to have one of my Ph.D. students (one who doesn't know anything about the experiment) run the study. That feels like it should be enough. The only person (me) who knows all the details (e.g., correct answers to the questions, assignments of participants to conditions) has no interaction with the participants, and the person who does all the talking to people (the Ph.D. student) doesn't know anything. Except, that last part is very unlikely to be true. In order for the Ph.D. student to run the study effectively, they need to have been briefed by me, the researcher. And, as it happens, the Ph.D. student also knows me, and knows a bit about my general beliefs about people and psychology (e.g., I tend to think humans are much smarter than psychologists give them credit for). As a result of all this, it's almost impossible for the experimenter to avoid knowing a little bit about what expectations I have. And even a little bit of knowledge can have an effect: suppose the experimenter accidentally conveys the fact that the participants are expected to do well in this task. Well, there's a thing called the \"Pygmalion effect\": if you expect great things of people, they'll rise to the occasion; but if you expect them to fail, they'll do that too. In other words, the expectations become a self-fulfilling prophesy.\n", "\n", "\n", "### Demand effects and reactivity\n", "\n", "When talking about experimenter bias, the worry is that the experimenter's knowledge or desires for the experiment are communicated to the participants, and that these effect people's behaviour {cite}`@Rosenthal1966`. However, even if you manage to stop this from happening, it's almost impossible to stop people from knowing that they're part of a psychological study. And the mere fact of knowing that someone is watching/studying you can have a pretty big effect on behaviour. This is generally referred to as ***reactivity*** or ***demand effects***. The basic idea is captured by the Hawthorne effect: people alter their performance because of the attention that the study focuses on them. The effect takes its name from a the \"Hawthorne Works\" factory outside of Chicago (see {cite}`Adair1984`). A study done in the 1920s looking at the effects of lighting on worker productivity at the factory turned out to be an effect of the fact that the workers knew they were being studied, rather than the lighting.\n", "\n", "To get a bit more specific about some of the ways in which the mere fact of being in a study can change how people behave, it helps to think like a social psychologist and look at some of the *roles* that people might adopt during an experiment, but might not adopt if the corresponding events were occurring in the real world:\n", "\n", "- The *good participant* tries to be too helpful to the researcher: he or she seeks to figure out the experimenter's hypotheses and confirm them.\n", "- The *negative participant* does the exact opposite of the good participant: he or she seeks to break or destroy the study or the hypothesis in some way.\n", "- The *faithful participant* is unnaturally obedient: he or she seeks to follow instructions perfectly, regardless of what might have happened in a more realistic setting.\n", "- The *apprehensive participant* gets nervous about being tested or studied, so much so that his or her behaviour becomes highly unnatural, or overly socially desirable.\n", "\n", "\n", "### Placebo effects\n", "\n", "The ***placebo effect*** is a specific type of demand effect that we worry a lot about. It refers to the situation where the mere fact of being treated causes an improvement in outcomes. The classic example comes from clinical trials: if you give people a completely chemically inert drug and tell them that it's a cure for a disease, they will tend to get better faster than people who aren't treated at all. In other words, it is people's belief that they are being treated that causes the improved outcomes, not the drug.\n", "\n", "### Situation, measurement and subpopulation effects\n", " \n", "In some respects, these terms are a catch-all term for \"all other threats to external validity\". They refer to the fact that the choice of subpopulation from which you draw your participants, the location, timing and manner in which you run your study (including who collects the data) and the tools that you use to make your measurements might all be influencing the results. Specifically, the worry is that these things might be influencing the results in such a way that the results won't generalise to a wider array of people, places and measures. \n", " \n", "### Fraud, deception and self-deception \n", "\n", ">*It is difficult to get a man to understand something, when his salary depends on his not understanding it.*\n", ">\n", ">-- Upton Sinclair\n", "\n", "\n", "One final thing that I feel like I should mention. While reading what the textbooks often have to say about assessing the validity of the study, I couldn't help but notice that they seem to make the assumption that the researcher is honest. I find this hilarious. While the vast majority of scientists are honest, in my experience at least, some are not. [^note6] Not only that, as I mentioned earlier, scientists are not immune to belief bias -- it's easy for a researcher to end up deceiving themselves into believing the wrong thing, and this can lead them to conduct subtly flawed research, and then hide those flaws when they write it up. So you need to consider not only the (probably unlikely) possibility of outright fraud, but also the (probably quite common) possibility that the research is unintentionally \"slanted\". I opened a few standard textbooks and didn't find much of a discussion of this problem, so here's my own attempt to list a few ways in which these issues can arise are:\n", "\n", "\n", "- ***Data fabrication***. Sometimes, people just make up the data. This is occasionally done with \"good\" intentions. For instance, the researcher believes that the fabricated data do reflect the truth, and may actually reflect \"slightly cleaned up\" versions of actual data. On other occasions, the fraud is deliberate and malicious. Some high-profile examples where data fabrication has been alleged or shown include Cyril Burt (a psychologist who is thought to have fabricated some of his data), Andrew Wakefield (who has been accused of fabricating his data connecting the MMR vaccine to autism) and Hwang Woo-suk (who falsified a lot of his data on stem cell research). \n", "- ***Hoaxes***. Hoaxes share a lot of similarities with data fabrication, but they differ in the intended purpose. A hoax is often a joke, and many of them are intended to be (eventually) discovered. Often, the point of a hoax is to discredit someone or some field. There's quite a few well known scientific hoaxes that have occurred over the years (e.g., Piltdown man) some of were deliberate attempts to discredit particular fields of research (e.g., the Sokal affair). \n", "- ***Data misrepresentation***. While fraud gets most of the headlines, it's much more common in my experience to see data being misrepresented. When I say this, I'm not referring to newspapers getting it wrong (which they do, almost always). I'm referring to the fact that often, the data don't actually say what the researchers think they say. My guess is that, almost always, this isn't the result of deliberate dishonesty, it's due to a lack of sophistication in the data analyses. For instance, think back to the example of Simpson's paradox that I discussed in the beginning of these notes. It's very common to see people present \"aggregated\" data of some kind; and sometimes, when you dig deeper and find the raw data yourself, you find that the aggregated data tell a different story to the disaggregated data. Alternatively, you might find that some aspect of the data is being hidden, because it tells an inconvenient story (e.g., the researcher might choose not to refer to a particular variable). There's a lot of variants on this; many of which are very hard to detect.\n", "- **_Study \"misdesign\"_**. Okay, this one is subtle. Basically, the issue here is that a researcher designs a study that has built-in flaws, and those flaws are never reported in the paper. The data that are reported are completely real, and are correctly analysed, but they are produced by a study that is actually quite wrongly put together. The researcher really wants to find a particular effect, and so the study is set up in such a way as to make it \"easy\" to (artifactually) observe that effect. One sneaky way to do this -- in case you're feeling like dabbling in a bit of fraud yourself -- is to design an experiment in which it's obvious to the participants what they're \"supposed\" to be doing, and then let reactivity work its magic for you. If you want, you can add all the trappings of double blind experimentation etc. It won't make a difference, since the study materials themselves are subtly telling people what you want them to do. When you write up the results, the fraud won't be obvious to the reader: what's obvious to the participant when they're in the experimental context isn't always obvious to the person reading the paper. Of course, the way I've described this makes it sound like it's always fraud: probably there are cases where this is done deliberately, but in my experience the bigger concern has been with unintentional misdesign. The researcher *believes* ... and so the study just happens to end up with a built in flaw, and that flaw then magically erases itself when the study is written up for publication.\n", "- **_Data mining & post hoc hypothesising_**. Another way in which the authors of a study can more or less lie about what they found is by engaging in what's referred to as \"data mining\". As we'll discuss later in the class, if you keep trying to analyse your data in lots of different ways, you'll eventually find something that \"looks\" like a real effect but isn't. This is referred to as \"data mining\". It used to be quite rare because data analysis used to take weeks, but now that everyone has very powerful statistical software on their computers, it's becoming very common. Data mining per se isn't \"wrong\", but the more that you do it, the bigger the risk you're taking. The thing that is wrong, and I suspect is very common, is *unacknowledged* data mining. That is, the researcher run every possible analysis known to humanity, finds the one that works, and then pretends that this was the only analysis that they ever conducted. Worse yet, they often \"invent\" a hypothesis after looking at the data, to cover up the data mining. To be clear: it's not wrong to change your beliefs after looking at the data, and to reanalyse your data using your new \"post hoc\" hypotheses. What is wrong (and, I suspect, common) is failing to acknowledge that you've done so. If you acknowledge that you did it, then other researchers are able to take your behaviour into account. If you don't, then they can't. And that makes your behaviour deceptive. Bad! \n", "- **_Publication bias & self-censoring_**. Finally, a pervasive bias is \"non-reporting\" of negative results. This is almost impossible to prevent. Journals don't publish every article that is submitted to them: they prefer to publish articles that find \"something\". So, if 20 people run an experiment looking at whether reading *Finnegans Wake* causes insanity in humans, and 19 of them find that it doesn't, which one do you think is going to get published? Obviously, it's the one study that did find that *Finnegans Wake* causes insanity [^note7]. This is an example of a *publication bias*: since no-one ever published the 19 studies that didn't find an effect, a naive reader would never know that they existed. Worse yet, most researchers \"internalise\" this bias, and end up *self-censoring* their research. Knowing that negative results aren't going to be accepted for publication, they never even try to report them. As a friend of mine says \"for every experiment that you get published, you also have 10 failures\". And she's right. The catch is, while some (maybe most) of those studies are failures for boring reasons (e.g. you stuffed something up) others might be genuine \"null\" results that you ought to acknowledge when you write up the \"good\" experiment. And telling which is which is often hard to do. A good place to start is a paper by {cite}`Ioannidis2005` with the depressing title \"Why most published research findings are false\". I'd also suggest taking a look at work by {cite}`Kuhberger2014` presenting statistical evidence that this actually happens in psychology.\n", "\n", "\n", "There's probably a lot more issues like this to think about, but that'll do to start with. What I really want to point out is the blindingly obvious truth that real world science is conducted by actual humans, and only the most gullible of people automatically assumes that everyone else is honest and impartial. Actual scientists aren't usually *that* naive, but for some reason the world likes to pretend that we are, and the textbooks we usually write seem to reinforce that stereotype.\n", "\n", "## Summary\n", "\n", "This chapter isn't really meant to provide a comprehensive discussion of psychological research methods: it would require another volume just as long as this one to do justice to the topic. However, in real life statistics and study design are tightly intertwined, so it's very handy to discuss some of the key topics. In this chapter, I've briefly discussed the following topics:\n", "\n", "\n", "- [Introduction to psychological measurement](measurement). What does it mean to operationalise a theoretical construct? What does it mean to have variables and take measurements?\n", "- [Scales of measurement and types of variables](scales). Remember that there are *two* different distinctions here: there's the difference between discrete and continuous data, and there's the difference between the four different scale types (nominal, ordinal, interval and ratio). \n", "- [Reliability of a measurement](reliability). If I measure the \"same\" thing twice, should I expect to see the same result? Only if my measure is reliable. But what does it mean to talk about doing the \"same\" thing? Well, that's why we have different types of reliability. Make sure you remember what they are.\n", "- [Terminology: predictors and outcomes](ivdv). What roles do variables play in an analysis? Can you remember the difference between predictors and outcomes? Dependent and independent variables? Etc. \n", "- [Experimental and non-experimental research designs](researchdesigns). What makes an experiment an experiment? Is it a nice white lab coat, or does it have something to do with researcher control over variables?\n", "- [Validity and its threats](validity). Does your study measure what you want it to? How might things go wrong? And is it my imagination, or was that a very long list of possible ways in which things can go wrong? \n", "\n", "\n", "\n", "All this should make clear to you that study design is a critical part of research methodology. I built this chapter from the classic little book by {cite}`campbell1963experimental`, but there are of course a large number of textbooks out there on research design. Spend a few minutes with your favourite search engine and you'll find dozens. \n", "\n", "[^note5]: The reason why I say that it's unmeasured is that if you *have* measured it, then you can use some fancy statistical tricks to deal with the confound. Because of the existence of these statistical solutions to the problem of confounds, we often refer to a confound that we have measured and dealt with as a *covariate*. Dealing with covariates is a topic for a more advanced course, but I thought I'd mention it in passing, since it's kind of comforting to at least know that this stuff exists.\n", "\n", "[^note6]: Some people might argue that if you're not honest then you're not a real scientist. Which does have some truth to it I guess, but that's disingenuous (google the \"No true Scotsman\" fallacy). The fact is that there are lots of people who are employed ostensibly as scientists, and whose work has all of the trappings of science, but who are outright fraudulent. Pretending that they don't exist by saying that they're not scientists is just childish.\n", "\n", "[^note7]: Clearly, the real effect is that only insane people would even try to read *Finnegans Wake.*" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.5" } }, "nbformat": 4, "nbformat_minor": 4 } { "cells": [ { "cell_type": "markdown", "id": "discrete-howard", "metadata": {}, "source": [ "(getting-started-with-python)=\n", "# Getting Started with Python" ] }, { "cell_type": "markdown", "id": "unable-university", "metadata": {}, "source": [ ">*Robots are nice to work with.*\n", ">\n", ">--Roger Zelazny [^note1]\n", "\n", "[^note1]: Source: *Dismal Light* (1968)." ] }, { "cell_type": "markdown", "id": "wicked-importance", "metadata": {}, "source": [ "In this chapter I'll discuss how to get started in Python. I will have very little to say about how to [download and install Python](https://www.python.org/downloads/), because there are so many different ways to do this, with new advances coming out so often, that almost anything I say will be out of date as soon as I push the publish button on this chapter. Instead, most of the chapter will be focused on getting you started typing Python commands. Our goal in this chapter is not to learn any statistical concepts: we're just trying to learn the basics of how Python works and get comfortable interacting with the system. To do this, we'll spend a bit of time using Python as a simple calculator, since that's arguably the easiest thing to do with Python. In doing so, you'll get a bit of a feel for what it's like to work in Python. From there I'll introduce some very basic programming ideas: in particular, I'll talk about the idea of defining *variables* to store information, and a few things that you can do with these variables. \n", "\n", "However, before going into any of the specifics, it's worth talking a little about why you might want to use Python at all. Given that you're reading this, you've probably got your own reasons. However, if those reasons are \"because that's what my stats class uses\", it might be worth explaining a little why your lecturer has chosen to use Python for the class. Of course, I don't really know why *other* people choose Python, so I'm really talking about why I use it.\n", "\n", "- It's sort of obvious, but worth saying anyway: doing your statistics on a computer is faster, easier and more powerful than doing statistics by hand. Computers excel at mindless repetitive tasks, and a lot of statistical calculations are both mindless and repetitive. For most people, the only reason to ever do statistical calculations with pencil and paper is for learning purposes. In my class I do occasionally suggest doing some calculations that way, but the only real value to it is pedagogical. It does help you to get a \"feel\" for statistics to do some calculations yourself, so it's worth doing it once. But only once!\n", "- Doing statistics in a spreadsheet (e.g., Microsoft Excel) is generally a bad idea in the long run. Although many people are likely to feel more familiar with them, spreadsheets are very limited in terms of what analyses they allow you do. Furthermore, because of \"features\" like auto-formatting, if you're not careful, [spreadsheet programs can really mess up your data](https://genomebiology.biomedcentral.com/articles/10.1186/s13059-016-1044-7). If you get into the habit of trying to do your real life data analysis using spreadsheets, then you've dug yourself into a very deep hole.\n", "- Avoiding proprietary software is a very good idea. There are a lot of commercial packages out there that you can buy, some of which I like and some of which I don't. They're usually very glossy in their appearance, and generally very powerful (much more powerful than spreadsheets). However, they're also very expensive: usually, the company sells \"student versions\" (versions that are but shadowy husks of the real thing) very cheaply; they sell full powered \"educational versions\" at a price that makes me wince; and they sell commercial licences with a staggeringly high price tag. The business model here is to suck you in during your student days, and then leave you dependent on their tools when you go out into the real world. It's hard to blame them for trying, but personally I'm not in favour of shelling out thousands of dollars if I can avoid it. And you can avoid it: if you make use of tools like Python that are open source and free, you never get trapped having to pay exorbitant licensing fees. \n", "- Something that you might not appreciate now, but will love later on if you do anything involving data analysis, is the fact that Python is highly extensible. When you download and install Python, you get all the basic \"modules\", and those are very powerful on their own. However, because Python is so open and so widely used, it's become something of a standard tool in statistics, and so lots of people write their own packages that extend the system. And these are freely available too. One of the consequences of this, I've noticed, is that if you look at people doing advanced work in statistical analysis, especially in the fields of machine learning and natural language processing, a *lot* of them use Python. In other words, if you learn how to do your basic statistics in Python, then you're a lot closer to being able to use the state of the art methods than you would be if you'd started out with a \"simpler\" system: so if you want to become a genuine expert in psychological data analysis, learning Python is a very good use of your time.\n", "- Related to the previous point: Python is a real programming language. As you get better at using Python for data analysis, you're also learning to program. To some people this might seem like a bad thing, but in truth, programming is a core research skill across a lot of the social and behavioural sciences. Think about how many surveys and experiments are done online, or presented on computers. Think about all those online social environments which you might be interested in studying; and maybe collecting data from in an automated fashion. Think about artificial intelligence systems, computer vision, \"data science\" and speech recognition. If any of these are things that you think you might want to be involved in -- as someone \"doing research in psychology\", that is -- you'll need to know a bit of programming. And if you don't already know how to program, then learning how to do statistics using Python is a nice way to start.\n", "\n", "Those are the main reasons I use Python. It's not without its flaws: although it is relatively easy to learn, as programming languages go, it is a lot harder to learn than Excel, and it has a few very annoying quirks to it that we're all pretty much stuck with, but on the whole I think the strengths outweigh the weakness.\n" ] }, { "cell_type": "markdown", "id": "medium-buddy", "metadata": {}, "source": [ "## Installing Python\n", "\n", "Ok, I said I wasn't going to say much about [installing Python](https://www.python.org/downloads/), and I'm not. If you are reading this text for a course, then presumably your instructor has a particular way they would like you to access Python. If not, it is not difficult to find instructions for installing Python on your machine. A further complication is the installation of \"modules\" or \"packages\". Much of the power of Python comes not only from the language itself, but from the massive library of freely availabe code which others have written and made available for all of us to use. Although Python has an extremely rich ecosystem of packages available, it has not yet (as of this writing) really settled on a single way to distribute these packages. This can be frustrating, when you need to install some bit of code to do your analysis. Again, there are a variety of solutions available, such as [Anaconda](https://www.anaconda.com), and your instructor can likely help you with this. In this text, I have very consciously tried to limit the number of packages needed to a few of the very most common needed for data analysis: `numpy`, `scipy`, `pingouin`, `pandas` and `seaborn`. These should all be relatively easy to access, and should be relatively stable.\n", "\n", "Alternatively, you could use a cloud interface, like Google CoLab, although be aware that this requires you to upload any data that you may be analyzing to Google, so this should never be used for any sensitive data of any kind, especially if your data has any sort of information that could be used to identify your participants. If at all in doubt, it is always safer to keep your data safely enrypted on your own machine.\n", "\n", "In addition to the various ways to install Python, there are also a variety of ways to send commands to Python. Although you may eventually want to simply write your Python code in a text file, and run it directly from the command line of your computer, a much easier way to interact with Python is either using a so-called IDE (Integrated Development Environment), which is a kind of software whose role in life is to make it easier for you to write Python code, or using an interactive web application like [Jupyter Notebooks](https://jupyter.org). Unless your instructor tells you otherwise, I strongly reccomend Jupyter Notebooks as the best way to get started writing Python code. This book was written in Jupyter Notebooks, and assembled using [Jupyter Book](https://jupyterbook.org/intro.html)" ] }, { "cell_type": "markdown", "id": "configured-minister", "metadata": {}, "source": [ "## Using the code in this book\n", "\n", "Most of the code in this book should be usable no matter what environment you are using. I have tried to make the blocks of code in this book as self-sufficient as possible, so that you can simply copy/paste into your own Python interface. In practice, this means I have (as much as I remember to) imported the necessary modules into each block of code, even when this wasn't necessary for the code to run, although here and there I have deviated from this, when I thought it was clear that I was showing a series of commands to be run right after each other, or when I just plain forgot.\n", "\n", "Although most of the time you should be able to copy and paste the code from this book, there is one exception: in order to make figures with captions that I can link to elsewhere in the text, I have used a function called `glue`. Anywhere you see a line of code with `glue` in it, you can just ignore it: it's just there to make this book work right, and doesn't have anything to do with the code you will need for analysing data. If it is in some of the code that you copy and paste into your own notebook, just delete those lines, because they will probably make Python complain." ] }, { "cell_type": "markdown", "id": "rising-advertiser", "metadata": {}, "source": [ "(firstcommand)=\n", "\n", "## Typing commands at the Python console\n", "\n", "One of the easiest things you can do with Python is use it as a simple calculator, so this is probably a good place to start. For instance, try typing `10 + 20`, and hitting enter.[^note2] When you do this, you've entered a ***command***, and Python will \"execute\" that command. If you are using Jupyter Notebooks, what you will see on screen will look something like this:\n", "\n", "[^note2]: Seriously. If you're in a position to do so, open up Python and start typing. The simple act of typing it rather than \"just reading\" makes a big difference. It makes the concepts more concrete, and it ties the abstract ideas (programming and statistics) to the actual context in which you need to use them. Statistics is something you *do*, not just something you read about in a textbook." ] }, { "cell_type": "code", "execution_count": 7, "id": "armed-store", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "30" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "10 + 20" ] }, { "cell_type": "markdown", "id": "knowing-locking", "metadata": {}, "source": [ "Not a lot of surprises here. But there's a few things worth talking about, even with such a simple example. Firstly, it's important that you understand how to read the extract. In this example, what *I* typed was the `10 + 20` part. Obviously, the correct answer to the sum `10 + 20` is `30`, and not surprisingly Python has printed that out as part of its response. But if you are using Jupyter Notebooks, you will probably also see something like `In [1]` and `Out [1]` part, which probably doesn't make a lot of sense to you right now. You're going to see that a lot. You can think of `Out [1]: 30` as if Python were saying \"the answer to the 1st question you asked is 30\". That's not quite the truth, but it's close enough for now[^note_inout] If you see `In [*]`, that means that Python is still working on whatever you asked it to do.\n", "\n", "[^note_inout]: If you want to learn more about these `In` and `Out` numbers, try typing `In` or `Out` into a cell in a jupyter notebook and running it. You will see that `In` is in fact a list of all the commands you have run, and `Out` is a Python dictionary with all the outputs from all the cells that have been run, with the \"out\" number as a key, and the output as the value." ] }, { "cell_type": "markdown", "id": "outer-nature", "metadata": {}, "source": [ "### Be very careful to avoid typos\n", "\n", "Before we go on to talk about other types of calculations that we can do with Python, there's a few other things I want to point out. The first thing is that, while Python is good software, it's still software. It's pretty stupid, and because it's stupid it can't handle typos. It takes it on faith that you meant to type *exactly* what you did type. For example, suppose that you hit the wrong key when trying to type `+`, and as a result your command ended up being `10 = 20` rather than `10 + 20`. Here's what happens:" ] }, { "cell_type": "code", "execution_count": 2, "id": "bound-carbon", "metadata": {}, "outputs": [ { "ename": "SyntaxError", "evalue": "cannot assign to literal here. Maybe you meant '==' instead of '='? (2682558333.py, line 1)", "output_type": "error", "traceback": [ "\u001b[0;36m Input \u001b[0;32mIn [2]\u001b[0;36m\u001b[0m\n\u001b[0;31m 10 = 20\u001b[0m\n\u001b[0m ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m cannot assign to literal here. Maybe you meant '==' instead of '='?\n" ] } ], "source": [ "10 = 20" ] }, { "cell_type": "markdown", "id": "separated-senate", "metadata": {}, "source": [ "What's happened here is that Python has attempted to interpret `10 = 20` as a command, and spits out an error message because the command doesn't make any sense to it. When a *human* looks at this, and then looks down at their keyboard and sees that `+` and `=` are on the same key (depending on what country your keyboard is from, of course), it's pretty obvious that the command was a typo. But Python doesn't know this, so it gets upset. And, if you look at it from its perspective, this makes sense. All that Python \"knows\" is that `10` is a legitimate number, `20` is a legitimate number, and `=` is a legitimate part of the language too. In other words, from its perspective this really does look like the user meant to type `10 = 20`, since all the individual parts of that statement are legitimate and it's too stupid to realise that this is probably a typo. Therefore, Python takes it on faith that this is exactly what you meant... it only \"discovers\" that the command is nonsense when it tries to follow your instructions, typo and all. And then it whinges, and spits out an error." ] }, { "cell_type": "markdown", "id": "recognized-fiber", "metadata": {}, "source": [ "Even more subtle is the fact that some typos won't produce errors at all, because they happen to correspond to \"well-formed\" Python commands. For instance, suppose that not only did I forget to hit the shift key when trying to type `10 + 20`, I also managed to press the key next to one I meant do. The resulting typo would produce the command `10 - 20`. Clearly, Python has no way of knowing that you meant to *add* 20 to 10, not *subtract* 20 from 10, so what happens this time is this:" ] }, { "cell_type": "code", "execution_count": 3, "id": "fuzzy-following", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-10" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "10 - 20" ] }, { "cell_type": "markdown", "id": "constitutional-organ", "metadata": {}, "source": [ "In this case, Python produces the right answer, but to the the wrong question. \n", "\n", "To some extent, I'm stating the obvious here, but it's important. The people who wrote Python are smart. You, the user, are smart. But Python itself is dumb. And because it's dumb, it has to be mindlessly obedient. It does *exactly* what you ask it to do. There is no equivalent to \"autocorrect\" in Python[^note_codecompletion], and for good reason. When doing advanced stuff -- and even the simplest of statistics is pretty advanced in a lot of ways -- it's dangerous to let a mindless automaton like Python try to overrule the human user. But because of this, it's your responsibility to be careful. Always make sure you type *exactly what you mean*. When dealing with computers, it's not enough to type \"approximately\" the right thing. In general, you absolutely *must* be precise in what you say to Python ... like all machines it is too stupid to be anything other than absurdly literal in its interpretation.\n", "\n", "[^note_codecompletion]: Ok, some IDE programs do offer code completion, so that you can begin typing code, and the IDE will suggest the Python code that it thinks you intend to type, but this still won't save you if type the wrong thing. And also it might not guess your intentions correctly. Even smart IDEs are still just dumb computer programs." ] }, { "cell_type": "markdown", "id": "strong-logic", "metadata": {}, "source": [ "### Python is (a teeny bit) flexible with spacing\n", "\n", "Of course, now that I've been so uptight about the importance of always being precise, I should point out that there are some exceptions. Or, more accurately, there are some situations in which Python does show a bit more flexibility than my previous description suggests. One thing Python is smart enough to do is ignore redundant spacing. What I mean by this is that, when I typed `10 + 20` before, I could equally have done this" ] }, { "cell_type": "code", "execution_count": 4, "id": "balanced-generic", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "30" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "10+ 20" ] }, { "cell_type": "markdown", "id": "toxic-sphere", "metadata": {}, "source": [ "or this" ] }, { "cell_type": "code", "execution_count": 5, "id": "banned-parish", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "30" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "10+20" ] }, { "cell_type": "markdown", "id": "respiratory-poster", "metadata": {}, "source": [ "and I would get exactly the same answer. However, by now the people reading this who are already familiar with Python are howling in anger at my characterization of Python as flexible with spacing, because there are other ways in which Python is *notoriously* inflexible when it comes to spaces. In certain (ok, in very many) situations, Python cares a *lot* about where there are and are not spaces, and how many spaces there are. We will get to these situations later, but for now just be aware that although Python is fairly liberal when it comes to spaces within a line of code, it is zealously conservative when it comes to indenting lines of code. There are strict rules for when and how lines of code are to be indented in Python, and Python will enforce these rules with an iron fist." ] }, { "cell_type": "markdown", "id": "informed-clarity", "metadata": {}, "source": [ "Whether you use `10+20` or `10 + 20` is a matter of preference. Personally, I prefer to put the spaces in, because I like the way it looks better, but I will probably write it both ways in this book. Sometimes I get lazy and leave the spaces out." ] }, { "cell_type": "markdown", "id": "worst-leonard", "metadata": {}, "source": [ "(arithmetic)=\n", "\n", "## Doing simple calculations with Python\n", "\n", "Okay, now that we've discussed some of the tedious details associated with typing Python commands, let's get back to learning how to use some of the most powerful and flexible software in the world as a \\$2 calculator. So far, all we know how to do is addition. Clearly, a calculator that only did addition would be a bit stupid, so I should tell you about how to perform other simple calculations using Python. But first, some more terminology. Addition is an example of an \"operation\" that you can perform (specifically, an arithmetic operation), and the ***operator*** that performs it is `+`. To people with a programming or mathematics background, this terminology probably feels pretty natural, but to other people it might feel like I'm trying to make something very simple (addition) sound more complicated than it is (by calling it an arithmetic operation). To some extent, that's true: if addition was the only operation that we were interested in, it'd be a bit silly to introduce all this extra terminology. However, as we go along, we'll start using more and more different kinds of operations, so it's probably a good idea to get the language straight now, while we're still talking about very familiar concepts like addition! " ] }, { "cell_type": "markdown", "id": "public-archive", "metadata": {}, "source": [ "### Adding, subtracting, multiplying and dividing\n", "\n", "So, now that we have the terminology, let's learn how to perform some arithmetic operations in Python. To that end, the table below lists the operators that correspond to the basic arithmetic we learned in primary school: addition, subtraction, multiplication and division." ] }, { "cell_type": "markdown", "id": "dependent-farming", "metadata": {}, "source": [ "| operation | operator | example input | example output |\n", "| :---------: | :------: | :-----------: | :------------: |\n", "| addition | + | 10 + 2 | 12 |\n", "| subtraction | - | 9 - 3 | 6 |\n", "| multiplication | * | 5 * 5 | 25 |\n", "| division | / | 10 / 3 | 3.333 |\n", "| power | ** | 5 ** 2 | 25 |\n" ] }, { "cell_type": "markdown", "id": "weekly-amino", "metadata": {}, "source": [ "As you can see, Python uses fairly standard symbols to denote each of the different operations you might want to perform: addition is done using the `+` operator, subtraction is performed by the `-` operator, and so on. So if I wanted to find out what 57 times 61 is (and who wouldn't?), I can use Python instead of a calculator, like so:" ] }, { "cell_type": "code", "execution_count": 6, "id": "static-breed", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3477" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "57 * 61" ] }, { "cell_type": "markdown", "id": "experienced-relief", "metadata": {}, "source": [ "So that's handy.\n", "\n", "### Taking powers\n", "\n", "\n", "The first four operations listed above are things we all learned in primary school, but they aren't the only arithmetic operations built into Python. There are three other arithmetic operations that I should probably mention: taking powers, doing integer division, and calculating a modulus. Of the three, the only one that is of any real importance for the purposes of this book is taking powers, so I'll discuss that one here.\n", "\n", "For those of you who can still remember your high school maths, this should be familiar. But for some people high school maths was a long time ago, and others of us didn't listen very hard in high school. It's not complicated. As I'm sure everyone will probably remember the moment they read this, the act of multiplying a number $x$ by itself $n$ times is called \"raising $x$ to the $n$-th power\". Mathematically, this is written as $x^n$. Some values of $n$ have special names: in particular $x^2$ is called $x$-squared, and $x^3$ is called $x$-cubed. So, the 4th power of 5 is calculated like this:\n", "\n", "$$\n", "5^4 = 5 \\times 5 \\times 5 \\times 5 \n", "$$" ] }, { "cell_type": "markdown", "id": "present-advice", "metadata": {}, "source": [ "One way that we could calculate $5^4$ in Python would be to type in the complete multiplication as it is shown in the equation above. That is, we could do this" ] }, { "cell_type": "code", "execution_count": 7, "id": "conscious-third", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "625" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "5 * 5 * 5 * 5" ] }, { "cell_type": "markdown", "id": "medical-examination", "metadata": {}, "source": [ "but it does seem a bit tedious. It would be very annoying indeed if you wanted to calculate $5^{15}$, since the command would end up being quite long. Therefore, to make our lives easier, we use the power operator instead. When we do that, our command to calculate $5^4$ goes like this:" ] }, { "cell_type": "code", "execution_count": 8, "id": "defensive-sullivan", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "625" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "5 ** 4" ] }, { "cell_type": "markdown", "id": "selected-optimum", "metadata": {}, "source": [ "Much easier.[^note3]\n", "\n", "[^note3]: In fact, Python has two other ways to calculate powers. We could also have written `pow(5,4)`, or imported the `math` module and then written `math.pow(5,4)`. If you try this, you will see that a key difference is that `math.pow` will always return a number with a decimal place. In computer speak, `math.pow()` converts the numbers you enter into it into \"floating-point values\". Just FYI." ] }, { "cell_type": "markdown", "id": "norman-afternoon", "metadata": {}, "source": [ "(bedmas)=\n", "\n", "### Doing calculations in the right order\n", "\n", "Okay. At this point, you know how to take one of the most powerful pieces of statistical software in the world, and use it as a \\$2 calculator. And as a bonus, you've learned a few very basic programming concepts. That's not nothing (you could argue that you've just saved yourself \\$2) but on the other hand, it's not very much either. In order to use Python more effectively, we need to introduce more programming concepts.\n", "\n", "In most situations where you would want to use a calculator, you might want to do multiple calculations. Python lets you do this, just by typing in longer commands. [^note4] In fact, we've already seen an example of this earlier, when I typed in `5 * 5 * 5 * 5`. However, let's try a slightly different example:\n", "\n", "[^note4]: If you're reading this with Python open, a good learning trick is to try typing in a few different variations on what I've done here. If you experiment with your commands, you'll quickly learn what works and what doesn't." ] }, { "cell_type": "code", "execution_count": 9, "id": "contemporary-prior", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "9" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "1 + 2 * 4" ] }, { "cell_type": "markdown", "id": "expressed-jason", "metadata": {}, "source": [ "Clearly, this isn't a problem for Python either. However, it's worth stopping for a second, and thinking about what Python just did. Clearly, since it gave us an answer of `9` it must have multiplied `2 * 4` (to get an interim answer of 8) and then added 1 to that. But, suppose it had decided to just go from left to right: if Python had decided instead to add `1+2` (to get an interim answer of 3) and then multiplied by 4, it would have come up with an answer of `12`. How did it know what to do?\n", "\n", "To answer this, you need to know the **_order of operations_** that Python uses. If you remember back to your high school maths classes, it's actually the same order that you got taught when you were at school: the \"**_BEDMAS_**\" order. That is, first calculate things inside **B**rackets `()`, then calculate **E**xponents `**`, then **D**ivision `/` and **M**ultiplication `*`, then **A**ddition `+` and **S**ubtraction `-`. So, to continue the example above, if we want to force Python to calculate the `1+2` part before the multiplication, all we would have to do is enclose it in brackets:" ] }, { "cell_type": "code", "execution_count": 10, "id": "subsequent-private", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "12" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(1 + 2) * 4 " ] }, { "cell_type": "markdown", "id": "partial-browse", "metadata": {}, "source": [ "This is a fairly useful thing to be able to do. The only other thing I should point out about order of operations is what to expect when you have two operations that have the same priority: that is, how does Python resolve ties? For instance, multiplication and division are actually the same priority, but what should we expect when we give Python a problem like `4 / 2 * 3` to solve? If it evaluates the multiplication first and then the division, it would calculate a value of two-thirds. But if it evaluates the division first it calculates a value of 6. The answer, in this case, is that Python goes from *left to right*, so in this case the division step would come first:" ] }, { "cell_type": "code", "execution_count": 11, "id": "ruled-marker", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "6.0" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "4 / 2 * 3" ] }, { "cell_type": "markdown", "id": "applicable-exchange", "metadata": {}, "source": [ "All of the above being said, it's helpful to remember that *brackets always come first*. So, if you're ever unsure about what order Python will do things in, an easy solution is to enclose the thing *you* want it to do first in brackets. There's nothing stopping you from typing `(4 / 2) * 3`. By enclosing the division in brackets we make it clear which thing is supposed to happen first. In this instance you wouldn't have needed to, since Python would have done the division first anyway, but when you're first starting out it's better to make sure Python does what you want!" ] }, { "cell_type": "markdown", "id": "wanted-attraction", "metadata": {}, "source": [ "(assign)=\n", "\n", "## Storing a number as a variable\n", "\n", "One of the most important things to be able to do in Python (or any programming language, for that matter) is to store information in **_variables_**. Variables in Python aren't exactly the same thing as the variables we talked about in the last chapter on research methods, but they are similar. At a conceptual level you can think of a variable as *label* for a certain piece of information, or even several different pieces of information. When doing statistical analysis in Python, all of your data (the variables you measured in your study) will be stored as variables in Python, but as well see later in the book you'll find that you end up creating variables for other things too. However, before we delve into all the messy details of data sets and statistical analysis, let's look at the very basics for how we create variables and work with them. " ] }, { "cell_type": "markdown", "id": "authorized-clearing", "metadata": {}, "source": [ "Since we've been working with numbers so far, let's start by creating variables to store our numbers. And since most people like concrete examples, let's invent one. Suppose I'm trying to calculate how much money I'm going to make from this book. There's several different numbers I might want to store. Firstly, I need to figure out how many copies I'll sell. This isn't exactly *Harry Potter*, so let's assume I'm only going to sell one copy per student in my class. That's 350 sales, so let's create a variable called `sales`. What I want to do is assign a **_value_** to my variable `sales`, and that value should be `350`. We do this by using the **_assignment operator_**, which is `=`. Here's how we do it:" ] }, { "cell_type": "code", "execution_count": 12, "id": "relative-richmond", "metadata": {}, "outputs": [], "source": [ "sales = 350" ] }, { "cell_type": "markdown", "id": "parental-filing", "metadata": {}, "source": [ "\n", "When you hit enter, Python doesn't print out any output. But behind the scenes, Python has created a new variable, with a range of properties. One of these is that it contains the value `350`. The simplest way to see what is currently stored in a variable is to just write the name of the variable, and hit enter:" ] }, { "cell_type": "code", "execution_count": 13, "id": "selected-generation", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "350" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sales" ] }, { "cell_type": "markdown", "id": "alpha-ukraine", "metadata": {}, "source": [ "It might seem obvious, but it is probably worth mentioning anyway, that direction matters here. Whatever is to the right of the `=` sign is assigned to the variable to the left of the `=` sign." ] }, { "cell_type": "markdown", "id": "small-malpractice", "metadata": {}, "source": [ "### Doing calculations using variables\n", "\n", "Okay, let's get back to my original story. In my quest to become rich, I've written this textbook. To figure out how good a strategy is, I've started creating some variables in Python. In addition to defining a `sales` variable that counts the number of copies I'm going to sell, I can also create a variable called `royalty`, indicating how much money I get per copy. Let's say that my royalties are about \\$7 per book:" ] }, { "cell_type": "code", "execution_count": 16, "id": "modified-constraint", "metadata": {}, "outputs": [], "source": [ "sales = 350\n", "royalty = 7" ] }, { "cell_type": "markdown", "id": "demanding-breakfast", "metadata": {}, "source": [ "The nice thing about variables (in fact, the whole point of having variables) is that we can do anything with a variable that we ought to be able to do with the information that it stores. That is, since Python allows me to multiply `350` by `7`" ] }, { "cell_type": "code", "execution_count": 17, "id": "falling-microphone", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2450" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "350 * 7" ] }, { "cell_type": "markdown", "id": "intelligent-manufacturer", "metadata": {}, "source": [ "it also allows me to multiply `sales` by `royalty`" ] }, { "cell_type": "code", "execution_count": 18, "id": "talented-indianapolis", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2450" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sales * royalty" ] }, { "cell_type": "markdown", "id": "western-handle", "metadata": {}, "source": [ "As far as Python is concerned, the `sales * royalty` command is the same as the `350 * 7` command. Not surprisingly, I can assign the output of this calculation to a new variable, which I'll call `revenue`. And when we do this, the new variable `revenue` gets the value `2450`. So let's do that, and then get Python to print out the value of `revenue` so that we can verify that it's done what we asked:" ] }, { "cell_type": "code", "execution_count": 19, "id": "falling-niagara", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2450\n" ] } ], "source": [ "revenue = sales * royalty\n", "print(revenue)" ] }, { "cell_type": "markdown", "id": "developmental-multimedia", "metadata": {}, "source": [ "That's fairly straightforward. A slightly more subtle thing we can do is reassign the value of my variable, based on its current value. For instance, suppose that one of my students (no doubt under the influence of psychotropic drugs) loves the book so much that he or she donates me an extra \\$550. The simplest way to capture this is by a command like this:" ] }, { "cell_type": "code", "execution_count": 18, "id": "undefined-desire", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3000" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "revenue = revenue + 550\n", "revenue" ] }, { "cell_type": "markdown", "id": "written-interference", "metadata": {}, "source": [ "In this calculation, Python has taken the old value of `revenue`[^note_print] (i.e., 2450) and added 550 to that value, producing a value of 3000. This new value is assigned to the `revenue` variable, overwriting its previous value. In any case, we now know that I'm expecting to make \\$3000 off this. Pretty sweet, I thinks to myself. Or at least, that's what I thinks until I do a few more calculations and work out what the implied hourly wage I'm making off this looks like. \n", "\n", "[^note_print]: Just as an aside, you may have noticed (of course you did!) that the first time I wrote `print(revenue)` and the second time I just wrote `revenue`. This works, because in jupyter notebooks, if you write the variable name by itself, it will output the contents of that variable, just as if you had asked it `print` the variable. This only works for the last variable in the cell though, so you if want to see the contents of several variables, you will have to use `print`" ] }, { "cell_type": "code", "execution_count": 34, "id": "prescribed-memorial", "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "id": "utility-korean", "metadata": {}, "outputs": [], "source": [ "\n" ] }, { "cell_type": "code", "execution_count": null, "id": "african-rocket", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.5" } }, "nbformat": 4, "nbformat_minor": 5 } { "cells": [ { "cell_type": "markdown", "id": "needed-confidentiality", "metadata": {}, "source": [ "# More Python Concepts" ] }, { "cell_type": "markdown", "id": "effective-flour", "metadata": {}, "source": [ "(mechanics)=\n", "\n", "> *Form follows function*\n", ">\n", ">-- Louis Sullivan" ] }, { "cell_type": "markdown", "id": "organizational-parks", "metadata": {}, "source": [ "In the chapter on [Getting started with Python](getting-started-with-python) our main goal was to, well, get started with Python. As we go through the book we'll run into a lot of new Python concepts, which I'll explain alongside the relevant data analysis concepts. However, there's still quite a few things that I need to talk about now, otherwise we'll run into problems when we start trying to work with data and do statistics. So that's the goal in this chapter: to build on the introductory stuff from the last chapter, to get you to the point that we can start using Python for statistics. Broadly speaking, the chapter comes in two parts. The first half of the chapter is devoted to the \"mechanics\" of Python: installing and loading packages, managing the workspace, navigating the file system, and loading and saving data. In the second half, I'll talk more about what kinds of variables exist in Python, and introduce three new kinds of variables: factors, data frames and formulas. I'll finish up by talking a little bit about the help documentation in Python as well as some other avenues for finding assistance. In general, I'm not trying to be comprehensive in this chapter, I'm trying to make sure that you've got the basic foundations needed to tackle the content that comes later in the book. However, a lot of the topics are revisited in more detail later. " ] }, { "cell_type": "markdown", "id": "blocked-first", "metadata": {}, "source": [ "## Using comments\n", "\n", "Before discussing any of the more complicated stuff, I want to introduce the **_comment_** character, `#`. It has a simple meaning: it tells Python to ignore everything else you've written on this line. You won't have much need of the `#` character immediately, but it's very useful later on when writing scripts. However, while you don't need to use it, I want to be able to include comments in my Python extracts. For instance, if you read this:\n" ] }, { "cell_type": "code", "execution_count": 1, "id": "abstract-vermont", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "8.539539450000001\n" ] } ], "source": [ "\n", "seeker = 3.1415 # create the first variable\n", "lover = 2.7183 # create the second variable\n", "keeper = seeker * lover # now multiply them to create a third one\n", "print(keeper) # print out the value of 'keeper'\n" ] }, { "cell_type": "markdown", "id": "headed-corpus", "metadata": {}, "source": [ "it's a lot easier to understand what I'm doing than if I just write this:" ] }, { "cell_type": "code", "execution_count": 2, "id": "mobile-latitude", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "8.539539450000001\n" ] } ], "source": [ "\n", "seeker = 3.1415\n", "lover = 2.7183\n", "keeper = seeker * lover\n", "print(keeper) \n" ] }, { "cell_type": "markdown", "id": "fleet-server", "metadata": {}, "source": [ "From now on, you'll start seeing `#` characters appearing in the code extracts, with some human-readable explanatory remarks next to them. These are still perfectly legitimate commands, since Python knows that it should ignore the `#` character and everything after it. But hopefully they'll help make things a little easier to understand." ] }, { "cell_type": "markdown", "id": "blocked-bread", "metadata": {}, "source": [ "(packageinstall)=\n", "\n", "## Installing and importing \n", "\n", "\n", "There is lots to love about Python as a programming language. Although it has its quirks and peculiarities like any language (programming or natural), it is relatively flexible and welcoming to newcomers, while still very, very powerful. But one of the best things about Python isn't even the language itself, it is the rich ecosystem of code written by other people that you can use to make Python do things for you. These **libraries** or **packages** [^note2] contain code that people have written to solve particular problems, and then kindly made available for other people, like you and me, so that we don't have to spend our time reinventing the wheel. By installing and importing libraries, you can achieve very complicated things with only a few lines of your own code, by standing on the shoulders of others. Just ask Cueball from the webcomic xkcd:[^note3]\n", "\n", "[^note2]: There are some subtle differences between libraries, packages, and modules, but we don't need to concern ourselves with these here, and I may well mix up these words in the text. The key thing is, they are bits of code that we need to import to make stuff happen in Python. \n", "[^note3]: https://xkcd.com/353/" ] }, { "cell_type": "markdown", "id": "technological-cholesterol", "metadata": {}, "source": [ "![xkcdpython](https://imgs.xkcd.com/comics/python.png)" ] }, { "cell_type": "markdown", "id": "complicated-death", "metadata": {}, "source": [ "When doing anything other than the very most basic forms of data analysis in Python, we will almost always need to use libraries. However, before we get started, there's a critical distinction that you need to understand, which is the difference between having a package **_installed_** on your computer, and having a package **_imported_** in Python. I do not have any idea how many Python libraries are available out there, but it is a lot. Thousands. If you install Python on your computer, you won't get all of them, just a handfull of the standard ones. Depending on how you install Python on your computer, you may have more or fewer libraries installed, but either way, there are thousands more out there that you do not currently have installed. So that's what installed means: it means \"it's on your computer somewhere\". The critical thing to remember is that just because something is on your computer doesn't mean Python can use it. In order for Python to be able to *use* one of your installed libraries, that library must also be \"imported\". Basically what it boils down to is this:\n", "\n", ">A library must be installed before it can be imported.\n", "\n", ">A library must be imported before it can be used.\n", "\n", "This two step process might seem a little odd at first, but the designers of Python had very good reasons to do it this way,[^note4] and you get the hang of it pretty quickly.\n", "\n", "I won't get into the details of installing libraries here, simply because it is too much for me to tackle. If you are using Python in an online enviroment, you may already have access to all the libraries mentioned in this book. If you are working with Python on your own computer, the exact details of how you install packages may vary. If you want to use Python on your own computer, and are just getting started, I recommend [Anaconda](https://www.anaconda.com/products/individual#Downloads) as a relatively easy way to install Python and get quick access to all the most common and important libraries.\n", "\n", "[^note4]: Basically, the reason is that there are thousands of libraries, and probably thousands of authors of libraries, and no-one really knows what all of them do. Keeping the installation separate from the loading minimizes the chances that two libraries will interact with each other in a nasty way, like using the same name for different things. " ] }, { "cell_type": "markdown", "id": "functional-russian", "metadata": {}, "source": [ "### What libraries does this book use?\n", "\n", "In this book, I have made a concerted effort to limit the number of libraries needed. Often you will find that you can use different libraries to achieve the same results, and sometimes one of these may suit your needs more than another. This is something that can make doing analysis by code rather than pointing and clicking in a dedicated statistics program a bit off-putting; in Excel, there is usually only one way to do things, while in Python, there are many. I think this is part of what makes doing statistics using code better, though: you can make your own informed choices, and do *exactly* the analysis you want to do; you don't have to accept some piece of software's default settings. However, the point of this book is to get you started doing data analysis and statistics in Python, not to show you all the different ways you could achieve the same goal, so in an effort to keep things simple, I have tried to limit the libraries used in this book to a few of the most important and most common ones for doing statistics with Python. The libraries we will return to again and again are: `numpy`, `pandas`, `matplotlib`, `seaborn`, `statistics`, `math`, and `pingouin`, but I may use others as well, as needed. Other powerful libraries that I won't make much use of in this book, but that you will probably end up using at some point if you continue to do statistics with Python include `scipy`, `statmodels`, and `scikit-learn`.\n", "\n", "### How will I import libraries in this book?\n", "\n", "Once you import a library into Python's active memory, you don't need to do it again. In writing this book, each chapter is a python file [^note5]. So, if I have imported e.g. `numpy` early in the chapter, I don't need to do it again in a later section of the chapter. But, normally I will, because I want the code snippets in this book to be as easy as possible to copy and paste into your own computer. If I don't put the `import` command at the top of the snippet, and you have not already imported the library, then you might copy and paste my code into your computer and get an error message. Then again, sometimes I might forget to put the import statement in, or I might think it should be obvious, or I might just get lazy, so make sure to keep an eye out for this!\n", "\n", "[^note5]: Well, actually it is an `.ipynb` file, but let's not bicker and argue about who killed who." ] }, { "cell_type": "markdown", "id": "exterior-columbus", "metadata": {}, "source": [ "(importinglibraries)=\n", "### Importing libraries\n", "\n", "Assuming you have the libraries you need installed on your computer, or can access them in the virtual Python environment you are using in your browser, you will need to import them before you can actually use them. So, for instance, if I want to find the sum of five numbers, I can write" ] }, { "cell_type": "code", "execution_count": 3, "id": "patient-lancaster", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "18" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "numbers = [4, 5, 1, 2, 6]\n", "sum(numbers)" ] }, { "cell_type": "markdown", "id": "tracked-nothing", "metadata": {}, "source": [ "because the authors of Python felt that adding numbers together was such a basic thing that there should be a built-in command for it. At least, I assume so. I don't really know what the authors of Python thought. But, oddly enough, Python doesn't have a built-in command called \"mean\". So if I want to know the mean of those same five numbers, I cannot just write" ] }, { "cell_type": "code", "execution_count": 4, "id": "federal-smith", "metadata": {}, "outputs": [ { "ename": "NameError", "evalue": "name 'mean' is not defined", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", "Input \u001b[0;32mIn [4]\u001b[0m, in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0m \u001b[43mmean\u001b[49m(numbers)\n", "\u001b[0;31mNameError\u001b[0m: name 'mean' is not defined" ] } ], "source": [ "mean(numbers)" ] }, { "cell_type": "markdown", "id": "artificial-interval", "metadata": {}, "source": [ "because Python doesn't know what `mean`, er, means. Luckily, we don't have to resort to first finding the sum and then dividing by the number of numbers, because there are libraries that _do_ have built-in commands for finding means. The `statistics` library is one. To use the commands in this library, we first have to `import` it. This gives us access to all the many useful commands in the `statistics` library, one of which is `mean`:" ] }, { "cell_type": "code", "execution_count": 5, "id": "hispanic-african", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3.6" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import statistics\n", "\n", "numbers = [4, 5, 1, 2, 6]\n", "\n", "statistics.mean(numbers)" ] }, { "cell_type": "markdown", "id": "weighted-federation", "metadata": {}, "source": [ "You probably noticed the `.` in the code above. This is the way we tell Python that we want to use a command called `mean` which is found inside the library `statistics`. Without the `.`, even though we have imported `statistics`, which has a command called `mean`, we still can't just write `mean(numbers)`. We have to tell Python where to look for this command. This all seems very cumbersome, but it's really not so bad, there are good reasons for doing it this way[^noteconflicts], and you will get used to it fairly quickly.\n", "\n", "[^noteconflicts]: Chiefly among them, this helps avoid unpleasant situations where we may have imported two different libraries that each contain a function with the same name, but which do different things, or worse yet, do different things but _appear_ to do the same thing, which could really get us in trouble!" ] }, { "cell_type": "markdown", "id": "italian-airfare", "metadata": {}, "source": [ "One of the ways in which Python is quite flexible is that it gives you some options in terms of how you import libraries. More precisely, you can:\n", "\n", "1. Choose to import only a portion of a library \n", "2. Rename libraries or portions of libraries when importing\n", "\n", "Let's say we don't want to import the entire `statistics` library — we only want the `mean` command. We can achieve this like this:" ] }, { "cell_type": "code", "execution_count": 6, "id": "greenhouse-specification", "metadata": {}, "outputs": [], "source": [ "from statistics import mean" ] }, { "cell_type": "markdown", "id": "japanese-drive", "metadata": {}, "source": [ "Why would we want to do this? Well, one good reason is that now we *can* simply write `mean(numbers)`; we no longer have to write out `statistics.mean(numbers)`:" ] }, { "cell_type": "code", "execution_count": 7, "id": "smart-animal", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3.6" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "numbers = [4, 5, 1, 2, 6]\n", "mean(numbers)" ] }, { "cell_type": "markdown", "id": "compound-malta", "metadata": {}, "source": [ "Is this the height of laziness? Maybe. But if you start writing the same thing over and over again, saving a few characters here and there is pretty sweet. And this brings us to the other import option: renaming libraries. It is common practice in Python to give libraries abbreviations when we import them. Many of the most common libraries have conventional abbreviations, although you could use anything you like. Thus, you will often see e.g." ] }, { "cell_type": "code", "execution_count": 8, "id": "careful-biography", "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import seaborn as sns" ] }, { "cell_type": "markdown", "id": "hearing-appreciation", "metadata": {}, "source": [ "This is very convenient, but be careful: if you e.g. import `numpy` as `np`, then Python will only recognize it as `np`, at least for the time your code is in Python's active memory. Also, although you can use whatever abbreviations you like, I highly recommend sticking to the conventional ones, for your sake and for the sake of others trying to read your code. It's kind of fun the first time to do something like" ] }, { "cell_type": "code", "execution_count": 9, "id": "advisory-mainstream", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3.6" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import statistics as why_you_gotta_be_so\n", "\n", "why_you_gotta_be_so.mean(numbers)" ] }, { "cell_type": "markdown", "id": "close-journal", "metadata": {}, "source": [ "but good code should be easy to read by yourself and others, and if you start playing too fast and loose with renaming, it starts to get less clear what's going on." ] }, { "cell_type": "markdown", "id": "consistent-certificate", "metadata": {}, "source": [ "## Listing the objects in active memory\n", "\n", "Let's suppose that you're reading through this book, and what you're doing is sitting down with it once a week and working through a whole chapter in each sitting. Not only that, you've been following my advice and typing in all these commands into Python. So far during this chapter, you'd have typed quite a few commands, although not all of them actually created variables.\n", "\n", "An important part of learning to program is to develop the ability to keep a mental model of what Python knows and doesn't know at any given time active in your mind. This sounds very abstract, and it is, but as you become more familar with coding I think you will see what I mean. I won't dwell on this here, but it may be useful to take a quick peak at what I mean. If you are working in e.g. a Jupyter Notebook (and I do suggest you do this, at least at first), then by typing `%who` you can see a list of all the variables that Python is currently aware of. So, in my case, I get the following:" ] }, { "cell_type": "code", "execution_count": 10, "id": "static-trinidad", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "keeper\t lover\t mean\t np\t numbers\t seeker\t sns\t statistics\t why_you_gotta_be_so\t \n", "\n" ] } ], "source": [ "%who" ] }, { "cell_type": "markdown", "id": "major-citation", "metadata": {}, "source": [ "Here we can see variables that we defined, like `keeper` and `lover`, and also libraries that we imported (and renamed), like `np` and `sns`, as well as the library `statistics` which I then ill-advisedly re-imported and renamed `why_you_gotta_be_so`. To see more details on these variables, we can type `%whos`" ] }, { "cell_type": "code", "execution_count": 11, "id": "excess-video", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Variable Type Data/Info\n", "-------------------------------------------\n", "keeper float 8.539539450000001\n", "lover float 2.7183\n", "mean function \n", "np module kages/numpy/__init__.py'>\n", "numbers list n=5\n", "seeker float 3.1415\n", "sns module ges/seaborn/__init__.py'>\n", "statistics module ython3.10/statistics.py'>\n", "why_you_gotta_be_so module ython3.10/statistics.py'>\n" ] } ], "source": [ "%whos" ] }, { "cell_type": "markdown", "id": "sudden-board", "metadata": {}, "source": [ "This tells us that e.g. `keeper` is a floating-point decimal number with the value 8.539539450000001, and shows us the true names of the objects we have renamed on import. These commands that start with a `%` sign, by the way, are called \"magic\" commands, and can only be used in environments like Jupyter, which support them. If you are not working in such an environment, you can use the command `dir()`, which achieves the same thing, but will also show you lots of information you probably aren't interested in at this stage." ] }, { "cell_type": "markdown", "id": "stunning-point", "metadata": {}, "source": [ "### Removing variables from the workspace\n", "\n", "Looking over that list of variables, it occurs to me that I really don't need them any more. I created them originally just to make a point, but they don't serve any useful purpose anymore, and now I want to get rid of them. I'll show you how to do this, but first I want to warn you -- there's no \"undo\" option for variable removal. Once a variable is removed, it's gone forever. Most of the time this doesn't matter, though, because we still have the code we used to create the variable, so unless you delete that too, you can always just re-run the cells in your Jupyter Notebook, and Python will re-make your variables for you, just like they were before. But quite clearly we have no need for these variables any more, so we can safely get rid of them by using the `del()` command." ] }, { "cell_type": "code", "execution_count": 12, "id": "suburban-shield", "metadata": {}, "outputs": [], "source": [ "del(keeper, lover)" ] }, { "cell_type": "markdown", "id": "wrong-cotton", "metadata": {}, "source": [ "With `%who` or `dir()` we can check that they are gone." ] }, { "cell_type": "code", "execution_count": 13, "id": "middle-major", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "mean\t np\t numbers\t seeker\t sns\t statistics\t why_you_gotta_be_so\t \n" ] } ], "source": [ "%who" ] }, { "cell_type": "markdown", "id": "developed-conclusion", "metadata": {}, "source": [ "If you want to remove all the variables in memory, and you are working in a Jupyter environment, then `%reset` is a handy way to do this, although I must say that in practice I rarely, if ever, have a need to remove variables from memory. There is usually no harm in them sitting around unused, and if you define a new variable with the same name as an old one, it will just write over the old variable with the same name." ] }, { "cell_type": "markdown", "id": "thirty-circumstances", "metadata": {}, "source": [ "(load)=\n", "## Loading and saving data\n", "\n", "\n", "There are several types of files that are likely to be relevant to us when doing data analysis, and there are two in particular that are especially important from the perspective of this book:\n", "\n", "\n", "- *Comma separated value (CSV) files* are those with a .csv file extension. These are just regular old text files, and they can be opened with almost any software. It's quite typical for people to store data in CSV files, precisely because they're so simple.\n", "- *Script files* are those with a .py file extension or an .ipynb extension. These aren't data files at all; rather, they're used to save a collection of commands that you want Python to execute later. They're just text files, but we won't make use of them until later.\n", " \n", "\n", "In this section I'll talk about how to import data from a CSV file. First though, we need to make a quick digression, and talk about file systems. I know this is not a very exciting topic, but it is absolutely _critical_ to doing data analysis. If you want to work with your data in Python, you need to be able to tell Python where the data is located." ] }, { "cell_type": "markdown", "id": "legendary-century", "metadata": {}, "source": [ "(filesystem)=\n", "### Filesystem paths\n", "\n", "In this section I describe the basic idea behind file locations and file paths. Regardless of whether you're using Window, macOS or Linux, every file on the computer is assigned a (fairly) human readable address, and every address has the same basic structure: it describes a *path* that starts from a *root* location, through as series of *folders* (or if you're an old-school computer user, *directories*), and finally ends up at the file. \n", "\n", "On a Windows computer the root is the physical drive [^notedrive] on which the file is stored, and for most home computers the name of the hard drive that stores all your files is C: and therefore most file paths on Windows begin with C:. After that come the folders, and on Windows the folder names are separated by a `\\` symbol. So, the complete path to this book on my Windows computer might be something like this:\n", "\n", "```\n", "C:\\Users\\ethan\\pythonbook\\book\\LSP.pdf\n", "```\n", "and what that *means* is that the book is called LSP.pdf, and it's in a folder called `book` which itself is in a folder called `ethan` which itself is ... well, you get the idea. On Linux, Unix and Mac OS systems, the addresses look a little different, but they're more or less identical in spirit. Instead of using the backslash, folders are separated using a forward slash, and unlike Windows, they don't treat the physical drive as being the root of the file system. So, the path to this book on my Mac might be something like this:\n", "\n", "```\n", "/Users/ethan/pythonbook/book/LSP.pdf\n", "```\n", "\n", "So that's what we mean by the \"path\" to a file, and before we move on, it is critical that you learn how to copy the path to a file on your computer so that you can paste it into Python. There are (again!) multiple ways to do this on the various operating systems, and it doesn't really matter which method you use. A quick search will lead you to many many online tutorials; just find a method that works for you, on your computer. Even if you are working in a virtual Python environment in the browser (such as Google Colab), you will need some concept of file structure to navigate to your data and scripts.\n", "\n", "[^notedrive]: Well, the partition, technically." ] }, { "cell_type": "markdown", "id": "experimental-craft", "metadata": {}, "source": [ "(loadingcsv)=\n", "\n", "### Loading data from CSV files into Python\n", "\n", "One quite commonly used data format is the humble \"comma separated value\" file, also called a CSV file, and usually bearing the file extension .csv. CSV files are just plain old-fashioned text files, and what they store is basically just a table of data. This is illustrated below, which shows a file called booksales.csv that I've created. As you can see, each row corresponds to a variable, and each row represents the book sales data for one month. The first row doesn't contain actual data though: it has the names of the variables.\n", "\n", " \n", " " ] }, { "cell_type": "markdown", "id": "twenty-surge", "metadata": {}, "source": [ "```{image} ../img/mechanics/booksalescsv.png\n", ":alt: booksales\n", ":width: 600px\n", ":align: center\n", "```" ] }, { "cell_type": "markdown", "id": "recovered-filling", "metadata": {}, "source": [ "As is often the case, there are many different ways to get the data from a CSV file into Python so that you can begin doing things with it. Here we will use the `pandas` library, which happens to have a handy command called `read_csv()` which does just what it says.\n", "\n", "We can't just read the data in willy-nilly, though. We need some place to put it. As you may have already guessed, we need to define a variable to put our data into. We haven't talked about variable types yet, and now is not the time, but let it suffice to say that there are different kinds of variables, and some of them can store structured data like the rows and columns in a CSV file. `pandas` calls this kind of variable a [\"dataframe\"](pandas). You can name your dataframe whatever you like, of course, but by convention they are often called \"df\", so we'll do that too. Thus:" ] }, { "cell_type": "code", "execution_count": 14, "id": "norwegian-drove", "metadata": {}, "outputs": [], "source": [ "# import pandas, and call it \"pd\" for short\n", "\n", "import pandas as pd\n", "\n", "# make a new dataframe variable, and use the \"read_csv\" command from the pandas library to put the contents\n", "# of the file located at \"/Users/ethan/Documents/GitHub/pythonbook/Data/booksales.csv\" in the dataframe df\n", "\n", "df = pd.read_csv(\"/Users/ethan/Documents/GitHub/pythonbook/Data/booksales.csv\")\n" ] }, { "cell_type": "markdown", "id": "included-clothing", "metadata": {}, "source": [ "Here I have put the full path into the parentheses following `pd.read_csv`, but often I prefer to save the path as a variable, and put that variable into the parentheses instead, like this:" ] }, { "cell_type": "code", "execution_count": 15, "id": "beneficial-basis", "metadata": {}, "outputs": [], "source": [ "import pandas as pd\n", "\n", "file = \"/Users/ethan/Documents/GitHub/pythonbook/Data/booksales.csv\"\n", "\n", "df = pd.read_csv(file)" ] }, { "cell_type": "markdown", "id": "responsible-prison", "metadata": {}, "source": [ "Either way works, but I think this looks nicer, and it also has an additional advantage: it makes the code more versatile. Right now we are just loading a single CSV, but if we want to load many CSV files, it might be useful to write a [loop](loops) that puts different paths into the same variable `file`. But that is a discussion for [another time](loops)." ] }, { "cell_type": "markdown", "id": "increased-reset", "metadata": {}, "source": [ "### Saving a dataframe as a CSV\n", "\n", "Sometimes we create new dataframes using Python. Maybe the CSV we loaded had lots of information that we don't need, or maybe we have loaded several CSV's in, taken a few columns of data from each one, and then combined these into a new dataframe. Or perhaps we have done some calculations on the original data and added a column with e.g. the sum of each row. If we want to save this new dataframe as a csv, `pandas` has a command for that as well:\n", "\n", "``df.to_csv('/Users/ethan/Desktop/my_file.csv')``\n", "\n", "Every `pandas` dataframe has the built-in ability to be exported as a CSV file. We just need to tell Python what the new file should be called, and where we want it to go in our filesystem. Pretty straightforward, really." ] }, { "cell_type": "markdown", "id": "sharp-lingerie", "metadata": {}, "source": [ "(useful)=\n", "## Useful things to know about variables\n", "\n", "In the chapter on [Getting started with Python](getting-started-with-python), I talked a about variables, how they're assigned and some of the things you can do with them, but there are a lot of additional complexities. That probably comes as no great surprise, and we can't go over all of these now, but this section is basically just a bunch of things that I want to briefly mention, that don't really fit in anywhere else. In short, now I'll talk about several different issues in this section, which are only loosely connected to one another." ] }, { "cell_type": "markdown", "id": "nonprofit-million", "metadata": {}, "source": [ "(types)=\n", "## Variable types\n", "\n", "As we've seen, Python allows you to store different kinds of data. We have seen variables that store text (_strings_), numbers (_integers_ or _floats_), and even whole datasets (_dataframes_). These are just three of the many different types of variable that Python can store. Other common variable types in Python include _dictionaries_, _lists_, and _tuples_. It's important that we remember what kind of information each variable stores (and even more important that Python remembers) since different kinds of variables allow you to do different things to them. For instance, if your variables have numerical information in them, then it's okay to add them together:" ] }, { "cell_type": "code", "execution_count": 16, "id": "addressed-effort", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "3\n" ] } ], "source": [ "x = 1 # a is a number\n", "y = 2 # b is a number\n", "z = x + y\n", "print(z)" ] }, { "cell_type": "markdown", "id": "furnished-tyler", "metadata": {}, "source": [ "If they contain character data (strings), Python will still let you add the variables, but the outcome might be unexpected:" ] }, { "cell_type": "code", "execution_count": 17, "id": "union-literature", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'12'" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = \"1\" # x is string, as indicated by the quotation marks\n", "y = \"2\" # y is string, as indicated by the quotation marks\n", "x + y " ] }, { "cell_type": "markdown", "id": "muslim-madison", "metadata": {}, "source": [ "To us, there isn't really that big a difference between 1 and \"1\", but to Python, these are entirely different classes of things." ] }, { "cell_type": "markdown", "id": "genuine-antenna", "metadata": {}, "source": [ "### Checking variable types\n", "\n", "Okay, let's suppose that I've forgotten what kind of data I stored in the variable `x` (which happens depressingly often). Python provides a function that will let us find out: `type()`" ] }, { "cell_type": "code", "execution_count": 18, "id": "gross-covering", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "str" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = \"hello world\" # x is text (aka a \"string\")\n", "\n", "type(x)" ] }, { "cell_type": "code", "execution_count": 1, "id": "orange-morgan", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "pandas.core.frame.DataFrame" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "file = \"/Users/ethan/Documents/GitHub/pythonbook/Data/booksales.csv\"\n", "\n", "x = pd.read_csv(file) # x is a dataframe\n", "\n", "type(x)" ] }, { "cell_type": "code", "execution_count": 20, "id": "divided-teaching", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "int" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = 100 # x is an integer\n", "type(x)" ] }, { "cell_type": "code", "execution_count": 1, "id": "rural-separation", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "float" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = 3.14 # x is a floating-point decimal\n", "type(x)" ] }, { "cell_type": "markdown", "id": "published-romance", "metadata": {}, "source": [ "Exciting, no? Trust me, you'll need to do this more often than you might think." ] }, { "cell_type": "markdown", "id": "e9094937", "metadata": {}, "source": [ "Since Python packages can add their own variable types (like the `pandas.core.frame.DataFrame` variable we saw above), there can be quite a few different variable types in Python. But it is worth spending a little time getting to know the standard variable types in Python, so here I'll introduce you to a few of the usual suspects: integers, floats, strings, dictionaries, lists, and sets. These variable types are like the basic building blocks in Python, and once you become familar with them and get to know their quirks, foibles, and properties, everything else in Python becomes so much easier." ] }, { "cell_type": "markdown", "id": "179a4c73", "metadata": {}, "source": [ "## Regular variables and collection variables\n", "\n", "There are two main types of variable in Python, which I will call \"regular variables\" and \"collection variables\". \n", "\n", "The regular variables are:\n", "\n", "1. Integers\n", "2. Floats\n", "3. Strings\n", "\n", "\n", "The collection variables are:\n", "\n", "1. Lists\n", "2. Tuples\n", "3. Sets\n", "4. Dictionaries\n", "\n", "As I [mentioned](loadingcsv), there are also other kinds of collection variables, such as `pandas` dataframes provide more structure and allow for more complex, organized data storage and manipulation. But we'll get to these in due time, as needed. For now, I want to introduce you to the basic set of variables in Python." ] }, { "cell_type": "markdown", "id": "7110a55a", "metadata": {}, "source": [ "## Regular variables" ] }, { "cell_type": "markdown", "id": "8a5f4d73", "metadata": {}, "source": [ "(ints)=\n", "\n", "### Intergers" ] }, { "cell_type": "markdown", "id": "69818339", "metadata": {}, "source": [ "We have already spent some time with one of Python's most common variable types, the interger. Integers are pretty straightforward: they are whole numbers, like 2, 7, 4392, 0, and -34. There's not so much more to say about them. You can do all [the usual sorts of things](arithmetic) with integers that you might expect: you can add them, subtract them, multiply them, etc. To give a some quick examples:" ] }, { "cell_type": "code", "execution_count": 91, "id": "27fd4eb3", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Total number of animals: 103\n" ] } ], "source": [ "cows = 75\n", "pigs = 21\n", "hens = 7\n", "\n", "animals_total = cows + pigs + hens\n", "\n", "print(\"Total number of animals:\", animals_total)" ] }, { "cell_type": "markdown", "id": "067484eb", "metadata": {}, "source": [ "(floats)=\n", "\n", "### Floats" ] }, { "cell_type": "markdown", "id": "37344b77", "metadata": {}, "source": [ "On their face, floats are also pretty straightforward: they're decimal fractions, like 0.234, 0.333, or -0.000003. Most of the time, this level of understanding is all you need. Occasionally, though, you will run into some confusing behavior with floats, so it is worth saying just a few more words about them. There are two issues with floats which we should at least be aware of:\n", "\n", "1. Computers store decimal fractions as base 2 fractions (binary), not base 10, the way we are used to thinking of them.\n", "2. There are more than one way to round fractions, and Python may not round fractions the way you expect.\n", "\n", "Let's start by looking at some decimal fractions:" ] }, { "cell_type": "code", "execution_count": 92, "id": "ee301748", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.1\n" ] } ], "source": [ "a = 1/10\n", "print(a)" ] }, { "cell_type": "code", "execution_count": 93, "id": "ad1dd487", "metadata": { "scrolled": true }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.3333333333333333\n" ] } ], "source": [ "a = 1/3\n", "print(a)" ] }, { "cell_type": "markdown", "id": "5675df77", "metadata": {}, "source": [ "Nothing odd here. In base 10 math, 1 can be equally divided into ten parts, and the answer is 0.1\n", "\n", "But to our computers, the answer to 1 divided by 10 is stored as a `float` variable, and stored in base 2. Which means, that to Python 1 divided by 10 is actually 0.0001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001 1001....\n", "\n", "That is, to the computer, 1/10 is actually an infinitely repeating series, just like the answer to 1/3 is in the base 10 math we are used to. So when Python shows us that 1/10 = 0.1, it is really just rounding up, so that the answer looks like what we expect to see. This means that decimal fractions are not as precise as we might think, and explains why the answer Python gives us when we add decimals together might be surprising:" ] }, { "cell_type": "code", "execution_count": 94, "id": "c214f90d", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.30000000000000004" ] }, "execution_count": 94, "metadata": {}, "output_type": "execute_result" } ], "source": [ ".1 + .1 + .1" ] }, { "cell_type": "markdown", "id": "70a053d0", "metadata": {}, "source": [ "This answer makes no sense if we think that 0.1 is _exactly_ 0.1, but makes more sense if we understand that to Python, 0.1 is just an approximation of the answer to 1/10. Now, most of the time this is not going to matter much, but it _could_ make a difference, so it is good to be aware of it.\n", "\n", "Now, here's the other thing to be aware of with floats in Python. Often when we have a number with a decimal, we want to round it. So rather than 1.5, we might want to round to a whole number:" ] }, { "cell_type": "code", "execution_count": 95, "id": "a647457b", "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/plain": [ "2" ] }, "execution_count": 95, "metadata": {}, "output_type": "execute_result" } ], "source": [ "round(1.5)" ] }, { "cell_type": "markdown", "id": "fc46e1da", "metadata": {}, "source": [ "But, although most of us who aren't mathematicians or computer scientists or whatever can go blissfully through life being unaware of it, there are many [different strategies for rounding numbers](https://en.wikipedia.org/wiki/Rounding). Python uses the one known as \"[rounding half to even](https://en.wikipedia.org/wiki/Rounding#Rounding_half_to_even)\", which means that when there is a tie, that is, when the number we want to round is exactly halfway between two integers, Python will round to the nearest _even_ number. Now, unlike 0.1, which as we saw above, is stored as an approximation of an infinitely long series in base 2, 0.5 is actually stored as a precise number in base 2[^base2.5], which means that even to Python, 1.5 is exactly halfway between 1 and 2, and 2.5 is halfway between 2 and 3. And, when deciding ties Python rounds to the neares even number, we can get some surprising results:\n", "\n", "[^base2.5]: 0.5 is actually stored as 0.1 in base 2. Ha ha! Isn't math fun?" ] }, { "cell_type": "code", "execution_count": 96, "id": "4c11d18a", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1.5 rounded is: 2\n", "2.5 rounded is: 2\n" ] } ], "source": [ "print('1.5 rounded is:',round(1.5))\n", "print('2.5 rounded is:',round(2.5))" ] }, { "cell_type": "markdown", "id": "b98938b5", "metadata": {}, "source": [ "Great, Python. Just great -_-" ] }, { "cell_type": "markdown", "id": "3cc2b76e", "metadata": {}, "source": [ "We musn't get off an any further tangents here, because most of the time we don't have to think about these bits of esoterica, but I will mention just one last oddity about floats that you might conceivably run into at some point. Let's say we add the following two numbers:" ] }, { "cell_type": "code", "execution_count": 98, "id": "11e24edc", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2.000000000000001" ] }, "execution_count": 98, "metadata": {}, "output_type": "execute_result" } ], "source": [ "1 + 1.000000000000001" ] }, { "cell_type": "markdown", "id": "2860c5b9", "metadata": {}, "source": [ "So far so good. Now say we put in just one more zero:" ] }, { "cell_type": "code", "execution_count": 99, "id": "e343b247", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2.0" ] }, "execution_count": 99, "metadata": {}, "output_type": "execute_result" } ], "source": [ "1 + 1.0000000000000001" ] }, { "cell_type": "markdown", "id": "0ad116e6", "metadata": {}, "source": [ "Ugh. Really? Yup. If your computer has a 64-bit processor (and these days, it probaby does), there is a hard limit on the number of decimal places it can store. If you like, play around with it yourself, and see where these limits lie. Just for \"fun\", here are a few last examples:" ] }, { "cell_type": "code", "execution_count": 100, "id": "59166bbc", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "2.0\n", "2.000000000000001\n", "2.000000000000001\n" ] } ], "source": [ "print(1 + 1.0000000000000001)\n", "print(1 + 1.000000000000001)\n", "print(1 + 1.0000000000000009)" ] }, { "cell_type": "markdown", "id": "4bbb2efd", "metadata": {}, "source": [ "Alright. Enough floats. Mostly they work the way you would expect, but if they don't, now you know some of the possible reasons. Time to move on to something more fun: strings!" ] }, { "cell_type": "markdown", "id": "756c61dd", "metadata": {}, "source": [ "(strings)=\n", "\n", "### Strings" ] }, { "cell_type": "markdown", "id": "24cf7241", "metadata": {}, "source": [ "Strings are variables that store characters, like \"g\", \"h\", \"a\", and \"p\", but also \"cat\", \"apple\", and \"supercalifragilisticexpialidocious\". Strings can also be things like \"$\", \",\", \";\" and \"?\". Somewhat confusingly, strings can also be things like \"3\", \"42\", and \"-8374.87\". Maybe it is helpful to think of strings as being one of the roles these symbols can play, just like a book could be a gift, a paperweight, a doorstop, or a weapon. Ok, maybe that's not so helpful after all. Let's try some examples instead:" ] }, { "cell_type": "code", "execution_count": 101, "id": "2f349b4d", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "I am a string\n" ] } ], "source": [ "a = \"I am a string\"\n", "print(a)" ] }, { "cell_type": "code", "execution_count": 102, "id": "1bd7a8c2", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "I am a string\n" ] } ], "source": [ "a = \"I am a \"\n", "b = \"string\"\n", "c = a + b\n", "print(c)" ] }, { "cell_type": "code", "execution_count": 103, "id": "13e22869", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "3\n" ] } ], "source": [ "# These numbers are integers\n", "a = 1\n", "b = 2\n", "print(a + b)" ] }, { "cell_type": "code", "execution_count": 104, "id": "f62bc5d3", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "12\n" ] } ], "source": [ "# These numbers are stringss\n", "a = '1'\n", "b = '2'\n", "print(a + b)" ] }, { "cell_type": "markdown", "id": "d1b4428a", "metadata": {}, "source": [ "So we can add strings together, but the result is different than when we add integers together. To Python, the numbers in the cell above are just the _characters_ 1 and 2, not the _values_ 1 and 2. So it just sticks them together, just like it would any other strings, if you add them.\n", "\n", "Now, while we can add two strings together, we can't multiply two strings by each other, or subtract, or divide. What would that even mean. We _can_ multiply strings by an interger, though:" ] }, { "cell_type": "code", "execution_count": 105, "id": "cb7606aa", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "catcatcatcatcat\n" ] } ], "source": [ "a = 'cat'\n", "print(a*5)" ] }, { "cell_type": "markdown", "id": "d498d933", "metadata": {}, "source": [ "There are many more fun things we can do with strings; many more than I will go into here. I will give a few examples, though, just to whet your appetite:" ] }, { "cell_type": "code", "execution_count": 119, "id": "3bc418af", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'A Long Time Ago In A Galaxy Far Far Away'" ] }, "execution_count": 119, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a = 'a long time ago in a galaxy far far away'\n", "a.title()" ] }, { "cell_type": "code", "execution_count": 107, "id": "6d7a5797", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'A LONG TIME AGO IN A GALAXY FAR FAR AWAY'" ] }, "execution_count": 107, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a.upper()" ] }, { "cell_type": "code", "execution_count": 108, "id": "c0213018", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "['a', 'long', 'time', 'ago', 'in', 'a', 'galaxy', 'far', 'far', 'away']" ] }, "execution_count": 108, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a.split()" ] }, { "cell_type": "code", "execution_count": 114, "id": "9854bdba", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "['', ' long time ', 'go in ', ' g', 'l', 'xy f', 'r f', 'r ', 'w', 'y']" ] }, "execution_count": 114, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a.split('a')" ] }, { "cell_type": "code", "execution_count": 115, "id": "33e4bf13", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'aX XlXoXnXgX XtXiXmXeX XaXgXoX XiXnX XaX XgXaXlXaXxXyX XfXaXrX XfXaXrX XaXwXaXy'" ] }, "execution_count": 115, "metadata": {}, "output_type": "execute_result" } ], "source": [ "'X'.join(a)" ] }, { "cell_type": "code", "execution_count": 120, "id": "d75da207", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'a long time ago in a suburb far far away'" ] }, "execution_count": 120, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a.replace('galaxy', 'suburb')" ] }, { "cell_type": "code", "execution_count": 111, "id": "9c30bf06", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'A long time Ago in A gAlAxy fAr fAr AwAy'" ] }, "execution_count": 111, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a.replace('a', 'A')" ] }, { "cell_type": "code", "execution_count": 112, "id": "fab35238", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'a_long_time_ago_in_a_galaxy_far_far_away'" ] }, "execution_count": 112, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a.replace(' ', '_')" ] }, { "cell_type": "markdown", "id": "25a4f5f3", "metadata": {}, "source": [ "Python is a fantastic tool for working with strings, which makes it the language of choice for very many people working with text data of any sort. But now it is time to move on to collection variables." ] }, { "cell_type": "markdown", "id": "5bd97ed1", "metadata": {}, "source": [ "(arguments)=\n", "## Arguments\n", "\n", "This might be as good a place as any to introduce the concept of _arguments_. Arguments in Python are not heated discussions about whether or not Die Hard is a Christmas movie[^diehard], or how to pronounce the first consonant in \"gif\"[^gif]. Arguments in Python are opportunities for us to give Python functions some information. As an example, above I used the the `replace` method. Let's try a different example:\n", "\n", "[^diehard]: It is if you want it to be.\n", "[^gif]: Yeah, right. You're not going to trick me into stating an opinion on that one!" ] }, { "cell_type": "code", "execution_count": 125, "id": "506dd081", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'@ long time @go in @ g@l@xy f@r f@r @w@y'" ] }, "execution_count": 125, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a = 'a long time ago in a galaxy far far away'\n", "\n", "a.replace('a', '@')" ] }, { "cell_type": "markdown", "id": "d66b6a65", "metadata": {}, "source": [ "The `replace` method has three possible arugments, and requires that at least the first two be used. Put differently, we cannot simply write" ] }, { "cell_type": "code", "execution_count": 126, "id": "66dfa0c9", "metadata": {}, "outputs": [ { "ename": "TypeError", "evalue": "replace expected at least 2 arguments, got 0", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)", "Input \u001b[0;32mIn [126]\u001b[0m, in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1\u001b[0m a \u001b[38;5;241m=\u001b[39m \u001b[38;5;124m'\u001b[39m\u001b[38;5;124ma long time ago in a galaxy far far away\u001b[39m\u001b[38;5;124m'\u001b[39m\n\u001b[0;32m----> 3\u001b[0m \u001b[43ma\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mreplace\u001b[49m\u001b[43m(\u001b[49m\u001b[43m)\u001b[49m\n", "\u001b[0;31mTypeError\u001b[0m: replace expected at least 2 arguments, got 0" ] } ], "source": [ "a = 'a long time ago in a galaxy far far away'\n", "\n", "a.replace()" ] }, { "cell_type": "markdown", "id": "a622164f", "metadata": {}, "source": [ "Because, as the error message says, `replace` expects us to at a minimum tell it what to replace, and what to replace it with. That seems fair enough, actually. How else is it supposed to know, if we don't tell it? In fact, `replace` has another, optional, argument, \"count\". If we use this third arguement, we can also tell `replace` how _many_ instances of the old thing we want to replace with the new. The default value is all of them, but if we wanted, we could e.g. only replace the first two instances of 'a' with '@':" ] }, { "cell_type": "code", "execution_count": 128, "id": "edbb475e", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'@ long time @go in a galaxy far far away'" ] }, "execution_count": 128, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a.replace('a', '@', 2)" ] }, { "cell_type": "markdown", "id": "6db606ee", "metadata": {}, "source": [ "How did I know that `replace` has three arguments? How will you know how many arguments other methods and functions have? Well, I knew because I searched the internet for \"python string replace method arguments\", and that's my advice to you, too. Until just now, I had no idea that there was a third argument available for `replace`, because I had never needed it for anything. So I just learned something too!" ] }, { "cell_type": "markdown", "id": "f955d076", "metadata": {}, "source": [ "## Collection variables\n", "\n", "Collection variables provide a way to, well, _collect_ information. We'll look at these in turn, starting with lists." ] }, { "cell_type": "markdown", "id": "initial-happiness", "metadata": {}, "source": [ "(lists)=\n", "\n", "### Lists\n", "\n", "Lists are just what they sound like: a single variable that contains a list of items. Just about any variable type you can think of can be listed in a Python list:" ] }, { "cell_type": "code", "execution_count": 2, "id": "geographic-sessions", "metadata": {}, "outputs": [], "source": [ "shopping = [\"apples\", \"pears\", \"bananas\"] # a list strings\n", "scores = [90, 65, 100, 82] # a list of integers\n", "mixed = [\"cats\", 7, 309.42] # a list of mixed strings, integers, and floats\n", "combined = [shopping, scores, mixed] # a list of lists!" ] }, { "cell_type": "code", "execution_count": 3, "id": "needed-boxing", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['apples', 'pears', 'bananas']\n", "[90, 65, 100, 82]\n", "['cats', 7, 309.42]\n", "[['apples', 'pears', 'bananas'], [90, 65, 100, 82], ['cats', 7, 309.42]]\n" ] } ], "source": [ "print(shopping)\n", "print(scores)\n", "print(mixed)\n", "print(combined)" ] }, { "cell_type": "markdown", "id": "armed-cable", "metadata": {}, "source": [ "Let's say I am so enamored with Python that I actually decided to keep my shopping list in a Python list. Seems unlikely, I know, but bear with me. Later, I realize I have forgotten what I wrote on the list. This does kind of sound like me, actually. To see the contents of the entire list, I can use `print()`, the way I did above. But let's say I only want to see the second item on the list. Python has a way to access specific items in lists, but it will seem strange at first!" ] }, { "cell_type": "markdown", "id": "indian-newfoundland", "metadata": {}, "source": [ "Let's take a look. To access an item in a list, we need to know its `index`, that is, its location in the list. We indicate an index with square brackets. In Python, this is known as [`slicing`](indexing). So to find the item with the index 2 in my shopping list, I can write `shopping[2]`, like so:" ] }, { "cell_type": "code", "execution_count": 24, "id": "capital-living", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'bananas'" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "shopping = [\"apples\", \"pears\", \"bananas\"]\n", "shopping[2]" ] }, { "cell_type": "markdown", "id": "d06cebd1", "metadata": {}, "source": [ "What?!?!? We asked for item 2, and Python gives us \"bananas\"? But \"bananas\" is the third item in the list?!?! What's going on?\n", "\n", "The simple answer for this is that Python uses \"zero-based indexing\": basically, Python starts counting at zero. So \"apples\" is in the zeroeth position, \"pears\" is in the first position, and \"bananas\" is in the second position. If it helps, maybe think of it like buildings in Europe that start with the ground floor, and then go up to the first floor, etc.\n", "\n", "Now just when you have started to get used to zero-indexing, try negative indexing on for size. We can also count backwards from the end of the list, by using negative indices such as `shopping[-2]` But be careful: when you use negative indexing, Python behaves the way you might have originally expected it to. Thus, `shopping[-1]` will return \"bananas\", `shopping[-2]` will give us \"pears\", etc. That's just how Python is.\n", "\n", "At some point I stumbled on the visualization below. Unfortunately, I can no longer remember where I found it, so I cannot attribute it, but you may find it helpful when trying to remember how indexing works in Python. But probably the best way is to just try it out, and pretty soon you'll get used to it." ] }, { "cell_type": "code", "execution_count": 1, "id": "ee79ed3b", "metadata": {}, "outputs": [], "source": [ "# +---+---+---+---+---+---+\n", "# | P | y | t | h | o | n |\n", "# +---+---+---+---+---+---+\n", "# 0 1 2 3 4 5 6\n", "# -6 -5 -4 -3 -2 -1" ] }, { "cell_type": "markdown", "id": "announced-objective", "metadata": {}, "source": [ "### Finding the length of a list\n", "\n", "One last thing on lists for now: it can often be useful to check how many items are in your list. With the toy examples we are using here, of course, it is easy to see how long the list is, because we just typed in the items ourselves. But in actual data analysis, we often deal with very long lists that contain an unknown number of items. In these cases we can use `len()` to check how long the list is:" ] }, { "cell_type": "code", "execution_count": 25, "id": "detected-september", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "len(shopping)" ] }, { "cell_type": "markdown", "id": "bea1c39c", "metadata": {}, "source": [ "(tuples)=\n", "\n", "## Tuples\n", "\n", "Tuples are similar to lists, but they have one important difference: you can mess around and change what is inside of lists, but you can't do that with tuples. In programmer-ese, we can say that lists are _mutable_, but tuples are _immutable_. Once you make one, you can't change it (unless you just make a new one, with the same name, in which case you're now technically dealing with a new tuple, so you didn't really change it all).\n", "\n", "While Python uses square brackets (`[ ]`) to indicate lists, it uses regular old parentheses plus a comma to indicate tuples. Let's take a look:" ] }, { "cell_type": "code", "execution_count": 44, "id": "ca9bc23b", "metadata": {}, "outputs": [], "source": [ "shopping_list = [\"apples\", \"pears\", \"bananas\"]\n", "shopping_tuple = (\"apples\", \"pears\", \"bananas\")" ] }, { "cell_type": "markdown", "id": "9f6e6be0", "metadata": {}, "source": [ "We can still get data out of tuples in the same way we get it out of lists, by slicing them, using square brackets and an index value:" ] }, { "cell_type": "code", "execution_count": 42, "id": "d51c60b1", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "apples\n" ] } ], "source": [ "print(shopping_tuple[0])" ] }, { "cell_type": "markdown", "id": "6f7deeab", "metadata": {}, "source": [ "Now, if I decide I don't want apples after all, but want grapes instead, I can just re-assign the new value to the appropriate position in the list, replacing \"apples\" with \"grapes\"" ] }, { "cell_type": "code", "execution_count": 8, "id": "9f55fa6a", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['grapes', 'pears', 'bananas']\n" ] } ], "source": [ "shopping_list[0] = 'grapes'\n", "print(shopping_list)" ] }, { "cell_type": "markdown", "id": "611405f0", "metadata": {}, "source": [ "Don't try that kind of funny business with a tuple, though. It won't have any of it:" ] }, { "cell_type": "code", "execution_count": 9, "id": "152908d1", "metadata": {}, "outputs": [ { "ename": "TypeError", "evalue": "'tuple' object does not support item assignment", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)", "Input \u001b[0;32mIn [9]\u001b[0m, in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0m shopping_tuple[\u001b[38;5;241m0\u001b[39m] \u001b[38;5;241m=\u001b[39m \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mgrapes\u001b[39m\u001b[38;5;124m'\u001b[39m\n\u001b[1;32m 2\u001b[0m \u001b[38;5;28mprint\u001b[39m(shopping_tuple)\n", "\u001b[0;31mTypeError\u001b[0m: 'tuple' object does not support item assignment" ] } ], "source": [ "shopping_tuple[0] = 'grapes'\n", "print(shopping_tuple)" ] }, { "cell_type": "markdown", "id": "1fad2fd9", "metadata": {}, "source": [ "One more detail, before we leave tuples. Remember I said that Python uses parentheses _and_ a comma to indicate that something is a tuple? I meant it. If we try to make a tuple that only has one item, you might think we could do it like this:" ] }, { "cell_type": "code", "execution_count": 13, "id": "a21fe766", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "str" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a = ('pears')\n", "type(a)" ] }, { "cell_type": "markdown", "id": "61f96df9", "metadata": {}, "source": [ "But, as you can see, Python does not think this is a tuple. It just sees it as a string. To make a tuple with only item, we still need the comma:" ] }, { "cell_type": "code", "execution_count": 14, "id": "1169d224", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "tuple" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a = ('pears',)\n", "type(a)" ] }, { "cell_type": "code", "execution_count": 32, "id": "172ee871", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "pears\n" ] } ], "source": [ "print(a[0])" ] }, { "cell_type": "markdown", "id": "2b9daa36", "metadata": {}, "source": [ "Ah. Now Python understands that we want to make a tuple." ] }, { "cell_type": "markdown", "id": "4c20aa1b", "metadata": {}, "source": [ "(dicts)=\n", "## Dictionaries\n", "\n", "Dictionaries are a very useful variable type in Python, but it may take you a while to appreciate their usefulness. Trust me, though. Once you begin to appreciate what dictionaries can do, you will learn to love them. But, like anything (or anyone), before you can truly love dictionaries, you have to get to know them.\n", "\n", "In essence, dictionaries are very simple. They consist of two parts: keys and values. You can think of the key as the word you look up in the dictionary, and the value as the definition. The syntax for defining a dictionary is `{key: value}`. A simple key-value pair could be `{'name': 'Ethan'}`." ] }, { "cell_type": "code", "execution_count": 20, "id": "56c875df", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Ethan\n" ] } ], "source": [ "me = {'name': 'Ethan'}\n", "print(me['name'])" ] }, { "cell_type": "markdown", "id": "9ae58736", "metadata": {}, "source": [ "Of course, most dictionaries will have more than one entry. If were to make a dictionary with information about my pet, for instance, I might write:" ] }, { "cell_type": "code", "execution_count": 46, "id": "896fa6b2", "metadata": {}, "outputs": [], "source": [ "my_pet = {'name': 'D2',\n", " 'species': 'cat',\n", " 'likes': 'sleeping',\n", " 'dislikes': 'loud noises'}" ] }, { "cell_type": "markdown", "id": "601dc998", "metadata": {}, "source": [ "Now, if I can't remember what kind of animal my pet is, I can always consult my dictionary:" ] }, { "cell_type": "code", "execution_count": 22, "id": "e18f224e", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'cat'" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "my_pet['species']" ] }, { "cell_type": "markdown", "id": "85c5f2b0", "metadata": {}, "source": [ "Dictionary values can be more than just a single string, though. Let's say I want to plan out my week using Python (because that would be a normal thing to do!) I can make a dictionary called `my_week`, and include entries for shopping and a to-do list. " ] }, { "cell_type": "code", "execution_count": 28, "id": "e8b342f3", "metadata": {}, "outputs": [], "source": [ "my_week = {'shopping': ['apples', 'pears', 'bananas'],\n", " 'todo': ('sleep', 'teach', 'feed the cat')}" ] }, { "cell_type": "code", "execution_count": 29, "id": "888ed3b2", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['apples', 'pears', 'bananas']\n" ] } ], "source": [ "print(my_week['shopping'])" ] }, { "cell_type": "code", "execution_count": 30, "id": "1c6ad264", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "('sleep', 'teach', 'feed the cat')\n" ] } ], "source": [ "print(my_week['todo'])" ] }, { "cell_type": "markdown", "id": "fc47394c", "metadata": {}, "source": [ "Notice that I have made the value for `shopping` a list, but the value for `todo` is a tuple. That might be because I want to keep open the option to change items in my shopping list, but want to keep my to-do list fixed. Or it might just be because I wanted to demonstrate that dictionary values can be all kinds of different variable types." ] }, { "cell_type": "markdown", "id": "b933f648", "metadata": {}, "source": [ "## Combining variable types\n", "\n", "We have already seen that while regular variables are either one type or another, collection variables can contain different types of variables. For example, in the section on [lists](lists), we saw that lists can also contain other lists. These collection variables are really quite flexible, so before we leave them, I just want to give a few more examples of variable type combinations:" ] }, { "cell_type": "markdown", "id": "805c33e0", "metadata": {}, "source": [ "#### Lists of lists" ] }, { "cell_type": "code", "execution_count": 34, "id": "6a30b405", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[['Barred Owl', 'Burrowing Owl', 'Barn Owl', 'Screech Owl', 'Spotted Owl'], ['Goldfinch', 'Crossbill', 'Redpoll', 'Grosbeak'], ['House Sparrow', 'Song Sparrow', 'White-Throated Sparrow', 'Captain Jack Sparrow']]\n" ] } ], "source": [ "owls = ['Barred Owl', 'Burrowing Owl', 'Barn Owl', 'Screech Owl', 'Spotted Owl']\n", "finches = ['Goldfinch', 'Crossbill', 'Redpoll', 'Grosbeak']\n", "sparrows = ['House Sparrow', 'Song Sparrow', 'White-Throated Sparrow', 'Captain Jack Sparrow']\n", "\n", "birds = [owls, finches, sparrows]\n", "print(birds)" ] }, { "cell_type": "markdown", "id": "5624972c", "metadata": {}, "source": [ "#### A dictionary of ....???" ] }, { "cell_type": "code", "execution_count": 39, "id": "ee66e607", "metadata": {}, "outputs": [], "source": [ "mishmosh = {'birds': birds,\n", " 'dictionaries': (my_week, my_pet)}" ] }, { "cell_type": "markdown", "id": "d4c4c73b", "metadata": {}, "source": [ "Now, this one is a collection of strange bedfellows, but Python isn't judgy. So, in `mishmosh`, I have two keys, `birds` and `dictionaries`. Why not? If we look in the dictionary, we can see that the value for the key `birds` is a list of lists:" ] }, { "cell_type": "code", "execution_count": 40, "id": "d8763b62", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[['Barred Owl', 'Burrowing Owl', 'Barn Owl', 'Screech Owl', 'Spotted Owl'],\n", " ['Goldfinch', 'Crossbill', 'Redpoll', 'Grosbeak'],\n", " ['House Sparrow',\n", " 'Song Sparrow',\n", " 'White-Throated Sparrow',\n", " 'Captain Jack Sparrow']]" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mishmosh['birds']" ] }, { "cell_type": "markdown", "id": "cb9f9909", "metadata": {}, "source": [ "and the value for the key `dictionaries` is a tuple of dictionaries!" ] }, { "cell_type": "code", "execution_count": 41, "id": "771f8e2d", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "({'shopping': ['apples', 'pears', 'bananas'],\n", " 'todo': ('sleep', 'teach', 'feed the cat')},\n", " {'name': 'D2',\n", " 'species': 'cat',\n", " 'likes': 'sleeping',\n", " 'dislikes': 'loud noises'})" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mishmosh['dictionaries']" ] }, { "cell_type": "markdown", "id": "96bc89e0", "metadata": {}, "source": [ "We'll leave it there for now, but hopefully you can see that by using these basic elements: regular variables and collection variables, you can store data in a structured way, so that you can access it when you need it. There is one last type of collection variable that will come in handy, and that is the set. Sets are the collection of _unique_ items in a collection of items. As an example, say every student in a class wrote down their favorite color, and we put them all into a list, like this:" ] }, { "cell_type": "markdown", "id": "d68095f0", "metadata": {}, "source": [ "(sets)=\n", "## Sets" ] }, { "cell_type": "code", "execution_count": 49, "id": "5142f24c", "metadata": {}, "outputs": [], "source": [ "our_colors = ['green', 'blue', 'red', 'blue', 'orange', 'yellow', 'green', 'blue', 'green', 'blue']" ] }, { "cell_type": "markdown", "id": "3be2806f", "metadata": {}, "source": [ "Since every student wrote down a color, if we find the length of the list, we will also find the number of students in the class:" ] }, { "cell_type": "code", "execution_count": 50, "id": "2791e533", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "10" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "len(our_colors)" ] }, { "cell_type": "markdown", "id": "ab4185ac", "metadata": {}, "source": [ "But some students wrote the same colors as other students. So if we want to find the unique colors chosen, ignoring in repeated colors, we can find the `set` of our list:" ] }, { "cell_type": "code", "execution_count": 51, "id": "bd1e16a1", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "{'blue', 'green', 'orange', 'red', 'yellow'}" ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "set(our_colors)" ] }, { "cell_type": "markdown", "id": "37f132f5", "metadata": {}, "source": [ "And by finding the length of the set, we can see how many unique colors were chosen:" ] }, { "cell_type": "code", "execution_count": 52, "id": "68e46639", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "5" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "len(set(our_colors))" ] }, { "cell_type": "markdown", "id": "2d889159", "metadata": {}, "source": [ "As another example, we could make a see how many unique characters make up a sentence:" ] }, { "cell_type": "code", "execution_count": 55, "id": "b4851c83", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "24" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "len(set('Python is the most wonderful programming language in the world and I love it so much!'))" ] }, { "cell_type": "markdown", "id": "26e368b1", "metadata": {}, "source": [ "In case you are wondering, here are the characters:" ] }, { "cell_type": "code", "execution_count": 56, "id": "66849e95", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "{'r', 'w', 'v', ' ', 'i', 'y', 'P', 'a', 'l', 'u', 'd', 'm', 'n', 'h', 'e', 'I', 'c', 'p', 't', 'g', 's', 'o', '!', 'f'}\n" ] } ], "source": [ "print(set('Python is the most wonderful programming language in the world and I love it so much!'))" ] }, { "cell_type": "markdown", "id": "a0b2417d", "metadata": {}, "source": [ "Notice that not only does Python count the blank space and the exclamation mark as characters, it also counts the upper and lower case I/i as separate characters. Just to keep you on your toes." ] }, { "cell_type": "markdown", "id": "66d7ffc6", "metadata": {}, "source": [ "(methods)=\n", "## Object methods\n", "\n", "We have already seen that different types of variables (or \"objects\") in Python have different properties. Add two integers together, and you will get the sum of those two numbers. Add two strings together, and Python will just stick those two strings together to make a longer string. The differences in variable properties go deeper, however. We have already seen an example of this with the string type. Remeber the following example?" ] }, { "cell_type": "code", "execution_count": 95, "id": "c66762ad", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "A Long Time Ago In A Galaxy Far Far Away\n" ] } ], "source": [ "a = 'a long time ago in a galaxy far far away'\n", "a = a.title()\n", "print(a)" ] }, { "cell_type": "markdown", "id": "cb96f6f2", "metadata": {}, "source": [ "Here, we created a string of words separated by spaces, all lowercase, and called it `a`. Then, by writing the name of the variable, `a`, followed by `.title()`, Python knew to convert the string to title case. We can also convert our variable `a`, now in title case, to upper case, by a similar method:" ] }, { "cell_type": "code", "execution_count": 96, "id": "ec616882", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "A LONG TIME AGO IN A GALAXY FAR FAR AWAY\n" ] } ], "source": [ "a = a.upper()\n", "print(a)" ] }, { "cell_type": "markdown", "id": "3510f151", "metadata": {}, "source": [ "`title()` and `upper()` are examples of special properties that all string variables possess. These special properties are called \"methods\". To see the full list of methods available for a variable in Python, we can use the function `dir()`. So, for example, since `a` is a string, we can see what methods are available for strings by writing `dir(a)`:" ] }, { "cell_type": "code", "execution_count": 97, "id": "857b5469", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "['__add__',\n", " '__class__',\n", " '__contains__',\n", " '__delattr__',\n", " '__dir__',\n", " '__doc__',\n", " '__eq__',\n", " '__format__',\n", " '__ge__',\n", " '__getattribute__',\n", " '__getitem__',\n", " '__getnewargs__',\n", " '__gt__',\n", " '__hash__',\n", " '__init__',\n", " '__init_subclass__',\n", " '__iter__',\n", " '__le__',\n", " '__len__',\n", " '__lt__',\n", " '__mod__',\n", " '__mul__',\n", " '__ne__',\n", " '__new__',\n", " '__reduce__',\n", " '__reduce_ex__',\n", " '__repr__',\n", " '__rmod__',\n", " '__rmul__',\n", " '__setattr__',\n", " '__sizeof__',\n", " '__str__',\n", " '__subclasshook__',\n", " 'capitalize',\n", " 'casefold',\n", " 'center',\n", " 'count',\n", " 'encode',\n", " 'endswith',\n", " 'expandtabs',\n", " 'find',\n", " 'format',\n", " 'format_map',\n", " 'index',\n", " 'isalnum',\n", " 'isalpha',\n", " 'isascii',\n", " 'isdecimal',\n", " 'isdigit',\n", " 'isidentifier',\n", " 'islower',\n", " 'isnumeric',\n", " 'isprintable',\n", " 'isspace',\n", " 'istitle',\n", " 'isupper',\n", " 'join',\n", " 'ljust',\n", " 'lower',\n", " 'lstrip',\n", " 'maketrans',\n", " 'partition',\n", " 'removeprefix',\n", " 'removesuffix',\n", " 'replace',\n", " 'rfind',\n", " 'rindex',\n", " 'rjust',\n", " 'rpartition',\n", " 'rsplit',\n", " 'rstrip',\n", " 'split',\n", " 'splitlines',\n", " 'startswith',\n", " 'strip',\n", " 'swapcase',\n", " 'title',\n", " 'translate',\n", " 'upper',\n", " 'zfill']" ] }, "execution_count": 97, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dir(a)" ] }, { "cell_type": "markdown", "id": "7f72d122", "metadata": {}, "source": [ "Yikes! You can safely ignore all the things with `__` in front of and after them, like `__add__`, and `__class__`. These aren't meant for us to meddle with! But even setting these aside, you can see that for strings the list is quite long, and to be honest with you, I don't even know what all of these do. Lists, on the other hand, don't have as many methods available to them:" ] }, { "cell_type": "code", "execution_count": 112, "id": "3a096f00", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "['__add__',\n", " '__class__',\n", " '__class_getitem__',\n", " '__contains__',\n", " '__delattr__',\n", " '__delitem__',\n", " '__dir__',\n", " '__doc__',\n", " '__eq__',\n", " '__format__',\n", " '__ge__',\n", " '__getattribute__',\n", " '__getitem__',\n", " '__gt__',\n", " '__hash__',\n", " '__iadd__',\n", " '__imul__',\n", " '__init__',\n", " '__init_subclass__',\n", " '__iter__',\n", " '__le__',\n", " '__len__',\n", " '__lt__',\n", " '__mul__',\n", " '__ne__',\n", " '__new__',\n", " '__reduce__',\n", " '__reduce_ex__',\n", " '__repr__',\n", " '__reversed__',\n", " '__rmul__',\n", " '__setattr__',\n", " '__setitem__',\n", " '__sizeof__',\n", " '__str__',\n", " '__subclasshook__',\n", " 'append',\n", " 'clear',\n", " 'copy',\n", " 'count',\n", " 'extend',\n", " 'index',\n", " 'insert',\n", " 'pop',\n", " 'remove',\n", " 'reverse',\n", " 'sort']" ] }, "execution_count": 112, "metadata": {}, "output_type": "execute_result" } ], "source": [ "b = [3, 6, 1231]\n", "dir(b)" ] }, { "cell_type": "markdown", "id": "faeb3115", "metadata": {}, "source": [ "We can access these special methods by writing the name of the variable, and then a `.` and then the name of the method. But we can only use the methods that are available for the type of variable we are working with. Lists, for instance, do not have a `title` method, so it's no good taking our list `b` and writing" ] }, { "cell_type": "code", "execution_count": 113, "id": "70a424e3", "metadata": {}, "outputs": [ { "ename": "AttributeError", "evalue": "'list' object has no attribute 'title'", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mAttributeError\u001b[0m Traceback (most recent call last)", "Input \u001b[0;32mIn [113]\u001b[0m, in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0m \u001b[43mb\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mtitle\u001b[49m()\n", "\u001b[0;31mAttributeError\u001b[0m: 'list' object has no attribute 'title'" ] } ], "source": [ "b.title()" ] }, { "cell_type": "markdown", "id": "12e4523f", "metadata": {}, "source": [ "(dynamictypes)=\n", "## Dynamic typing\n", "\n", "In some respects, Python is a very flexible and user-friendly language. One of the ways that Python shows its friendly face to us is the way variables are defined. In some languages, you need to say up front what kind of variable you want to make. If you want to define a variable `b` that contains a the number 342, then you might need to write something like `int b = 342`. In Python, we can just write `b = 342`, and then Python does the work of figuring out that `b` is an integer variable. If we later assign something else to `b`, e.g. `b = 'zebrafish`, Python doesn't object. It just says \"Hey, I guess `b` is a string now. Cool!\" This flexibility is called \"Dynamic Typing\" and it can make working with Python easier, but can also get us in trouble, because sometime we _think_ we know what type our variable is, but we are wrong. Luckily, you can always check with `type()`" ] }, { "cell_type": "code", "execution_count": 116, "id": "d762004a", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "int" ] }, "execution_count": 116, "metadata": {}, "output_type": "execute_result" } ], "source": [ "b = 342\n", "type(b)" ] }, { "cell_type": "code", "execution_count": 117, "id": "3ddb5bc1", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "str" ] }, "execution_count": 117, "metadata": {}, "output_type": "execute_result" } ], "source": [ "b = 'zebrafish'\n", "type(b)" ] }, { "cell_type": "markdown", "id": "675b1d4d", "metadata": {}, "source": [ "(indexing)=\n", "## Indexing and slicing\n", "\n", "It's no good having data stored in a collection variable if we can't get it out again. We have already seen some examples of using indices and slices to retrieve data, but before we end the chapter, let's just see a few more. Remember our list of birds:" ] }, { "cell_type": "code", "execution_count": 59, "id": "668d0585", "metadata": {}, "outputs": [], "source": [ "owls = ['Barred Owl', 'Burrowing Owl', 'Barn Owl', 'Screech Owl', 'Spotted Owl']\n", "finches = ['Goldfinch', 'Crossbill', 'Redpoll', 'Grosbeak']\n", "sparrows = ['House Sparrow', 'Song Sparrow', 'White-Throated Sparrow', 'Captain Jack Sparrow']\n", "\n", "birds = [owls, finches, sparrows]\n" ] }, { "cell_type": "markdown", "id": "f74ccdea", "metadata": {}, "source": [ "We already know how to find the owl in position 2 (keeping in mind that we start counting at 0):" ] }, { "cell_type": "code", "execution_count": 60, "id": "39a3be63", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'Barn Owl'" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "owls[2]" ] }, { "cell_type": "markdown", "id": "37207809", "metadata": {}, "source": [ "Or the final owl in the list:" ] }, { "cell_type": "code", "execution_count": 61, "id": "159634d4", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'Spotted Owl'" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "owls[-1]" ] }, { "cell_type": "markdown", "id": "69cd68b0", "metadata": {}, "source": [ "But the variable `birds` is a list of lists." ] }, { "cell_type": "code", "execution_count": 91, "id": "60a100b6", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[['Barred Owl', 'Burrowing Owl', 'Barn Owl', 'Screech Owl', 'Spotted Owl'],\n", " ['Goldfinch', 'Crossbill', 'Redpoll', 'Grosbeak'],\n", " ['House Sparrow',\n", " 'Song Sparrow',\n", " 'White-Throated Sparrow',\n", " 'Captain Jack Sparrow']]" ] }, "execution_count": 91, "metadata": {}, "output_type": "execute_result" } ], "source": [ "birds" ] }, { "cell_type": "markdown", "id": "f35f5aa2", "metadata": {}, "source": [ "So to dig the data out of `birds`, we need to go two levels deep with our slices. To get 'Redpoll', for example, we need to find the item in position 2 of list 1:" ] }, { "cell_type": "code", "execution_count": 92, "id": "6166dbda", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Redpoll\n" ] } ], "source": [ "print(birds[1][2])" ] }, { "cell_type": "markdown", "id": "c4e9ca4a", "metadata": {}, "source": [ "Our variable `mishmsoh` was even more complicated, but the same principle still applies. We just need to work our way down through the layers, until we get to the data we want." ] }, { "cell_type": "code", "execution_count": 93, "id": "130cf45a", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "{'birds': [['Barred Owl',\n", " 'Burrowing Owl',\n", " 'Barn Owl',\n", " 'Screech Owl',\n", " 'Spotted Owl'],\n", " ['Goldfinch', 'Crossbill', 'Redpoll', 'Grosbeak'],\n", " ['House Sparrow',\n", " 'Song Sparrow',\n", " 'White-Throated Sparrow',\n", " 'Captain Jack Sparrow']],\n", " 'dictionaries': ({'shopping': ['apples', 'pears', 'bananas'],\n", " 'todo': ('sleep', 'teach', 'feed the cat')},\n", " {'name': 'D2',\n", " 'species': 'cat',\n", " 'likes': 'sleeping',\n", " 'dislikes': 'loud noises'})}" ] }, "execution_count": 93, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mishmosh = {'birds': birds,\n", " 'dictionaries': (my_week, my_pet)}\n", "mishmosh" ] }, { "cell_type": "markdown", "id": "72b09a7f", "metadata": {}, "source": [ "So, if I am still having a hard time remembering what kind of pet I have, all I need to do is find the information in the variable:" ] }, { "cell_type": "code", "execution_count": 75, "id": "4091c051", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "My pet is a cat\n", "His name is D2\n", "He likes sleeping\n" ] } ], "source": [ "print('My pet is a', mishmosh['dictionaries'][1]['species'])\n", "print('His name is', mishmosh['dictionaries'][1]['name'])\n", "print('He likes', mishmosh['dictionaries'][1]['likes'])" ] }, { "cell_type": "markdown", "id": "181425a0", "metadata": {}, "source": [ "If this seems like an enormous amount of work to go to just to store details about my cat, well, I don't blame you. But just file these variable types away somewhere in your mind, because they will all be very useful later on as you progress with Python. And yes, I did name my cat D2." ] }, { "cell_type": "markdown", "id": "4893a787", "metadata": {}, "source": [ "(slices)=\n", "## Slices\n", "\n", "I have mentioned the concept of slicing before, but now it is time to talk about it in a bit more depth. Sometimes we want to access a portion of our data. For example, we might want only part of a string, or part of a list. Say I go to the store every day, and every day I write down how much I spend." ] }, { "cell_type": "code", "execution_count": 76, "id": "5e9193b9", "metadata": {}, "outputs": [], "source": [ "spent = [2,5,345,4,6,12,432,6,575,43,64,32,234,23,6,3,565,345,43,233,12,865,45645,45,332,453,346,23,234,3,324,1]" ] }, { "cell_type": "markdown", "id": "b9c39ca5", "metadata": {}, "source": [ "We can _slice_ the data, to find any grouping of these numbers that we like. For example, the first five:" ] }, { "cell_type": "code", "execution_count": 77, "id": "4b7f2700", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[2, 5, 345, 4, 6]" ] }, "execution_count": 77, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spent[0:5]" ] }, { "cell_type": "markdown", "id": "202c3e58", "metadata": {}, "source": [ "The slice uses square brackets[^namespacepolution] to indicate which data we want. By now, we are familiar enough with Python to know that Python starts counting at zero, so it makes sense[^sense] that if we want the first 5 numbers from our list, then we will need to start with the number in the zeroeth position. But why does the slice end with the number in the fifth? Because, wouldn't that be the fourth number, if we start counting at zero? Yes, it would, except that Python slices _end_ with the item in the position indicated by the second number _not including that number_! So when we write `[0:5]`, Python starts counting at 0, then includes everything up to one less than the final number. Sometimes I think the authors of Python are just messing with us. But if you play around with it a bit, you'll soon get the hang of it.\n", "\n", "[^namespacepolution]: Yes, square brackets, just like lists use square brackets. Haha! Isn't it fun when the same symbol means different things, and you just have to know what it means in a given context? Of course it is!\n", "\n", "[^sense]: I use the term \"sense\" broadly, here.\n", "\n" ] }, { "cell_type": "markdown", "id": "83b244b3", "metadata": {}, "source": [ "We can also get the last five items:" ] }, { "cell_type": "code", "execution_count": 79, "id": "e60ed0ba", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[23, 234, 3, 324, 1]" ] }, "execution_count": 79, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spent[-5:]" ] }, { "cell_type": "markdown", "id": "966d7005", "metadata": {}, "source": [ "Here we are counting backwards to five positions from the end, and then going forwards to the end.\n", "\n", "Of course, we could also take a section from the middle:" ] }, { "cell_type": "code", "execution_count": 81, "id": "b7141a0a", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[6, 12, 432, 6, 575, 43]" ] }, "execution_count": 81, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spent[4:10]" ] }, { "cell_type": "markdown", "id": "3ea7069e", "metadata": {}, "source": [ "The general form for slices is `[start:end:step]`, where `step` indicates how many positions to jump each time. If you don't indicate any value for `step`, Python assumes it is one. If you don't put any value for `end`, then Python assumes it is the same as the value for `start`. We can combine these to find:\n", "\n", "#### A single value" ] }, { "cell_type": "code", "execution_count": 84, "id": "7633186c", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "6" ] }, "execution_count": 84, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spent[4]" ] }, { "cell_type": "markdown", "id": "85ce643a", "metadata": {}, "source": [ " #### A range of values between `start` and `end`" ] }, { "cell_type": "code", "execution_count": 87, "id": "05e92396", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[6, 12, 432, 6, 575, 43]" ] }, "execution_count": 87, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spent[4:10]" ] }, { "cell_type": "markdown", "id": "dc5202ee", "metadata": {}, "source": [ "#### Values from the nth position to the end" ] }, { "cell_type": "code", "execution_count": 88, "id": "b8b00eb3", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[3, 565, 345, 43, 233, 12, 865, 45645, 45, 332, 453, 346, 23, 234, 3, 324, 1]" ] }, "execution_count": 88, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spent[15:]" ] }, { "cell_type": "markdown", "id": "f31bed41", "metadata": {}, "source": [ "#### Every nth item within a range\n", "\n", "Here, for example, we find every seventh item between item 0 and 17:" ] }, { "cell_type": "code", "execution_count": 89, "id": "2dfa8126", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[2, 6, 6]" ] }, "execution_count": 89, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spent[0:17:7]" ] }, { "cell_type": "markdown", "id": "74381e50", "metadata": {}, "source": [ "And here we find every fourth item in the entire list:" ] }, { "cell_type": "code", "execution_count": 90, "id": "3d5a728f", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[2, 6, 575, 234, 565, 12, 332, 234]" ] }, "execution_count": 90, "metadata": {}, "output_type": "execute_result" } ], "source": [ "spent[::4]" ] }, { "cell_type": "markdown", "id": "869ec741", "metadata": {}, "source": [ "Now that we have seen how to use the different data types to store data, and indices and slices to get data out, it is time for a break. Storing data in collections like lists, tuples, and dictionaries, together with accessing data using indices and slices gives you enormous flexibility, but it takes practice to get used to it. Slicing in particular, I find, gives people (including me!) trouble, so I really do recommend spending some time making lists of data, and slicing them in every way you can think of, until you get the feeling for how it works." ] }, { "cell_type": "markdown", "id": "handy-timing", "metadata": {}, "source": [ "\n", "## Summary\n", "\n", "This chapter continued where [the previous chapter](getting-started-with-python) left off. The focus was still primarily on introducing basic Python concepts, but this time at least you can see how those concepts are related to data analysis:\n", "\n", "- [Installing and importing libraries](packageinstall). Knowing how to extend the functionality of Python by installing and using libraries is critical to becoming an effective Python user\n", "- [Loading and saving data](load). Finally, we encountered actual data files. Loading and saving data is obviously a crucial skill, this was dealt with here.\n", "- [Filesystem paths](filesystem). This section dealt very briefly with the concept of filesystems. Mostly, I just told you what a filesystem path is, and left it up to you to figure out how to find paths on your machine.\n", "- [Useful things to know about variables](useful). In this section, we talked about variable types, with a special focus on lists and dataframes. There is much more to say about these topics, but hopefully this is enough to get you started.\n", "\n", "\n", "Taken together, the chapter on [Getting Started with Python](getting-started-with-python) and [More Python Concepts](mechanics) provide enough of a background that you can finally get started doing some statistics! Yes, there's a lot more Python concepts that you ought to know (and we'll talk about some of them in the chapters on [Data Wrangling](datawrangling) and [Basic Programming](programming), but I think that we've talked quite enough about programming for the moment. It's time to see how your experience with programming can be used to do some data analysis..." ] }, { "cell_type": "code", "execution_count": null, "id": "indoor-electron", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.5" } }, "nbformat": 4, "nbformat_minor": 5 } { "cells": [ { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(descriptives)=\n", "# Descriptive statistics\n", "\n", "Any time that you get a new data set to look at, one of the first tasks that you have to do is find ways of summarising the data in a compact, easily understood fashion. This is what **_descriptive statistics_** (as opposed to inferential statistics) is all about. In fact, to many people the term \"statistics\" is synonymous with descriptive statistics. It is this topic that we'll consider in this chapter, but before going into any details, let's take a moment to get a sense of why we need descriptive statistics. To do this, let's load the `afl_finalists.csv` and `afl_margins.csv` files. Don't worry about the Python code for now; we'll get back to that. For now, we'll focus on the data." ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [], "source": [ "\n", "import pandas as pd\n", "\n", "afl_finalists = pd.read_csv('https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/afl_finalists.csv')\n", "afl_margins = pd.read_csv('https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/afl_margins.csv')" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "There are two variables here, `afl_finalists` and `afl_margins`. We'll focus a bit on these two variables in this chapter, so I'd better tell you what they are. Unlike most of data sets in this book, these are actually real data, relating to the Australian Football League (AFL) [^note1] The `afl_margins` variable contains the winning margin (number of points) for all 176 home and away games played during the 2010 season. The `afl_finalists` variable contains the names of all 400 teams that played in all 200 finals matches played during the period 1987 to 2010. Let's have a look at the `afl_margins` variable:\n", "\n", "[^note1]: Note for non-Australians: the AFL is an Australian rules football competition. You don't need to know anything about Australian rules in order to follow this section." ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " afl.margins\n", "0 56\n", "1 31\n", "2 56\n", "3 8\n", "4 32\n", ".. ...\n", "171 28\n", "172 38\n", "173 29\n", "174 10\n", "175 10\n", "\n", "[176 rows x 1 columns]\n" ] } ], "source": [ "print(afl_margins)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "\n", "This output doesn't make it easy to get a sense of what the data are actually saying. Just \"looking at the data\" isn't a terribly effective way of understanding data. In order to get some idea about what's going on, we need to calculate some descriptive statistics (this chapter) and draw some nice pictures (next chapter). Since the descriptive statistics are the easier of the two topics, I'll start with those, but nevertheless I'll show you a histogram of the `afl_margins` data, since it should help you get a sense of what the data we're trying to describe actually look like. But for what it's worth, this histogram was generated using the `histplot()` function from the `seaborn` package. We'll talk a lot more about how to draw histograms in [](DrawingGraphs). For now, it's enough to look at the histogram and note that it provides a fairly interpretable representation of the `afl_margins` data." ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "\n", "import seaborn as sns\n", "\n", "ax = sns.histplot(afl_margins)\n", "\n", "ax.set(xlabel =\"Winning Margin\", \n", " ylabel = \"Frequency\")\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} afl_fig\n", ":figwidth: 600px\n", ":name: fig-AFL-Margins\n", "\n", "A histogram of the AFL 2010 winning margin data (the `afl_margins` variable). As you might expect, the larger the margin the less frequently you tend to see it.\n", "```\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(central-tendency)=\n", "## Measures of central tendency\n", "\n", "Athough drawing pictures of the data, as I did in {numref}`fig-AFL-Margins` is an excellent way to convey the \"gist\" of what the data is trying to tell you, it's often extremely useful to try to condense the data into a few simple \"summary\" statistics. In most situations, the first thing that you'll want to calculate is a measure of **_central tendency_**. That is, you'd like to know something about where the \"average\" or \"middle\" of your data lies. The two most commonly used measures are the mean, median and mode; occasionally people will also report a trimmed mean. I'll explain each of these in turn, and then discuss when each of them is useful.\n", "\n", "### The mean\n", "\n", "The **_mean_** of a set of observations is just a normal, old-fashioned average: add all of the values up, and then divide by the total number of values. The first five AFL margins were 56, 31, 56, 8 and 32, so the mean of these observations is just:\n", "\n", "\n", "$\\frac{56 + 31 + 56 + 8 + 32}{5} = \\frac{183}{5} = 36.60$\n", "\n", "\n", "Of course, this definition of the mean isn't news to anyone: averages (i.e., means) are used so often in everyday life that this is pretty familiar stuff. However, since the concept of a mean is something that everyone already understands, I'll use this as an excuse to start introducing some of the mathematical notation that statisticians use to describe this calculation, and talk about how the calculations would be done in Python. \n", "\n", "The first piece of notation to introduce is $N$, which we'll use to refer to the number of observations that we're averaging (in this case $N = 5$). Next, we need to attach a label to the observations themselves. It's traditional to use $X$ for this, and to use subscripts to indicate which observation we're actually talking about. That is, we'll use $X_1$ to refer to the first observation, $X_2$ to refer to the second observation, and so on, all the way up to $X_N$ for the last one. Or, to say the same thing in a slightly more abstract way, we use $X_i$ to refer to the $i$-th observation. Just to make sure we're clear on the notation, the following table lists the 5 observations in the `afl_margins` variable, along with the mathematical symbol used to refer to it, and the actual value that the observation corresponds to:\n", "\n", "|the observation |its symbol |the observed value |\n", "|:----------------------|:----------|:------------------|\n", "|winning margin, game 1 |$X_1$ |56 points |\n", "|winning margin, game 2 |$X_2$ |31 points |\n", "|winning margin, game 3 |$X_3$ |56 points |\n", "|winning margin, game 4 |$X_4$ |8 points |\n", "|winning margin, game 5 |$X_5$ |32 points |" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Okay, now let's try to write a formula for the mean. By tradition, we use $\\bar{X}$ as the notation for the mean. So the calculation for the mean could be expressed using the following formula: \n", "\n", "$\\bar{X} = \\frac{X_1 + X_2 + ... + X_{N-1} + X_N}{N}$\n", "\n", "This formula is entirely correct, but it's terribly long, so we make use of the **_summation symbol_** $\\scriptstyle\\sum$ to shorten it.[^note2] If I want to add up the first five observations, I could write out the sum the long way, $X_1 + X_2 + X_3 + X_4 +X_5$ or I could use the summation symbol to shorten it to this: \n", "\n", "\n", "$\\sum_{i=1}^5 X_i$\n", "\n", "\n", "Taken literally, this could be read as \"the sum, taken over all $i$ values from 1 to 5, of the value $X_i$\". But basically, what it means is \"add up the first five observations\". In any case, we can use this notation to write out the formula for the mean, which looks like this: \n", "\n", "\n", "$\\bar{X} = \\frac{1}{N} \\sum_{i=1}^N X_i$\n", "\n", "\n", "In all honesty, I can't imagine that all this mathematical notation helps clarify the concept of the mean at all. In fact, it's really just a fancy way of writing out the same thing I said in words: add all the values up, and then divide by the total number of items. However, that's not really the reason I went into all that detail. My goal was to try to make sure that everyone reading this book is clear on the notation that we'll be using throughout the book: $\\bar{X}$ for the mean, $\\scriptstyle\\sum$ for the idea of summation, $X_i$ for the $i$th observation, and $N$ for the total number of observations. We're going to be re-using these symbols a fair bit, so it's important that you understand them well enough to be able to \"read\" the equations, and to be able to see that it's just saying \"add up lots of things and then divide by another thing\".\n", "\n", "### Calculating the mean in Python\n", "\n", "Okay that's the maths, how do we get the magic computing box to do the work for us? If you really wanted to, you could do this calculation directly in Python. For the first 5 AFL scores, do this just by typing it in as if Python were a calculator...\n", "\n", "[^note2]: The choice to use $\\Sigma$ to denote summation isn't arbitrary: it's the Greek upper case letter sigma, which is the analogue of the letter S in that alphabet. Similarly, there's an equivalent symbol used to denote the multiplication of lots of numbers: because multiplications are also called \"products\", we use the $\\Pi$ symbol for this; the Greek upper case pi, which is the analogue of the letter P." ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "36.6" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "(56 + 31 + 56 + 8 + 32) / 5\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "... in which case Python outputs the answer 36.6, just as if it were a calculator. However, that's not the only way to do the calculations, and when the number of observations starts to become large, it's easily the most tedious. Besides, in almost every real world scenario, you've already got the actual numbers stored in a variable of some kind, just like we have with the `afl_margins` variable. Under those circumstances, what you want is a function that will just add up all the values stored in a numeric vector. That's what the `sum()` function does. If we want to add up all 176 winning margins in the data set, we can do so using the following command:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "6213" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "margins = afl_margins['afl.margins']\n", "\n", "sum(margins)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "If we only want the sum of the first five observations, then we can use square brackets to pull out only the first five elements of the vector. So the command would now be:" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0 56\n", "1 31\n", "2 56\n", "3 8\n", "4 32\n", "Name: afl.margins, dtype: int64" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "margins[0:5]" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Observant readers will have noticed that to get the first 5 elements we need to ask for elements 0 through 5, which seems to make no sense whatsoever. Python can be weird like that. I am **not** going to get into this here, but I talked about it in the section on [lists](lists) and went into more detail in the section on [slices](slices), and will probably mention it again in the section on [pulling out the contents of a data frame](indexingdataframes). To calculate the mean, we now tell Python to divide the output of this summation by five, so the command that we need to type now becomes the following:" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "36.6" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sum(margins[0:5])/5\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Or, we could just ask Python for the mean, without further ado:" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "36.6" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "margins[0:5].mean()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Now, as you may remember (you do remember, right?) I spent some time whinging back in [the previous chapter](importinglibraries) about how the Python developers were really mean because they gave us a built-in function for adding but not for averaging, and I made a big deal about how we had to import `statistics.mean`. So why, I hear you ask, can we all of sudden calculate means _without_ importing `statistics.mean`?" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "As you may have already guessed[^smartiepants], this is because the data we are working with are stored in a [`pandas` dataframe](pandas), which means we are dealing with a different variable type here. We can confirm this, by checking what kind of variable `margins` actually is: \n", "\n", "[^smartiepants]: Or already knew, in which case you are probaby skipping this section and won't read this anyway" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "pandas.core.series.Series" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "type(margins)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Ah. `margins` is not a list, or an integer, or any of the other usual suspects. It is a special kind of type that is defined by the `pandas` package, and like all variable types, it has its own set of special [methods](methods) that we can access using the `.` syntax. If you have been following along and entering the code on your own computer, then you can write `dir(margins)`, and you will see that `pandas.core.series.Series` variables have a long list of methods associated with them, and amongst them is both `sum` and `mean`." ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "If we were to convert `margins` to a different variable type, say, a list, then we could no longer use `.mean()`:" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "ename": "AttributeError", "evalue": "'list' object has no attribute 'mean'", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mAttributeError\u001b[0m Traceback (most recent call last)", "Input \u001b[0;32mIn [24]\u001b[0m, in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1\u001b[0m a \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mlist\u001b[39m(margins)\n\u001b[0;32m----> 2\u001b[0m \u001b[43ma\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mmean\u001b[49m()\n", "\u001b[0;31mAttributeError\u001b[0m: 'list' object has no attribute 'mean'" ] } ], "source": [ "a = list(margins)\n", "a.mean()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "In this case, we would have to take the longer route, by either converting our list `a` to a variable type with a `mean` method, such as a `pandas` series, or by importing `statistics.mean` and using that directly on the list:" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "35.30113636363637" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import statistics\n", "statistics.mean(margins)\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "I said that Python was powerful. I never said it was easy, did I? Don't worry, it gets easier with time[^numb]\n", "\n", "[^numb]: Or, perhaps, you just get numb to the pain." ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Here's what we would do to calculate the mean for only the first five observations:" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "36.6" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "statistics.mean(margins[0:5])" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(median)=\n", "\n", "\n", "### The median\n", "\n", "The second measure of central tendency that people use a lot is the **_median_**, and it's even easier to describe than the mean. The median of a set of observations is just the middle value. As before let's imagine we were interested only in the first 5 AFL winning margins: 56, 31, 56, 8 and 32. To figure out the median, we sort these numbers into ascending order: \n", "\n", "$$\n", "8, 31, \\mathbf{32}, 56, 56\n", "$$\n", "From inspection, it's obvious that the median value of these 5 observations is 32, since that's the middle one in the sorted list (I've put it in bold to make it even more obvious). Easy stuff. But what should we do if we were interested in the first 6 games rather than the first 5? Since the sixth game in the season had a winning margin of 14 points, our sorted list is now \n", "\n", "$$\n", "8, 14, \\mathbf{31}, \\mathbf{32}, 56, 56\n", "$$\n", "and there are *two* middle numbers, 31 and 32. The median is defined as the average of those two numbers, which is of course 31.5. As before, it's very tedious to do this by hand when you've got lots of numbers. To illustrate this, here's what happens when you use Python to sort all 176 winning margins. First, I'll use the `sort_values` method to display the winning margins in increasing numerical order. [^note3]\n", "\n", "[^note3]: `sort_values` is a *method* that belong to `pandas` *objects*. We'll discuss `pandas` more [later](pandas), and you are already somewhat familiar with the [concept of methods](methods) in Python. For now, the important thing is that it works!" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " afl.margins\n", "165 29\n", "173 29\n", "150 29\n", "117 30\n", "1 31\n", "4 32\n", "123 32\n", "136 33" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "sorted_margins = afl_margins.sort_values(by = 'afl.margins')\n", "sorted_margins[84:92]\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "If we peek at the middle of these sorted values, we can see that the middle values are 30 and 31, so the median winning margin for 2010 was 30.5 points. In real life, of course, no-one actually calculates the median by sorting the data and then looking for the middle value. In real life, we use the `median` command[^nomoremethodsnotes]:\n", "\n", "[^nomoremethodsnotes]: By now, you are hopefully getting used to the \"dot\" syntax, where we use a `.` to call an object [method](methods). These distinctions don't really matter so much at this point anyway, so I'm just going to go ahead and start calling things \"commands\". However, for the sake of thoroughness (or something) I will point out that if your data are not in an object like a `pandas` series that has a built-in `median` method, you can also do `import statistics` and then write `statistics.median(margins)` and get the same result." ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "30.5" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "margins.median()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "which outputs the median value of 30.5. \n", "\n", "\n", "### Mean or median? What's the difference?" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "\n", "```{figure} ../img/descriptives2/meanmedian.png\n", ":name: fig-meanmedian\n", ":width: 600px\n", ":align: center\n", "\n", "An illustration of the difference between how the mean and the median should be interpreted. The mean is basically the “centre of gravity” of the data set: if you imagine that the histogram of the data is a solid object, then the point on which you could balance it (as if on a see-saw) is the mean. In contrast, the median is the middle observation. Half of the observations are smaller, and half of the observations are larger.\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Knowing how to calculate means and medians is only a part of the story. You also need to understand what each one is saying about the data, and what that implies for when you should use each one. This is illustrated in {numref}`fig-meanmedian`. The mean is kind of like the \"centre of gravity\" of the data set, whereas the median is the \"middle value\" in the data. What this implies, as far as which one you should use, depends a little on what type of data you've got and what you're trying to achieve. As a rough guide:\n", " \n", "- If your data are nominal scale, you probably shouldn't be using either the mean or the median. Both the mean and the median rely on the idea that the numbers assigned to values are meaningful. If the numbering scheme is arbitrary, then it's probably best to use the [](mode) instead. \n", "- If your data are ordinal scale, you're more likely to want to use the median than the mean. The median only makes use of the order information in your data (i.e., which numbers are bigger), but doesn't depend on the precise numbers involved. That's exactly the situation that applies when your data are ordinal scale. The mean, on the other hand, makes use of the precise numeric values assigned to the observations, so it's not really appropriate for ordinal data.\n", "- For interval and ratio scale data, either one is generally acceptable. Which one you pick depends a bit on what you're trying to achieve. The mean has the advantage that it uses all the information in the data (which is useful when you don't have a lot of data), but it's very sensitive to extreme values, as we'll see when we look at [trimmed means](trimmed_mean). " ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Let's expand on that last part a little. One consequence is that there's systematic differences between the mean and the median when the histogram is asymmetric (or \"skewed\": for more detail, see the section on [skew and kurtosis](skew-and-kurtosis)). This is illustrated in {numref}`fig-meanmedian`, above. Notice that the median (right hand side) is located closer to the \"body\" of the histogram, whereas the mean (left hand side) gets dragged towards the \"tail\" (where the extreme values are). To give a concrete example, suppose Bob (income \\$50,000), Kate (income \\$60,000) and Jane (income \\$65,000) are sitting at a table: the average income at the table is \\$58,333 and the median income is \\$60,000. Then Bill sits down with them (income \\$100,000,000). The average income has now jumped to \\$25,043,750 but the median rises only to \\$62,500. If you're interested in looking at the overall income at the table, the mean might be the right answer; but if you're interested in what counts as a typical income at the table, the median would be a better choice here." ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(housingpriceexample)=\n", "### A real life example\n", "\n", "To try to get a sense of why you need to pay attention to the differences between the mean and the median, let's consider a real life example. Since I tend to mock journalists for their poor scientific and statistical knowledge, I should give credit where credit is due. This is from an excellent article on the ABC news website [^note4] 24 September, 2010:\n", "\n", ">Senior Commonwealth Bank executives have travelled the world in the past couple of weeks with a presentation showing how Australian house prices, and the key price to income ratios, compare favourably with similar countries. \"Housing affordability has actually been going sideways for the last five to six years,\" said Craig James, the chief economist of the bank's trading arm, CommSec.\n", "\n", "This probably comes as a huge surprise to anyone with a mortgage, or who wants a mortgage, or pays rent, or isn't completely oblivious to what's been going on in the Australian housing market over the last several years. Back to the article:\n", "\n", ">CBA has waged its war against what it believes are housing doomsayers with graphs, numbers and international comparisons. In its presentation, the bank rejects arguments that Australia's housing is relatively expensive compared to incomes. It says Australia's house price to household income ratio of 5.6 in the major cities, and 4.3 nationwide, is comparable to many other developed nations. It says San Francisco and New York have ratios of 7, Auckland's is 6.7, and Vancouver comes in at 9.3.\n", "\n", "More excellent news! Except, the article goes on to make the observation that...\n", "\n", ">Many analysts say that has led the bank to use misleading figures and comparisons. If you go to page four of CBA's presentation and read the source information at the bottom of the graph and table, you would notice there is an additional source on the international comparison -- Demographia. However, if the Commonwealth Bank had also used Demographia's analysis of Australia's house price to income ratio, it would have come up with a figure closer to 9 rather than 5.6 or 4.3\n", "\n", "That's, um, a rather serious discrepancy. One group of people say 9, another says 4-5. Should we just split the difference, and say the truth lies somewhere in between? Absolutely not: this is a situation where there is a right answer and a wrong answer. Demographia are correct, and the Commonwealth Bank is incorrect. As the article points out\n", "\n", ">[An] obvious problem with the Commonwealth Bank's domestic price to income figures is they compare average incomes with median house prices (unlike the Demographia figures that compare median incomes to median prices). The median is the mid-point, effectively cutting out the highs and lows, and that means the average is generally higher when it comes to incomes and asset prices, because it includes the earnings of Australia's wealthiest people. To put it another way: the Commonwealth Bank's figures count Ralph Norris' multi-million dollar pay packet on the income side, but not his (no doubt) very expensive house in the property price figures, thus understating the house price to income ratio for middle-income Australians.\n", "\n", "Couldn't have put it better myself. The way that Demographia calculated the ratio is the right thing to do. The way that the Bank did it is incorrect. As for why an extremely quantitatively sophisticated organisation such as a major bank made such an elementary mistake, well... I can't say for sure, since I have no special insight into their thinking, but the article itself does happen to mention the following facts, which may or may not be relevant:\n", "\n", ">[As] Australia's largest home lender, the Commonwealth Bank has one of the biggest vested interests in house prices rising. It effectively owns a massive swathe of Australian housing as security for its home loans as well as many small business loans.\n", "\n", "My, my. " ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(trimmed_mean)=\n", "### Trimmed mean \n", "\n", "One of the fundamental rules of applied statistics is that the data are messy. Real life is never simple, and so the data sets that you obtain are never as straightforward as the statistical theory says. [^note5] This can have awkward consequences. To illustrate, consider this rather strange looking data set: \n", "\n", "$$\n", "-100,2,3,4,5,6,7,8,9,10\n", "$$\n", "If you were to observe this in a real life data set, you'd probably suspect that something funny was going on with the $-100$ value. It's probably an **_outlier_**, a value that doesn't really belong with the others. You might consider removing it from the data set entirely, and in this particular case I'd probably agree with that course of action. In real life, however, you don't always get such cut-and-dried examples. For instance, you might get this instead: \n", "\n", "$$\n", "-15,2,3,4,5,6,7,8,9,12\n", "$$\n", "The $-15$ looks a bit suspicious, but not anywhere near as much as that $-100$ did. In this case, it's a little trickier. It *might* be a legitimate observation, it might not.\n", "\n", "When faced with a situation where some of the most extreme-valued observations might not be quite trustworthy, the mean is not necessarily a good measure of central tendency. It is highly sensitive to one or two extreme values, and is thus not considered to be a **_robust_** measure. One remedy that we've seen is to use the median. A more general solution is to use a \"trimmed mean\". To calculate a trimmed mean, what you do is \"discard\" the most extreme examples on both ends (i.e., the largest and the smallest), and then take the mean of everything else. The goal is to preserve the best characteristics of the mean and the median: just like a median, you aren't highly influenced by extreme outliers, but like the mean, you \"use\" more than one of the observations. Generally, we describe a trimmed mean in terms of the percentage of observation on either side that are discarded. So, for instance, a 10% trimmed mean discards the largest 10% of the observations *and* the smallest 10% of the observations, and then takes the mean of the remaining 80% of the observations. Not surprisingly, the 0% trimmed mean is just the regular mean, and the 50% trimmed mean is the median. In that sense, trimmed means provide a whole family of central tendency measures that span the range from the mean to the median.\n", "\n", "\n", "For our toy example above, we have 10 observations, and so a 10% trimmed mean is calculated by ignoring the largest value (i.e., `12`) and the smallest value (i.e., `-15`) and taking the mean of the remaining values. First, let's enter the data\n", "\n", "[^note4]: www.abc.net.au/news/stories/2010/09/24/3021480.htm\n", "\n", "[^note5]: Or at least, the basic statistical theory -- these days there is a whole subfield of statistics called *robust statistics* that tries to grapple with the messiness of real data and develop theory that can cope with it." ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [], "source": [ "\n", "dataset = [-15,2,3,4,5,6,7,8,9,12]\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Next, let's calculate means and medians. Since our data is now in a regular old list, and not in a dataframe, we can't use the `.mean()` and `.median()` [methods](methods), so we'll just go the old-school route, and `import` our old friend `statistics`:" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "4.1" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import statistics\n", "statistics.mean(dataset)" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "5.5" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "statistics.median(dataset)\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "That's a fairly substantial difference, but I'm tempted to think that the mean is being influenced a bit too much by the extreme values at either end of the data set, especially the $-15$ one. So let's just try trimming the mean a bit. If I take a 10% trimmed mean [^note6], we'll drop the extreme values on either side, and take the mean of the rest: \n", "\n", "[^note6]: Here I use the `stats` function from the `scipy` module. But `stats` is picky: it only wants to deal with data in a certain format called `numpy arrays`. So, to give it what it wants, we also need to import `numpy`, and then convert our data into an `array`. Also, I only imported part of the `scipy` module (you can do that) and renamed the `numpy` module (you can do that too). For a refresher on these technicalities, flip back a few pages to the section on [importing libraries](importinglibraries)." ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "5.5" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import numpy as np\n", "from scipy import stats\n", "\n", "dataset2 = np.array(dataset)\n", "\n", "stats.trim_mean(dataset2, 0.1)\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "which in this case gives exactly the same answer as the median. Note that, to get a 10% trimmed mean you write `trim = .1`, not `trim = 10`. In any case, let's finish up by calculating the 5% trimmed mean for the `afl_margins` data " ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "33.75" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dataset3 = np.array(margins)\n", "stats.trim_mean(dataset3, 0.05)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(mode)= \n", "### Mode\n", "\n", "The mode of a sample is very simple: it is the value that occurs most frequently. To illustrate the mode using the AFL data, let's examine a different aspect to the data set. Who has played in the most finals? The `afl_finalists` data contains the name of every team that played in any AFL final from 1987-2010, so let's have a look at it. To do this we will use the `head()` method. `head()` is a method that can be used when the data is contained in a `pandas` `dataframe` object (which ours is). It can be useful when you're working with data with a lot of rows, since you can use it to tell you how many rows to return. There have been a lot of finals in this period so printing afl_finalists using `print(afl_finalists)` will just fill the screen. The command below tells Python we just want the first 25 rows of the dataframe." ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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24Collingwood
\n", "
" ], "text/plain": [ " afl.finalists\n", "0 Hawthorn\n", "1 Melbourne\n", "2 Carlton\n", "3 Melbourne\n", "4 Hawthorn\n", "5 Carlton\n", "6 Melbourne\n", "7 Carlton\n", "8 Hawthorn\n", "9 Melbourne\n", "10 Melbourne\n", "11 Hawthorn\n", "12 Melbourne\n", "13 Essendon\n", "14 Hawthorn\n", "15 Geelong\n", "16 Geelong\n", "17 Hawthorn\n", "18 Collingwood\n", "19 Melbourne\n", "20 Collingwood\n", "21 West Coast\n", "22 Collingwood\n", "23 Essendon\n", "24 Collingwood" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "afl_finalists.head(n=25)\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "There are actually 400 entries (aren't you glad we didn't print them all?). We *could* read through all 400, and count the number of occasions on which each team name appears in our list of finalists, thereby producing a **_frequency table_**. However, that would be mindless and boring: exactly the sort of task that computers are great at. So let's use the `value_counts()` method to do this task for us[^onelineortwo]:\n", "\n", "[^onelineortwo]: `value_counts()` is a method available to us because our data is in a `pandas` dataframe. In the example below, I have made a new variable called `finalists` which contains the `afl.finalists` column from the `afl_finalists` dataframe. Of course, we could just do `value_counts()` directly on the `pandas` column like this: `afl_finalists['afl.finalists'].value_counts()`. The result is the same." ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Geelong 39\n", "West Coast 38\n", "Essendon 32\n", "Melbourne 28\n", "Collingwood 28\n", "North Melbourne 28\n", "Hawthorn 27\n", "Carlton 26\n", "Adelaide 26\n", "Sydney 26\n", "Brisbane 25\n", "St Kilda 24\n", "Western Bulldogs 24\n", "Port Adelaide 17\n", "Richmond 6\n", "Fremantle 6\n", "Name: afl.finalists, dtype: int64" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "finalists = afl_finalists['afl.finalists']\n", "finalists.value_counts()\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Now that we have our frequency table, we can just look at it and see that, over the 24 years for which we have data, Geelong has played in more finals than any other team. Thus, the mode of the `finalists` data is `\"Geelong\"`. If we want to extract the mode without inspecting the table, we can use the `statistics.mode` function to tell us which team has most often played in the finals." ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'Geelong'" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "statistics.mode(finalists)\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "If we want to find the number of finals they have played in, we can e.g. first extract the frequencies with `value_counts` and then find the largest value with `max`." ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "39" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "freq = finalists.value_counts()\n", "freq.max()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Taken together, we observe that Geelong (39 finals) played in more finals than any other team during the 1987-2010 period. \n", "\n", "One last point to make with respect to the mode. While it's generally true that the mode is most often calculated when you have nominal scale data (because means and medians are useless for those sorts of variables), there are some situations in which you really do want to know the mode of an ordinal, interval or ratio scale variable. For instance, let's go back to thinking about our `afl_margins` variable. This variable is clearly ratio scale (if it's not clear to you, it may help to re-read [](scales)), and so in most situations the mean or the median is the measure of central tendency that you want. But consider this scenario... a friend of yours is offering a bet. They pick a football game at random, and (without knowing who is playing) you have to guess the *exact* margin. If you guess correctly, you win \\$50. If you don't, you lose \\$1. There are no consolation prizes for \"almost\" getting the right answer. You have to guess exactly the right margin. [^note7] For this bet, the mean and the median are completely useless to you. It is the mode that you should bet on. So, we calculate this modal value\n", "\n", "[^note7]: This is called a \"0-1 loss function\", meaning that you either win (1) or you lose (0), with no middle ground." ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "statistics.mode(margins)" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "8" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "freq = margins.value_counts()\n", "freq.max()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "So the 2010 data suggest you should bet on a 3 point margin, and since this was observed in 8 of the 176 game (4.5% of games) the odds are firmly in your favour. " ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(variability)=\n", "\n", "## Measures of variability\n", "\n", "The statistics that we've discussed so far all relate to *central tendency*. That is, they all talk about which values are \"in the middle\" or \"popular\" in the data. However, central tendency is not the only type of summary statistic that we want to calculate. The second thing that we really want is a measure of the **_variability_** of the data. That is, how \"spread out\" are the data? How \"far\" away from the mean or median do the observed values tend to be? For now, let's assume that the data are interval or ratio scale, so we'll continue to use the `afl_margins` data. We'll use this data to discuss several different measures of spread, each with different strengths and weaknesses. " ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(range)=\n", "### Range\n", "\n", "The **_range_** of a variable is very simple: it's the biggest value minus the smallest value. For the AFL winning margins data, the maximum value is 116, and the minimum value is 0. We can calculate these values in Python using the `max()` and `min()` functions:\n", "\n", "\n", "`margins.max()` \n", "`margins.min()`\n", "\n", "\n", "where I've omitted the output because it's not interesting.\n", "\n", "Although the range is the simplest way to quantify the notion of \"variability\", it's one of the worst. Recall from our discussion of the mean that we want our summary measure to be robust. If the data set has one or two extremely bad values in it, we'd like our statistics not to be unduly influenced by these cases. If we look once again at our toy example of a data set containing very extreme outliers... \n", "\n", "$$\n", "-100,2,3,4,5,6,7,8,9,10\n", "$$\n", "... it is clear that the range is not robust, since this has a range of 110, but if the outlier were removed we would have a range of only 8." ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(iqr)=\n", "### Interquartile range\n", "\n", "The **_interquartile range_** (IQR) is like the range, but instead of calculating the difference between the biggest and smallest value, it calculates the difference between the 25th quantile and the 75th quantile. Probably you already know what a **_quantile_** is (they're more commonly called percentiles), but if not: the 10th percentile of a data set is the smallest number $x$ such that 10% of the data is less than $x$. In fact, we've already come across the idea: the median of a data set is its 50th quantile / percentile! The `numpy` module actually provides you with a way of calculating quantiles, using the (surprise, surprise) `quantile()` function. Let's use it to calculate the median AFL winning margin:" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "30.5" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "import numpy as np\n", "np.quantile(margins, 0.5)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "And not surprisingly, this agrees with the answer that we saw earlier with the `median()` function. Now, we can actually input lots of quantiles at once, by specifying which quantiles we want. So lets do that, and get the 25th and 75th percentile:" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([12.75, 50.5 ])" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "np.quantile(margins, [0.25, .75])\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "And, by noting that $50.5 - 12.75 = 37.75$, we can see that the interquartile range for the 2010 AFL winning margins data is 37.75. Of course, that seems like too much work to do all that typing, and luckily we don't have to, since the kind folks at `scipy` have already done the work for us and provided us with the `stats.iqr` function, which will give us what we want." ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "37.75" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "from scipy import stats\n", "stats.iqr(margins)\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "While it's obvious how to interpret the range, it's a little less obvious how to interpret the IQR. The simplest way to think about it is like this: the interquartile range is the range spanned by the \"middle half\" of the data. That is, one quarter of the data falls below the 25th percentile, one quarter of the data is above the 75th percentile, leaving the \"middle half\" of the data lying in between the two. And the IQR is the range covered by that middle half." ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(aad)=\n", "\n", "### Mean absolute deviation\n", "\n", "The two measures we've looked at so far, the range and the interquartile range, both rely on the idea that we can measure the spread of the data by looking at the quantiles of the data. However, this isn't the only way to think about the problem. A different approach is to select a meaningful reference point (usually the mean or the median) and then report the \"typical\" deviations from that reference point. What do we mean by \"typical\" deviation? Usually, the mean or median value of these deviations! In practice, this leads to two different measures, the \"mean absolute deviation (from the mean)\" and the \"median absolute deviation (from the median)\". From what I've read, the measure based on the median seems to be used in statistics, and does seem to be the better of the two, but to be honest I don't think I've seen it used much in psychology. The measure based on the mean does occasionally show up in psychology though. In this section I'll talk about the first one, and I'll come back to talk about the second one later.\n", "\n", "Since the previous paragraph might sound a little abstract, let's go through the **_mean absolute deviation_** from the mean a little more slowly. One useful thing about this measure is that the name actually tells you exactly how to calculate it. Let's think about our AFL winning margins data, and once again we'll start by pretending that there's only 5 games in total, with winning margins of 56, 31, 56, 8 and 32. Since our calculations rely on an examination of the deviation from some reference point (in this case the mean), the first thing we need to calculate is the mean, $\\bar{X}$. For these five observations, our mean is $\\bar{X} = 36.6$. The next step is to convert each of our observations $X_i$ into a deviation score. We do this by calculating the difference between the observation $X_i$ and the mean $\\bar{X}$. That is, the deviation score is defined to be $X_i - \\bar{X}$. For the first observation in our sample, this is equal to $56 - 36.6 = 19.4$. Okay, that's simple enough. The next step in the process is to convert these deviations to absolute deviations. We do this by converting any negative values to positive ones. Mathematically, we would denote the absolute value of $-3$ as $|-3|$, and so we say that $|-3| = 3$. We use the absolute value function here because we don't really care whether the value is higher than the mean or lower than the mean, we're just interested in how *close* it is to the mean. To help make this process as obvious as possible, the table below shows these calculations for all five observations:" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "| $i$ (which game) | $X_i$ (value) | $X_i - \\bar{X}$ (deviation from mean) | $\\|X_i - \\bar{X}\\|$ (absolute deviation) |\n", "|--------------|----------|--------------------------|--------------------|\n", "| 1 | 56 | 19.4 | 19.4 |\n", "| 2 | 31 | -5.6 | 5.6 |\n", "| 3 | 56 | 19.4 | 19.4 |\n", "| 4 | 8 | -28.6 | 28.6 |\n", "| 5 | 32 | -4.6 | 4.6 |" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Now that we have calculated the absolute deviation score for every observation in the data set, all that we have to do to calculate the mean of these scores. Let's do that:\n", "\n", "$$\n", "\n", "\\frac{19.4 + 5.6 + 19.4 + 28.6 + 4.6}{5} = 15.52\n", "\n", "$$\n", "\n", "And we're done. The mean absolute deviation for these five scores is 15.52. \n", "\n", "However, while our calculations for this little example are at an end, we do have a couple of things left to talk about. First, we should really try to write down a proper mathematical formula. But in order do to this I need some mathematical notation to refer to the mean absolute deviation. Irritatingly, \"mean absolute deviation\" and \"median absolute deviation\" have the same acronym (MAD), which leads to a certain amount of ambiguity. To make matters worse, packages that include functions to calculate these things for you _**both use the abbreviation MAD even though they mean different things!**_ Sigh. What I'll do is use AAD instead, short for *average* absolute deviation. Now that we have some unambiguous notation, here's the formula that describes what we just calculated:\n", "\n", "$$\n", "\n", "AAD\\mbox{}(X) = \\frac{1}{N} \\sum_{i = 1}^N |X_i - \\bar{X}|\n", "\n", "$$\n", "\n", "The last thing we need to talk about is how to calculate AAD in Python. One possibility would be to do everything using low level commands, laboriously following the same steps that I used when describing the calculations above. However, that's pretty tedious. You'd end up with a series of commands that might look like this:" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "15.52\n" ] } ], "source": [ "from statistics import mean\n", "\n", "X = [56, 31, 56, 8, 32] # 1. enter the data\n", "X_bar = mean(X) # 2. find the mean of the data\n", "AD = [] # 3. find the absolute value of the difference between\n", "for i in X: # each value and the mean and add it to the list AD \n", " AD.append(abs((i-X_bar))) \n", "AAD = mean(AD) # 4. find the mean of the absolute values\n", "print(AAD)\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Each of those commands is pretty simple, but there's just too many of them. And because I find that to be too much typing, I suggest using the `pandas` method `mad()` to make life easier. There is one important thing to notice, however: `pandas` will want the data to be in a different format, namely the simply and elegantly named `pandas.core.series.Series` format. Haha, just kidding. At least about the simple and elegant name. The format is great, though, and can do lots of things that ordinary lists can't, so it is worth putting up with the name. To find the AAD, we can just do:" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "15.52" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "data = pd.Series( [56, 31, 56, 8, 32])\n", "data.mad()\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Ok, I lied again. There is more than one thing to notice (when isn't there?). `pandas` calls this the \"MAD\". And, as we have seen, this is _also_ the acronym for the _median absolute difference_. To find the median absolute difference, we can use the `robust` module from the `statsmodels` package, which has a function called, you guessed it, `mad`. Except this time the \"m\" in \"MAD\" stand for \"median\" and not \"mean\". Haha, isn't naming fun?\n", "\n", "We'll talk more about [median absolute deviation](mad) below. For now, just be careful to be aware of which one you are using, and why, and everything will be ok!" ] }, { "cell_type": "code", "execution_count": 45, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Pandas MAD is the mean absolute deviation: 15.52\n", "Statsmodels MAD is the median absolute deviation: 24.0\n" ] } ], "source": [ "import pandas as pd\n", "import numpy as np\n", "from statsmodels import robust\n", "\n", "data = pd.Series( [56, 31, 56, 8, 32])\n", "pandas_mad = data.mad()\n", "\n", "data = np.array([56, 31, 56, 8, 32])\n", "statsmodels_mad = robust.mad(data, c = 1)\n", "\n", "print(\"Pandas MAD is the mean absolute deviation:\", pandas_mad)\n", "print(\"Statsmodels MAD is the median absolute deviation:\", statsmodels_mad)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(var)=\n", "### Variance \n", "\n", "\n", "Although mean absolute deviation measure has its uses, it's not the best measure of variability to use. From a purely mathematical perspective, there are some solid reasons to prefer squared deviations rather than absolute deviations. If we do that, we obtain a measure is called the **_variance_**, which has a lot of really nice statistical properties that I'm going to ignore, [^note8] and one massive psychological flaw that I'm going to make a big deal out of in a moment. The variance of a data set $X$ is sometimes written as $\\mbox{Var}(X)$ but it's more commonly denoted $s^2$ (the reason for this will become clearer shortly). The formula that we use to calculate the variance of a set of observations is as follows:\n", "\n", "$$\n", "\\mbox{Var}(X) = \\frac{1}{N} \\sum_{i=1}^N \\left( X_i - \\bar{X} \\right)^2\n", "$$\n", "\n", "\n", "As you can see, it's basically the same formula that we used to calculate the mean absolute deviation, except that instead of using \"absolute deviations\" we use \"squared deviations\". It is for this reason that the variance is sometimes referred to as the \"mean square deviation\".\n", "\n", "Now that we've got the basic idea, let's have a look at a concrete example. Once again, let's use the first five AFL games as our data. If we follow the same approach that we took last time, we end up with the following table:\n", "\n", "| which game | value | deviation from mean | squared deviation |\n", "| :--------: | :---: | :-----------------: | :---------------: |\n", "| 1 | 56 | 19.4 | 376.36 |\n", "| 2 | 31 | -5.6 | 31.36 |\n", "| 3 | 56 | 19.4 | 376.36 |\n", "| 4 | 8 | -28.6 | 817.96 |\n", "| 5 | 32 | -4.6 | 21.16 |\n", "\n", "The same table again, translated into Math-ese, looks like this:\n", "\n", "| *i* | $X_i$ | $X_i - \\bar{X}$ | $(X_i - \\bar{X}$)$^2$ |\n", "| :--------: | :---: | :-----------------: | :-----------------------------------------: \n", "| 1 | 56 | 19.4 ㅤㅤ | 376.36 ㅤㅤ|\n", "| 2 | 31 | -5.6 ㅤㅤ | 31.36 ㅤㅤ|\n", "| 3 | 56 | 19.4 ㅤㅤ | 376.36 ㅤㅤ|\n", "| 4 | 8 | -28.6 ㅤㅤ | 817.96 ㅤㅤ|\n", "| 5 | 32 | -4.6 ㅤㅤ | 21.16 ㅤㅤ|\n", "\n", "That last column contains all of our squared deviations, so all we have to do is average them. If we do that by typing all the numbers into Python by hand...\n", "\n", "[^note8]: Well, I will very briefly mention the one that I think is coolest, for a very particular definition of \"cool\", that is. Variances are *additive*. Here's what that means: suppose I have two variables $X$ and $Y$, whose variances are $\\mbox{Var}(X)$ and $\\mbox{Var}(Y)$ respectively. Now imagine I want to define a new variable $Z$ that is the sum of the two, $Z = X+Y$. As it turns out, the variance of $Z$ is equal to $\\mbox{Var}(X) + \\mbox{Var}(Y)$. This is a *very* useful property, but it's not true of the other measures that I talk about in this section." ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "324.64" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "( 376.36 + 31.36 + 376.36 + 817.96 + 21.16 ) / 5\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "... we end up with a variance of 324.64. Exciting, isn't it? For the moment, let's ignore the burning question that you're all probably thinking (i.e., what the heck does a variance of 324.64 actually mean?) and instead talk a bit more about how to do the calculations in Python, because this will reveal something very weird.\n", "\n", "As always, we want to avoid having to type in a whole lot of numbers ourselves, and fortunately the `statistics` module provides a function called `variance` which saves us the trouble. And as it happens, we have the values lying around in the variable `data`, which we created in the previous section. With this in mind, we can just calculate the variance of `data` by using the following command" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "405" ] }, "execution_count": 47, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import statistics\n", "\n", "statistics.variance(data)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "and you get the same... no, wait... you get a completely different answer. That’s just weird. Is Python broken? Is this a typo? Are Danielle and Ethan idiots?\n", "As it happens, the answer is no[^note9]. It's not a typo, and Python is not making a mistake. To get a feel for what's happening, let's stop using the tiny data set containing only 5 data points, and switch to the full set of 176 games that we've got stored in our `afl_margins` vector. First, let's calculate the variance by using the formula that I described above:\n", "\n", "[^note9]: With the possible exception of the third question." ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "675.9718168904958" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import statistics\n", "m = statistics.mean(afl_margins['afl.margins']) #Find the mean of afl.margins\n", "v = [] #Create an empty list\n", "for n in afl_margins['afl.margins']: #Look at each entry in afl.margins\n", " squared_error = (n-m)**2 #Find the squared difference between each item and the mean\n", " v.append(squared_error) #Put each squared in the list v\n", "var = statistics.mean(v) #Find the mean of v (mean of the squared errors)\n", "var" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Now let's use the `statistics.variance()` function:" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "679.834512987013" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import statistics\n", "\n", "var = statistics.variance(afl_margins['afl.margins'])\n", "\n", "var" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Hm. These two numbers are very similar this time. That seems like too much of a coincidence to be a mistake. And of course it isn't a mistake. In fact, it's very simple to explain what Python is doing here, but slightly trickier to explain *why* Python is doing it. So let's start with the \"what\". What Python is doing is evaluating a slightly different formula to the one I showed you above. Instead of averaging the squared deviations, which requires you to divide by the number of data points $N$, Python has chosen to divide by $N-1$. In other words, the formula that Python is using is this one " ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\n", "\\frac{1}{N-1} \\sum_{i=1}^N \\left( X_i - \\bar{X} \\right)^2\n", "\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "So that's the *what*. The real question is *why* Python is dividing by $N-1$ and not by $N$. After all, the variance is supposed to be the *mean* squared deviation, right? So shouldn't we be dividing by $N$, the actual number of observations in the sample? Well, yes, we should. However, as we'll discuss in the section on [estimation](estimation), there's a subtle distinction between \"describing a sample\" and \"making guesses about the population from which the sample came\". Up to this point, it's been a distinction without a difference. Regardless of whether you're describing a sample or drawing inferences about the population, the mean is calculated exactly the same way. Not so for the variance, or the standard deviation, or for many other measures besides. What I outlined to you initially (i.e., take the actual average, and thus dividing by $N$) assumes that you literally intend to calculate the variance of the sample. Most of the time, however, you're not terribly interested in the sample *in and of itself*. Rather, the sample exists to tell you something about the world. If so, you're actually starting to move away from calculating a \"sample statistic\", and towards the idea of estimating a \"population parameter\". However, I'm getting ahead of myself. For now, let's just take it on faith that Python knows what it's doing, and we'll revisit the question later on when we talk about [estimation](estimation). \n", "\n", "By the way, if you _do_ want to calculate variance and divide by $N$ and not $N-1$, Python does a have a way to do this as well; you just need to ask for `pvariance()` instead of `variance()`:" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "statistics.pvariance divides by N: 675.9718168904959\n", "statistics.variance divide by N-1: 679.834512987013\n" ] } ], "source": [ "population_variance = statistics.pvariance(afl_margins['afl.margins'])\n", "sample_variance = statistics.variance(afl_margins['afl.margins'])\n", "\n", "print(\"statistics.pvariance divides by N: \", population_variance)\n", "print(\"statistics.variance divide by N-1: \", sample_variance)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Okay, one last thing. This section so far has read a bit like a mystery novel. I've shown you how to calculate the variance, described the weird \"$N-1$\" thing that Python does and hinted at the reason why it's there, but I haven't mentioned the single most important thing... how do you *interpret* the variance? Descriptive statistics are supposed to describe things, after all, and right now the variance is really just a gibberish number. Unfortunately, the reason why I haven't given you the human-friendly interpretation of the variance is that there really isn't one. This is the most serious problem with the variance. Although it has some elegant mathematical properties that suggest that it really is a fundamental quantity for expressing variation, it's completely useless if you want to communicate with an actual human... variances are completely uninterpretable in terms of the original variable! All the numbers have been squared, and they don't mean anything anymore. This is a huge issue. For instance, according to the table I presented earlier, the margin in game 1 was \"376.36 points-squared higher than the average margin\". This is *exactly* as stupid as it sounds; and so when we calculate a variance of 324.64, we're in the same situation. I've watched a lot of footy games, and never has anyone referred to \"points squared\". It's *not* a real unit of measurement, and since the variance is expressed in terms of this gibberish unit, it is totally meaningless to a human.\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(sd)=\n", "### Standard deviation\n", "\n", "Okay, suppose that you like the idea of using the variance because of those nice mathematical properties that I haven't talked about, but -- since you're a human and not a robot -- you'd like to have a measure that is expressed in the same units as the data itself (i.e., points, not points-squared). What should you do? The solution to the problem is obvious: take the square root of the variance, known as the **_standard deviation_**, also called the \"root mean squared deviation\", or RMSD. This solves our problem fairly neatly: while nobody has a clue what \"a variance of 324.68 points-squared\" really means, it's much easier to understand \"a standard deviation of 18.01 points\", since it's expressed in the original units. It is traditional to refer to the standard deviation of a sample of data as $s$, though \t\"sd\" and \"std dev.\" are also used at times. Because the standard deviation is equal to the square root of the variance, you probably won't be surprised to see that the formula is:\n", "\n", "$$\n", "s = \\sqrt{ \\frac{1}{N} \\sum_{i=1}^N \\left( X_i - \\bar{X} \\right)^2 }\n", "$$\n", "\n", "and the function that we use to calculate it is `stdev()`. However, as you might have guessed from our discussion of the variance, what Python actually calculates is slightly different to the formula given above. Just like the we saw with the variance, what Python calculates is a version that divides by $N-1$ rather than $N$. For reasons that will make sense when we return to this topic in the chapter on [estimation](estimation), I'll refer to this new quantity as $\\hat\\sigma$ (read as: \"sigma hat\"), and the formula for this is \n", "\n", "$$\n", "\\hat\\sigma = \\sqrt{ \\frac{1}{N-1} \\sum_{i=1}^N \\left( X_i - \\bar{X} \\right)^2 }\n", "$$\n", "\n", "With that in mind, calculating standard deviations in Python is simple:" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "26.073636359108274" ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "statistics.stdev(margins)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Interpreting standard deviations is slightly more complex. Because the standard deviation is derived from the variance, and the variance is a quantity that has little to no meaning that makes sense to us humans, the standard deviation doesn't have a simple interpretation. As a consequence, most of us just rely on a simple rule of thumb: in general, you should expect 68% of the data to fall within 1 standard deviation of the mean, 95% of the data to fall within 2 standard deviation of the mean, and 99.7% of the data to fall within 3 standard deviations of the mean. This rule tends to work pretty well most of the time, but it's not exact: it's actually calculated based on an *assumption* that the histogram is symmetric and \"bell shaped\". [^note10] As you can tell from looking at the AFL winning margins histogram in {numref}`fig-aflsd`, this isn't exactly true of our data! Even so, the rule is approximately correct. As it turns out, 65.3% of the AFL margins data fall within one standard deviation of the mean. This is shown visually in {numref}`fig-aflsd`.\n", "\n", "[^note10]: Strictly, the assumption is that the data are *normally* distributed, which is an important concept that we'll discuss more in the chapter [](probability), and will turn up over and over again later in the book." ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "\n", "```{figure} ../img/descriptives2/figure_2.5_mean_SD.png\n", ":name: fig-aflsd\n", ":width: 600px\n", ":align: center\n", "\n", "\n", "An illustration of the standard deviation, applied to the AFL winning margins data. The shaded bars in the histogram show how much of the data fall within one standard deviation of the mean. In this case, 65.3% of the data set lies within this range, which is pretty consistent with the \\\"approximately 68% rule\\\" discussed in the main text.\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(mad)=\n", "### Median absolute deviation\n", "\n", "The last measure of variability that I want to talk about is the **_median absolute deviation_** (MAD). The basic idea behind MAD is very simple: it's just the median of the absolute deviations from the median of the data. Find the distance of each data point from the median of all the data points (ignoring the signs), and then take the median of that.\n", "\n", "This has a straightforward interpretation: every observation in the data set lies some distance away from the typical value (the median). So the MAD is an attempt to describe a *typical deviation from a typical value* in the data set. It wouldn't be unreasonable to interpret the MAD value of 19.5 for our AFL data by saying something like this:\n", "\n", ">The median winning margin in 2010 was 30.5, indicating that a typical game involved a winning margin of about 30 points. However, there was a fair amount of variation from game to game: the MAD value was 19.5, indicating that a typical winning margin would differ from this median value by about 19-20 points.\n", "\n", "As you'd expect, Python has a method for calculating MAD. It is in the `robust` object from the `statsmodels` package, and you will be shocked no doubt to hear that it's called `mad()`. However, it's a little bit more complicated than the functions that we've been using previously. If you want to use it to calculate MAD in the exact same way that I have described it above, the command that you need to use specifies two arguments: the data set itself `x`, and a `constant` that I'll explain in a moment. For our purposes, the constant is 1, so our command becomes" ] }, { "cell_type": "code", "execution_count": 52, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "19.5" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import numpy as np\n", "from statsmodels import robust\n", "\n", "robust.mad(margins, c=1)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Apart from the weirdness of having to type that `c = 1` part, this is pretty straightforward.\n", "\n", "Okay, so what exactly is this `c = 1` argument? I won't go into all the details here, but here's the gist. Although the \"raw\" MAD value that I've described above is completely interpretable on its own terms, that's not actually how it's used in a lot of real world contexts. Instead, what happens a lot is that the researcher *actually* wants to calculate the standard deviation. However, in the same way that the mean is very sensitive to extreme values, the standard deviation is vulnerable to the exact same issue. So, in much the same way that people sometimes use the median as a \"robust\" way of calculating \"something that is like the mean\", it's not uncommon to use MAD as a method for calculating \"something that is like the standard deviation\". Unfortunately, the *raw* MAD value doesn't do this. Our raw MAD value is 19.5, and our standard deviation was 26.07. However, what some clever person has shown is that, under certain assumptions (the assumption again being that the data are normally-distributed!), you can multiply the raw MAD value by 1.4826 and obtain a number that is directly comparable to the standard deviation. As a consequence, the default value of `constant` is 1.4826, and so when you use the `mad()` command without manually setting a value, here's what you get:" ] }, { "cell_type": "code", "execution_count": 53, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "28.91074326085924" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "robust.mad(margins)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "I should point out, though, that if you want to use this \"corrected\" MAD value as a robust version of the standard deviation, you really are relying on the assumption that the data are (or at least, are \"supposed to be\" in some sense) symmetric and basically shaped like a bell curve. That's really *not* true for our `afl_margins` data, so in this case I wouldn't try to use the MAD value this way.\n", "\n", "\n", "### Which measure to use?\n", "\n", "We've discussed quite a few measures of spread (range, IQR, MAD, variance and standard deviation), and hinted at their strengths and weaknesses. Here's a quick summary:\n", "\n", "\n", "- [*Range*](range). Gives you the full spread of the data. It's very vulnerable to outliers, and as a consequence it isn't often used unless you have good reasons to care about the extremes in the data.\n", "- [*Interquartile range*](iqr). Tells you where the \"middle half\" of the data sits. It's pretty robust, and complements the median nicely. This is used a lot.\n", "- [*Mean absolute deviation*](aad). Tells you how far “on average” the observations are from the mean. It’s very interpretable, but has a few minor issues (not discussed here) that make it less attractive to statisticians than the standard deviation. Used sometimes, but not often.\n", "- [*Variance*](var). Tells you the average squared deviation from the mean. It's mathematically elegant, and is probably the \"right\" way to describe variation around the mean, but it's completely uninterpretable because it doesn't use the same units as the data. Almost never used except as a mathematical tool; but it's buried \"under the hood\" of a very large number of statistical tools.\n", "- [*Standard deviation*](sd). This is the square root of the variance. It's fairly elegant mathematically, and it's expressed in the same units as the data so it can be interpreted pretty well. In situations where the mean is the measure of central tendency, this is the default. This is by far the most popular measure of variation. \n", "- [*Median absolute deviation*](mad). The typical (i.e., median) deviation from the median value. In the raw form it's simple and interpretable; in the corrected form it's a robust way to estimate the standard deviation, for some kinds of data sets. Not used very often, but it does get reported sometimes.\n", "\n", "\n", "In short, the IQR and the standard deviation are easily the two most common measures used to report the variability of the data; but there are situations in which the others are used. I've described all of them in this book because there's a fair chance you'll run into most of these somewhere." ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(skew-and-kurtosis)=\n", "## Skew and kurtosis\n", "\n", "There are two more descriptive statistics that you will sometimes see reported in the psychological literature, known as skew and kurtosis. In practice, neither one is used anywhere near as frequently as the measures of central tendency and variability that we've been talking about. Skew is pretty important, so you do see it mentioned a fair bit; but I've actually never seen kurtosis reported in a scientific article to date. " ] }, { "cell_type": "code", "execution_count": 54, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import pandas as pd\n", "import seaborn as sns\n", "from matplotlib import pyplot as plt\n", "\n", "\n", "# load some data\n", "url = 'https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/skewdata.csv'\n", "\n", "\n", "df_skew = pd.read_csv(url)\n", "\n", "\n", "fig, axes = plt.subplots(1, 3, figsize=(15, 5))\n", "\n", "ax1 = sns.histplot(data = df_skew.loc[df_skew['Skew'] == 'NegSkew'], x = 'Values', binwidth = 0.02, ax=axes[0])\n", "ax2 = sns.histplot(data = df_skew.loc[df_skew['Skew'] == 'NoSkew'], x = 'Values', binwidth = 0.02, ax=axes[1])\n", "ax3 = sns.histplot(data = df_skew.loc[df_skew['Skew'] == 'PosSkew'], x = 'Values', binwidth = 0.02, ax=axes[2])\n", "\n", "axes[0].set_title(\"Negative Skew\")\n", "axes[1].set_title(\"No Skew\")\n", "axes[2].set_title(\"Positive Skew\")\n", "\n", "for ax in axes:\n", " ax.set(xticklabels=[])\n", " ax.set(yticklabels=[])\n", " ax.set(xlabel=None)\n", " ax.set(ylabel=None)\n", " ax.tick_params(bottom=False)\n", " ax.tick_params(left=False)\n", "\n", "sns.despine()\n", " " ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} skew_fig\n", ":figwidth: 600px\n", ":name: fig-skew\n", "\n", "An illustration of skewness. On the left we have a negatively skewed data set (skewness = -.93), in the middle we have a data set with no skew (technically, skewness = .006), and on the right we have a positively skewed data set (skewness = .93).\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Since it's the more interesting of the two, let's start by talking about the **_skewness_**. Skewness is basically a measure of asymmetry, and the easiest way to explain it is by drawing some pictures. As {numref}`fig-skew` illustrates, if the data tend to have a lot of extreme small values (i.e., the lower tail is \"longer\" than the upper tail) and not so many extremely large values (left panel), then we say that the data are *negatively skewed*. On the other hand, if there are more extremely large values than extremely small ones (right panel) we say that the data are *positively skewed*. That's the qualitative idea behind skewness. The actual formula for the skewness of a data set is as follows\n", "\n", "$$\n", "\\mbox{skewness}(X) = \\frac{1}{N \\hat{\\sigma}^3} \\sum_{i=1}^N (X_i - \\bar{X})^3\n", "$$\n", "\n", "where $N$ is the number of observations, $\\bar{X}$ is the sample mean, and $\\hat{\\sigma}$ is the standard deviation (the \"divide by $N-1$\" version, that is). Luckily, `pandas`\n", "already knows how to calculate skew:" ] }, { "cell_type": "code", "execution_count": 55, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.7804075289401982" ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "margins.skew(axis = 0, skipna = True)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Not surprisingly, it turns out that the AFL winning margins data is fairly skewed.\n", "\n", "\n", "The final measure that is sometimes referred to, though very rarely in practice, is the **_kurtosis_** of a data set. Put simply, kurtosis is a measure of the \"pointiness\" of a data set, as illustrated in {numref}`fig-kurtosis`." ] }, { "cell_type": "code", "execution_count": 56, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import numpy as np \n", "import seaborn as sns \n", "from scipy import stats \n", "import matplotlib.pyplot as plt\n", "\n", "# load some data\n", "\n", "file1 = 'https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/kurtosisdata.csv'\n", "file2 = 'https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/kurtosisdata_ncurve.csv'\n", "\n", "\n", "\n", "df_kurtosis = pd.read_csv(file1)\n", "\n", "# define a normal distribution with a mean of 0 and a standard deviation of 1\n", "mu = 0\n", "sigma = 1\n", "x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)\n", "y = stats.norm.pdf(x, mu, sigma)\n", "\n", "platykurtic = df_kurtosis.loc[df_kurtosis[\"Kurtosis\"] == \"Platykurtic\"]\n", "mesokurtic = df_kurtosis.loc[df_kurtosis[\"Kurtosis\"] == \"Mesokurtic\"]\n", "leptokurtic = df_kurtosis.loc[df_kurtosis[\"Kurtosis\"] == \"Leptokurtic\"]\n", "\n", "\n", "fig, axes = plt.subplots(1, 3, figsize=(15, 5))\n", "\n", "ax1 = sns.histplot(data=platykurtic, x = \"Values\", binwidth=.5, ax=axes[0])\n", "ax2 = sns.histplot(data=mesokurtic, x = \"Values\", binwidth=.5, ax=axes[1])\n", "ax3 = sns.histplot(data=leptokurtic, x = \"Values\", binwidth=.5, ax=axes[2])\n", "\n", "\n", "\n", "#ax2 = ax.twinx()\n", "sns.lineplot(x=x,y=y*40000, ax=ax1, color='black')\n", "sns.lineplot(x=x,y=y*40000, ax=ax2, color='black')\n", "sns.lineplot(x=x,y=y*40000, ax=ax3, color='black')\n", "\n", "\n", "axes[0].set_title(\"Platykurtic\\n\\\"too flat\\\"\")\n", "axes[1].set_title(\"Mesokurtic\\n\\\"just right\\\"\")\n", "axes[2].set_title(\"Leptokurtic\\n\\\"too pointy\\\"\")\n", "\n", "for ax in axes:\n", " ax.set_xlim(-6,6)\n", " ax.set_ylim(0,25000)\n", " ax.set(xticklabels=[])\n", " ax.set(yticklabels=[])\n", " ax.set(xlabel=None)\n", " ax.set(ylabel=None)\n", " ax.tick_params(bottom=False)\n", " ax.tick_params(left=False)\n", " \n", "\n", "sns.despine()\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} kurtosis_fig\n", ":figwidth: 600px\n", ":name: fig-kurtosis\n", "\n", "An illustration of kurtosis. On the left, we have a “platykurtic” data set (kurtosis = -.95), meaning that the data set is “too flat”. In the middle we have a “mesokurtic” data set (kurtosis is almost exactly 0), which means that the pointiness of the data is just about right. Finally, on the right, we have a “leptokurtic” data set (kurtosis = 2.12) indicating that the data set is “too pointy”. Note that kurtosis is measured with respect to a normal curve (black line).\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "By convention, we say that the \"normal curve\" (black lines) has zero kurtosis, so the pointiness of a data set is assessed relative to this curve. In this Figure, the data on the left are not pointy enough, so the kurtosis is negative and we call the data *platykurtic*. The data on the right are too pointy, so the kurtosis is positive and we say that the data is *leptokurtic*. But the data in the middle are just pointy enough, so we say that it is *mesokurtic* and has kurtosis zero. This is summarised in the table below:\n", "\n", "|informal term |technical name |kurtosis value |\n", "|:------------------|:--------------|:--------------|\n", "|too flat |platykurtic |negative |\n", "|just pointy enough |mesokurtic |zero |\n", "|too pointy |leptokurtic |positive |\n", "\n", "\n", "The equation for kurtosis is pretty similar in spirit to the formulas we've seen already for the variance and the skewness; except that where the variance involved squared deviations and the skewness involved cubed deviations, the kurtosis involves raising the deviations to the fourth power: [^note_kurtosis]\n", "\n", "$$\n", "\\mbox{kurtosis}(X) = \\frac{1}{N \\hat\\sigma^4} \\sum_{i=1}^N \\left( X_i - \\bar{X} \\right)^4 - 3\n", "$$\n", "\n", "I know, it's not terribly interesting to me either. To make things worse, there are several different formulae for calculating kurtosis, so different statistics packages may give you different results, depending on which formula they use. For instance, if we were to do this for the AFL margins, these three different methods give three different results:\n", "\n", "[^note_kurtosis]: The \"$-3$\" part is something that statisticians tack on to ensure that the normal curve has kurtosis zero. It looks a bit stupid, just sticking a \"-3\" at the end of the formula, but there are good mathematical reasons for doing this." ] }, { "cell_type": "code", "execution_count": 57, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Pandas: 0.10109718805638757\n", "Fischer: 0.06434955786516161\n", "Pearson: 3.0643495578651616\n" ] } ], "source": [ "print(\"Pandas: \", margins.kurtosis())\n", "print(\"Fischer: \",stats.kurtosis(margins, fisher=True))\n", "print(\"Pearson: \",stats.kurtosis(margins, fisher=False))" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Take your pick, I guess? `pandas` actually also calculates Fischer kurtosis, but `stats.kurtosis(margins, fisher=True)` adds a \"bias correction\" by default, while the `pandas` version doesn't. `stats.kurtosis(margins, fisher=True, bias=False)`will get you the same thing as the `pandas` version. Ugh. If you want to assess the kurtosis of the data, you could probably do worse than just plotting the data and using your eyeballs." ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "## Getting an overall summary of a variable\n", "\n", "Up to this point in the chapter I've explained several different summary statistics that are commonly used when analysing data, along with specific functions that you can use in Python to calculate each one. However, it's kind of annoying to have to separately calculate means, medians, standard deviations, skews etc. Wouldn't it be nice if Python had some helpful functions that would do all these tedious calculations at once? Something that *describes* the data? Maybe something like `describe()`, perhaps? Why yes, yes it would. So much so that this very function exists, available as a method for `pandas` objects.\n", "\n", "\n", "\n", "### \"Describing\" a variable\n", "\n", "The `describe()` method is an easy thing to use, but a tricky thing to understand in full, since it's a generic function. The basic idea behind the `describe()` method is that it prints out some useful information about whatever object (i.e., variable, as far as we're concerned) you ask it to describe. As a consequence, the behaviour of the `describe()` function differs quite dramatically depending on the class of the object that you give it. Let's start by giving it a *numeric* object:" ] }, { "cell_type": "code", "execution_count": 58, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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afl.margins
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" ], "text/plain": [ " afl.margins\n", "count 176.000000\n", "mean 35.301136\n", "std 26.073636\n", "min 0.000000\n", "25% 12.750000\n", "50% 30.500000\n", "75% 50.500000\n", "max 116.000000" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "afl_margins.describe()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "For numeric variables, we get a whole bunch of useful descriptive statistics. It gives us the minimum and maximum values (i.e., the range), the first and third quartiles (25th and 75th percentiles; i.e., the IQR), the mean and the median. In other words, it gives us a pretty good collection of descriptive statistics related to the central tendency and the spread of the data.\n", "\n", "Okay, what about if we feed it a logical vector instead? Let's say I want to know something about how many \"blowouts\" there were in the 2010 AFL season. I operationalise the concept of a blowout as a game in which the winning margin exceeds 50 points. Let's create a logical variable `blowouts` in which the $i$-th element is `TRUE` if that game was a blowout according to my definition:" ] }, { "cell_type": "code", "execution_count": 59, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
afl.marginsblowouts
056True
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" ], "text/plain": [ " afl.margins blowouts\n", "0 56 True\n", "1 31 False\n", "2 56 True\n", "3 8 False\n", "4 32 False" ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "afl_margins['blowouts'] = np.where(afl_margins['afl.margins'] > 50, True, False)\n", "afl_margins.head()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "So that's what the `blowouts` variable looks like. Now let's ask Python to `describe()` this data: " ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "count 176\n", "unique 2\n", "top False\n", "freq 132\n", "Name: blowouts, dtype: object" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "afl_margins['blowouts'].describe()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "In this context, `describe` gives us the total number of games (176), the number of categories for those games (2, either blowout or not a blowout), the most common category (False, that is, not a blowout), and a count for the more common category. A little cryptic, but not entirely unreasonable. \n", "\n", "\n", "\n", "### \"Describing\" a data frame\n", "\n", "Okay what about data frames? When you `describe()` a dataframe, it produces a slightly condensed summary of each variable inside the data frame (as long as you specify that you want `'all'` the variables). To give you a sense of how this can be useful, let's try this for a new data set, one that you've never seen before. The data is stored in the `clinical_trial_data.csv` file, and we'll use it a lot in a [later chapter](ANOVA) when we discuss analysis of variance (you can find a complete description of the data at the start of that chapter). Let's load it, and see what we've got:" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
drugtherapymood_gain
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1placebono.therapy0.3
2placebono.therapy0.1
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" ], "text/plain": [ " drug therapy mood_gain\n", "0 placebo no.therapy 0.5\n", "1 placebo no.therapy 0.3\n", "2 placebo no.therapy 0.1\n", "3 anxifree no.therapy 0.6\n", "4 anxifree no.therapy 0.4" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "file = 'https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/clinical_trial_data.csv'\n", "\n", "df_clintrial = pd.read_csv(file)\n", "df_clintrial.head()\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Our dataframe `df_clintrial` contains three variables, `drug`, `therapy` and `mood_gain`. Presumably then, this data is from a clinical trial of some kind, in which people were administered different drugs, and the researchers looked to see what the drugs did to their mood. Let's see if the `describe()` function sheds a little more light on this situation:" ] }, { "cell_type": "code", "execution_count": 62, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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drugtherapymood_gain
count181818.000000
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" ], "text/plain": [ " drug therapy mood_gain\n", "count 18 18 18.000000\n", "unique 3 2 NaN\n", "top placebo no.therapy NaN\n", "freq 6 9 NaN\n", "mean NaN NaN 0.883333\n", "std NaN NaN 0.533854\n", "min NaN NaN 0.100000\n", "25% NaN NaN 0.425000\n", "50% NaN NaN 0.850000\n", "75% NaN NaN 1.300000\n", "max NaN NaN 1.800000" ] }, "execution_count": 62, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_clintrial.describe(include = 'all')" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "If we want to `describe` the entire dataframe, we need to add the argument `include = 'all'`. This gives us information on all of the of columns, but this is still rather limited. I mean, I guess I learned something about this data, but if we want to really understand these data, we will have to use other tools to investigate them. That is what the rest of this book is about.\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(zcores)=\n", "## Standard scores\n", "\n", "\n", "Suppose my friend is putting together a new questionnaire intended to measure \"grumpiness\". The survey has 50 questions, which you can answer in a grumpy way or not. Across a big sample (hypothetically, let's imagine a million people or so!) the data are fairly normally distributed, with the mean grumpiness score being 17 out of 50 questions answered in a grumpy way, and the standard deviation is 5. In contrast, when I take the questionnaire, I answer 35 out of 50 questions in a grumpy way. So, how grumpy am I? One way to think about it would be to say that I have grumpiness of 35/50, so you might say that I'm 70% grumpy. But that's a bit weird, when you think about it. If my friend had phrased her questions a bit differently, people might have answered them in a different way, so the overall distribution of answers could easily move up or down depending on the precise way in which the questions were asked. So, I'm only 70% grumpy *with respect to this set of survey questions*. Even if it's a very good questionnaire, this isn't very a informative statement. \n", "\n", "A simpler way around this is to describe my grumpiness by comparing me to other people. Shockingly, out of my friend's sample of 1,000,000 people, only 159 people were as grumpy as me (that's not at all unrealistic, frankly), suggesting that I'm in the top 0.016% of people for grumpiness. This makes much more sense than trying to interpret the raw data. This idea -- that we should describe my grumpiness in terms of the overall distribution of the grumpiness of humans -- is the qualitative idea that standardisation attempts to get at. One way to do this is to do exactly what I just did, and describe everything in terms of percentiles. However, the problem with doing this is that \"it's lonely at the top\". Suppose that my friend had only collected a sample of 1000 people (still a pretty big sample for the purposes of testing a new questionnaire, I'd like to add), and this time gotten a mean of 16 out of 50 with a standard deviation of 5, let's say. The problem is that almost certainly, not a single person in that sample would be as grumpy as me.\n", "\n", "\n", "However, all is not lost. A different approach is to convert my grumpiness score into a **_standard score_**, also referred to as a $z$-score. The standard score is defined as the number of standard deviations above the mean that my grumpiness score lies. To phrase it in \"pseudo-maths\" the standard score is calculated like this:\n", "\n", "$$\n", "\\mbox{standard score} = \\frac{\\mbox{raw score} - \\mbox{mean}}{\\mbox{standard deviation}}\n", "$$ \n", "\n", "In actual maths, the equation for the $z$-score is\n", "\n", "$$\n", "z_i = \\frac{X_i - \\bar{X}}{\\hat\\sigma}\n", "$$\n", "\n", "So, going back to the grumpiness data, we can now transform Dan's raw grumpiness into a standardised grumpiness score. I haven't discussed how to compute $z$-scores, explicitly, but you can probably guess. For a dataset d, an easy way to find a z-score is to import `stats` from `scipy` and then do: `stats.zscore(d)`. If the mean is 17 and the standard deviation is 5 then my standardised grumpiness score would be \n", "\n", "$$\n", "z = \\frac{35 - 17}{5} = 3.6\n", "$$\n", "\n", "Technically, because I'm calculating means and standard deviations from a sample of data, but want to talk about my grumpiness relative to a population, what I'm actually doing is *estimating* a $z$ score. However, since we haven't talked about [estimation](estimation) yet I think it's best to ignore this subtlety, especially as it makes very little difference to our calculations.\n", "\n", "To interpret this value, recall the rough heuristic that I provided in the section on [standard deviation](sd), in which I noted that 99.7% of values are expected to lie within 3 standard deviations of the mean. So the fact that my grumpiness corresponds to a $z$ score of 3.6 indicates that I'm very grumpy indeed. " ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "In addition to allowing you to interpret a raw score in relation to a larger population (and thereby allowing you to make sense of variables that lie on arbitrary scales), standard scores serve a second useful function. Standard scores can be compared to one another in situations where the raw scores can't. Suppose, for instance, my friend also had another questionnaire that measured extraversion using a 24 items questionnaire. The overall mean for this measure turns out to be 13 with standard deviation 4, and I scored a 2. As you can imagine, it doesn't make a lot of sense to try to compare my raw score of 2 on the extraversion questionnaire to my raw score of 35 on the grumpiness questionnaire. The raw scores for the two variables are \"about\" fundamentally different things, so this would be like comparing apples to oranges.\n", "\n", "What about the standard scores? Well, this is a little different. If we calculate the standard scores, we get $z = (35-17)/5 = 3.6$ for grumpiness and $z = (2-13)/4 = -2.75$ for extraversion. These two numbers *can* be compared to each other. [^note11] I'm much less extraverted than most people ($z = -2.75$) and much grumpier than most people ($z = 3.6$): but the extent of my unusualness is much more extreme for grumpiness (since 3.6 is a bigger number than 2.75). Because each standardised score is a statement about where an observation falls *relative to its own population*, it *is* possible to compare standardised scores across completely different variables. \n", "\n", "[^note11]: Though some caution is usually warranted. It's not always the case that one standard deviation on variable A corresponds to the same \"kind\" of thing as one standard deviation on variable B. Use common sense when trying to determine whether or not the $z$ scores of two variables can be meaningfully compared." ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(correlations)=\n", "## Correlations\n", "\n", "Up to this point we have focused entirely on how to construct descriptive statistics for a single variable. What we haven't done is talked about how to describe the relationships *between* variables in the data. To do that, we want to talk mostly about the **_correlation_** between variables. But first, we need some data.\n", "\n", "### The data \n", "\n", "After spending so much time looking at the AFL data, I'm starting to get bored with sports. Instead, let's turn to a topic close to every parent's heart: sleep. The following data set is fictitious, but based on real events. Suppose I'm curious to find out how much my infant son's sleeping habits affect my mood. Let's say that I can rate my grumpiness very precisely, on a scale from 0 (not at all grumpy) to 100 (grumpy as a very, very grumpy old man). And, lets also assume that I've been measuring my grumpiness, my sleeping patterns and my son's sleeping patterns for quite some time now. Let's say, for 100 days. And, being a nerd, I've saved the data as a file called `parenthood.csv`. If we load the data..." ] }, { "cell_type": "code", "execution_count": 63, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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dan_sleepbaby_sleepdan_grumpday
07.5910.18561
17.9111.66602
25.147.92823
37.719.61554
46.689.75675
\n", "
" ], "text/plain": [ " dan_sleep baby_sleep dan_grump day\n", "0 7.59 10.18 56 1\n", "1 7.91 11.66 60 2\n", "2 5.14 7.92 82 3\n", "3 7.71 9.61 55 4\n", "4 6.68 9.75 67 5" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "file = 'https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/parenthood.csv'\n", "parenthood = pd.read_csv(file)\n", "\n", "parenthood.head()\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "... we see that the file contains a single data frame called `parenthood`, which contains four variables `dan_sleep`, `baby_sleep`, `dan_grump` and `day`. Next, I'll calculate some basic descriptive statistics:" ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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dan_sleepbaby_sleepdan_grumpday
count100.000000100.000000100.00000100.000000
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std1.0158842.07423210.0496729.011492
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25%6.2925006.42500057.0000025.750000
50%7.0300007.95000062.0000050.500000
75%7.7400009.63500071.0000075.250000
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\n", "
" ], "text/plain": [ " dan_sleep baby_sleep dan_grump day\n", "count 100.000000 100.000000 100.00000 100.000000\n", "mean 6.965200 8.049200 63.71000 50.500000\n", "std 1.015884 2.074232 10.04967 29.011492\n", "min 4.840000 3.250000 41.00000 1.000000\n", "25% 6.292500 6.425000 57.00000 25.750000\n", "50% 7.030000 7.950000 62.00000 50.500000\n", "75% 7.740000 9.635000 71.00000 75.250000\n", "max 9.000000 12.070000 91.00000 100.000000" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "parenthood.describe()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Finally, to give a graphical depiction of what each of the three interesting variables looks like, {numref}`fig-grump` plots histograms. \n", "\n" ] }, { "cell_type": "code", "execution_count": 65, "metadata": { "tags": [ "hide-outuput", "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "from matplotlib import pyplot as plt\n", "\n", "dan_grump = parenthood['dan_grump']\n", "dan_sleep = parenthood['dan_sleep']\n", "baby_sleep = parenthood['baby_sleep']\n", "\n", "fig, axes = plt.subplots(1, 3, figsize=(15, 5), sharey=True)\n", "fig.suptitle('Sleep Data')\n", "\n", "# My grumpiness\n", "sns.histplot(dan_grump, ax=axes[0])\n", "axes[0].set_title(dan_grump.name)\n", "\n", "# My sleep\n", "sns.histplot(dan_sleep, ax=axes[1])\n", "axes[1].set_title(dan_sleep.name)\n", "\n", "# Baby's sleep\n", "sns.histplot(baby_sleep, ax=axes[2])\n", "axes[2].set_title(baby_sleep.name);\n", "\n", "for ax in axes:\n", " ax.set(yticklabels=[])\n", " ax.set(ylabel=None)\n", " ax.tick_params(bottom=False)\n", " ax.tick_params(left=False)\n", " \n", "sns.despine()\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} grump_fig\n", ":figwidth: 650px\n", ":name: fig-grump\n", "\n", "Histograms for the three interesting variables in the parenthood data set.\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "One thing to note: just because Python can calculate dozens of different statistics doesn't mean you should report all of them. If I were writing this up for a report, I'd probably pick out those statistics that are of most interest to me (and to my readership), and then put them into a nice, simple table like the one in the table below. [^note12] Notice that when I put it into a table, I gave everything \"human readable\" names. This is always good practice. Notice also that I'm not getting enough sleep. This isn't good practice, but other parents tell me that it's standard practice. \n", "\n", "|variable |min |max |mean |median |std. dev |IQR |\n", "|:-----------------------|:----|:-----|:-----|:------|:--------|:----|\n", "|Dan's grumpiness |41 |91 |63.71 |62 |10.05 |14 |\n", "|Dan's hours slept |4.84 |9 |6.97 |7.03 |1.02 |1.45 |\n", "|Dan's son's hours slept |3.25 |12.07 |8.05 |7.95 |2.07 |3.21 |\n", "\n", "[^note12]: Actually, even that table is more than I'd bother with. In practice most people pick *one* measure of central tendency, and *one* measure of variability only." ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(correlation)=\n", "\n", "### The strength and direction of a relationship\n" ] }, { "cell_type": "code", "execution_count": 66, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "fig, axes = plt.subplots(1, 2, figsize=(15, 5), sharey=True)\n", "fig.suptitle('Sleepy, grumpy scatterplots')\n", "\n", "sns.scatterplot(x = dan_sleep, y = dan_grump, ax = axes[0])\n", "fig.axes[0].set_title(\"Dan\")\n", "fig.axes[0].set_xlabel(\"Sleep\")\n", "fig.axes[0].set_ylabel(\"My grumpiness\")\n", "\n", "sns.scatterplot(x = baby_sleep, y = dan_grump, ax = axes[1])\n", "fig.axes[1].set_title(\"Baby\")\n", "fig.axes[1].set_xlabel(\"Sleep\")\n", "fig.axes[1].set_ylabel(\"My grumpiness\")\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} sleep_scatter_fig1\n", ":figwidth: 600px\n", ":name: fig-sleep_scatter1\n", "\n", "Scatterplots showing the relationship between dan.sleep and dan.grump (left) and the rela-\n", "tionship between baby.sleep and dan.grump (right).\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "We can draw scatterplots to give us a general sense of how closely related two variables are. Ideally though, we might want to say a bit more about it than that. For instance, let's compare the relationship between `dan_sleep` and `dan_grump` with that between `baby_sleep` and `dan_grump` {numref}`fig-sleep_scatter1`. When looking at these two plots side by side, it's clear that the relationship is *qualitatively* the same in both cases: more sleep equals less grump! However, it's also pretty obvious that the relationship between `dan_sleep` and `dan_grump` is *stronger* than the relationship between `baby_sleep` and `dan_grump`. The plot on the left is \"neater\" than the one on the right. What it feels like is that if you want to predict what my mood is, it'd help you a little bit to know how many hours my son slept, but it'd be *more* helpful to know how many hours I slept. \n", "\n", "In contrast, let's consider {numref}`fig-sleep_scatter2`. If we compare the scatterplot of \"`baby_sleep` v `dan_grump`\" to the scatterplot of `baby_sleep` v `dan_sleep`, the overall strength of the relationship is the same, but the direction is different. That is, if my son sleeps more, I get *more* sleep (positive relationship, but if he sleeps more then I get *less* grumpy (negative relationship)." ] }, { "cell_type": "code", "execution_count": 67, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "fig, axes = plt.subplots(1, 2, figsize=(15, 5), sharey=False) # y axes are now on different scales, so sharey=False\n", "fig.suptitle('Sleepier, grumpier scatterplots')\n", "\n", "sns.scatterplot(x = dan_sleep, y = dan_grump, ax = axes[0])\n", "fig.axes[0].set_xlabel(\"Baby's sleep\")\n", "fig.axes[0].set_ylabel(\"My grumpiness\")\n", "\n", "sns.scatterplot(x = baby_sleep, y = dan_sleep, ax = axes[1])\n", "fig.axes[1].set_xlabel(\"Baby's sleep\")\n", "fig.axes[1].set_ylabel(\"My sleep\")\n", "\n", "sns.despine()\n", "#glue(\"sleep_scatter-fig2\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} sleep_scatter_fig2\n", ":figwidth: 600px\n", ":name: fig-sleep_scatter2\n", "\n", "Scatterplots showing the relationship between baby_sleep and dan_grump (left), as compared\n", "to the relationship between baby.sleep and dan.sleep (right).\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "### The correlation coefficient\n", "\n", "We can make these ideas a bit more explicit by introducing the idea of a **_correlation coefficient_** (or, more specifically, Pearson's correlation coefficient), which is traditionally denoted by $r$. The correlation coefficient between two variables $X$ and $Y$ (sometimes denoted $r_{XY}$), which we'll define more precisely in the next section, is a measure that varies from $-1$ to $1$. When $r = -1$ it means that we have a perfect negative relationship, and when $r = 1$ it means we have a perfect positive relationship. When $r = 0$, there's no relationship at all. If you look at {numref}`fig-corrs`, you can see several plots showing what different correlations look like." ] }, { "cell_type": "code", "execution_count": 68, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "import seaborn as sns\n", "\n", "mean = [0, 0]\n", "cov = [[1, 0], [0, 1]]\n", "r = [0, .33, .66, 1]\n", "rneg = [0, -.33, -.66, -1]\n", "\n", "fig, axes = plt.subplots(4, 2, figsize=(15, 15), sharey=False)\n", "\n", "for s, val in enumerate(r):\n", " cov = [[1, val], [val, 1]]\n", " x, y = np.random.multivariate_normal(mean, cov, 100).T\n", " sns.scatterplot(x=x,y=y, ax = axes[s,0])\n", " axes[s,0].set_title('r = ' + str(val))\n", "\n", "for s, val in enumerate(rneg):\n", " cov = [[1, val], [val, 1]]\n", " x, y = np.random.multivariate_normal(mean, cov, 100).T\n", " sns.scatterplot(x=x,y=y, ax = axes[s,1])\n", " axes[s,1].set_title('r = ' + str(val))\n", "\n", "\n", "sns.despine()\n", " \n", "#glue(\"corrs-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} corrs-fig\n", ":figwidth: 600px\n", ":name: fig-corrs\n", "\n", "Illustration of the effect of varying the strength and direction of a correlation. In the left hand column, the correlations are 0, .33, .66 and 1. In the right hand column, the correlations are 0, -.33, -.66 and -1.\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "The formula for the Pearson's correlation coefficient can be written in several different ways. I think the simplest way to write down the formula is to break it into two steps. Firstly, let's introduce the idea of a **_covariance_**. The covariance between two variables $X$ and $Y$ is a generalisation of the notion of the variance; it's a mathematically simple way of describing the relationship between two variables that isn't terribly informative to humans:\n", "\n", "$$\n", "\\mbox{Cov}(X,Y) = \\frac{1}{N-1} \\sum_{i=1}^N \\left( X_i - \\bar{X} \\right) \\left( Y_i - \\bar{Y} \\right)\n", "$$\n", "\n", "Because we're multiplying (i.e., taking the \"product\" of) a quantity that depends on $X$ by a quantity that depends on $Y$ and then averaging [^note_covariance], you can think of the formula for the covariance as an \"average cross product\" between $X$ and $Y$. The covariance has the nice property that, if $X$ and $Y$ are entirely unrelated, then the covariance is exactly zero. If the relationship between them is positive (in the sense shown in {numref}`fig-corrs` then the covariance is also positive; and if the relationship is negative then the covariance is also negative. In other words, the covariance captures the basic qualitative idea of correlation. Unfortunately, the raw magnitude of the covariance isn't easy to interpret: it depends on the units in which $X$ and $Y$ are expressed, and worse yet, the actual units that the covariance itself is expressed in are really weird. For instance, if $X$ refers to the `dan_sleep` variable (units: hours) and $Y$ refers to the `dan_grump` variable (units: grumps), then the units for their covariance are \"hours $\\times$ grumps\". And I have no freaking idea what that would even mean. \n", "\n", "The Pearson correlation coefficient $r$ fixes this interpretation problem by standardising the covariance, in pretty much the exact same way that the $z$-score standardises a raw score: by dividing by the standard deviation. However, because we have two variables that contribute to the covariance, the standardisation only works if we divide by both standard deviations (this is an oversimplification, but it'll do for our purposes) In other words, the correlation between $X$ and $Y$ can be written as follows:\n", "\n", "$$\n", "r_{XY} = \\frac{\\mbox{Cov}(X,Y)}{ \\hat{\\sigma}_X \\ \\hat{\\sigma}_Y}\n", "$$\n", "\n", "By doing this standardisation, not only do we keep all of the nice properties of the covariance discussed earlier, but the actual values of $r$ are on a meaningful scale: $r= 1$ implies a perfect positive relationship, and $r = -1$ implies a perfect negative relationship. I'll expand a little more on this point later, in section on [interpreting correlations](interpreting-correlations). But before I do, let's look at how to calculate correlations in Python.\n", "\n", "[^note_covariance]: Just like we saw with the variance and the standard deviation, in practice we divide by $N-1$ rather than $N$." ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "### Calculating correlations in Python\n", "\n", "\n", "\n", "Calculating correlations in Python can be done using the `corr()` method. The simplest way to use the command is to specify two input arguments `x` and `y`, each one corresponding to one of the variables. The following extract illustrates the basic usage of the function: [^note13] \n", "\n", "[^note13]: If you are reading this after having already completed the chapter on [hypothesis testing](hypothesis-testing) you might be wondering about hypothesis tests for correlations. This can be done with `pingouin.corr` (or `scipy.stats.pearsonr`, or `scipy.stats.spearmanr`).\n" ] }, { "cell_type": "code", "execution_count": 69, "metadata": {}, "outputs": [], "source": [ "x = parenthood['dan_sleep']\n", "y = parenthood['dan_grump']" ] }, { "cell_type": "code", "execution_count": 70, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-0.9033840374657269" ] }, "execution_count": 70, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x.corr(y)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "However, the `corr()` function is a bit more powerful than this simple example suggests. For example, you can also calculate a complete \"correlation matrix\", between all pairs of variables in the data frame:" ] }, { "cell_type": "code", "execution_count": 71, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " dan_sleep baby_sleep dan_grump day\n", "dan_sleep 1.000000 0.627949 -0.903384 -0.098408\n", "baby_sleep 0.627949 1.000000 -0.565964 -0.010434\n", "dan_grump -0.903384 -0.565964 1.000000 0.076479\n", "day -0.098408 -0.010434 0.076479 1.000000" ] }, "execution_count": 71, "metadata": {}, "output_type": "execute_result" } ], "source": [ "parenthood.corr()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(interpreting-correlations)=\n", "### Interpreting a correlation\n", " \n", "Naturally, in real life you don't see many correlations of 1. So how should you interpret a correlation of, say $r= .4$? The honest answer is that it really depends on what you want to use the data for, and on how strong the correlations in your field tend to be. A friend of mine in engineering once argued that any correlation less than $.95$ is completely useless (I think he was exaggerating, even for engineering). On the other hand there are real cases -- even in psychology -- where you should really expect correlations that strong. For instance, one of the benchmark data sets used to test theories of how people judge similarities is so clean that any theory that can't achieve a correlation of at least $.9$ really isn't deemed to be successful. However, when looking for (say) elementary correlates of intelligence (e.g., inspection time, response time), if you get a correlation above $.3$ you're doing very very well. In short, the interpretation of a correlation depends a lot on the context. That said, the rough guide in {numref}`table-corr-interpretation` is pretty typical." ] }, { "cell_type": "code", "execution_count": 72, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "text/html": [ "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
CorrelationStrengthDirection
-1.0 to -0.9Very strongNegative
-0.9 to -0.7StrongNegative
-0.7 to -0.4ModerateNegative
-0.4 to -0.2WeakNegative
-0.2 to 0NegligibleNegative
0 to 0.2 NegligiblePositive
0.2 to 0.4WeakPositive
.4 to 0.7ModeratePositive
0.7 to 0.9StrongPositive
0.9 to 1.0Very strongPositive
\n" ], "text/plain": [ "" ] }, "execution_count": 72, "metadata": {}, "output_type": "execute_result" } ], "source": [ "correlation = [\"-1.0 to -0.9\", \"-0.9 to -0.7\", \"-0.7 to -0.4\", \n", " \"-0.4 to -0.2\", \"-0.2 to 0\", \"0 to 0.2 \", \"0.2 to 0.4\",\n", " \".4 to 0.7\", \"0.7 to 0.9\", \"0.9 to 1.0\"]\n", "strength = [\"Very strong\", \"Strong\", \"Moderate\", \"Weak\", \"Negligible\", \"Negligible\", \"Weak\",\n", " \"Moderate\", \"Strong\", \"Very strong\"]\n", "direction = [\"Negative\"]*5 + [\"Positive\"]*5\n", "df = pd.DataFrame(\n", " {'Correlation': correlation,\n", " 'Strength': strength,\n", " 'Direction': direction\n", " }) \n", "df.style.hide(axis='index')\n", "#glue(\"corr-interpretation-table\", df, display=True)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} corr-interpretation-table\n", ":figwidth: 600px\n", ":name: table-corr-interpretation\n", "\n", "A rough guide to interpreting correlations. Note that I say a rough guide. There aren’t hard and fast rules for what counts as strong or weak relationships. It depends on the context.\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "However, something that can never be stressed enough is that you should *always* look at the scatterplot before attaching any interpretation to the data. A correlation might not mean what you think it means. The classic illustration of this is \"Anscombe's Quartet\" {cite}`Anscombe1973`, which is a collection of four data sets. Each data set has two variables, an $X$ and a $Y$. For all four data sets the mean value for $X$ is 9 and the mean for $Y$ is 7.5. The, standard deviations for all $X$ variables are almost identical, as are those for the the $Y$ variables. And in each case the correlation between $X$ and $Y$ is $r = 0.816$. You can verify this yourself, like this:" ] }, { "cell_type": "code", "execution_count": 73, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.81642051634484\n", "0.8162365060002427\n", "0.8162867394895982\n", "0.8165214368885031\n" ] } ], "source": [ "x = [10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5]\n", "y1 = [8.04, 6.95, 7.58, 8.81, 8.33, 9.96, 7.24, 4.26, 10.84, 4.82, 5.68]\n", "y2 = [9.14, 8.14, 8.74, 8.77, 9.26, 8.10, 6.13, 3.10, 9.13, 7.26, 4.74]\n", "y3 = [7.46, 6.77, 12.74, 7.11, 7.81, 8.84, 6.08, 5.39, 8.15, 6.42, 5.73]\n", "x4 = [8, 8, 8, 8, 8, 8, 8, 19, 8, 8, 8]\n", "y4 = [6.58, 5.76, 7.71, 8.84, 8.47, 7.04, 5.25, 12.50, 5.56, 7.91, 6.89]\n", "\n", "df = pd.DataFrame(\n", " {'x': x,\n", " 'y1': y1,\n", " 'y2': y2,\n", " 'y3': y3,\n", " 'x4': x4,\n", " 'y4': y4\n", " })\n", "\n", "print(df['x'].corr(df['y1']))\n", "print(df['x'].corr(df['y2']))\n", "print(df['x'].corr(df['y3']))\n", "print(df['x4'].corr(df['y4']))" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "and so on. \n", "\n", "You'd think that these four data sets would look pretty similar to one another. They do not. If we draw scatterplots of $X$ against $Y$ for all four variables, as shown in {numref}`fig-anscombe` we see that all four of these are *spectacularly* different to each other. " ] }, { "cell_type": "code", "execution_count": 74, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "\n", "fig, axes = plt.subplots(2, 2, figsize=(15, 10), sharey=True)\n", "fig.suptitle('Anscombe\\'s quartet')\n", "\n", "sns.scatterplot(x = df['x'], y = df['y1'], ax = axes[0,0])\n", "sns.scatterplot(x = df['x'], y = df['y2'], ax = axes[0,1])\n", "sns.scatterplot(x = df['x'], y = df['y3'], ax = axes[1,0])\n", "sns.scatterplot(x = df['x4'], y = df['y4'], ax = axes[1,1])\n", "\n", "sns.despine()\n", "\n", "#glue(\"anscombe-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} anscombe-fig\n", ":figwidth: 600px\n", ":name: fig-anscombe\n", "\n", "Anscombe’s quartet. All four of these data sets have a Pearson correlation of r “ .816, but they are qualitatively different from one another.\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "The lesson here, which so very many people seem to forget in real life is \"*always graph your raw data*\". This will be the focus of the chapter on [](DrawingGraphs).\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "The Pearson correlation coefficient is useful for a lot of things, but it does have shortcomings. One issue in particular stands out: what it actually measures is the strength of the *linear* relationship between two variables. In other words, what it gives you is a measure of the extent to which the data all tend to fall on a single, perfectly straight line. Often, this is a pretty good approximation to what we mean when we say \"relationship\", and so the Pearson correlation is a good thing to calculation. Sometimes, it isn't. \n", "\n", "One very common situation where the Pearson correlation isn't quite the right thing to use arises when an increase in one variable $X$ really is reflected in an increase in another variable $Y$, but the nature of the relationship isn't necessarily linear. An example of this might be the relationship between effort and reward when studying for an exam. If you put in zero effort ($X$) into learning a subject, then you should expect a grade of 0% ($Y$). However, a little bit of effort will cause a *massive* improvement: just turning up to lectures means that you learn a fair bit, and if you just turn up to classes, and scribble a few things down so your grade might rise to 35%, all without a lot of effort. However, you just don't get the same effect at the other end of the scale. As everyone knows, it takes *a lot* more effort to get a grade of 90% than it takes to get a grade of 55%. What this means is that, if I've got data looking at study effort and grades, there's a pretty good chance that Pearson correlations will be misleading. \n", "\n", "\n", "To illustrate, consider the data plotted in {numref}`fig-rankcorr`, showing the relationship between hours worked and grade received for 10 students taking some class. The curious thing about this -- highly fictitious -- data set is that increasing your effort *always* increases your grade. It might be by a lot or it might be by a little, but increasing effort will never decrease your grade. The data are stored in `effort.csv`:" ] }, { "cell_type": "code", "execution_count": 75, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " hours grade\n", "0 2 13\n", "1 76 91\n", "2 40 79\n", "3 6 14\n", "4 16 21\n", "5 28 74\n", "6 27 47\n", "7 59 85\n", "8 46 84\n", "9 68 88" ] }, "execution_count": 75, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "file = 'https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/effort.csv'\n", "\n", "effort = pd.read_csv(file)\n", "\n", "effort" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "If we run a standard Pearson correlation, it shows a strong relationship between hours worked and grade received," ] }, { "cell_type": "code", "execution_count": 76, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.9094019658612525" ] }, "execution_count": 76, "metadata": {}, "output_type": "execute_result" } ], "source": [ "effort['hours'].corr(effort['grade'])" ] }, { "cell_type": "code", "execution_count": 77, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "\n", "fig, ax = plt.subplots()\n", "sns.regplot(x='hours', y='grade', data=effort, ci=None, ax=ax)\n", "sns.lineplot(x='hours', y='grade', data=effort, color = \"black\", ci=None, ax=ax)\n", "ax.lines[0].set_linestyle(\"--\")\n", "\n", "sns.despine()\n", "\n", "#glue(\"rankcorr-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} rankcorr-fig\n", ":figwidth: 600px\n", ":name: fig-rankcorr\n", "\n", "The relationship between hours worked and grade received, for a toy data set consisting of only 10 students (each dot corresponds to one student). The dashed line through the middle shows the linear relationship between the two variables. This produces a strong Pearson correlation of $r = .91$. However, the interesting thing to note here is that there's actually a perfect monotonic relationship between the two variables: in this toy example at least, increasing the hours worked always increases the grade received, as illustrated by the solid line. This is reflected in a Spearman correlation of $rho = 1$. With such a small data set, however, it's an open question as to which version better describes the actual relationship involved.\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "\n", "but this doesn't actually capture the observation that increasing hours worked *always* increases the grade. There's a sense here in which we want to be able to say that the correlation is *perfect* but for a somewhat different notion of what a \"relationship\" is. What we're looking for is something that captures the fact that there is a perfect **_ordinal relationship_** here. That is, if student 1 works more hours than student 2, then we can guarantee that student 1 will get the better grade. That's not what a correlation of $r = .91$ says at all.\n", "\n", "How should we address this? Actually, it's really easy: if we're looking for ordinal relationships, all we have to do is treat the data as if it were ordinal scale! So, instead of measuring effort in terms of \"hours worked\", lets rank all 10 of our students in order of hours worked. That is, student 1 did the least work out of anyone (2 hours) so they get the lowest rank (rank = 1). Student 4 was the next laziest, putting in only 6 hours of work in over the whole semester, so they get the next lowest rank (rank = 2). Notice that I'm using \"rank = 1\" to mean \"low rank\". Sometimes in everyday language we talk about \"rank = 1\" to mean \"top rank\" rather than \"bottom rank\". So be careful: you can rank \"from smallest value to largest value\" (i.e., small equals rank 1) or you can rank \"from largest value to smallest value\" (i.e., large equals rank 1). In this case, I'm ranking from smallest to largest, because that is how Python does it. But in real life, it's really easy to forget which way you set things up, so you have to put a bit of effort into remembering! \n", "\n", "Okay, so let's have a look at our students when we rank them from worst to best in terms of effort and reward: \n", "\n", "\n", "| | rank (hours worked) | rank (grade received) |\n", "| :--------: | :-----------------: | :-------------------: |\n", "| student 1 | 1 | 1 |\n", "| student 2 | 10 | 10 |\n", "| student 3 | 6 | 6 |\n", "| student 4 | 2 | 2 |\n", "| student 5 | 3 | 3 |\n", "| student 6 | 5 | 5 |\n", "| student 7 | 4 | 4 |\n", "| student 8 | 8 | 8 |\n", "| student 9 | 7 | 7 |\n", "| student 10 | 9 | 9 |\n", "\n", "\n", "Hm. These are *identical*. The student who put in the most effort got the best grade, the student with the least effort got the worst grade, etc. We can get Python to construct new variables with these rankings using the `rank()` method, and specifiying which columns in our dataframe we want to rank, like this:" ] }, { "cell_type": "code", "execution_count": 78, "metadata": {}, "outputs": [], "source": [ "ranked_hours = effort['hours'].rank()\n", "ranked_grades = effort['grade'].rank()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "As the table above shows, these two rankings are identical, so if we now correlate them we get a perfect relationship:" ] }, { "cell_type": "code", "execution_count": 79, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.9999999999999999" ] }, "execution_count": 79, "metadata": {}, "output_type": "execute_result" } ], "source": [ "ranked_hours.corr(ranked_grades)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "What we've just re-invented is **_Spearman's rank order correlation_**, usually denoted $\\rho$ to distinguish it from the Pearson correlation $r$. We can calculate Spearman's $\\rho$ using R in two different ways. Firstly we could do it the way I just showed, using the `rank()` function to construct the rankings, and then calculate the Pearson correlation on these ranks. However, that's way too much effort to do every time. It's much easier to just specify the `method` argument of the `corr()` method. [^note14] Since we are skipping the extra step of \"manually\" creating new, ranked variables, we can just operate directly on the dataframe columns:\n", "\n", "[^note14]: Yikes, that's two uses of the word method, and they mean different things. Sigh. Language is hard, and things get confusing when code-switching between human language and computer language!" ] }, { "cell_type": "code", "execution_count": 80, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.9999999999999999" ] }, "execution_count": 80, "metadata": {}, "output_type": "execute_result" } ], "source": [ "effort['hours'].corr(effort['grade'], method=\"spearman\")" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "The default value of the `method` argument is `\"pearson\"`, which is why we didn't have to specify it earlier on when we were doing Pearson correlations, although we could have, for extra clarity. " ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(missing)=\n", "## Handling missing values\n", "\n", "There's one last topic that I want to discuss briefly in this chapter, and that's the issue of **_missing data_**. Real data sets very frequently turn out to have missing values: perhaps someone forgot to fill in a particular survey question, for instance. Missing data can be the source of a lot of tricky issues, most of which I'm going to gloss over. However, at a minimum, you need to understand the basics of handling missing data in Python. \n", "\n", "\n", "### The single variable case\n", "\n", "Let's start with the simplest case, in which you're trying to calculate descriptive statistics for a single variable which has missing data. In Python, this means that there will be `nan` (\"not a number\") values in your data vector. Let's create a variable like that:" ] }, { "cell_type": "code", "execution_count": 81, "metadata": {}, "outputs": [], "source": [ "partial = [10, 20, float('nan'), 30]" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Let's assume that you want to calculate the mean of this variable. By default, Python assumes that you want to calculate the mean using all four elements of this vector, which is probably the safest thing for a dumb automaton to do, but it's rarely what you actually want. Why not? Well, remember that although `nan` stands for \"not a number\", the more accurate interpretation of `nan` here is \"There should be a number here, but I don't know what that number is\". This means that `1 + nan = nan`: if I add 1 to some number that I don't know (i.e., the `nan`) then the answer is *also* a number that I don't know. As a consequence, if you don't explicitly tell Python to ignore the `nan` values, and the data set does have missing values, then the output will itself be a missing value. If I try to calculate the mean of the `partial` vector, without doing anything about the missing value, here's what happens:" ] }, { "cell_type": "code", "execution_count": 82, "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/plain": [ "nan" ] }, "execution_count": 82, "metadata": {}, "output_type": "execute_result" } ], "source": [ "statistics.mean(partial)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Technically correct, but deeply unhelpful. There are a few ways to deal with this. The first is to use methods from `numpy`. `numpy` has a collection of methods to calculate nan-friendly versions of your favorite descriptive statistics (although correlation is a special case, more on that below), such as `nanmean`, `nanmedian`, `nanpercentile`, `nanmax`, `nanmin`, `nansum`, `nanstd`, etc. So:" ] }, { "cell_type": "code", "execution_count": 83, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "20.0\n", "20.0\n", "8.16496580927726\n" ] } ], "source": [ "import numpy as np\n", "\n", "\n", "print(np.nanmean(partial))\n", "\n", "print(np.nanmedian(partial))\n", "\n", "print(np.nanstd(partial))\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "This is great! Now we can get the descriptive statistics we want, without those pesky `nan`s getting in the way. Still, it is a little tedious to need to remember to to use the `np.nan` version of these functions when dealing with data containing `nan`s (which real data sets often do). As a bit of a consolation, `pandas` dataframes can already calculate these statistics for us, and ignore `nan`s by default. So, if we put our data in a `pandas` dataframe, we don't need to worry about it:" ] }, { "cell_type": "code", "execution_count": 84, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " var1 var2 var3\n", "0 10.0 10.0 NaN\n", "1 20.0 NaN 12.0\n", "2 NaN NaN 18.0\n", "3 30.0 35.0 27.0" ] }, "execution_count": 84, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "df = pd.DataFrame(\n", " {'var1': partial,\n", " 'var2': [10, float('nan'), float('nan'), 35],\n", " 'var3': [float('nan'), 12, 18, 27]\n", " }) \n", "\n", "df" ] }, { "cell_type": "code", "execution_count": 85, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "20.0\n", "20.0\n", "10.0\n" ] } ], "source": [ "print(df['var1'].mean())\n", "print(df['var1'].median())\n", "print(df['var1'].std())" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "This is also great, but there is one little niggling problem. `var1` in our dataframe is the same as the `partial` variable we defined earlier. The mean and median values look the same as when we used the `numpy` methods, but the standard deviation is little bit different. What's up with that? As it turns out, `numpy` and `pandas` calculate standard deviation in slightly different ways: `numpy` uses $N$ in the demoninator, while `pandas` uses the \"unbiased estimator\" $N-1$ in the demoniator. To make `numpy` behave like `pandas`, we need to pass the `ddof=1` argument to `nanmean()`, like so:" ] }, { "cell_type": "code", "execution_count": 86, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "10.0\n", "10.0\n" ] } ], "source": [ "print(df['var1'].std())\n", "print(np.nanstd(partial, ddof=1))" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "and all is well! " ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Notice that the mean is `20` (i.e., `60 / 3`) and *not* `15`. When Python ignores an `nan` value, it genuinely ignores it. In effect, the calculation above is identical to what you'd get if you asked for the mean of the three-element vector `[10, 20, 30]`. This is also why the mean and the median are the same in this case.\n", "\n", "\n", "### Missing values in pairwise calculations\n", "\n", "I mentioned earlier that correlation is a special case. It doesn't have an `np.nancorr` argument, because the story becomes a lot more complicated when more than one variable is involved. To explore this, lets look at the data in `parenthood2.csv`. This is just like the `parenthood` data from before, but with some missing values introducd. Maybe I was just too tired some mornings to record the baby's hours of sleep, or to measure my own grumpiness. It happens." ] }, { "cell_type": "code", "execution_count": 87, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " dan_sleep baby_sleep dan_grump day\n", "0 7.59 NaN 56.0 1\n", "1 7.91 11.66 60.0 2\n", "2 5.14 7.92 82.0 3\n", "3 7.71 9.61 55.0 4\n", "4 6.68 9.75 NaN 5" ] }, "execution_count": 87, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "file = 'https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/parenthood2.csv'\n", "\n", "parenthood2 = pd.read_csv(file)\n", "parenthood2.head()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "We can see some of those pesky `nan`s right in the first 5 rows, and if we `describe()` our data, we can get a feeling for how many there are:" ] }, { "cell_type": "code", "execution_count": 88, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " dan_sleep baby_sleep dan_grump day\n", "count 91.000000 89.000000 92.000000 100.000000\n", "mean 6.976923 8.114494 63.152174 50.500000\n", "std 1.020409 2.046821 9.851574 29.011492\n", "min 4.840000 3.250000 41.000000 1.000000\n", "25% 6.285000 6.460000 56.000000 25.750000\n", "50% 7.030000 8.200000 61.000000 50.500000\n", "75% 7.785000 9.610000 70.250000 75.250000\n", "max 9.000000 12.070000 89.000000 100.000000" ] }, "execution_count": 88, "metadata": {}, "output_type": "execute_result" } ], "source": [ "parenthood2.describe()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "By looking in the `count` row, we can see that out of the 100 days for which we have data, there are 9 days missing for `dan_sleep`, 11 days missing for `baby_sleep` and eight days missing for `dan_grump`. Suppose what I would like is a correlation matrix. And let's also suppose that I don't bother to tell Python how to handle those missing values. Here's what happens:" ] }, { "cell_type": "code", "execution_count": 89, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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dan_sleepbaby_sleepdan_grumpday
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" ], "text/plain": [ " dan_sleep baby_sleep dan_grump day\n", "dan_sleep 1.000000 0.614723 -0.903442 -0.076797\n", "baby_sleep 0.614723 1.000000 -0.567803 0.058309\n", "dan_grump -0.903442 -0.567803 1.000000 0.005833\n", "day -0.076797 0.058309 0.005833 1.000000" ] }, "execution_count": 89, "metadata": {}, "output_type": "execute_result" } ], "source": [ "parenthood2.corr()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "This actually looks pretty good! We *know* there are `nan`s in the data, but `pandas` seems to deal with them handily. This is not untrue, but there is a small but important detail to be aware of. When it encounters data with `nan`s, `pandas` only looks at the pair of variables that it's trying to correlate when determining what to drop. So, for instance, since the only missing value for observation 1 of `parenthood2` is for `baby_sleep`, Python will only drop observation 1 when `baby_sleep` is one of the variables involved, and keeps observation 1 when trying to correlate e.g. `dan.sleep` and `dan_grump`. If we want to simply ignore *all* rows that contain a `nan`, we need to tell `pandas` to drop them, like this:" ] }, { "cell_type": "code", "execution_count": 90, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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dan_sleepbaby_sleepdan_grumpday
dan_sleep1.0000000.639498-0.8995150.061329
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" ], "text/plain": [ " dan_sleep baby_sleep dan_grump day\n", "dan_sleep 1.000000 0.639498 -0.899515 0.061329\n", "baby_sleep 0.639498 1.000000 -0.586561 0.145558\n", "dan_grump -0.899515 -0.586561 1.000000 -0.068166\n", "day 0.061329 0.145558 -0.068166 1.000000" ] }, "execution_count": 90, "metadata": {}, "output_type": "execute_result" } ], "source": [ "parenthood2.dropna().corr()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "By checking the length of `parenthood2` and `parenthood2.dropna()`, we can see that using `dropna()` removes 27 entire rows from our data:" ] }, { "cell_type": "code", "execution_count": 91, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "100\n", "73\n" ] } ], "source": [ "print(len(parenthood2))\n", "print(len(parenthood2.dropna()))" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "The results from `parenthood2.corr()` and `parenthood2.dropna().corr()` are similar, but not quite the same. The two approaches have different strengths and weaknesses. The \"pairwise\" approach, the one in which Python only drops observations when they are involved in the comparison at hand, has the advantage that it keeps more observations, so you're making use of more of your data and (as we'll discuss in tedious detail in the chapter on [estimation](estimation)) this improves the reliability of your estimated correlation. On the other hand, it means that every correlation in your correlation matrix is being computed from a slightly different set of observations, which can be awkward when you want to compare the different correlations that you've got. \n", " \n", "So which method should you use? It depends a lot on *why* you think your values are missing, and probably depends a little on how paranoid you are. For instance, if you think that the missing values were \"chosen\" completely randomly [^note15] then you'll probably want to use the pairwise method. If you think that missing data are a cue to thinking that the whole observation might be rubbish (e.g., someone just selecting arbitrary responses in your questionnaire), but that there's no pattern to which observations are \"rubbish\", then it's probably safer to keep only those observations that are complete. If you think there's something systematic going on, in that some observations are more likely to be missing than others, then you have a much trickier problem to solve, and one that is beyond the scope of this book.\n", "\n", "[^note15]: The technical term here is \"missing completely at random\" (often written MCAR for short). Makes sense, I suppose, but it does sound ungrammatical to me." ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "## Summary\n", "\n", "Calculating some basic descriptive statistics is one of the very first things you do when analysing real data, and descriptive statistics are much simpler to understand than inferential statistics, so like every other statistics textbook I've started with descriptives. In this chapter, we talked about the following topics:\n", "\n", "\n", "- [*Measures of central tendency*](central-tendency). Broadly speaking, central tendency measures tell you where the data are. There's three measures that are typically reported in the literature: the mean, median and mode.\n", "- [*Measures of variability*](variability). In contrast, measures of variability tell you about how \"spread out\" the data are. The key measures are: range, standard deviation, interquartile reange \n", "- [*Standard scores*](zcores). The $z$-score is a slightly unusual beast. It's not quite a descriptive statistic, and not quite an inference. Make sure you understood this section: it'll come up again later. \n", "- [*Correlations*](correlations). Want to know how strong the relationship is between two variables? Calculate a correlation.\n", "- [*Missing data*](missing). Dealing with missing data is one of those frustrating things that data analysts really wish the didn't have to think about. In real life it can be hard to do well. For the purpose of this book, we only touched on the basics in this section.\n", "\n", "In the next section we'll move on to a discussion of how to draw pictures! Everyone loves a pretty picture, right? But before we do, I want to end on an important point. A traditional first course in statistics spends only a small proportion of the class on descriptive statistics, maybe one or two lectures at most. The vast majority of the lecturer's time is spent on inferential statistics, because that's where all the hard stuff is. That makes sense, but it hides the practical everyday importance of choosing good descriptives. With that in mind..." ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "## Epilogue: Good descriptive statistics are descriptive!\n", "\n", "\n", ">*The death of one man is a tragedy.\n", ">The death of millions is a statistic.*\n", ">\n", ">-- Josef Stalin, Potsdam 1945\n", "\n", "\n", "\n", ">*950,000 -- 1,200,000*\n", ">\n", ">-- Estimate of Soviet repression deaths, \n", "> 1937-1938 (Ellman, 2002) {cite}`Ellman2002`\n", "\n", "Stalin's infamous quote about the statistical character death of millions is worth giving some thought. The clear intent of his statement is that the death of an individual touches us personally and its force cannot be denied, but that the deaths of a multitude are incomprehensible, and as a consequence mere statistics, more easily ignored. I'd argue that Stalin was half right. A statistic is an abstraction, a description of events beyond our personal experience, and so hard to visualise. Few if any of us can imagine what the deaths of millions is \"really\" like, but we can imagine one death, and this gives the lone death its feeling of immediate tragedy, a feeling that is missing from Ellman's cold statistical description.\n", "\n", "Yet it is not so simple: without numbers, without counts, without a description of what happened, we have *no chance* of understanding what really happened, no opportunity event to try to summon the missing feeling. And in truth, as I write this, sitting in comfort on a Saturday morning, half a world and a whole lifetime away from the Gulags, when I put the Ellman estimate next to the Stalin quote a dull dread settles in my stomach and a chill settles over me. The Stalinist repression is something truly beyond my experience, but with a combination of statistical data and those recorded personal histories that have come down to us, it is not entirely beyond my comprehension. Because what Ellman's numbers tell us is this: over a two year period, Stalinist repression wiped out the equivalent of every man, woman and child currently alive in the city where I live. Each one of those deaths had it's own story, was it's own tragedy, and only some of those are known to us now. Even so, with a few carefully chosen statistics, the scale of the atrocity starts to come into focus. \n", "\n", "Thus it is no small thing to say that the first task of the statistician and the scientist is to summarise the data, to find some collection of numbers that can convey to an audience a sense of what has happened. This is the job of descriptive statistics, but it's not a job that can be told solely using the numbers. You are a data analyst, not a statistical software package. Part of your job is to take these *statistics* and turn them into a *description*. When you analyse data, it is not sufficient to list off a collection of numbers. Always remember that what you're really trying to do is communicate with a human audience. The numbers are important, but they need to be put together into a meaningful story that your audience can interpret. That means you need to think about framing. You need to think about context. And you need to think about the individual events that your statistics are summarising. " ] } ], "metadata": { "celltoolbar": "Tags", "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.5" } }, "nbformat": 4, "nbformat_minor": 4 } { "cells": [ { "cell_type": "markdown", "id": "metric-scanner", "metadata": {}, "source": [ "(DrawingGraphs)=\n", "# Drawing Graphs" ] }, { "cell_type": "markdown", "id": "raising-signature", "metadata": {}, "source": [ ">*Above all else show the data.*\n", ">\n", ">--Edward Tufte[^note1]\n", "\n", "Visualising data is one of the most important tasks facing the data analyst. It's important for two distinct but closely related reasons. Firstly, there's the matter of drawing \"presentation graphics\": displaying your data in a clean, visually appealing fashion makes it easier for your reader to understand what you're trying to tell them. Equally important, perhaps even more important, is the fact that drawing graphs helps *you* to understand the data. To that end, it's important to draw \"exploratory graphics\" that help you learn about the data as you go about analysing it. These points might seem pretty obvious, but I cannot count the number of times I've seen people forget them.\n", "\n", "[^note1]: The origin of this quote is Tufte's lovely book *The Visual Display of Quantitative Information*." ] }, { "cell_type": "markdown", "id": "related-outside", "metadata": {}, "source": [ "```{figure} ../img/graphics/snow_ghost_map2.png\n", ":name: fig-ghostmap\n", ":width: 600px\n", ":align: center\n", "\n", "A stylised redrawing of John Snow’s original cholera map. Each small dot represents the location of a cholera case, and each large circle shows the location of a well. As the plot makes clear, the cholera outbreak is centred very closely on the Broad St pump.\n", "\n", "```" ] }, { "cell_type": "markdown", "id": "black-torture", "metadata": {}, "source": [ "To give a sense of the importance of this chapter, I want to start with a classic illustration of just how powerful a good graph can be. To that end, {numref}`fig-ghostmap` shows a redrawing of one of the most famous data visualisations of all time: John Snow's 1854 map of cholera deaths. The map is elegant in its simplicity. In the background we have a street map, which helps orient the viewer. Over the top, we see a large number of small dots, each one representing the location of a cholera case. The larger symbols show the location of water pumps, labelled by name. Even the most casual inspection of the graph makes it very clear that the source of the outbreak is almost certainly the Broad Street pump. Upon viewing this graph, Dr Snow arranged to have the handle removed from the pump, ending the outbreak that had killed over 500 people. Such is the power of a good data visualisation." ] }, { "cell_type": "markdown", "id": "imperial-match", "metadata": {}, "source": [ "The goals in this chapter are twofold: firstly, to discuss several fairly standard graphs that we use a lot when analysing and presenting data, and secondly, to show you how to create these graphs in Python. The graphs themselves tend to be pretty straightforward, so in that respect this chapter is pretty simple. Where people usually struggle is learning how to produce graphs, and especially, learning how to produce good graphs.[^note2] Fortunately, learning how to draw graphs in Python is reasonably simple, as long as you're not too picky about what your graph looks like. What I mean when I say this is that Python has a lot of good graphing functions, and most of the time you can produce a clean, high-quality graphic without having to learn very much about the low-level details of how Python handles graphics. Unfortunately, on those occasions when you do want to do something non-standard, or if you need to make highly specific changes to the figure, you actually do need to learn a fair bit about these details; and those details are both complicated and boring. With that in mind, the structure of this chapter is as follows: I'll start out by giving you a very quick overview of how graphics work in Python. I'll then discuss several different kinds of graph and how to draw them, as well as showing the basics of how to customise these plots. In a future version of this book, I intend to finish this chapter off by talking about what makes a good or a bad graph, but I haven't yet had the time to write that section.\n", "\n", "[^note2]: I should add that this isn't unique to Python. Like everything in Python, there's a pretty steep learning curve to learning how to draw graphs, and like always there's a massive payoff at the end in terms of the quality of what you can produce. But to be honest, I've seen the same problems show up regardless of what system people use. I suspect that the hardest thing to do is to force yourself to take the time to think deeply about what your graphs are doing. I say that in full knowledge that only about half of my graphs turn out as well as they ought to. Understanding what makes a good graph is easy: actually designing a good graph is *hard*." ] }, { "cell_type": "markdown", "id": "dedicated-voluntary", "metadata": {}, "source": [ "(graphics)=\n", "\n", "## An overview of Python graphics\n", "\n", "Reduced to its simplest form, you can think of a Python graphic as being much like a painting. You start out with an empty canvas. Every time you use a graphics function, it paints some new things onto your canvas. Later on, you can paint more things over the top if you want; but just like painting, you can't \"undo\" your strokes. If you make a mistake, you have to throw away your painting and start over. Fortunately, this is way more easy to do when using Python than it is when painting a picture in real life: you delete the plot and then type a new set of commands.[^note3] This way of thinking about drawing graphs is referred to as the **_painter's model_**. So far, this probably doesn't sound particularly complicated, and for the vast majority of graphs you'll want to draw it's exactly as simple as it sounds. Much like painting in real life, the headaches usually start when we dig into details. To see why, I'll expand this \"painting metaphor\" a bit further just to show you the basics of what's going on under the hood, but before I do I want to stress that you really don't need to understand all these complexities in order to draw graphs. I'd been using Python for years before I even realised that most of these issues existed! However, I don't want you to go through the same pain I went through every time I inadvertently discovered one of these things, so here's a quick overview.\n", "\n", "When you paint a picture, you need to paint it **with** something. Maybe you want to do an oil painting, but maybe you want to use watercolour. And, generally speaking, you pretty much have to pick one or the other. The analog to this in Python is a plotting library. A plotting library is a collection of commands about what to draw and where to draw it. In scientific plotting in Python, this usually comes down to something called [_matplotlib_](https://matplotlib.org). `Matplotlib` is a very powerful collection of paints and brushes that allows you to be quite creative and make some lovely and customized figures, but it can be a real pain to work with at times. Fortunately, there are ways to ease the pain. In this book, we will mostly produce figures with the library [_seaborn_](https://seaborn.pydata.org), whose purpose is to make it easier to produce statistical figures with `matplotlib`. `Seaborn` greatly simplifies the process of data visualization in Python, and this is why we will rely on it so heavily. `Seaborn`'s simplicity is both its strength and its weakness, however, and sooner or later, you will probably need to dip into `matplotlib` to make your painting just the way you want it. Luckily for us, because `seaborn` is built on top of `matplotlib`, using one doesn't rule out the other. In keeping with our painting analogy, perhaps we can think of `seaborn` a bit like a stencil, that lets us quickly make a nice image. But there's nothing to stop us from coloring outside the lines if we want to.\n", "\n", "Thirdly, a painting is usually done in a particular **style**. Maybe it's a still life, maybe it's an impressionist piece, or maybe you're trying to annoy me by pretending that cubism is a legitimate artistic style. Regardless, each artistic style imposes some overarching aesthetic and perhaps even constraints on what can (or should) be painted using that style. `Seaborn` has an opinion about how plots should look, and how they should be created; it has a nice, clean style that I personally like, but it definitely is a style.\n", "\n", "At this point, I think we've covered more than enough background material. The point that I'm trying to make by providing this discussion isn't to scare you with all these horrible details, but rather to try to convey to you the fact that Python doesn't really provide a single coherent graphics system. Instead, Python itself provides a platform, and different people have built different graphical tools using that platform. As a consequence of this fact, there's (once again!) many different ways to achieve the goal of drawing graphs with Python. At this stage you don't need to understand these complexities, but it's useful to know that they're there. For now, I think we can be happy with a simpler view of things: to the extent possible, we'll draw pictures using `seaborn`, and dip into `matplotlib` as needed.\n", "\n", "So let's start painting.\n", "\n", "[^note3]: Of course, if you are using something like a Jupyter notebook to write your code (and you are doing that, right?), this is even easier, since all you have to do is edit your script and then run it again.\n" ] }, { "cell_type": "markdown", "id": "ceramic-joyce", "metadata": {}, "source": [ "(introplotting)=\n", "\n", "## An introduction to plotting\n", "\n", "Before I discuss any specialised graphics, let's start by drawing a few very simple graphs just to get a feel for what it's like to draw pictures using Python and `seaborn`. To that end, let's start by importing `seaborn`, and create a small list called `fibonacci` that contains a few numbers we'd like Python to draw for us. The customary abbreviation for `seaborn` is `sns`, so let's use that. We'll use our list `fibonacci` to provide the values for the y-axis, but of course, we'll need something for the x-axis as well. Since `fibonacci` contains 7 digits, we'll create another list, `x`, which has increasing integers from 1 to 7. Then we'll use `seaborn` to plot the x and y axes against each other:" ] }, { "cell_type": "code", "execution_count": 10, "id": "arctic-gates", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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eVMHLJ1KCDzW2fVjS70i62fYZSU9GxNPlTtWTuyR9QdIb3bVjSfpqRHy3vJF6doukZ7vPqtckHYmI1LffDYitkl68eK2gTZL+PiKOljtSz74s6VB36eG0pC+WPE/PbN8o6V5Jf1j4vqt+GyEA4MpYQgGApAg4ACRFwAEgKQIOAEkRcABIioADQFIEHACS+j9Y5D4OVzVKtAAAAABJRU5ErkJggg==\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "\n", "fibonacci = [1,1,2,3,5,8,13]\n", "x = [1,2,3,4,5,6,7]\n", "\n", "# By the way, instead of x = [1,2,3,4,5,6,7] we could have also written\n", "#x = range(1,len(fibonacci)+1)\n", "# This achieves the same thing, and could be useful e.g. if we didn't know how long the fibonacci list was.\n", "\n", "sns.scatterplot(x = x, y = fibonacci)" ] }, { "cell_type": "markdown", "id": "third-meaning", "metadata": {}, "source": [ "### A little color\n", "\n", "Already, we can see `seaborn`'s opinion expressing itself in a variety of subtle ways. For example, it has chosen a size and color for the points in the scatterplot. These seem a little small to me, and while blue is a great color, I am partial to orange. So.." ] }, { "cell_type": "code", "execution_count": 11, "id": "prescribed-valuable", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "sns.scatterplot(x = x, y = fibonacci, s = 300, color = 'orange')" ] }, { "cell_type": "markdown", "id": "impressed-preserve", "metadata": {}, "source": [ "### Title and axis labels\n", "\n", "This is great, but a good figure should have a title, and axis labels. You would think this was straightforward. After all, shouldn't all figures have these things by default? After all, we're not making [Bezos charts](https://twitter.com/jsnell/status/481863414180896769) here, right? It's not quite as easy as you'd expect, but it's not so bad, either. The easiest way to add titles and labels to your figure is to first put your figure into a variable, and then `set` the title and axis labels of that variable. Sounds complicated, but it's ok, once you get used to it. By convention, the variable for storing the figure is called `ax`.[^note4]\n", "\n", "Once the figure is safely stored inside of `ax`[^note5], we can use `.set` to set the title and axis labels:\n", "\n", "[^note4]: `ax` for \"axis\", I assume. I think this name harkens back to a different programming language, MATLAB, which inspired `pyplot`, which is part of `matplotlib`, which is... well, anyway... it's often called `ax`, ok?\n", "\n", "[^note5]: Actually, you can also apply `set` directly to the initial command creating the figure, like this: `sns.scatterplot(x = x, y = fibonacci, s = 400, color = 'orange').set(title = 'My first plot', xlabel = 'My x-axis', ylabel='My y-axis')` but most of the time when preparing a figure it is useful to store it as a variable, so you might as well get in the habit now." ] }, { "cell_type": "code", "execution_count": 12, "id": "dying-emerald", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[Text(0.5, 1.0, 'My first plot'),\n", " Text(0.5, 0, 'My x-axis'),\n", " Text(0, 0.5, 'My y-axis')]" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "ax = sns.scatterplot(x = x, y = fibonacci, s = 300, color = 'orange')\n", "ax.set(title = 'My first plot', xlabel = 'My x-axis', ylabel='My y-axis')" ] }, { "cell_type": "markdown", "id": "positive-persian", "metadata": {}, "source": [ "### Font size and related matters\n", "\n", "Now that we have a title and axis labels, it would be nice to change the size of the text. At least, my weak eyes wouldn't mind bumping it up a little bit. As always, there is more than one way to do this, but a simple approach is to use what `seaborn` calls \"contexts\". Remember, `seaborn` is all about providing a quick way to make an attractive graph without messing around too much with `matplotlib`. So while you _can_ dive in and set everything manually, using the `matplotlib` guts that `seaborn` tidly covers over, the `seaborn` way is to just to just say \"Hey, I want to make a plot for my presentation, please make it look ok.\" This is what `context` does. With the `context` command, you tell `seaborn` if your figure is going to remain in a jupyter notebook, or whether it is for some other purpose. Your options are: \"paper\", \"notebook\", \"talk\", and \"poster\". You can then supply `seaborn` with a `font_scale`, which is basically a number telling `seaborn` how much larger than the default font size (1) you would like your text to be. `seaborn` then adjusts things and tries to make it work.\n", "\n", "An important side-note on context: if you are working in jupyter notebook like the one I am writing this book in, once you set the `seaborn` context, it will continue to use that context within the notebook, unless you change it. This could be good or bad, depending on what you want, but it is worth keeping in mind, if you plots don't look like you expect." ] }, { "cell_type": "code", "execution_count": 13, "id": "baking-shoulder", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[Text(0.5, 1.0, 'My first plot'),\n", " Text(0.5, 0, 'My x-axis'),\n", " Text(0, 0.5, 'My y-axis')]" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "sns.set_context(\"notebook\", font_scale=1.5)\n", "ax = sns.scatterplot(x = x, y = fibonacci, s = 300, color = 'orange')\n", "ax.set(title = 'My first plot', \n", " xlabel = 'My x-axis', \n", " ylabel='My y-axis')" ] }, { "cell_type": "markdown", "id": "respiratory-starter", "metadata": {}, "source": [ "Notice that not only did `seaborn` change the font size for the title and labels, but it also increased the size of the tick-marks on the axes, and changed the size of the numbers on the axes as well, with the added consequence that while we used to have numbers from one to seven on the x-axis, we now only have numbers 2, 4, and 6. `seaborn` has opinions about what looks good, remember, and its opinions might or might now suit your needs. Most of the time, `seaborn` will make reasonable decisions, and will make you a decent-looking plot, but you do need to keep an eye on what it is up to. Luckily, if it makes a choice that you don't like, you can always dip down into `matplotlib` and touch things up to get them just the way you want. Going through all the options in `matplotlib` is beyond the scope of this book though, and there are many good resources out there to help you; mostly if you run into a problem, you just need to do some internet searching, and you will find someone who had a similar problem, and received good (and possibly contradictory!) advice on sites like [stackoverflow](https://stackoverflow.com). Sooner or later, this is where all coders end up in their search for answers to coding problems. Goodness knows this Python translation of Learning Statistics with R would never have happened without copious consulation of stackoverflow and similar sites." ] }, { "cell_type": "markdown", "id": "orange-input", "metadata": {}, "source": [ "### Open the box\n", "\n", "`seaborn` has more customization options that you will probably want to know about, but many of them are more applicable to more complex figures, so I'll wait to mention them until later. For now, though, I just want to show you one more option: removing the top and right lines of the box that `seaborn` draws around the data. To me, these make the data feel cramped, and I'd rather give them room to breathe and flourish. So I like to use the `despine` command to remove these, and give the data a little elbow room. `seaborn` has its opinions, and I have mine!" ] }, { "cell_type": "code", "execution_count": 14, "id": "golden-virtue", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "sns.set_context(\"notebook\", font_scale=1.5)\n", "ax = sns.scatterplot(x = x, y = fibonacci, s = 300, color = 'orange')\n", "ax.set(title = 'My first plot', \n", " xlabel = 'My x-axis', \n", " ylabel='My y-axis')\n", "sns.despine(top = True, right = True)" ] }, { "cell_type": "markdown", "id": "complex-anatomy", "metadata": {}, "source": [ "## Plotting more complex data\n", "\n", "At this point, we have seen how to make a fairly attractive plot (if I may say so myself) using `seaborn`. However, ususally the data we want to plot are more complicated than this simple series of numbers. `seaborn` has many good options for plotting complex data, but it is happiest if those data are stored in a structured format called a \"dataframe\". We talked about dataframes back in the chapter called [More Python concepts](mechanics) and we'll talk more extensively about them in the chapter on [Data Wrangling](datawrangling) and will meet them again and again throughout this book. A dataframe is a kind of structured data with columns and rows, just a like a spreadsheet, and having your data organized in a dataframe will make working with `seaborn` much easier. We'll use `pandas` to organize our data in dataframes. So, let's grab some data, put it in a dataframe, and talk more seriously about some real life graphics that you'll want to draw. I have stored the data online, and since `pandas` can read data straight from an internet URL (very handy!) we'll grab it from there:\n", "\n" ] }, { "cell_type": "code", "execution_count": 18, "id": "colored-zimbabwe", "metadata": {}, "outputs": [], "source": [ "import pandas as pd\n", "\n", "afl_margins = pd.read_csv('https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/afl_margins.csv')" ] }, { "cell_type": "markdown", "id": "dying-controversy", "metadata": {}, "source": [ "Just to remind ourselves what the data look like, we can use `head` to take a peek:" ] }, { "cell_type": "code", "execution_count": 19, "id": "enclosed-vacation", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
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" ], "text/plain": [ " afl.margins\n", "0 56\n", "1 31\n", "2 56\n", "3 8\n", "4 32" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "afl_margins.head()" ] }, { "cell_type": "markdown", "id": "alpha-signature", "metadata": {}, "source": [ "(histograms)=\n", "### Histograms\n", "\n", "We can begin with the humble **_histogram_**. Histograms are one of the simplest and most useful ways of visualising data. They make most sense when you have interval or ratio scale (e.g., the `afl_margins` data from the chapter on [descriptive statistics](descriptives)) and what you want to do is get an overall impression of the data. You probably already know how histograms work, since they're so widely used, but for the sake of completeness I'll describe them. All you do is divide up the possible values into **_bins_**, and then count the number of observations that fall within each bin. This count is referred to as the frequency of the bin, and is displayed as a bar: in the AFL winning margins data, there are 33 games in which the winning margin was less than 10 points, and it is this fact that is represented by the height of the leftmost bar in the figure below. Drawing this histogram with `seaborn` is pretty straightforward. The function you need to use is called `histplot`, and it has pretty reasonable default settings.\n", "\n", "By the way, I am going to change the `seaborn` context back to the default settings here, so that my figures will look as consistent as possible with yours, if you are following along at home." ] }, { "cell_type": "code", "execution_count": 20, "id": "immediate-bride", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "sns.set_context(\"notebook\", font_scale = 1)\n", "\n", "sns.histplot(data = afl_margins, x=\"afl.margins\")" ] }, { "cell_type": "markdown", "id": "infrared-gregory", "metadata": {}, "source": [ "Although this image would need some cleaning up in order to make a good presentation graphic (i.e., one you'd include in a report), it nevertheless does a pretty good job of describing the data. In fact, the big strength of a histogram is that (properly used) it does show the entire spread of the data, so you can get a pretty good sense about what it looks like. The downside to histograms is that they aren't very compact: unlike some of the other plots I'll talk about it's hard to cram 20-30 histograms into a single image without overwhelming the viewer. And of course, if your data are nominal scale (e.g., the `afl_finalists` data) then histograms are useless.\n" ] }, { "cell_type": "markdown", "id": "inclusive-circumstances", "metadata": {}, "source": [ "The main subtlety that you need to be aware of when drawing histograms is determining where the \"breaks\" that separate bins should be located, and (relatedly) how many bins there should be. In the figure above, you can see that `seaborn` has made pretty sensible choices all by itself: the breaks are located at 0, 10, 20, ... 120, which is exactly what I would have done had I been forced to make a choice myself. On the other hand, consider the following two histograms, which have divided the data into fewer and more bins, respectively:" ] }, { "cell_type": "code", "execution_count": 21, "id": "acute-chain", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Text(0.5, 1.0, 'Too many bins!')" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "from matplotlib import pyplot as plt\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 5))\n", "\n", "ax1 = sns.histplot(data = afl_margins, x=\"afl.margins\", bins = 3, ax=axes[0])\n", "ax2 = sns.histplot(data = afl_margins, x=\"afl.margins\", bins = 116, ax=axes[1])\n", "\n", "axes[0].set_title(\"Too few bins!\")\n", "axes[1].set_title(\"Too many bins!\")\n" ] }, { "cell_type": "markdown", "id": "later-belize", "metadata": {}, "source": [ "In the plot to the right, the bins are only 1 point wide. As a result, although the plot is very informative (it displays the entire data set with no loss of information at all!) the plot is very hard to interpret, and feels quite cluttered. On the other hand, the plot to the left has a bin width of 50 points, and has the opposite problem: it's very easy to \"read\" this plot, but it doesn't convey a lot of information. One gets the sense that this histogram is hiding too much. In short, the way in which you specify the number of bins has a big effect on what the histogram looks like, so it's important to make sure you choose the breaks sensibly. In general `seaborn` does a pretty good job of selecting the breaks on its own, but nevertheless it's usually a good idea to play around with the bins a bit to see what happens." ] }, { "cell_type": "markdown", "id": "brazilian-catalyst", "metadata": {}, "source": [ "(boxplots)=\n", "### Boxplots\n", "\n", "A great alternative to histograms is the **_boxplot_**, sometimes called a \"box and whiskers\" plot. Like histograms, they're most suited to interval or ratio scale data. The idea behind a boxplot is to provide a simple visual depiction of the median, the interquartile range, and the range of the data. And because they do so in a fairly compact way, boxplots have become a very popular statistical graphic, especially during the exploratory stage of data analysis when you're trying to understand the data yourself. Let's have a look at how they work, again using the `afl_margins` data as our example. Firstly, let's actually calculate these numbers ourselves using the `describe()` function:" ] }, { "cell_type": "code", "execution_count": 10, "id": "manufactured-count", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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afl.margins
count176.000000
mean35.301136
std26.073636
min0.000000
25%12.750000
50%30.500000
75%50.500000
max116.000000
\n", "
" ], "text/plain": [ " afl.margins\n", "count 176.000000\n", "mean 35.301136\n", "std 26.073636\n", "min 0.000000\n", "25% 12.750000\n", "50% 30.500000\n", "75% 50.500000\n", "max 116.000000" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "afl_margins.describe()" ] }, { "cell_type": "markdown", "id": "english-drain", "metadata": {}, "source": [ "So how does a boxplot capture these numbers? The easiest way to describe what a boxplot looks like is just to draw one. The function for doing this in `seaborn` is (surprise, surprise) `boxplot()`." ] }, { "cell_type": "code", "execution_count": 22, "id": "fewer-winning", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "sns.boxplot(data = afl_margins, y = 'afl.margins')" ] }, { "cell_type": "markdown", "id": "expensive-smoke", "metadata": {}, "source": [ "When you look at this plot, this is how you should interpret it: the line in the middle of the box is the median; the box itself spans the range from the 25th percentile to the 75th percentile, and the \"whiskers\" extend to 1.5 times the interquartile range. The two points above the top whisker represent \"extreme values\", that is, points that fall outside the range of the whiskers.\n", "\n", "One small note on the code above: since we are now drawing from data in a dataframe, rather then simply using lists as our data, like we did for the first scatterplots with the `fibonacci` data, there are two changes to the way we tell `seaborn` to make a plot. First, we specify the dataframe that stores the data by writing `data = afl_margins`. Second, we told `seaborn` which column in `afl_margins` to look in for the data by using the name of the column in quotes: `y = 'afl.margins'`. " ] }, { "cell_type": "markdown", "id": "secondary-colonial", "metadata": {}, "source": [ "In the example above, we made a vertical boxplot, which in this case resulted in a fairly squat-looking boxplot. As an alternative, we could display the data on the x-axis instead, which will result in a horizontal boxplot." ] }, { "cell_type": "code", "execution_count": 23, "id": "rational-understanding", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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"text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "sns.boxplot(data = afl_margins, x = 'afl.margins')" ] }, { "cell_type": "markdown", "id": "whole-backing", "metadata": {}, "source": [ "Because the boxplot automatically separates out those observations that lie within a certain range, people often use them as an informal method for detecting outliers: observations that are “suspiciously” distant from the rest of the data. Here’s an example. Suppose that I’d drawn the boxplot for the AFL margins data, and it came up looking like the one below. It’s pretty clear that something funny is going on with one of the observations. Apparently, there was one game in which the margin was over 300 points! That doesn’t sound right to me. Now that I’ve become suspicious, it’s time to look a bit more closely at the data." ] }, { "cell_type": "code", "execution_count": 24, "id": "systematic-offering", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "afl_margins.loc[177] = 327\n", "sns.boxplot(data = afl_margins, x = 'afl.margins')" ] }, { "cell_type": "code", "execution_count": 25, "id": "designed-revolution", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " afl.margins\n", "177 327" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "afl_margins[afl_margins['afl.margins'] > 300]" ] }, { "cell_type": "markdown", "id": "abroad-charger", "metadata": {}, "source": [ "Aha! Game 177 had a margin of 327. Now, in this case this should come as no surprise to me, since I added this data point myself with the code `afl_margins.loc[177] = 327` above. But if we play along for a minute, then a game with a margin of 327 definitely doesn't sound right. So then I go back to the original data source (the internet!) and I discover that the actual margin of that game was 33 points. Now it's pretty clear what happened. Someone must have typed in the wrong number. Easily fixed, just by typing `afl_margins.loc[177] = 33`. While this might seem like a silly example, I should stress that this kind of thing actually happens a lot. Real world data sets are often riddled with stupid errors, especially when someone had to type something into a computer at some point. In fact, there's actually a name for this phase of data analysis, since in practice it can waste a huge chunk of our time: **_data cleaning_**. It involves searching for typos, missing data and all sorts of other obnoxious errors in raw data files." ] }, { "cell_type": "markdown", "id": "appropriate-message", "metadata": {}, "source": [ "What about the real data? There is still that *other* extreme point at 116. Does the value of 116 constitute a funny observation not? Possibly. As it turns out the game in question was Fremantle v Hawthorn, and was played in round 21 (the second last home and away round of the season). Fremantle had already qualified for the final series. Since the outcome of the game was therefore irrelevant to them, they team decided to rest several of their star players. As a consequence, Fremantle went into the game severely underpowered. In contrast, Hawthorn had started the season very poorly but had ended on a massive winning streak, and for them a win could secure a place in the finals. With the game played on Hawthorn's home turf[^note7] and with so many unusual factors at play, it is perhaps no surprise that Hawthorn annihilated Fremantle by 116 points. Two weeks later, however, the two teams met again in an elimination final on Fremantle's home ground, and Fremantle won comfortably by 30 points.[^note8]\n", "\n", "So, should we exclude the game from subsequent analyses? If this were a psychology experiment rather than an AFL season, I'd be quite tempted to exclude it because there's pretty strong evidence that Fremantle weren't really trying very hard: and to the extent that my research question is based on an assumption that participants are genuinely trying to do the task. On the other hand, in a lot of studies we're actually interested in seeing the full range of possible behaviour, and that includes situations where people decide not to try very hard: so excluding that observation would be a bad idea. In the context of the AFL data, a similar distinction applies. If I'd been trying to make tips about who would perform well in the finals, I would have (and in fact did) disregard the Round 21 massacre, because it's way too misleading. On the other hand, if my interest is solely in the home and away season itself, I think it would be a shame to throw away information pertaining to one of the most distinctive (if boring) games of the year. In other words, the decision about whether to include outliers or exclude them depends heavily on *why* you think the data look they way they do, and what you want to use the data *for*. Statistical tools can provide an automatic method for suggesting candidates for deletion, but you really need to exercise good judgment here. As I've said before, Python is a mindless automaton. It doesn't watch the footy, so it lacks the broader context to make an informed decision. You are *not* a mindless automaton, so you should exercise judgment: if the outlier looks legitimate to you, then keep it. In any case, let's return to our discussion of how to draw boxplots.\n", "\n", "[^note7]: Sort of. The game was played in Launceston, which is a de facto home away from home for Hawthorn.\n", "[^note8]: Contrast this situation with the next largest winning margin in the data set, which was Geelong's 108 point demolition of Richmond in round 6 at their home ground, Kardinia Park. Geelong have been one of the most dominant teams over the last several years, a period during which they strung together an incredible 29-game winning streak at Kardinia Park. Richmond have been useless for several years. This is in no meaningful sense an outlier. Geelong have been winning by these margins (and Richmond losing by them) for quite some time. Frankly I'm surprised that the result wasn't more lopsided: as happened to Melbourne in 2011 when Geelong won by a modest 186 points." ] }, { "cell_type": "markdown", "id": "romance-billy", "metadata": {}, "source": [ "(multipleboxplots)=\n", "\n", "### Drawing multiple boxplots\n", "\n", "One last thing. What if you want to draw multiple boxplots at once? Suppose, for instance, I wanted separate boxplots showing the AFL margins not just for 2010, but for every year between 1987 and 2010. To do that, the first thing we'll have to do is find the data. These are stored in the `aflsmall2.csv` file. So let's load it and take a quick peek at what's inside:" ] }, { "cell_type": "code", "execution_count": 27, "id": "brown-short", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " margin year\n", "0 33 1987\n", "1 59 1987\n", "2 45 1987\n", "3 91 1987\n", "4 39 1987" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df = pd.read_csv('https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/afl2small.csv')\n", "df.head()" ] }, { "cell_type": "markdown", "id": "excellent-engagement", "metadata": {}, "source": [ "With `df.head` we can peek at the first few rows of the dataframe, and see that indeed it has one column with game margins, and another column with the year the game was played. The first five are indeed from 1987. `pandas` also lets us peek at the _last_ few rows, with `tail`:" ] }, { "cell_type": "code", "execution_count": 28, "id": "overall-madagascar", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " margin year\n", "4291 5 2010\n", "4292 41 2010\n", "4293 24 2010\n", "4294 0 2010\n", "4295 56 2010" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.tail()" ] }, { "cell_type": "markdown", "id": "systematic-treaty", "metadata": {}, "source": [ "and these games are from 2010, also as we might expect, so everything looks good. Incidently, these data are arranged in what is known as _long_ format, with all the margins in one column, and a second column indicating the year. This is as opposed to _wide_ format, in which there would be a separate column for every year. We'll return to these later in the chapter on [data wrangling](datawrangling).\n", "\n", "To plot these data, we just need to call up `boxplot()` once again, give `seaborn` the name of our dataframe, set the x-axis to \"year\" and the y-axis to \"margin\"." ] }, { "cell_type": "code", "execution_count": 17, "id": "functional-investing", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "ax = sns.boxplot(x = 'year', y = 'margin', data = df)" ] }, { "cell_type": "markdown", "id": "level-nomination", "metadata": {}, "source": [ "Clearly, there are things we could do to touch this figure up, but at least it gives a sense of why it's sometimes useful to choose boxplots instead of histograms. Even before taking the time to turn this basic output into something more readable, it's possible to get a good sense of what the data look like from year to year without getting overwhelmed with too much detail. Now imagine what would have happened if I'd tried to cram 24 histograms into this space: no chance at all that the reader is going to learn anything useful." ] }, { "cell_type": "markdown", "id": "least-advancement", "metadata": {}, "source": [ "### Alternatives to boxplots\n", "\n", "Boxplots are a very effective way to summarize data visually, but there are other methods that are worth pointing out. In the figure below, the same data (AFL game margins in the years 2005-2010) represented in three different ways. In the first panel, we see the boxplot, which we already know and love. The middle panel simply plots the data points. All of them. This approach, `stripplot`, shifts (or, as we say in the scientific plotting business, \"jitters\") the points slightly on the x-axis, so that they don't overlap each other. In the `stripplot`, we can't see exactly where the median or quartiles lie, for instance, like we can in the boxplot, but we can actually see all the data, which I tend to be in favor of. To me, in the boxplots, it sort of looks like all the data points lie inside the box, even though I know that of course this is not true. Then again, although the random jitter insures that the points don't all lie on top of each other, it is still a little difficult to get a feeling for how where most of the points lie. The third plot, in the panel to the right, attempts to solve this problem. This is a `violinplot`, in which the width of the colored area indicates the _density_ of data points. Thus, by looking at where the figure is wide and where it is thin, we can get an intuitive feeling for where most of the points lie. \n", "\n", "#### A word of caution on violin plots\n", "An important point to stress is that while every part of the boxplot and stripplot represent exact features of the data, the shape of the violinplot is an estimate (a kernal density estimate, if you want to get techinical), that attempts to show the underlying distribution that produced the data. This will of course always be a best guess, since the true distribution is unknowable, in most cases. Because violin plots show distribution estimates, they cannot be interpreted in exactly the same way as, say, a boxplot. A glance at the y-axes in the figure below shows why. While the boxplots and strip plots share an identical y axis, the violin plot's y-axis is different. If you look at the bottom, for instance, you will see that it extends below zero, and a winning margin of e.g. -1 is nonsensical. But this is because kernal density estimators don't know anything about how games are decided in the Australian Football League. They just look at the data, and make a best guess about the distribution that these data were sampled from. So violin plots are very informative, especially in cases where they actually are violin-shaped, indicating that the data may come from a distribution with more than one peak, you have to remember what you are looking at, and not confuse them with curvy boxplots. Perhaps to alleviate this problem, the default setting in `seaborn` is to draw a little mini boxplot inside the violin plot.\n", "\n" ] }, { "cell_type": "code", "execution_count": 29, "id": "lyric-carol", "metadata": {}, "outputs": [], "source": [ "df = pd.read_csv('https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/afl2small.csv')\n", "df = df[df['year'] > 2004]" ] }, { "cell_type": "code", "execution_count": 30, "id": "clean-mason", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "\n", "fig, axes = plt.subplots(1, 3, figsize=(15, 5))\n", "\n", "ax1 = sns.boxplot(x = 'year', y = 'margin', data = df, ax=axes[0])\n", "ax2 = sns.stripplot(x = 'year', y = 'margin', data = df, ax=axes[1])\n", "ax3 = sns.violinplot(x = 'year', y = 'margin', data = df, ax=axes[2])\n", "\n", "ax2.set(ylabel = '')\n", "ax3.set(ylabel = '')\n", "\n", "for ax in axes:\n", " ax.spines['right'].set_visible(False)\n", " ax.spines['top'].set_visible(False)" ] }, { "cell_type": "markdown", "id": "deadly-quick", "metadata": {}, "source": [ "One final alternative to the plain boxplot that I will show you here is to _overlay_ a strip plot on top of a boxplot. Using this method, you may get the best of both worlds. One potential issue with overlaying plots, though, is that they can obscure each other, concealing information. In the figure below, the panel to the left shows boxplots and the underlying data points in lovely color, but this makes it hard to see the data points when they are on top of the boxes. In the middle panel, this problem is solved by making the data points black. But now it is kind of hard to see the outliers indicated by the boxplots. The panel to the right is the best of the three, in my opinion, because you can see the box indiciating the quartiles, you can see the median, you can see the outliers, and you can see the data points. I think, though, that if I were preparing this plot for publication, I would remove the black diamonds from the boxplots showing the outliers, because you can see the actual data points right next to them, and it just gets confusing. To me, the outliers no longer look like outiers, they just look like weird data points. The moral? Data visualization is hard!" ] }, { "cell_type": "code", "execution_count": 31, "id": "tired-macedonia", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "properties = {\n", " 'boxprops':{'facecolor':'none', 'edgecolor':'black'},\n", " 'medianprops':{'color':'black'},\n", " 'whiskerprops':{'color':'black'},\n", " 'capprops':{'color':'black'}\n", "}\n", "\n", "fig, axes = plt.subplots(1, 3, figsize=(15, 5))\n", "\n", "\n", "ax1 = sns.boxplot(x = 'year', y = 'margin', data = df, ax=axes[0])\n", "ax1 = sns.stripplot(x = 'year', y = 'margin', data=df, ax=axes[0])\n", "\n", "ax2 = sns.boxplot(x = 'year', y = 'margin', data = df, ax=axes[1])\n", "ax2 = sns.stripplot(x = 'year', y = 'margin', data=df, color = 'black', ax=axes[1])\n", "\n", "ax3 = sns.boxplot(x = 'year', y = 'margin', data = df, **properties, ax=axes[2])\n", "ax3 = sns.stripplot(x = 'year', y = 'margin', data=df, ax=axes[2])\n", "\n", "axes[0].set_title(\"Where are my points?\")\n", "axes[1].set_title(\"Where are my outliers?\")\n", "axes[2].set_title(\"Is this better?\")\n", "\n", "\n", "ax2.set(ylabel = '')\n", "ax3.set(ylabel = '')\n", "\n", "for ax in axes:\n", " ax.spines['right'].set_visible(False)\n", " ax.spines['top'].set_visible(False)" ] }, { "cell_type": "markdown", "id": "israeli-thesis", "metadata": {}, "source": [ "(scatterplots)=\n", "\n", "## Scatterplots\n", "\n", "**_Scatterplots_** are a simple but effective tool for visualising data. We've [already seen scatterplots](introplotting) in this chapter, when drawing the `fibonacci` variable as a collection of dots. However, for the purposes of this section I have a slightly different notion in mind. Instead of just plotting one variable, what I want to do with my scatterplot is display the relationship between *two* variables, like we saw with the figures in the [section on correlation](correlation) It's this latter application that we usually have in mind when we use the term \"scatterplot\". In this kind of plot, each observation corresponds to one dot: the horizontal location of the dot plots the value of the observation on one variable, and the vertical location displays its value on the other variable. In many situations you don't really have a clear opinions about what the *causal* relationship is (e.g., does A cause B, or does B cause A, or does some other variable C control both A and B). If that's the case, it doesn't really matter which variable you plot on the x-axis and which one you plot on the y-axis. However, in many situations you do have a pretty strong idea which variable you think is most likely to be causal, or at least you have some suspicions in that direction. If so, then it's conventional to plot the cause variable on the x-axis, and the effect variable on the y-axis. With that in mind, let's look at how to draw scatterplots in R, using the same `parenthood` data set (i.e. `parenthood.csv`) that I used when introducing the idea of correlations." ] }, { "cell_type": "code", "execution_count": 32, "id": "classical-holiday", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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dan_sleepbaby_sleepdan_grumpday
07.5910.18561
17.9111.66602
25.147.92823
37.719.61554
46.689.75675
\n", "
" ], "text/plain": [ " dan_sleep baby_sleep dan_grump day\n", "0 7.59 10.18 56 1\n", "1 7.91 11.66 60 2\n", "2 5.14 7.92 82 3\n", "3 7.71 9.61 55 4\n", "4 6.68 9.75 67 5" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "import pandas as pd\n", "\n", "df = pd.read_csv('https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/parenthood.csv')\n", "df.head()" ] }, { "cell_type": "markdown", "id": "conscious-welsh", "metadata": {}, "source": [ "Suppose my goal is to draw a scatterplot displaying the relationship between the amount of sleep that I get (`dan.sleep`) and how grumpy I am the next day (`dan.grump`). As you might expect given our earlier use of `scatterplot()` to display the `fibonacci` data, the function that we use is the `scatterplot()` function. Simple!" ] }, { "cell_type": "code", "execution_count": 34, "id": "dangerous-obligation", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "ax = sns.scatterplot(x = 'dan_sleep', y = 'dan_grump', data = df)" ] }, { "cell_type": "markdown", "id": "younger-speech", "metadata": {}, "source": [ "This does show a nice clear relationship between hours of sleep and grumpiness, but it would be nice to make that relationship even clearer with a line through the middle of the points. This can be easily done by replacing `scatterplot` with `regplot`. Everything else stays the same:" ] }, { "cell_type": "code", "execution_count": 35, "id": "metropolitan-practitioner", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "ax = sns.regplot(x = 'dan_sleep', y = 'dan_grump', data = df)" ] }, { "cell_type": "markdown", "id": "liberal-chicken", "metadata": {}, "source": [ "In this case, the line fits the data quite closely. In fact, we could easily see the relationship even without the line. Nevertheless, it is still just a model: the _exact_, true relationship between hours of sleep and grumpiness level is probably unknowable, so the slope of the line represents a best guess, given the data available. This uncertainty is represented by the faint translucent shaded area around the line. This area represents the [confidence interval](ci) of the model. In this case, the shaded area is quite narrow because the model (the line) does a pretty good job of representing the data. In other cases, this shaded area will be larger." ] }, { "cell_type": "markdown", "id": "metric-commission", "metadata": {}, "source": [ "(bargraphs)=\n", "\n", "## Bar graphs\n", "\n", "Another form of graph that you often want to plot is the **_bar graph_**. To illustrate the use of the function, I'll use the `finalists` variable that I introduced in the section on [descriptive statistics](descriptives). What I want to do is draw a bar graph that displays the number of finals that each team has played in over the time spanned by the `afl` data set." ] }, { "cell_type": "code", "execution_count": 36, "id": "baking-building", "metadata": {}, "outputs": [], "source": [ "import pandas as pd\n", "\n", "df = pd.read_csv('https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/afl_finalists.csv')" ] }, { "cell_type": "code", "execution_count": 42, "id": "described-sterling", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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TeamFinals
0Geelong39
1West Coast38
2Essendon32
3Melbourne28
4Collingwood28
5North Melbourne28
6Hawthorn27
7Carlton26
8Adelaide26
9Sydney26
10Brisbane25
11St Kilda24
12Western Bulldogs24
13Port Adelaide17
14Richmond6
15Fremantle6
\n", "
" ], "text/plain": [ " Team Finals\n", "0 Geelong 39\n", "1 West Coast 38\n", "2 Essendon 32\n", "3 Melbourne 28\n", "4 Collingwood 28\n", "5 North Melbourne 28\n", "6 Hawthorn 27\n", "7 Carlton 26\n", "8 Adelaide 26\n", "9 Sydney 26\n", "10 Brisbane 25\n", "11 St Kilda 24\n", "12 Western Bulldogs 24\n", "13 Port Adelaide 17\n", "14 Richmond 6\n", "15 Fremantle 6" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "\n", "# count up the number of times each team has been in the finals\n", "finalists = df['afl.finalists'].value_counts()\n", "\n", "# convert the result to a dataframe\n", "df2 = finalists.to_frame()\n", "\n", "# convert the row names to a column of data\n", "df2.index.name = 'Team'\n", "df2.reset_index(inplace=True)\n", "\n", "# rename columns with clearer names\n", "df2.columns = ['Team', 'Finals']\n", "\n", "df2" ] }, { "cell_type": "markdown", "id": "monthly-nowhere", "metadata": {}, "source": [ "By now, if you have been following along, you can probably guess the syntax yourself. Specify the x data, specify the y data, and specify the data source:" ] }, { "cell_type": "code", "execution_count": 43, "id": "greatest-silver", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "ax = sns.barplot(x = 'Team', y = 'Finals', data = df2)" ] }, { "cell_type": "markdown", "id": "selected-suspect", "metadata": {}, "source": [ "Lovely, right? Pretty colors! Unfortunately, the team's names are entirely illegible. Sometimes people try to deal with this sort of problem by rotating the x-axis tick labels 45 degrees so that they don't run all over each other. That is an option, but an easy fix to this problem that you should always keep in mind is to simply switch the x an y axes:" ] }, { "cell_type": "code", "execution_count": 44, "id": "considerable-chapel", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "ax = sns.barplot(x = 'Finals', y = 'Team', data = df2)" ] }, { "cell_type": "markdown", "id": "continuing-graphics", "metadata": {}, "source": [ "There, now isn't that better? I haven't been touching up these plots much in this chapter, because I want to focus on how easy is can be to generate nice simple plots, but in this case that y-axis label is just staring me in the face, now that I have turned the bars onto their sides. Normally I would insist that a figure should always have clear axis labels, but presumably, if we are putting this in a paper or presentation, the audience already knows that we are talking about AFL teams, and we could also put this info in the title instead. So let's do a little makeover:" ] }, { "cell_type": "code", "execution_count": 54, "id": "better-lambda", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "ax = sns.barplot(x = 'Finals', y = 'Team', data = df2)\n", "ax.set(title = 'Number of Finals per Team', xlabel = 'Finals', ylabel='')\n", "sns.despine(top=True, right=True)" ] }, { "cell_type": "markdown", "id": "fabulous-programming", "metadata": {}, "source": [ "Now we're talking! A nice clear figure, that is pretty and easy to read. Yum!" ] }, { "cell_type": "markdown", "id": "resident-discovery", "metadata": {}, "source": [ "(saveimage)=\n", "\n", "## Saving image files\n", "\n", "Hold on, you might be thinking. What's the good of being able to draw pretty pictures in Python if I can't save them and send them to friends to brag about how awesome my data is? How do I save the picture? A (fairly) easy solution is to use the `savefig` command from the `pyplot` library in `matplotlib`. I say fairly, because without a few tweaks, it may give you less than desirable results. In particular, I would like to draw your attention to the arguements `facecolor` and `bbox_inches` in the last line of the code below. Without setting `facecolor` to \"white\" and `bbox_inches` to \"tight\", the image is saved with a dark background outside of the main area of the figure, and most of the team names get cut off on the margins. Why these settings aren't the default I can't imagine, but there is probably some good reason. In any case, using 'savefig' you can export your figure in a variety of formats. Here I have used the .png format." ] }, { "cell_type": "code", "execution_count": 55, "id": "german-dance", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "from matplotlib import pyplot as plt\n", "\n", "ax = sns.barplot(x = 'Finals', y = 'Team', data = df2)\n", "ax.set(title = 'Number of Finals per Team', xlabel = 'Finals', ylabel='')\n", "sns.despine(top=True, right=True)\n", "sns.set_context(\"paper\", font_scale=1.5)\n", "\n", "\n", "plt.savefig('/Users/ethan/Desktop/AFL.png', facecolor='white', bbox_inches='tight')" ] }, { "cell_type": "markdown", "id": "composite-aberdeen", "metadata": {}, "source": [ "## Enough for now\n", "\n", "I could go on and on. I really enjoy data visualization because it helps me understand my data, and I have been known to spend waaaay too much time trying to craft the perfect graph when I probably should be getting on with writing the paper. What I have given here is only a quick introducton to some of the most common graph types, and a few options for how to configure them. Much, much, much more is possible, either directly through `seaborn`, or combined with the power of `matplotlib` and other graphics libraries. But alas, the rest of this book isn't going to get written if I stay here all day, so I'll leave it there for now. \"Perfection\", as a co-author recently informed me, kindly, yet firmly, \"is the enemy of the Good\". Or in this case, the Good-Enough." ] }, { "cell_type": "code", "execution_count": null, "id": "coupled-colon", "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "id": "c33af0d0", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.5" } }, "nbformat": 4, "nbformat_minor": 5 } { "cells": [ { "cell_type": "markdown", "id": "41340fc3", "metadata": {}, "source": [ "(datawrangling)=\n", "\n", "# Data Wrangling" ] }, { "cell_type": "markdown", "id": "6d4493e2", "metadata": {}, "source": [ ">*The garden of life never seems to confine itself to the plots philosophers have\n", "laid out for its convenience. Maybe a few more tractors would do the trick.*\n", ">\n", ">--Roger Zelazny [^note1]\n", "\n", "[^note1]: The quote comes from *Home is the Hangman*, published in 1975.]\n" ] }, { "cell_type": "markdown", "id": "6dc6fdd3", "metadata": {}, "source": [ "This is a somewhat strange chapter, even by my standards. My goal in this chapter is to talk a bit more honestly about the realities of working with data than you'll see anywhere else in the book. The problem with real world data sets is that they are *messy*. Very often the data file that you start out with doesn't have the variables stored in the right format for the analysis you want to do. Sometimes might be a lot of missing values in your data set. Sometimes you only want to analyse a subset of the data. Et cetera. In other words, there's a lot of **_data manipulation_** that you need to do, just to get all your data set into the format that you need it. The purpose of this chapter is to provide a basic introduction to all these pragmatic topics. Although the chapter is motivated by the kinds of practical issues that arise when manipulating real data, I'll stick with the practice that I've adopted through most of the book and rely on very small, toy data sets that illustrate the underlying issue. Because this chapter is essentially a collection of \"tricks\" and doesn't tell a single coherent story, it may be useful to start with a list of topics:\n", "\n", "- [Dataframes](pandas)\n", "- [Tabulating data](freqtables)\n", "- [Transforming or recoding a variable](transform)\n", "- [Some useful mathematical functions](mathfunc)\n", "- [Slicing](slicing) and dicing\n", "- [Extracting a subset of a data frame](subsets)\n", "- [Sorting, flipping or merging data sets](manipulations)\n", "- [Reshaping a data frame](reshaping)\n", "\n", "\n", "I may dump more trick and tips here later, and I'm really only scratching the surface of several fairly different and important topics. My advice, as usual, is to read through the chapter once and try to follow as much of it as you can. Don't worry too much if you can't grasp it all at once, especially the later sections. The rest of the book is only lightly reliant on this chapter, so you can get away with just understanding the basics. However, what you'll probably find is that later on you'll need to flick back to this chapter in order to understand some of the concepts that I refer to here." ] }, { "cell_type": "markdown", "id": "082f5340", "metadata": {}, "source": [ "(pandas)=\n", "\n", "## Dataframes" ] }, { "cell_type": "markdown", "id": "3a206092", "metadata": {}, "source": [ "We've already used the `pandas` package [here](descriptives) and [there](DrawingGraphs), and [even over here](loadingcsv), but now it's time to look more closely at `pandas` dataframes. \n", "\n", "In order to understand why we use dataframes, it helps to try to see what problem the solve. So let's imagine a little scenario in which I collected some data from nine participants. Let's say I divded the participants in two groups (\"test\" and \"control\"), and gave them a task. I then recorded their score on the task, as well as the time it took them to complete the task. I also noted down how old they were.\n", "\n", "the data look like this:" ] }, { "cell_type": "code", "execution_count": 127, "id": "a2d4ed3d", "metadata": {}, "outputs": [], "source": [ "age = [17, 19, 21, 37, 18, 19, 47, 18, 19]\n", "score = [12, 10, 11, 15, 16, 14, 25, 21, 29]\n", "rt = [3.552, 1.624, 6.431, 7.132, 2.925, 4.662, 3.634, 3.635, 5.234]\n", "group = [\"test\", \"test\", \"test\", \"test\", \"test\", \"control\", \"control\", \"control\", \"control\"]" ] }, { "cell_type": "markdown", "id": "387d9ee8", "metadata": {}, "source": [ "So there are four variables in active memory: `age`, `rt`, `group` and `score`. And it just so happens that all four of them are the same size (i.e., they're all lists with 9 elements). Aaaand it just so happens that `age[0]` corresponds to the age of the first person, and `rt[0]` is the response time of that very same person, etc. In other words, you and I both know that all four of these variables correspond to the *same* data set, and all four of them are organised in exactly the same way. \n", "\n", "However, Python *doesn't* know this! As far as it's concerned, there's no reason why the `age` variable has to be the same length as the `rt` variable; and there's no particular reason to think that `age[1]` has any special relationship to `score[1]` any more than it has a special relationship to `score[4]`. In other words, when we store everything in separate variables like this, Python doesn't know anything about the relationships between things. It doesn't even really know that these variables actually refer to a proper data set. The data frame fixes this: if we store our variables inside a data frame, we're telling Python to treat these variables as a single, fairly coherent data set. \n", "\n", "To see how they do this, let's create one. So how do we create a data frame? One way we've already seen: if we use `pandas` to [import our data from a CSV file](loadingcsv), it will store it as a data frame. A second way is to create it directly from some existing lists using the `pandas.Dataframe()` function. All you have to do is type a list of variables that you want to include in the data frame. The output is, well, a data frame. So, if I want to store all four variables from my experiment in a data frame called `df` I can do so like this[^notedict]:\n", "\n", "[^notedict]: Although it really doesn't matter at this point, you may have noticed a new symbol here: the \"curly brackets\" or \"curly braces\". Python uses these to indicate yet another variable type: the dictionary. Here we are using the dictionary variable type in passing to feed our lists into a `pandas` dataframe." ] }, { "cell_type": "code", "execution_count": 128, "id": "9e62ecb9", "metadata": {}, "outputs": [], "source": [ "import pandas as pd\n", "\n", "df = pd.DataFrame(\n", " {'age': age,\n", " 'score': score,\n", " 'rt': rt,\n", " 'group': group\n", " })" ] }, { "cell_type": "code", "execution_count": 129, "id": "13084ec3", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " age score rt group\n", "0 17 12 3.552 test\n", "1 19 10 1.624 test\n", "2 21 11 6.431 test\n", "3 37 15 7.132 test\n", "4 18 16 2.925 test\n", "5 19 14 4.662 control\n", "6 47 25 3.634 control\n", "7 18 21 3.635 control\n", "8 19 29 5.234 control" ] }, "execution_count": 129, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df" ] }, { "cell_type": "markdown", "id": "0152344e", "metadata": {}, "source": [ "Note that `df` is a completely self-contained variable. Once you've created it, it no longer depends on the original variables from which it was constructed. That is, if we make changes to the original `age` variable, it will *not* lead to any changes to the age data stored in `df`. " ] }, { "cell_type": "markdown", "id": "3c9de9d8", "metadata": {}, "source": [ "(indexingdataframes)=\n", "### Pulling out the contents of a data frame" ] }, { "cell_type": "markdown", "id": "af51cc62", "metadata": {}, "source": [ "Let's take another look at our dataframe. We have created a dataframe called `df`, which contains all of our data for \"The Very Exciting Psychology Experiment\". Each row contains the data for one participant, so we can see that e.g. the first participant (in row zero, because Python!) was 17 years old, had a score of 12, responded in 3.552 seconds, and was placed in the test group. That's great, but how do we get this information out again? After all, there's no point in storing information if you don't use it, and there's no way to use information if you can't access it. So let's talk a bit about how to pull information out of a data frame. \n", "\n", "The first thing we might want to do is pull out one of our stored variables, let's say `score`. To access the data in the `score` column by the column name, we can write:" ] }, { "cell_type": "code", "execution_count": 130, "id": "f73718ce", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0 12\n", "1 10\n", "2 11\n", "3 15\n", "4 16\n", "5 14\n", "6 25\n", "7 21\n", "8 29\n", "Name: score, dtype: int64" ] }, "execution_count": 130, "metadata": {}, "output_type": "execute_result" } ], "source": [ "score_data = df['score']\n", "score_data" ] }, { "cell_type": "markdown", "id": "c29ad60c", "metadata": {}, "source": [ "Pretty easy, right? We could also choose to ask for only data from e.g. the first 4 particpants. To do this, we write:" ] }, { "cell_type": "code", "execution_count": 131, "id": "c7728465", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0 12\n", "1 10\n", "2 11\n", "3 15\n", "Name: score, dtype: int64" ] }, "execution_count": 131, "metadata": {}, "output_type": "execute_result" } ], "source": [ "score_data = df['score'][0:4]\n", "score_data" ] }, { "cell_type": "markdown", "id": "e89bab44", "metadata": {}, "source": [ "As always, we have to be very careful about the numbering, and things are even more confusing than I have let on, because what we are doing here is what Python calls *slicing* the data, and slice numbers work a little differently than index numbers. To get a slice of data from the first to the fourth rows, we need to write `[0:4]`, rather than `[0:3]`, because when slicing the data, we need to specify the start and end point of the slice, but the end point _does not include the value specified as the end._ Is this confusing? Yes, I think so! In any case, this is the way Python behaves, and we just need to get used to it. The best way to get the hang of it just to practice slicing a bunch of data, until you learn how to get the results you want." ] }, { "cell_type": "markdown", "id": "e6cd1d16", "metadata": {}, "source": [ "What if we want to get data from a row instead? In this case, we will use the loc attribute of a pandas dataframe, and use a number instead of name (i.e., no quotation marks), like this:" ] }, { "cell_type": "code", "execution_count": 132, "id": "1ef78341", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "age 21\n", "score 11\n", "rt 6.431\n", "group test\n", "Name: 2, dtype: object" ] }, "execution_count": 132, "metadata": {}, "output_type": "execute_result" } ], "source": [ "score_data = df.loc[2]\n", "score_data" ] }, { "cell_type": "markdown", "id": "b792e7c9", "metadata": {}, "source": [ "Now we have what we need to get the data for columns and rows. Great! Unfortunately, there is one more thing I should mention[^note7]. If you look at the contents of score_data above, you will see that it is still not just the data: it also has information about the data, including which column and row it came from. And if we use type() to check, we can see that it is yet another variable type: this time, a pandas.core.series.Series. Yikes!\n", "[^note7]: Actually, there are lots more things I should mention, but now is not the time. Working with dataframes takes practice, and there are some catches, but it's worth the effort!" ] }, { "cell_type": "code", "execution_count": 133, "id": "81361a2d", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "pandas.core.series.Series" ] }, "execution_count": 133, "metadata": {}, "output_type": "execute_result" } ], "source": [ "type(score_data)" ] }, { "cell_type": "markdown", "id": "f2796cc5", "metadata": {}, "source": [ "Luckily, it's not too hard to get the raw data out of a `pandas` series. The simplest way is to just turn it into a list variable, using the command `list()`:" ] }, { "cell_type": "code", "execution_count": 134, "id": "3e04a254", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[21, 11, 6.431, 'test']" ] }, "execution_count": 134, "metadata": {}, "output_type": "execute_result" } ], "source": [ "my_row = list(score_data)\n", "my_row" ] }, { "cell_type": "markdown", "id": "4dc3b386", "metadata": {}, "source": [ "If you want to get fancy, you can combine these steps, and do it all in one go:" ] }, { "cell_type": "code", "execution_count": 135, "id": "26d74e68", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[21, 11, 6.431, 'test']\n", "[12, 10, 11, 15, 16, 14, 25, 21, 29]\n" ] } ], "source": [ "my_row = list(df.loc[2])\n", "my_column = list(df['score'])\n", "print(my_row)\n", "print(my_column)" ] }, { "cell_type": "markdown", "id": "1167fcc4", "metadata": {}, "source": [ "### Some more dataframe tips for the road\n", "\n", "One problem that sometimes comes up in practice is that you forget what you called all your variables. To get a list of the column names, you can use the command:" ] }, { "cell_type": "code", "execution_count": 136, "id": "0e9e9528", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "['age', 'score', 'rt', 'group']" ] }, "execution_count": 136, "metadata": {}, "output_type": "execute_result" } ], "source": [ "list(df)" ] }, { "cell_type": "markdown", "id": "7d17c8d9", "metadata": {}, "source": [ "We can easily check to see how many rows and columns our dataframe has using `.shape`" ] }, { "cell_type": "code", "execution_count": 137, "id": "e00dffa7", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(9, 4)" ] }, "execution_count": 137, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.shape" ] }, { "cell_type": "markdown", "id": "b6b31fb0", "metadata": {}, "source": [ "The first number gives us the number of rows, and the second number is the number of columns. Our dataframe `df` is 9 rows long, and 4 columns wide." ] }, { "cell_type": "markdown", "id": "0bd8a8a7", "metadata": {}, "source": [ "Sometimes dataframes can be very large, and we just want to peek at them, to check what they look like, without data scrolling endlessly over the screen. The dataframe attribute head() is useful for this. By default it shows the first 5 lines of the dataframe:" ] }, { "cell_type": "code", "execution_count": 76, "id": "7479347e", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " age score rt group\n", "0 17 12 3.552 test\n", "1 19 10 1.624 test\n", "2 21 11 6.431 test\n", "3 37 15 7.132 test\n", "4 18 16 2.925 test" ] }, "execution_count": 76, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.head()" ] }, { "cell_type": "markdown", "id": "6e25f0de", "metadata": {}, "source": [ "And if you want to see the last rows of the dataframe? `tail()` has got you covered:" ] }, { "cell_type": "code", "execution_count": 77, "id": "f8198539", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
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" ], "text/plain": [ " age score rt group\n", "4 18 16 2.925 test\n", "5 19 14 4.662 control\n", "6 47 25 3.634 control\n", "7 18 21 3.635 control\n", "8 19 29 5.234 control" ] }, "execution_count": 77, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.tail()" ] }, { "cell_type": "markdown", "id": "83d07d92", "metadata": {}, "source": [ "Finally, if you just want to get all of your data out of the dataframe and into a list, then .values.tolist() will do the job, giving you a list of lists, with each item in the list containing the data for a single row:" ] }, { "cell_type": "code", "execution_count": 78, "id": "515ab30d", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[[17, 12, 3.552, 'test'],\n", " [19, 10, 1.624, 'test'],\n", " [21, 11, 6.431, 'test'],\n", " [37, 15, 7.132, 'test'],\n", " [18, 16, 2.925, 'test'],\n", " [19, 14, 4.662, 'control'],\n", " [47, 25, 3.634, 'control'],\n", " [18, 21, 3.635, 'control'],\n", " [19, 29, 5.234, 'control']]" ] }, "execution_count": 78, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.values.tolist()" ] }, { "cell_type": "markdown", "id": "c077249a", "metadata": {}, "source": [ "(freqtables)=\n", "\n", "## Tabulating and cross-tabulating data\n", "\n", "A very common task when analysing data is the construction of frequency tables, or cross-tabulation of one variable against another. There are several functions that you can use in Python for that purpose.\n", "\n", "Let's start with a simple example. As the father of a small child, I naturally spend a lot of time watching TV shows like *In the Night Garden*, and I have transcribed a short section of the dialogue. Let's make a `pandas` dataframe with two variables, `speaker` and `utterance`. When we take a look at the data, it becomes very clear what happened to my sanity. " ] }, { "cell_type": "code", "execution_count": 79, "id": "a9ae015f", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " speaker utterance\n", "0 upsy-daisy pip\n", "1 upsy-daisy pip\n", "2 upsy-daisy onk\n", "3 upsy-daisy onk\n", "4 tombliboo ee\n", "5 tombliboo oo\n", "6 makka-pakka pip\n", "7 makka-pakka pip\n", "8 makka-pakka onk\n", "9 makka-pakka onk" ] }, "execution_count": 79, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "data = {'speaker':[\"upsy-daisy\", \"upsy-daisy\", \"upsy-daisy\", \"upsy-daisy\", \"tombliboo\", \"tombliboo\", \"makka-pakka\", \"makka-pakka\",\n", " \"makka-pakka\", \"makka-pakka\"],\n", " 'utterance':[\"pip\", \"pip\", \"onk\", \"onk\", \"ee\", \"oo\", \"pip\", \"pip\", \"onk\", \"onk\"]}\n", "\n", "df = pd.DataFrame(data, columns=['speaker','utterance'])\n", "\n", "df\n" ] }, { "cell_type": "markdown", "id": "8246c955", "metadata": {}, "source": [ "With these as my data, one task I might find myself needing to do is construct a frequency count of the number of utterances each character produces during the show. As usual, there are more than one way to achieve this, but the `crosstab` method from `pandas` provides an easy way to do this:" ] }, { "cell_type": "code", "execution_count": 80, "id": "3ad7842a", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ "col_0 count\n", "speaker \n", "makka-pakka 4\n", "tombliboo 2\n", "upsy-daisy 4" ] }, "execution_count": 80, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pd.crosstab(index = df[\"speaker\"], columns = \"count\")" ] }, { "cell_type": "markdown", "id": "c7feeddf", "metadata": {}, "source": [ "The output here shows a column called \"speaker\", and a column called \"count\". In the \"speaker\" column, we can see the names of all the speakers, and in the \"count\" column, we can see the number of utterances for each speaker. In other words, it’s a frequency table. Notice that we set the argument `columns` to \"count\". If instead we want to cross-tabulate the speakers with the utterances, we can set `columns` to the \"utterances\" column in the dataframe:" ] }, { "cell_type": "code", "execution_count": 81, "id": "a28e45ff", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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utteranceeeonkoopipAll
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makka-pakka02024
tombliboo10102
upsy-daisy02024
All141410
\n", "
" ], "text/plain": [ "utterance ee onk oo pip All\n", "speaker \n", "makka-pakka 0 2 0 2 4\n", "tombliboo 1 0 1 0 2\n", "upsy-daisy 0 2 0 2 4\n", "All 1 4 1 4 10" ] }, "execution_count": 81, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pd.crosstab(index=df[\"speaker\"], columns=df[\"utterance\"],margins=True)" ] }, { "cell_type": "markdown", "id": "01dfe8d9", "metadata": {}, "source": [ "### Converting a table of counts to a table of proportions\n", "\n", "The tabulation commands discussed so far all construct a table of raw frequencies: that is, a count of the total number of cases that satisfy certain conditions. However, often you want your data to be organised in terms of proportions rather than counts. This could be as a proportion of the row totals or the column totals. Currently, these are both just called \"All\", so let's first save the output of our crosstab to a variable, and rename the row and column totals to \"rowtotals\" and \"coltotals\"." ] }, { "cell_type": "code", "execution_count": 82, "id": "56def5ea", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
eeonkoopiprowtotals
makka-pakka02024
tombliboo10102
upsy-daisy02024
coltotals141410
\n", "
" ], "text/plain": [ " ee onk oo pip rowtotals\n", "makka-pakka 0 2 0 2 4\n", "tombliboo 1 0 1 0 2\n", "upsy-daisy 0 2 0 2 4\n", "coltotals 1 4 1 4 10" ] }, "execution_count": 82, "metadata": {}, "output_type": "execute_result" } ], "source": [ "tabs = pd.crosstab(index=df[\"speaker\"], columns=df[\"utterance\"],margins=True)\n", "\n", "tabs.columns = list(tabs.columns)[0:-1] + ['rowtotals']\n", "tabs.index = list(tabs.index)[0:-1] + ['coltotals']\n", "\n", "tabs" ] }, { "cell_type": "markdown", "id": "09f553e5", "metadata": {}, "source": [ "Before we go on, it might be worthwhile looking at the steps used to rename the final column and row, because they give us some important information about the way `pandas` dataframes work. We saw before that you can get a list of the columns in your dataframe by using `list`:" ] }, { "cell_type": "code", "execution_count": 83, "id": "9072bcf9", "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/plain": [ "['ee', 'onk', 'oo', 'pip', 'rowtotals']" ] }, "execution_count": 83, "metadata": {}, "output_type": "execute_result" } ], "source": [ "list(tabs)" ] }, { "cell_type": "code", "execution_count": 84, "id": "f4be771f", "metadata": {}, "outputs": [ { "ename": "SyntaxError", "evalue": "invalid syntax (584976377.py, line 1)", "output_type": "error", "traceback": [ "\u001b[0;36m Input \u001b[0;32mIn [84]\u001b[0;36m\u001b[0m\n\u001b[0;31m Now we see that we can also get the names of the columns of our dataframe `tabs` by writing `tabs.columns`\u001b[0m\n\u001b[0m ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n" ] } ], "source": [ "Now we see that we can also get the names of the columns of our dataframe `tabs` by writing `tabs.columns`\n" ] }, { "cell_type": "code", "execution_count": 85, "id": "43fb4e15", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Index(['ee', 'onk', 'oo', 'pip', 'rowtotals'], dtype='object')" ] }, "execution_count": 85, "metadata": {}, "output_type": "execute_result" } ], "source": [ "tabs.columns" ] }, { "cell_type": "code", "execution_count": 86, "id": "a0768508", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "pandas.core.indexes.base.Index" ] }, "execution_count": 86, "metadata": {}, "output_type": "execute_result" } ], "source": [ "type(tabs.columns)" ] }, { "cell_type": "markdown", "id": "595db186", "metadata": {}, "source": [ "If we check to see what kind of object this is, we can see that it is a `pandas.core.indexes.base.Index` object. Isn't that a nice name? Furthermore, we can convert this object to a list in the usual way:" ] }, { "cell_type": "code", "execution_count": 87, "id": "46422890", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "['ee', 'onk', 'oo', 'pip', 'rowtotals']" ] }, "execution_count": 87, "metadata": {}, "output_type": "execute_result" } ], "source": [ "list(tabs.columns)" ] }, { "cell_type": "markdown", "id": "279fd2af", "metadata": {}, "source": [ "and now that it is a list, we can do the usual sorts of things that we can with lists, such as replace items in the list with other items. This allows us make changes to the column headers, and then we can re-assign our list to be the new column headers of the dataframe. Hoo boy! Finally, note that the dataframe also has an index of row names, called `.index`. So in our case, this is `tabs.index`. Sometimes these row names will be actual names, such as is the case in our dataframe `tabs`. Other times, like in the dataframe `df` from above, the `index` of the dataframe will just be numbers:" ] }, { "cell_type": "code", "execution_count": 88, "id": "cacf5be5", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]" ] }, "execution_count": 88, "metadata": {}, "output_type": "execute_result" } ], "source": [ "list(df.index)" ] }, { "cell_type": "markdown", "id": "73033f91", "metadata": {}, "source": [ "Either way, names or numbers, each `pandas` dataframe will have a `columns` and an `index` index.\n", "\n", "Now, after that fascinating digression into the structure of dataframes, back to our project which, as you surely recall, was converting our table of counts to a table of proportions. Having renamed the final column and row, now we can divide the entire frequency table by the totals in each column:" ] }, { "cell_type": "code", "execution_count": 89, "id": "b9be6b0b", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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eeonkoopiprowtotals
makka-pakka0.00.50.00.50.4
tombliboo1.00.01.00.00.2
upsy-daisy0.00.50.00.50.4
coltotals1.01.01.01.01.0
\n", "
" ], "text/plain": [ " ee onk oo pip rowtotals\n", "makka-pakka 0.0 0.5 0.0 0.5 0.4\n", "tombliboo 1.0 0.0 1.0 0.0 0.2\n", "upsy-daisy 0.0 0.5 0.0 0.5 0.4\n", "coltotals 1.0 1.0 1.0 1.0 1.0" ] }, "execution_count": 89, "metadata": {}, "output_type": "execute_result" } ], "source": [ "tabs/tabs.loc['coltotals']" ] }, { "cell_type": "markdown", "id": "c2e885cd", "metadata": {}, "source": [ "The columns sum to one, so we can see that makka-pakka and upsy-daisy each produced 40% of the utterances, while tombliboo only produced 20%. We can also see the proportion of characters associated with each utterance. For instance, whenever the utterance “ee” is made (in this data set), 100% of the time it’s a Tombliboo saying it. \n", "\n", "The procedure to obtain the row-wise proportion, the procedure is slightly different. I'm _not_ going to get into it now. It just _is_, ok?" ] }, { "cell_type": "code", "execution_count": 90, "id": "b51f934b", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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eeonkoopiprowtotals
makka-pakka0.00.50.00.51.0
tombliboo0.50.00.50.01.0
upsy-daisy0.00.50.00.51.0
coltotals0.10.40.10.41.0
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" ], "text/plain": [ " ee onk oo pip rowtotals\n", "makka-pakka 0.0 0.5 0.0 0.5 1.0\n", "tombliboo 0.5 0.0 0.5 0.0 1.0\n", "upsy-daisy 0.0 0.5 0.0 0.5 1.0\n", "coltotals 0.1 0.4 0.1 0.4 1.0" ] }, "execution_count": 90, "metadata": {}, "output_type": "execute_result" } ], "source": [ "tabs.div(tabs[\"rowtotals\"], axis=0)" ] }, { "cell_type": "markdown", "id": "6b623616", "metadata": {}, "source": [ "Each row now sums to one, but that’s not true for each column. What we’re looking at here is the proportions of utterances made by each character. In other words, 50% of Makka-Pakka’s utterances are “pip”, and the other 50% are “onk”." ] }, { "cell_type": "markdown", "id": "0078295c", "metadata": {}, "source": [ "(transform)=\n", "\n", "## Transforming and recoding a variable\n", "\n", "It's not uncommon in real world data analysis to find that one of your variables isn't quite equivalent to the variable that you really want. For instance, it's often convenient to take a continuous-valued variable (e.g., age) and break it up into a smallish number of categories (e.g., younger, middle, older). At other times, you may need to convert a numeric variable into a different numeric variable (e.g., you may want to analyse at the absolute value of the original variable). In this section I'll describe a few key tricks that you can make use of to do this.\n", "\n", "### Creating a transformed variable\n", "\n", "The first trick to discuss is the idea of **_transforming_** a variable. Taken literally, *anything* you do to a variable is a transformation, but in practice what it usually means is that you apply a relatively simple mathematical function to the original variable, in order to create new variable that either (a) provides a better way of describing the thing you're actually interested in or (b) is more closely in agreement with the assumptions of the statistical tests you want to do. Since -- at this stage -- I haven't talked about statistical tests or their assumptions, I'll show you an example based on the first case. \n", "\n", "To keep the explanation simple, the variable we'll try to transform isn't inside a data frame, though in real life it almost certainly would be. However, I think it's useful to start with an example that doesn't use data frames because it illustrates the fact that you already know how to do variable transformations. To see this, let's go through an example. Suppose I've run a short study in which I ask 10 people a single question: \n", "\n", ">On a scale of 1 (strongly disagree) to 7 (strongly agree), to what extent do you agree with the proposition that \"Dinosaurs are awesome\"?\n", "\n", "The data look like this:" ] }, { "cell_type": "code", "execution_count": 91, "id": "fe3c0acd", "metadata": {}, "outputs": [], "source": [ "data = [1, 7, 3, 4, 4, 4, 2, 6, 5, 5]" ] }, { "cell_type": "markdown", "id": "6fd05ced", "metadata": {}, "source": [ "However, if you think about it, this isn't the best way to represent these responses. Because of the fairly symmetric way that we set up the response scale, there's a sense in which the midpoint of the scale should have been coded as 0 (no opinion), and the two endpoints should be $+3$ (strong agree) and $-3$ (strong disagree). By recoding the data in this way, it's a bit more reflective of how we really think about the responses. The recoding here is trivially easy: we just subtract 4 from the raw scores. Since these data are in a list, we can use a \"list comprehension\" to step through each element in the list, and subtract 4 from it:" ] }, { "cell_type": "code", "execution_count": 92, "id": "62dc57fe", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[-3, 3, -1, 0, 0, 0, -2, 2, 1, 1]" ] }, "execution_count": 92, "metadata": {}, "output_type": "execute_result" } ], "source": [ "data = [1, 7, 3, 4, 4, 4, 2, 6, 5, 5]\n", "data = [x-4 for x in data]\n", "data" ] }, { "cell_type": "markdown", "id": "fc54decb", "metadata": {}, "source": [ "If your data is in a `numpy array` rather than a `list`, it is even easier: just subtract 4 from array, and Python takes care of the rest:" ] }, { "cell_type": "code", "execution_count": 93, "id": "023da02d", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([-3, 3, -1, 0, 0, 0, -2, 2, 1, 1])" ] }, "execution_count": 93, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import numpy as np\n", "data = np.array([1, 7, 3, 4, 4, 4, 2, 6, 5, 5])\n", "data = data - 4\n", "data" ] }, { "cell_type": "markdown", "id": "06c4467e", "metadata": {}, "source": [ "One reason why it might be useful to center the data is that there are a lot of situations where you might prefer to analyse the *strength* of the opinion separately from the *direction* of the opinion. We can do two different transformations on this variable in order to distinguish between these two different concepts. Firstly, to compute an `opinion_strength` variable, we want to take the absolute value of the centred data (using the `abs()` function that we've seen previously), like so:" ] }, { "cell_type": "code", "execution_count": 94, "id": "dc9bf153", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([3, 3, 1, 0, 0, 0, 2, 2, 1, 1])" ] }, "execution_count": 94, "metadata": {}, "output_type": "execute_result" } ], "source": [ "data = np.array([1, 7, 3, 4, 4, 4, 2, 6, 5, 5])\n", "data = data -4\n", "data = abs(data)\n", "data" ] }, { "cell_type": "markdown", "id": "9373fe22", "metadata": {}, "source": [ "Secondly, to compute a variable that contains only the direction of the opinion and ignores the strength, we can use the `numpy.sign()` method to do this. This method is really simple: all negative numbers are converted to $-1$, all positive numbers are converted to $1$ and zero stays as $0$. So, when we apply `numpy.sign()` to our data we obtain the following:" ] }, { "cell_type": "code", "execution_count": 95, "id": "2ae63fd4", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([-1, 1, -1, 0, 0, 0, -1, 1, 1, 1])" ] }, "execution_count": 95, "metadata": {}, "output_type": "execute_result" } ], "source": [ "data = np.array([1, 7, 3, 4, 4, 4, 2, 6, 5, 5])\n", "data = data - 4\n", "data = np.sign(data)\n", "data" ] }, { "cell_type": "markdown", "id": "f300821a", "metadata": {}, "source": [ "And we're done. We now have three shiny new variables, all of which are useful transformations of the original likert data. Before moving on, you might be curious to see what these calculations look like if the data had started out in a data frame. So, we can put our data in a dataframe, in a column called \"scores\"..." ] }, { "cell_type": "code", "execution_count": 96, "id": "0d952c99", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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scores
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" ], "text/plain": [ " scores\n", "0 1\n", "1 7\n", "2 3\n", "3 4\n", "4 4\n", "5 4\n", "6 2\n", "7 6\n", "8 5\n", "9 5" ] }, "execution_count": 96, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "df = pd.DataFrame(\n", " {'scores': np.array([1, 7, 3, 4, 4, 4, 2, 6, 5, 5])\n", " })\n", "\n", "df" ] }, { "cell_type": "markdown", "id": "9274ccda", "metadata": {}, "source": [ "... and then do some calculations:" ] }, { "cell_type": "code", "execution_count": 97, "id": "6ee5bb08", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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scorescenteredopinion_strengthopinion_direction
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" ], "text/plain": [ " scores centered opinion_strength opinion_direction\n", "0 1 -3 3 -1\n", "1 7 3 3 1\n", "2 3 -1 1 -1\n", "3 4 0 0 0\n", "4 4 0 0 0\n", "5 4 0 0 0\n", "6 2 -2 2 -1\n", "7 6 2 2 1\n", "8 5 1 1 1\n", "9 5 1 1 1" ] }, "execution_count": 97, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df['centered'] = df['scores']-4\n", "df['opinion_strength'] = abs(df['centered'])\n", "df['opinion_direction'] = np.sign(df['scores']-4)\n", "df" ] }, { "cell_type": "markdown", "id": "89fc291d", "metadata": {}, "source": [ "In other words, even though the data are now columns in a dataframe, we can use exactly the same means to calculate new variable. Even better, we can simply create new columns willy-nilly within the same dataframe, so we can keep everything together, all neat and tidy." ] }, { "cell_type": "markdown", "id": "9e9cce82", "metadata": {}, "source": [ "### Cutting a numeric variable into categories\n", "\n", "One pragmatic task that arises more often than you'd think is the problem of cutting a numeric variable up into discrete categories. For instance, suppose I'm interested in looking at the age distribution of people at a social gathering:" ] }, { "cell_type": "code", "execution_count": 98, "id": "4bf39578", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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age
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" ], "text/plain": [ " age\n", "0 60\n", "1 58\n", "2 24\n", "3 26\n", "4 34\n", "5 42\n", "6 31\n", "7 30\n", "8 33\n", "9 2\n", "10 9" ] }, "execution_count": 98, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#age = [60,58,24,26,34,42,31,30,33,2,9]\n", "import pandas as pd\n", "df = pd.DataFrame(\n", " {'age': np.array([60,58,24,26,34,42,31,30,33,2,9])\n", " })\n", "\n", "df" ] }, { "cell_type": "markdown", "id": "0aa536e7", "metadata": {}, "source": [ "In some situations it can be quite helpful to group these into a smallish number of categories. For example, we could group the data into three broad categories: young (0-20), adult (21-40) and older (41-60). This is a quite coarse-grained classification, and the labels that I've attached only make sense in the context of this data set (e.g., viewed more generally, a 42 year old wouldn't consider themselves as \"older\").\n", "\n", "As it happens, `pandas` has a convenient method called `cut` for grouping data in this way:" ] }, { "cell_type": "code", "execution_count": 99, "id": "0805c352", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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agecategories
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\n", "
" ], "text/plain": [ " age categories\n", "0 60 older\n", "1 58 older\n", "2 24 adult\n", "3 26 adult\n", "4 34 adult\n", "5 42 older\n", "6 31 adult\n", "7 30 adult\n", "8 33 adult\n", "9 2 young\n", "10 9 young" ] }, "execution_count": 99, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df['categories'] = pd.cut(x = df['age'], bins = [0,20,40,60], labels = ['young', 'adult', 'older'])\n", "df" ] }, { "cell_type": "markdown", "id": "fd9793de", "metadata": {}, "source": [ "Note that there are four numbers in the `bins` argument, but only three labels in the `labels` argument; this is because the `cut()` function requires that you specify the *edges* of the categories rather than the mid-points. In any case, now that we've done this, we can use the `cut()` function to assign each observation to one of these three categories. There are several arguments to the `cut()` function, but the three that we need to care about are:\n", "\n", "- `x`. The variable that needs to be categorised. \n", "- `bins`. This is either a vector containing the locations of the breaks separating the categories, or a number indicating how many categories you want.\n", "- `labels`. The labels attached to the categories. This is optional: if you don't specify this Python will attach a boring label showing the range associated with each category." ] }, { "cell_type": "markdown", "id": "b1ac2d9a", "metadata": {}, "source": [ "In the example above, I made all the decisions myself, but if you want to you can delegate a lot of the choices to Python. For instance, if you want you can specify the *number* of categories you want, rather than giving explicit ranges for them, and you can allow Python to come up with some labels for the categories. To give you a sense of how this works, have a look at the following example:" ] }, { "cell_type": "code", "execution_count": 100, "id": "22a99d74", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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agecategories
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" ], "text/plain": [ " age categories\n", "0 60 (40.667, 60.0]\n", "1 58 (40.667, 60.0]\n", "2 24 (21.333, 40.667]\n", "3 26 (21.333, 40.667]\n", "4 34 (21.333, 40.667]\n", "5 42 (40.667, 60.0]\n", "6 31 (21.333, 40.667]\n", "7 30 (21.333, 40.667]\n", "8 33 (21.333, 40.667]\n", "9 2 (1.942, 21.333]\n", "10 9 (1.942, 21.333]" ] }, "execution_count": 100, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df['categories'] = pd.cut(x = df['age'], bins = 3)\n", "df" ] }, { "cell_type": "markdown", "id": "b7a0564b", "metadata": {}, "source": [ "With this command, I've asked for three categories, but let Python make the choices for where the boundaries should be. All of the important information can be extracted by looking at the tabulated data:" ] }, { "cell_type": "code", "execution_count": 101, "id": "29ab763f", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
col_0count
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" ], "text/plain": [ "col_0 count\n", "categories \n", "(1.942, 21.333] 2\n", "(21.333, 40.667] 6\n", "(40.667, 60.0] 3" ] }, "execution_count": 101, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pd.crosstab(index = df[\"categories\"], columns = \"count\")" ] }, { "cell_type": "markdown", "id": "b5d157e8", "metadata": {}, "source": [ "This output takes a little bit of interpretation, but it's not complicated. What Python has done is determined that the lowest age category should run from 1.94 years up to 21.3 years, the second category should run from 21.3 years to 40.7 years, and so on. These labels are not nearly as easy on the eyes as our \"young, adult, and older\" categories, so it's usually a good idea to specify your own, meaningful labels to the categories.\n", "\n", "Before moving on, I should take a moment to talk a little about the mechanics of the `cut()` function. Notice that Python has tried to divide the `age` variable into three roughly equal sized bins. Unless you specify the particular breaks you want, that's what it will do. But suppose you want to divide the `age` variable into three categories of different size, but with approximately identical numbers of people. How would you do that? Well, if that's the case, then what you want to do is have the breaks correspond to the 0th, 33rd, 66th and 100th percentiles of the data. One way to do this would be to calculate those values using the `np.quantile()` function and then use those quantiles as input to the `cut()` function. That's pretty easy to do, but it does take a couple of lines to type. So instead, the `pandas` library has a function called `qCut()` that does exactly this:" ] }, { "cell_type": "code", "execution_count": 102, "id": "744277f7", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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agecategories
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" ], "text/plain": [ " age categories\n", "0 60 (33.6, 60.0]\n", "1 58 (33.6, 60.0]\n", "2 24 (1.999, 27.2]\n", "3 26 (1.999, 27.2]\n", "4 34 (33.6, 60.0]\n", "5 42 (33.6, 60.0]\n", "6 31 (27.2, 33.6]\n", "7 30 (27.2, 33.6]\n", "8 33 (27.2, 33.6]\n", "9 2 (1.999, 27.2]\n", "10 9 (1.999, 27.2]" ] }, "execution_count": 102, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df['categories'] = pd.qcut(x = df['age'], q = [0, .33, .66, 1] )\n", "df" ] }, { "cell_type": "markdown", "id": "4b81b7ed", "metadata": {}, "source": [ "Notice the difference in the boundaries that the `qcut()` method selects. The first and third categories now span an age range of about 25 years each, whereas the middle category has shrunk to a span of only 6 years. There are some situations where this is genuinely what you want (that's why I wrote the function!), but in general you should be careful. Usually the numeric variable that you're trying to cut into categories is already expressed in meaningful units (i.e., it's interval scale), but if you cut it into unequal bin sizes then it's often very difficult to attach meaningful interpretations to the resulting categories. \n", "\n", "More generally, regardless of whether you're using the original `cut()` method or the `qcut()` version, it's important to take the time to figure out whether or not the resulting categories make any sense at all in terms of your research project. If they don't make any sense to you as meaningful categories, then any data analysis that uses those categories is likely to be just as meaningless. More generally, in practice I've noticed that people have a very strong desire to carve their (continuous and messy) data into a few (discrete and simple) categories; and then run analysis using the categorised data instead of the original one.[^note2] I wouldn't go so far as to say that this is an inherently bad idea, but it does have some fairly serious drawbacks at times, so I would advise some caution if you are thinking about doing it. \n", "\n", "[^note2]: If you've read further into the book, and are re-reading this section, then a good example of this would be someone choosing to do an ANOVA using age categories as the grouping variable, instead of running a regression using `age` as a predictor. There are sometimes good reasons for do this: for instance, if the relationship between `age` and your outcome variable is highly non-linear, and you aren't comfortable with trying to run non-linear regression! However, unless you really do have a good rationale for doing this, it's best not to. It tends to introduce all sorts of other problems (e.g., the data will probably violate the normality assumption), and you can lose a lot of power." ] }, { "cell_type": "markdown", "id": "47ad4274", "metadata": {}, "source": [ "(mathfunc)=\n", "\n", "## A few more mathematical functions and operations" ] }, { "cell_type": "markdown", "id": "a5cb3ef7", "metadata": {}, "source": [ "(rounding)=\n", "### Rounding\n", "\n", "As you might expect, Python can round numbers with decimals to whole numbers. As you might also have come to expect by now, Python will not necessarily do it the way you expect! Take a look below:" ] }, { "cell_type": "code", "execution_count": 157, "id": "e3b4bac9", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "4\n", "4\n", "5\n" ] } ], "source": [ "print(round(4.4))\n", "print(round(4.5))\n", "print(round(4.51))" ] }, { "cell_type": "markdown", "id": "b5f6b989", "metadata": {}, "source": [ "Python rounds 4.4 to 4 and 4.51 to 5. However, notice what it does with 4.5: it rounds to this to 4 as well. This is just one of many ways to round numbers, so Python is not doing anything wrong; you just have to be aware that this is what it will do.\n", "\n", "You can add a second argument to indicate the number of decimal places desired:" ] }, { "cell_type": "code", "execution_count": 158, "id": "ef1d4372", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3.7773" ] }, "execution_count": 158, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a = 3.777298672345782376823578287355\n", "round (a, 5)" ] }, { "cell_type": "markdown", "id": "a350393e", "metadata": {}, "source": [ "### Row-wise means of a dataframe" ] }, { "cell_type": "code", "execution_count": null, "id": "a152141e", "metadata": {}, "outputs": [], "source": [ "df['mean'] = df.mean(axis = 1)" ] }, { "cell_type": "code", "execution_count": null, "id": "8b3782d7", "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "id": "62518b4c", "metadata": {}, "source": [ "(slicing)=\n", "## Slicing and dicing your data" ] }, { "cell_type": "markdown", "id": "a8240464", "metadata": {}, "source": [ "### slicing a list:\n", "\n", "\"Slicing\" is the most common way to get chunks of data out of a list. In theory, it is quite simple. The syntax is as follows: listname[first:last:step_size]. The first number in the square brackets is where the slice should start, the second is where it ends, and the third indicates the size of any jumps that should be made (e.g. take every 2nd or every 3rd item). If you put nothing in this third argument, it defaults to one.\n", "\n", "However, counting can be very tricky in Python! In my experience, even when you think you understand how it works, you can still get tripped up. The best thing to do is just take a list of data and start slicing it as many ways as you can think of, until you get a feeling for how it works." ] }, { "cell_type": "code", "execution_count": 138, "id": "77009a1e", "metadata": {}, "outputs": [], "source": [ "age = [17, 19, 21, 37, 18, 19, 47, 18, 19]" ] }, { "cell_type": "markdown", "id": "a1f882e0", "metadata": {}, "source": [ "Return the first four elements in the list:" ] }, { "cell_type": "code", "execution_count": 143, "id": "2df393bf", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[17, 19, 21, 37]" ] }, "execution_count": 143, "metadata": {}, "output_type": "execute_result" } ], "source": [ "age[0:4]" ] }, { "cell_type": "markdown", "id": "7db922a7", "metadata": {}, "source": [ "Return the last four elements in the list" ] }, { "cell_type": "code", "execution_count": 149, "id": "ba7b13ae", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[19, 47, 18, 19]" ] }, "execution_count": 149, "metadata": {}, "output_type": "execute_result" } ], "source": [ "age[-4:]" ] }, { "cell_type": "markdown", "id": "6ed0a329", "metadata": {}, "source": [ "A different way to get the last four elements in the list:" ] }, { "cell_type": "code", "execution_count": 144, "id": "cc4bfbcd", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[19, 47, 18, 19]" ] }, "execution_count": 144, "metadata": {}, "output_type": "execute_result" } ], "source": [ "age[5:len(age)]" ] }, { "cell_type": "markdown", "id": "9bd6ff5c", "metadata": {}, "source": [ "Return every second element, starting with the first" ] }, { "cell_type": "code", "execution_count": 145, "id": "8e730e99", "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/plain": [ "[17, 21, 18, 47, 19]" ] }, "execution_count": 145, "metadata": {}, "output_type": "execute_result" } ], "source": [ "age[::2]" ] }, { "cell_type": "markdown", "id": "b4c6a924", "metadata": {}, "source": [ "Return every second element, starting with second" ] }, { "cell_type": "code", "execution_count": 150, "id": "6bb873e3", "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/plain": [ "[19, 37, 19, 18]" ] }, "execution_count": 150, "metadata": {}, "output_type": "execute_result" } ], "source": [ "age[1::2]" ] }, { "cell_type": "markdown", "id": "9181027c", "metadata": {}, "source": [ "### \"popping\" items out of a list\n", "\n", "Sometimes it is useful to remove an item from a list. In Python, lists have a property called `.pop()` that allow you to do just that. Nothing against spinach, but there is one item in the list of fruits below that isn't like the others. So let's `.pop` it out:" ] }, { "cell_type": "code", "execution_count": 166, "id": "cbed1992", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "['apples', 'pears', 'bananas', 'strawberries', 'grapes']" ] }, "execution_count": 166, "metadata": {}, "output_type": "execute_result" } ], "source": [ "fruits = ['apples', 'pears', 'bananas', 'spinach', 'strawberries', 'grapes']\n", "fruits.pop(3)\n", "fruits" ] }, { "cell_type": "markdown", "id": "ca26cfcd", "metadata": {}, "source": [ "(subsets)=\n", "## Extracting a subset of a dataframe" ] }, { "cell_type": "code", "execution_count": null, "id": "f2bb3f14", "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": 66, "id": "e02441e3", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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agescorertgroup
017123.552test
119101.624test
221116.431test
337157.132test
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519144.662control
647253.634control
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" ], "text/plain": [ " age score rt group\n", "0 17 12 3.552 test\n", "1 19 10 1.624 test\n", "2 21 11 6.431 test\n", "3 37 15 7.132 test\n", "4 18 16 2.925 test\n", "5 19 14 4.662 control\n", "6 47 25 3.634 control\n", "7 18 21 3.635 control\n", "8 19 29 5.234 control" ] }, "execution_count": 66, "metadata": {}, "output_type": "execute_result" } ], "source": [ "age = [17, 19, 21, 37, 18, 19, 47, 18, 19]\n", "score = [12, 10, 11, 15, 16, 14, 25, 21, 29]\n", "rt = [3.552, 1.624, 6.431, 7.132, 2.925, 4.662, 3.634, 3.635, 5.234]\n", "group = [\"test\", \"test\", \"test\", \"test\", \"test\", \"control\", \"control\", \"control\", \"control\"]\n", "\n", "import pandas as pd\n", "\n", "df = pd.DataFrame(\n", " {'age': age,\n", " 'score': score,\n", " 'rt': rt,\n", " 'group': group\n", " })\n", "\n", "df" ] }, { "cell_type": "code", "execution_count": 64, "id": "bcc24417", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
agescorertgroup
017123.552test
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337157.132test
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" ], "text/plain": [ " age score rt group\n", "0 17 12 3.552 test\n", "1 19 10 1.624 test\n", "2 21 11 6.431 test\n", "3 37 15 7.132 test\n", "4 18 16 2.925 test" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_test = df[df['group'] == 'test']\n", "df_test" ] }, { "cell_type": "code", "execution_count": 65, "id": "f1cb4c2c", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
agescorertgroup
519144.662control
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" ], "text/plain": [ " age score rt group\n", "5 19 14 4.662 control\n", "6 47 25 3.634 control\n", "7 18 21 3.635 control\n", "8 19 29 5.234 control" ] }, "execution_count": 65, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_control = df[df['group'] == 'control']\n", "df_control" ] }, { "cell_type": "code", "execution_count": 69, "id": "c11c3a6f", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
agescorertgroup
337157.132test
647253.634control
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" ], "text/plain": [ " age score rt group\n", "3 37 15 7.132 test\n", "6 47 25 3.634 control" ] }, "execution_count": 69, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_old = df[df['age']> 21] \n", "df_old" ] }, { "cell_type": "code", "execution_count": null, "id": "6bb077de", "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": 71, "id": "57b57f0b", "metadata": { "scrolled": false }, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
agescorertgroup
119101.624test
418162.925test
519144.662control
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\n", "
" ], "text/plain": [ " age score rt group\n", "1 19 10 1.624 test\n", "4 18 16 2.925 test\n", "5 19 14 4.662 control\n", "7 18 21 3.635 control\n", "8 19 29 5.234 control" ] }, "execution_count": 71, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_youngish = df[(df['age'] < 21 ) & (df['age'] > 17)]\n", "df_youngish" ] }, { "cell_type": "code", "execution_count": 73, "id": "6bd2ace6", "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
agescorertgroup
337157.132test
647253.634control
\n", "
" ], "text/plain": [ " age score rt group\n", "3 37 15 7.132 test\n", "6 47 25 3.634 control" ] }, "execution_count": 73, "metadata": {}, "output_type": "execute_result" } ], "source": [ "old_and_slow = df[(df['age'] > 21) & (df['rt'] > 3)]\n", "old_and_slow" ] }, { "cell_type": "code", "execution_count": 74, "id": "ecab448b", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
agescorertgroup
647253.634control
\n", "
" ], "text/plain": [ " age score rt group\n", "6 47 25 3.634 control" ] }, "execution_count": 74, "metadata": {}, "output_type": "execute_result" } ], "source": [ "old_and_slow_control = df[(df['age'] > 21) & (df['rt'] > 3) & (df['group'] == 'control')]\n", "old_and_slow_control" ] }, { "cell_type": "markdown", "id": "1140d164", "metadata": {}, "source": [ "(manipulations)=\n", "## Sorting, flipping, and merging dataframes" ] }, { "cell_type": "markdown", "id": "4ef43266", "metadata": {}, "source": [ "### Sorting dataframes" ] }, { "cell_type": "code", "execution_count": 86, "id": "cb293c4e", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
agescorertgroup
017123.552test
119101.624test
221116.431test
337157.132test
418162.925test
519144.662control
647253.634control
718213.635control
819295.234control
\n", "
" ], "text/plain": [ " age score rt group\n", "0 17 12 3.552 test\n", "1 19 10 1.624 test\n", "2 21 11 6.431 test\n", "3 37 15 7.132 test\n", "4 18 16 2.925 test\n", "5 19 14 4.662 control\n", "6 47 25 3.634 control\n", "7 18 21 3.635 control\n", "8 19 29 5.234 control" ] }, "execution_count": 86, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "\n", "age = [17, 19, 21, 37, 18, 19, 47, 18, 19]\n", "score = [12, 10, 11, 15, 16, 14, 25, 21, 29]\n", "rt = [3.552, 1.624, 6.431, 7.132, 2.925, 4.662, 3.634, 3.635, 5.234]\n", "group = [\"test\", \"test\", \"test\", \"test\", \"test\", \"control\", \"control\", \"control\", \"control\"]\n", "\n", "df = pd.DataFrame(\n", " {'age': age,\n", " 'score': score,\n", " 'rt': rt,\n", " 'group': group\n", " })\n", "\n", "df" ] }, { "cell_type": "markdown", "id": "fe899ac5", "metadata": {}, "source": [ "#### Sorting by a column" ] }, { "cell_type": "code", "execution_count": 89, "id": "a6a26168", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
agescorertgroup
017123.552test
418162.925test
718213.635control
119101.624test
519144.662control
819295.234control
221116.431test
337157.132test
647253.634control
\n", "
" ], "text/plain": [ " age score rt group\n", "0 17 12 3.552 test\n", "4 18 16 2.925 test\n", "7 18 21 3.635 control\n", "1 19 10 1.624 test\n", "5 19 14 4.662 control\n", "8 19 29 5.234 control\n", "2 21 11 6.431 test\n", "3 37 15 7.132 test\n", "6 47 25 3.634 control" ] }, "execution_count": 89, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_sorted = df.sort_values(by=['age'])\n", "df_sorted" ] }, { "cell_type": "markdown", "id": "d1e84966", "metadata": {}, "source": [ "#### Sorting from largest to smallest" ] }, { "cell_type": "code", "execution_count": 92, "id": "20f35a44", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
agescorertgroup
647253.634control
337157.132test
221116.431test
119101.624test
519144.662control
819295.234control
418162.925test
718213.635control
017123.552test
\n", "
" ], "text/plain": [ " age score rt group\n", "6 47 25 3.634 control\n", "3 37 15 7.132 test\n", "2 21 11 6.431 test\n", "1 19 10 1.624 test\n", "5 19 14 4.662 control\n", "8 19 29 5.234 control\n", "4 18 16 2.925 test\n", "7 18 21 3.635 control\n", "0 17 12 3.552 test" ] }, "execution_count": 92, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_sorted = df.sort_values(by=['age'], ascending = False)\n", "df_sorted" ] }, { "cell_type": "markdown", "id": "63387a1a", "metadata": {}, "source": [ "#### Sorting by multiple columns" ] }, { "cell_type": "code", "execution_count": 93, "id": "8a7a654f", "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
agescorertgroup
017123.552test
418162.925test
718213.635control
119101.624test
519144.662control
819295.234control
221116.431test
337157.132test
647253.634control
\n", "
" ], "text/plain": [ " age score rt group\n", "0 17 12 3.552 test\n", "4 18 16 2.925 test\n", "7 18 21 3.635 control\n", "1 19 10 1.624 test\n", "5 19 14 4.662 control\n", "8 19 29 5.234 control\n", "2 21 11 6.431 test\n", "3 37 15 7.132 test\n", "6 47 25 3.634 control" ] }, "execution_count": 93, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_sorted = df.sort_values(by=['age', 'score'])\n", "df_sorted" ] }, { "cell_type": "markdown", "id": "f37604cd", "metadata": {}, "source": [ "### Flipping (transposing) a dataframe" ] }, { "cell_type": "code", "execution_count": 104, "id": "1f570292", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
time.1time.2time.3time.4time.5
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210020202020
3100100100100100
\n", "
" ], "text/plain": [ " time.1 time.2 time.3 time.4 time.5\n", "0 100 80 0 0 0\n", "1 100 100 90 30 10\n", "2 100 20 20 20 20\n", "3 100 100 100 100 100" ] }, "execution_count": 104, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "df_cakes = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/cakes.csv\")\n", "df_cakes" ] }, { "cell_type": "code", "execution_count": 105, "id": "4ec6c432", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
0123
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" ], "text/plain": [ " 0 1 2 3\n", "time.1 100 100 100 100\n", "time.2 80 100 20 100\n", "time.3 0 90 20 100\n", "time.4 0 30 20 100\n", "time.5 0 10 20 100" ] }, "execution_count": 105, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_cakes_flipped = df_cakes.transpose()\n", "df_cakes_flipped" ] }, { "cell_type": "markdown", "id": "7c21aa4a", "metadata": {}, "source": [ "An important point to recognise is that transposing a data frame is not always a sensible thing to do: in fact, I’d go so far as to argue that it’s usually not sensible. It depends a lot on whether the “cases” from your original data frame would make sense as variables, and to think of each of your original “variables” as cases. Still, there are some situations where it is useful to flip your data frame, so it’s nice to know that you can do it. A lot of statistical tools make the assumption that the rows of your data frame (or matrix) correspond to observations, and the columns correspond to the variables. That’s not unreasonable, of course, since that is a pretty standard convention. However, think about our cakes example here. This is a situation where you might want do an analysis of the different cakes (i.e. cakes as variables, time points as cases), but equally you might want to do an analysis where you think of the times as being the things of interest (i.e., times as variables, cakes as cases). If so, then it’s useful to know how to flip a data frame around." ] }, { "cell_type": "markdown", "id": "296b8f79", "metadata": {}, "source": [ "### Joining dataframes\n", "\n", "Maybe we got our cake data in two batches (haha, sorry!) [^notesorry]. First we recorded times 1-3, and then later recorded times 4-5 in a separate dataframe, so that they looked like this: \n", "\n", "[^notesorry]: Not sorry." ] }, { "cell_type": "code", "execution_count": null, "id": "a78bc987", "metadata": {}, "outputs": [], "source": [ "first_three = df_cakes.loc[:, ['time.1', 'time.2', 'time.3']] \n", "last_two = df_cakes.loc[:, ['time.4', 'time.5']]" ] }, { "cell_type": "code", "execution_count": 121, "id": "28115704", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
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" ], "text/plain": [ " time.4 time.5\n", "0 0 0\n", "1 30 10\n", "2 20 20\n", "3 100 100" ] }, "execution_count": 122, "metadata": {}, "output_type": "execute_result" } ], "source": [ "last_two" ] }, { "cell_type": "markdown", "id": "bd438a34", "metadata": {}, "source": [ "What we want to do is `.join()` those suckers back together:" ] }, { "cell_type": "code", "execution_count": 123, "id": "330fcfa9", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
time.1time.2time.3time.4time.5
010080000
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210020202020
3100100100100100
\n", "
" ], "text/plain": [ " time.1 time.2 time.3 time.4 time.5\n", "0 100 80 0 0 0\n", "1 100 100 90 30 10\n", "2 100 20 20 20 20\n", "3 100 100 100 100 100" ] }, "execution_count": 123, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_joined = first_three.join(last_two)\n", "df_joined" ] }, { "cell_type": "markdown", "id": "b2d537b0", "metadata": {}, "source": [ "### Concatenating dataframes\n", "\n", "A similar situation might have occured with our Very Exciting Psychology Experiment[^notetrademark]. We saved the data for the test group in one dataframe, and the data for the control group in a different dataframe, so that they look like this:\n", "\n", "[^notetrademark]: Registered trademark, patent pending." ] }, { "cell_type": "code", "execution_count": 125, "id": "ea145e96", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
agescorertgroup
017123.552test
119101.624test
221116.431test
337157.132test
418162.925test
\n", "
" ], "text/plain": [ " age score rt group\n", "0 17 12 3.552 test\n", "1 19 10 1.624 test\n", "2 21 11 6.431 test\n", "3 37 15 7.132 test\n", "4 18 16 2.925 test" ] }, "execution_count": 125, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_test = df[df['group'] == 'test']\n", "df_control = df[df['group'] == 'control']\n", "df_test" ] }, { "cell_type": "code", "execution_count": 124, "id": "9e25525b", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
agescorertgroup
519144.662control
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819295.234control
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" ], "text/plain": [ " age score rt group\n", "5 19 14 4.662 control\n", "6 47 25 3.634 control\n", "7 18 21 3.635 control\n", "8 19 29 5.234 control" ] }, "execution_count": 124, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_control" ] }, { "cell_type": "markdown", "id": "66f0b78f", "metadata": {}, "source": [ "Just like the cake data from before, we want to squish these data together into one dataframe, but unlike the cake data, we want to put one dataframe _on top_ of the other one. After all, the columns are all the same, it's just that the data are for two different groups. Luckily, we _did_ remember to add a column to each dataframe recording which group the data were from, so all we need to do is stack these two datarames together. This is called _concatenating_ the data, and we can accomplish it with `.concat()`:" ] }, { "cell_type": "code", "execution_count": 126, "id": "842a621c", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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agescorertgroup
017123.552test
119101.624test
221116.431test
337157.132test
418162.925test
519144.662control
647253.634control
718213.635control
819295.234control
\n", "
" ], "text/plain": [ " age score rt group\n", "0 17 12 3.552 test\n", "1 19 10 1.624 test\n", "2 21 11 6.431 test\n", "3 37 15 7.132 test\n", "4 18 16 2.925 test\n", "5 19 14 4.662 control\n", "6 47 25 3.634 control\n", "7 18 21 3.635 control\n", "8 19 29 5.234 control" ] }, "execution_count": 126, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_concatenated = pd.concat([df_test, df_control])\n", "df_concatenated" ] }, { "cell_type": "markdown", "id": "a893579f", "metadata": {}, "source": [ "(reshaping)=\n", "## Reshaping a dataframe" ] }, { "cell_type": "markdown", "id": "8957e8b7", "metadata": {}, "source": [ "One of the most annoying tasks that you need to undertake on a regular basis is that of reshaping a data frame. Framed in the most general way, reshaping the data means taking the data in whatever format it’s given to you, and converting it to the format you need it. Of course, if we’re going to characterise the problem that broadly, then about half of this chapter can probably be thought of as a kind of reshaping. So we’re going to have to narrow things down a little bit. I'm going to begin with two of the most common ways we need to reshape data: moving from wide-format data to long-format data, and moving from long-format data to wide-format data." ] }, { "cell_type": "markdown", "id": "522acccd", "metadata": {}, "source": [ "### Reshaping from wide to long format\n", "\n", "The most common format in which you might obtain data is as a “case by variable” layout, commonly\n", "known as the wide form of the data. To get a sense of what I’m talking about, consider an experiment in which we are interested in the different effects that alcohol and and caffeine have on people’s working memory capacity (WMC). We recruit 10 participants, and measure their WMC under three different conditions: a “no drug” condition, in which they are not under the influence of either caffeine or alcohol, a “caffeine” condition, in which they are under the inflence of caffeine, and an “alcohol” condition, in which... well, you can probably guess. Ideally, I suppose, there would be a fourth condition in which both drugs are administered, but for the sake of simplicity let’s ignore that. The `drugs` data frame gives you a sense of what kind of data you might observe in an experiment like this:" ] }, { "cell_type": "code", "execution_count": 167, "id": "6a4bca1a", "metadata": {}, "outputs": [], "source": [ "import pandas as pd\n", "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/drugs1.csv\")" ] }, { "cell_type": "code", "execution_count": 168, "id": "e4301889", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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89female5.26.27.2
910female6.27.47.8
\n", "
" ], "text/plain": [ " id gender alcohol caffeine no.drug\n", "0 1 female 3.7 3.7 3.9\n", "1 2 female 6.4 7.3 7.9\n", "2 3 female 4.6 7.4 7.3\n", "3 4 male 6.4 7.8 8.2\n", "4 5 female 4.9 5.2 7.0\n", "5 6 male 5.4 6.6 7.2\n", "6 7 male 7.9 7.9 8.9\n", "7 8 male 4.1 5.9 4.5\n", "8 9 female 5.2 6.2 7.2\n", "9 10 female 6.2 7.4 7.8" ] }, "execution_count": 168, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df" ] }, { "cell_type": "markdown", "id": "b4665207", "metadata": {}, "source": [ "This is a data set in “wide form”, in which each participant corresponds to a single row. We have two variables that are characteristics of the subject (i.e., their id number and their gender) and three variables that refer their performance in one of the three testing conditions (alcohol, caffeine or no drug). Because all of the testing conditions (i.e., the three drug types) are applied to all participants, drug type is an example of a **within-subject** factor. This is a case that can be fairly easily handled by `pandas` `.melt()` method:" ] }, { "cell_type": "code", "execution_count": 169, "id": "d9dc9d16", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
idgendervariablevalue
01femalealcohol3.7
12femalealcohol6.4
23femalealcohol4.6
34malealcohol6.4
45femalealcohol4.9
56malealcohol5.4
67malealcohol7.9
78malealcohol4.1
89femalealcohol5.2
910femalealcohol6.2
101femalecaffeine3.7
112femalecaffeine7.3
123femalecaffeine7.4
\n", "
" ], "text/plain": [ " id gender variable value\n", "0 1 female alcohol 3.7\n", "1 2 female alcohol 6.4\n", "2 3 female alcohol 4.6\n", "3 4 male alcohol 6.4\n", "4 5 female alcohol 4.9\n", "5 6 male alcohol 5.4\n", "6 7 male alcohol 7.9\n", "7 8 male alcohol 4.1\n", "8 9 female alcohol 5.2\n", "9 10 female alcohol 6.2\n", "10 1 female caffeine 3.7\n", "11 2 female caffeine 7.3\n", "12 3 female caffeine 7.4" ] }, "execution_count": 169, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_long = pd.melt(df, id_vars=[ 'id', 'gender'])\n", "df_long.head(13)" ] }, { "cell_type": "markdown", "id": "1828cf78", "metadata": {}, "source": [ "Sometimes, though, we want to do something a little more complex. For example, what if we didn't just measure working memory capacity (WMC), but we also measured their reaction time (RT). These data might look something like this:" ] }, { "cell_type": "code", "execution_count": 170, "id": "7b061b32", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
idgenderWMC_alcoholWMC_caffeineWMC_no.drugRT_alcoholRT_caffeineRT_no.drug
01female3.73.73.9488236371
12female6.47.37.9607376349
23female4.67.47.3643226412
34male6.47.88.2684206252
45female4.95.27.0593262439
\n", "
" ], "text/plain": [ " id gender WMC_alcohol WMC_caffeine WMC_no.drug RT_alcohol \\\n", "0 1 female 3.7 3.7 3.9 488 \n", "1 2 female 6.4 7.3 7.9 607 \n", "2 3 female 4.6 7.4 7.3 643 \n", "3 4 male 6.4 7.8 8.2 684 \n", "4 5 female 4.9 5.2 7.0 593 \n", "\n", " RT_caffeine RT_no.drug \n", "0 236 371 \n", "1 376 349 \n", "2 226 412 \n", "3 206 252 \n", "4 262 439 " ] }, "execution_count": 170, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/drugs.csv\")\n", "df.head()" ] }, { "cell_type": "markdown", "id": "f9fa4d81", "metadata": {}, "source": [ "We want to reshape the data from wide format to long format, but we want to group the WMC data in one column, and the RT data in another column. `pandas` has a solution for this as well, and it's called `wide_to_long()`. I wonder why they picked that name? \n", "\n", "`wide_to_long` is pretty powerful, but it has some niggling details that you have to get right, otherwise you will find yourself sitting sadly, wondering about your life choices. Basically, to get `wide_to_long` to do your bidding, you need to specify _at least_ four things: \n", "\n", "1. the dataframe\n", "2. some \"stubnames\"\n", "3. one or more \"id variables\"\n", "4. the name of the \"sub-observation variable\". \n", "\n", "Ugh. Plus, you might need some more things too. Let's take a look. First I'll show you the command that we need, and then I'll go through what it all means." ] }, { "cell_type": "code", "execution_count": 6, "id": "7d210e5e", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
WMCRT
idgenderdrug
1femalealcohol3.7488
caffeine3.7236
no.drug3.9371
2femalealcohol6.4607
caffeine7.3376
no.drug7.9349
3femalealcohol4.6643
caffeine7.4226
no.drug7.3412
4malealcohol6.4684
\n", "
" ], "text/plain": [ " WMC RT\n", "id gender drug \n", "1 female alcohol 3.7 488\n", " caffeine 3.7 236\n", " no.drug 3.9 371\n", "2 female alcohol 6.4 607\n", " caffeine 7.3 376\n", " no.drug 7.9 349\n", "3 female alcohol 4.6 643\n", " caffeine 7.4 226\n", " no.drug 7.3 412\n", "4 male alcohol 6.4 684" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "df_long = pd.wide_to_long(df, stubnames = ['WMC','RT'], \n", " i=['id','gender'], \n", " j='drug', \n", " sep = '_', \n", " suffix = '.+')\n", "df_long.head(10)" ] }, { "cell_type": "markdown", "id": "1d78ef7b", "metadata": {}, "source": [ "First of all we need the name of the dataframe in wide format that we want to convert to long format. That's easy enough. Next, we need the `stubnames`. If you look at the column headers in the original, wide data, you will notice that each of the data columns has a prefix: either WMC or RT, to indicate whether the column shows working memory capacity data or reaction time data. These prefixes are the \"stubs\", and `wide_to_long` will use them to group the data. Your columns _have to have some kind of grouping prefix_ for `wide_to_long` to work!\n", "\n", "Next, we have the argument `i`. This is where we put the id variables. These variable remain consistent for each participant throughout the experiment. Participant 1's RT may change as a result of intaking caffeine or alchol, but presumably her gender does not.\n", "\n", "Next up, we have the argument `j`. Here we put the name that of the column where `wide_to_long` will put the _labels_ that followed the prefix. In our case, these labels are \"alchol\", \"caffeine\", and \"no.drug\", so a logical choice for `j` might be \"drug\". \n", "\n", "Now, depending on our data, we might be able to stop here. These are the four required arguments to `wide_to_long`. But in this case, we need two more variables. \n", "\n", "The first of these is `sep`. This is where we can indicate if there is some character that has been used to separate the prefix from the label in the original column names. We used an underscore `_` to separate e.g. \"WMC\" from \"alchol\", so we can specify this here. Now, technically, we _could_ have just used \"WMC_\" as our stubname instead of \"WMC\", and left off the `sep` argument, but in that case our new columns would be called \"WMC_\" and \"RT_\", and that wouldn't look quite as prof, now would it?\n", "\n", "The second extra argument we need is `suffix`. By default, `wide_to_long` expects the labels to be numbers. I can't really imagine why the `pandas` authors would build in this assumption, but there it is. I'm sure they had their reasons. In any case, our labels are \"alcohol\", \"caffeine\", and \"no.drug\" are letters, not numbers, so we have to tell `pandas` to look for non-numeric labels. In the `suffix` argument we can enter a regular expression that searches for the pattern that we are interested in. Regular expressions deserve (and have received) entire books devoted entirely to their explanation, so we won't go further into this here. Suffice it to say that entering `.+` in the `suffix` argument will allow us to search for whatever might come after our separator `_`.\n", "\n", "There is one final step before our data is truly usable. Because `pandas` tries to preserve the idex information from the original dataframe, we end up with a somewhat odd-looking structure:" ] }, { "cell_type": "code", "execution_count": 7, "id": "95696b68", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
WMCRT
idgenderdrug
1femalealcohol3.7488
caffeine3.7236
no.drug3.9371
2femalealcohol6.4607
\n", "
" ], "text/plain": [ " WMC RT\n", "id gender drug \n", "1 female alcohol 3.7 488\n", " caffeine 3.7 236\n", " no.drug 3.9 371\n", "2 female alcohol 6.4 607" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_long.head(4)" ] }, { "cell_type": "markdown", "id": "a7b48c07", "metadata": {}, "source": [ "This is called a MultiIndex. It is quite clear for a human to read, but it is cumbersome if we want to do further calculations with our data, which we probably do. The solution is to thow out the old index information, and reset the index, like so:" ] }, { "cell_type": "code", "execution_count": 8, "id": "92c510c4", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
idgenderdrugWMCRT
01femalealcohol3.7488
11femalecaffeine3.7236
21femaleno.drug3.9371
32femalealcohol6.4607
42femalecaffeine7.3376
52femaleno.drug7.9349
63femalealcohol4.6643
73femalecaffeine7.4226
83femaleno.drug7.3412
94malealcohol6.4684
104malecaffeine7.8206
114maleno.drug8.2252
125femalealcohol4.9593
135femalecaffeine5.2262
145femaleno.drug7.0439
\n", "
" ], "text/plain": [ " id gender drug WMC RT\n", "0 1 female alcohol 3.7 488\n", "1 1 female caffeine 3.7 236\n", "2 1 female no.drug 3.9 371\n", "3 2 female alcohol 6.4 607\n", "4 2 female caffeine 7.3 376\n", "5 2 female no.drug 7.9 349\n", "6 3 female alcohol 4.6 643\n", "7 3 female caffeine 7.4 226\n", "8 3 female no.drug 7.3 412\n", "9 4 male alcohol 6.4 684\n", "10 4 male caffeine 7.8 206\n", "11 4 male no.drug 8.2 252\n", "12 5 female alcohol 4.9 593\n", "13 5 female caffeine 5.2 262\n", "14 5 female no.drug 7.0 439" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_long = df_long.reset_index()\n", "df_long.head(15)" ] }, { "cell_type": "markdown", "id": "0eeabf21", "metadata": {}, "source": [ "Ah. Now we have our wide format data in a nice long format." ] }, { "cell_type": "markdown", "id": "c80d3c9f", "metadata": {}, "source": [ "### Reshaping from long to wide format\n", "\n" ] }, { "cell_type": "markdown", "id": "cdce1d4a", "metadata": {}, "source": [ "Going the other way, from long to wide, is also not too hard, although it is still not quite as straightforward as one might wish. On the other hand it still far, far better than copy-pasting columns in Excel, which is a recipe for disaster. Trust me. I've been there.\n", "\n", "To go from long to wide, we can start with our shiny new long format dataframe `df_long` and use `.pivot` to \"swivel\" our long-format columns into a wider format. `.pivot()` takes three critical arguments: index, columns, and values.\n", "\n", "The \"index\" column keeps track of which data belongs with which: very important! In our case, we have a column called ` id` which contains a participant id-number for each participant, so we'll use that as our index. Then we know that in the wide dataframe, the right data will still go with the right participant.\n", "\n", "Next we have the \"columns\" argument. Here we can use `drug` to make new columns called \"alcohol\", \"caffeine\", and \"no.drug\". There is one more level of categorization in our data, however: we have two measurements: \"WMC\" and \"RT\". So we can use the `values` argument to gather together this information as well, so that each value ends up in the right row and column." ] }, { "cell_type": "code", "execution_count": 47, "id": "c457416a", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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genderWMCRT
drugalcoholcaffeineno.drugalcoholcaffeineno.drugalcoholcaffeineno.drug
id
1femalefemalefemale3.73.73.9488236371
2femalefemalefemale6.47.37.9607376349
3femalefemalefemale4.67.47.3643226412
4malemalemale6.47.88.2684206252
5femalefemalefemale4.95.27.0593262439
6malemalemale5.46.67.2492230464
7malemalemale7.97.98.9690259327
8malemalemale4.15.94.5486230305
9femalefemalefemale5.26.27.2686273327
10femalefemalefemale6.27.47.8645240498
\n", "
" ], "text/plain": [ " gender WMC RT \\\n", "drug alcohol caffeine no.drug alcohol caffeine no.drug alcohol caffeine \n", "id \n", "1 female female female 3.7 3.7 3.9 488 236 \n", "2 female female female 6.4 7.3 7.9 607 376 \n", "3 female female female 4.6 7.4 7.3 643 226 \n", "4 male male male 6.4 7.8 8.2 684 206 \n", "5 female female female 4.9 5.2 7.0 593 262 \n", "6 male male male 5.4 6.6 7.2 492 230 \n", "7 male male male 7.9 7.9 8.9 690 259 \n", "8 male male male 4.1 5.9 4.5 486 230 \n", "9 female female female 5.2 6.2 7.2 686 273 \n", "10 female female female 6.2 7.4 7.8 645 240 \n", "\n", " \n", "drug no.drug \n", "id \n", "1 371 \n", "2 349 \n", "3 412 \n", "4 252 \n", "5 439 \n", "6 464 \n", "7 327 \n", "8 305 \n", "9 327 \n", "10 498 " ] }, "execution_count": 47, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_wide = pd.pivot(df_long, index=['id'], columns='drug', values=['gender', 'WMC', 'RT'])\n", "df_wide" ] }, { "cell_type": "markdown", "id": "8e6ee7c0", "metadata": {}, "source": [ "And voilà! Our data have been shifted back into a format that nearly resembles how it started. There are some differences, though. Once again, `pandas` has used a \"MultiIndex\" to keep track of our data. As before, this is pretty easy to read, but less easy, perhaps, to work with. Remember, in our original data, WMC and RT were indicated as prefixes to the drug names, but this coupling was severed when we used these prefixes as \"stubs\" in the wide to long conversion. Let's put them back where the belong!\n", "\n", "One way to do this is to split our new wide dataframe into three separate dataframes: one for WMC, one for RT, and one for gender. Then we can rename the columns with the appropriate prefixes before reassembling them into one.\n", "\n", "We can start by selecting only the WMC data:" ] }, { "cell_type": "code", "execution_count": 48, "id": "8044eeb7", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
drugalcoholcaffeineno.drug
id
13.73.73.9
26.47.37.9
34.67.47.3
46.47.88.2
54.95.27.0
65.46.67.2
77.97.98.9
84.15.94.5
95.26.27.2
106.27.47.8
\n", "
" ], "text/plain": [ "drug alcohol caffeine no.drug\n", "id \n", "1 3.7 3.7 3.9\n", "2 6.4 7.3 7.9\n", "3 4.6 7.4 7.3\n", "4 6.4 7.8 8.2\n", "5 4.9 5.2 7.0\n", "6 5.4 6.6 7.2\n", "7 7.9 7.9 8.9\n", "8 4.1 5.9 4.5\n", "9 5.2 6.2 7.2\n", "10 6.2 7.4 7.8" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_WMC = df_wide['WMC']\n", "df_WMC" ] }, { "cell_type": "markdown", "id": "9f30d815", "metadata": {}, "source": [ "We can access the column names using the `.columns` method, then use a list comprehension to add the \"WMC_\" prefix again:" ] }, { "cell_type": "code", "execution_count": 49, "id": "689d8285", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " WMC_alcohol WMC_caffeine WMC_no.drug\n", "id \n", "1 3.7 3.7 3.9\n", "2 6.4 7.3 7.9\n", "3 4.6 7.4 7.3\n", "4 6.4 7.8 8.2\n", "5 4.9 5.2 7.0\n", "6 5.4 6.6 7.2\n", "7 7.9 7.9 8.9\n", "8 4.1 5.9 4.5\n", "9 5.2 6.2 7.2\n", "10 6.2 7.4 7.8" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_WMC.columns = ['WMC_' + col for col in df_WMC.columns]\n", "df_WMC " ] }, { "cell_type": "markdown", "id": "65c33ee8", "metadata": {}, "source": [ "We can do the same for the RT data..." ] }, { "cell_type": "code", "execution_count": 50, "id": "b7eb7168", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " RT_alcohol RT_caffeine RT_no.drug\n", "id \n", "1 488 236 371\n", "2 607 376 349\n", "3 643 226 412\n", "4 684 206 252\n", "5 593 262 439\n", "6 492 230 464\n", "7 690 259 327\n", "8 486 230 305\n", "9 686 273 327\n", "10 645 240 498" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_RT = df_wide['RT']\n", "df_RT.columns = ['RT_' + col for col in df_RT.columns]\n", "df_RT" ] }, { "cell_type": "markdown", "id": "59f02226", "metadata": {}, "source": [ "The gender data are a little different, because here we just have three columns of repeated data:" ] }, { "cell_type": "code", "execution_count": 53, "id": "fab080ac", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ "drug alcohol caffeine no.drug\n", "id \n", "1 female female female\n", "2 female female female\n", "3 female female female\n", "4 male male male\n", "5 female female female\n", "6 male male male\n", "7 male male male\n", "8 male male male\n", "9 female female female\n", "10 female female female" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_gender = df_wide['gender']\n", "df_gender" ] }, { "cell_type": "markdown", "id": "db2a151d", "metadata": {}, "source": [ "We really only need one of these, and since all columns contain the same information, we could choose any of them. We can use \"chained\" slicing to drill down to the \"alcohol\" column. Then, we can rename this column \"gender\":" ] }, { "cell_type": "code", "execution_count": 56, "id": "507745a7", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " gender\n", "id \n", "1 female\n", "2 female\n", "3 female\n", "4 male\n", "5 female\n", "6 male\n", "7 male\n", "8 male\n", "9 female\n", "10 female" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_gender = pd.DataFrame(df_wide['gender']['alcohol'])\n", "df_gender.columns = ['gender']\n", "df_gender" ] }, { "cell_type": "markdown", "id": "d1c0be40", "metadata": {}, "source": [ "Now that we have three dataframe (df_WMC, df_RT, and df_gender) that all share the same index (`id`) we can join them back together, using the `.join()` method. First we can tack df_WMC onto df_gender:" ] }, { "cell_type": "code", "execution_count": 58, "id": "e3e152a5", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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4male6.47.88.2
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" ], "text/plain": [ " gender WMC_alcohol WMC_caffeine WMC_no.drug\n", "id \n", "1 female 3.7 3.7 3.9\n", "2 female 6.4 7.3 7.9\n", "3 female 4.6 7.4 7.3\n", "4 male 6.4 7.8 8.2\n", "5 female 4.9 5.2 7.0\n", "6 male 5.4 6.6 7.2\n", "7 male 7.9 7.9 8.9\n", "8 male 4.1 5.9 4.5\n", "9 female 5.2 6.2 7.2\n", "10 female 6.2 7.4 7.8" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_wide = df_gender.join(df_WMC, on = 'id', how = 'left')\n", "df_wide" ] }, { "cell_type": "markdown", "id": "5d59ecce", "metadata": {}, "source": [ "Then we can attach `df_RT` to the right of the new `df_wide`:" ] }, { "cell_type": "code", "execution_count": 59, "id": "391e1de6", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " gender WMC_alcohol WMC_caffeine WMC_no.drug RT_alcohol RT_caffeine \\\n", "id \n", "1 female 3.7 3.7 3.9 488 236 \n", "2 female 6.4 7.3 7.9 607 376 \n", "3 female 4.6 7.4 7.3 643 226 \n", "4 male 6.4 7.8 8.2 684 206 \n", "5 female 4.9 5.2 7.0 593 262 \n", "6 male 5.4 6.6 7.2 492 230 \n", "7 male 7.9 7.9 8.9 690 259 \n", "8 male 4.1 5.9 4.5 486 230 \n", "9 female 5.2 6.2 7.2 686 273 \n", "10 female 6.2 7.4 7.8 645 240 \n", "\n", " RT_no.drug \n", "id \n", "1 371 \n", "2 349 \n", "3 412 \n", "4 252 \n", "5 439 \n", "6 464 \n", "7 327 \n", "8 305 \n", "9 327 \n", "10 498 " ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_wide = df_wide.join(df_RT, on = 'id', how = 'left')\n", "df_wide" ] }, { "cell_type": "markdown", "id": "7faa9603", "metadata": {}, "source": [ "As a final touch, we'll `.reset()` the index to flatten out the MultiIndex, and we are back to where we started:" ] }, { "cell_type": "code", "execution_count": 60, "id": "a1a6e561", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " id gender WMC_alcohol WMC_caffeine WMC_no.drug RT_alcohol RT_caffeine \\\n", "0 1 female 3.7 3.7 3.9 488 236 \n", "1 2 female 6.4 7.3 7.9 607 376 \n", "2 3 female 4.6 7.4 7.3 643 226 \n", "3 4 male 6.4 7.8 8.2 684 206 \n", "4 5 female 4.9 5.2 7.0 593 262 \n", "5 6 male 5.4 6.6 7.2 492 230 \n", "6 7 male 7.9 7.9 8.9 690 259 \n", "7 8 male 4.1 5.9 4.5 486 230 \n", "8 9 female 5.2 6.2 7.2 686 273 \n", "9 10 female 6.2 7.4 7.8 645 240 \n", "\n", " RT_no.drug \n", "0 371 \n", "1 349 \n", "2 412 \n", "3 252 \n", "4 439 \n", "5 464 \n", "6 327 \n", "7 305 \n", "8 327 \n", "9 498 " ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df_wide = df_wide.reset_index()\n", "df_wide" ] }, { "cell_type": "code", "execution_count": null, "id": "952b25e8", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.5" } }, "nbformat": 4, "nbformat_minor": 5 } { "cells": [ { "cell_type": "markdown", "id": "4b3a0580", "metadata": {}, "source": [ "(programming)=\n", "\n", "# Basic Programming" ] }, { "cell_type": "markdown", "id": "5ce78e2b", "metadata": {}, "source": [ ">*Machine dreams hold a special vertigo.*\n", ">\n", ">--William Gibson[^note1] \n", "\n", "[^note1]: The quote comes from *Count Zero* (1986)" ] }, { "cell_type": "markdown", "id": "f6826f95", "metadata": {}, "source": [ "Up to this point in the book I've tried hard to avoid using the word \"programming\" too much because -- at least in my experience -- it's a word that can cause a lot of fear. For one reason or another, programming (like mathematics and statistics) is often perceived by people on the \"outside\" as a black art, a magical skill that can be learned only by some kind of super-nerd. I think this is a shame. It's certainly true that advanced programming is a very specialised skill: several different skills actually, since there's quite a lot of different kinds of programming out there. However, the *basics* of programming aren't all that hard, and you can accomplish a lot of very impressive things just using those basics. \n", "\n", "With that in mind, the goal of this chapter is to discuss a few basic programming concepts and how to apply them in Python. However, before I do, I want to make one further attempt to point out just how non-magical programming really is, via one very simple observation: *you already know how to do it*. Stripped to its essentials, programming is nothing more (and nothing less) than the process of writing out a bunch of instructions that a computer can understand. To phrase this slightly differently, when you write a computer program, you need to write it in a **_programming language_** that the computer knows how to interpret. Python is one such language. Although I've been having you type all your commands at the command prompt, and all the commands in this book so far have been shown as if that's what I were doing, it's also quite possible (and as you'll see shortly, shockingly easy) to write a program using these Python commands. In other words, if this is the first time reading this book, then you're only one short chapter away from being able to legitimately claim that you can program in Python, albeit at a beginner's level. " ] }, { "cell_type": "markdown", "id": "980bcd36", "metadata": {}, "source": [ "(scripts)=\n", "\n", "## Scripts\n", "\n", "\n", "Computer programs come in quite a few different forms: the kind of program that we're mostly interested in from the perspective of everyday data analysis using Python is known as a **_script_**. The idea behind a script is that, instead of typing your commands into the Python console one at a time, instead you write them all in a text file, or in a \"notebook\", if you are using Jupyter Notebooks. Then, once you've finished writing them and saved the file, you can get Python to execute all the commands at once. In a moment I'll show you exactly how this is done, but first I'd better explain why you should care.\n", "\n", "\n", "### Why use scripts?\n", "\n", "Before discussing scripting and programming concepts in any more detail, it's worth stopping to ask why you should bother. After all, if you look at the Python commands that I've used everywhere else this book, you'll notice that they're all formatted as if I were typing them at the command line. Outside this chapter you won't actually see any scripts. Do not be fooled by this. The reason that I've done it that way is purely for pedagogical reasons. My goal in this book is to teach statistics and to teach Python. To that end, what *I've* needed to do is chop everything up into tiny little slices: each section tends to focus on one kind of statistical concept, and only a smallish number of Python functions. As much as possible, I want you to see what each function does in isolation, one command at a time. By forcing myself to write everything as if it were being typed at the command line, it imposes a kind of discipline on me: it *prevents* me from piecing together lots of commands into one big script. From a teaching (and learning) perspective I think that's the right thing to do... but from a *data analysis* perspective, it is not. When you start analysing real world data sets, you will rapidly find yourself needing to write scripts.\n", "\n", "\n", "To understand why scripts are so very useful, it may be helpful to consider the drawbacks to typing commands directly at the command prompt. The approach that we've been adopting so far, in which you type commands one at a time, and Python sits there patiently in between commands, is referred to as the **_interactive_** style. Doing your data analysis this way is rather like having a conversation ... a very annoying conversation between you and your data set, in which you and the data aren't directly speaking to each other, and so you have to rely on Python to pass messages back and forth. This approach makes a lot of sense when you're just trying out a few ideas: maybe you're trying to figure out what analyses are sensible for your data, or maybe just you're trying to remember how the various Python functions work, so you're just typing in a few commands until you get the one you want. In other words, the interactive style is very useful as a tool for exploring your data. However, it has a number of drawbacks:\n", "\n", "- *It's hard to save your work effectively*. You can save the workspace, so that later on you can load any variables you created. You can save your plots as images. And you can even save the history or copy the contents of the Python console to a file. Taken together, all these things let you create a reasonably decent record of what you did. But it does leave a lot to be desired. It seems like you ought to be able to save a single file that Python could use (in conjunction with your raw data files) and reproduce everything (or at least, everything interesting) that you did during your data analysis.\n", "\n", "- *It's annoying to have to go back to the beginning when you make a mistake.* Suppose you've just spent the last two hours typing in commands. Over the course of this time you've created lots of new variables and run lots of analyses. Then suddenly you realise that there was a nasty typo in the first command you typed, so all of your later numbers are wrong. Now you have to fix that first command, and then spend another hour or so combing through the Python history to try and recreate what you did.\n", "\n", "\n", "- *You can't leave notes for yourself*. Sure, you can scribble down some notes on a piece of paper, or even save a Word document that summarises what you did. But what you really want to be able to do is write down an English translation of your Python commands, preferably right \"next to\" the commands themselves. That way, you can look back at what you've done and actually remember what you were doing. In the simple exercises we've engaged in so far, it hasn't been all that hard to remember what you were doing or why you were doing it, but only because everything we've done could be done using only a few commands, and you've never been asked to reproduce your analysis six months after you originally did it! When your data analysis starts involving hundreds of variables, and requires quite complicated commands to work, then you really, really need to leave yourself some notes to explain your analysis to, well, yourself. \n", "\n", "- *It's nearly impossible to reuse your analyses later, or adapt them to similar problems*. Suppose that, sometime in January, you are handed a difficult data analysis problem. After working on it for ages, you figure out some really clever tricks that can be used to solve it. Then, in September, you get handed a really similar problem. You can sort of remember what you did, but not very well. You'd like to have a clean record of what you did last time, how you did it, and why you did it the way you did. Something like that would really help you solve this new problem.\n", "\n", "- *It's hard to do anything except the basics.* There's a nasty side effect of these problems. Typos are inevitable. Even the best data analyst in the world makes a lot of mistakes. So the chance that you'll be able to string together dozens of correct Python commands in a row are very small. So unless you have some way around this problem, you'll never really be able to do anything other than simple analyses. \n", "\n", "- *It's difficult to share your work other people.* Because you don't have this nice clean record of what Python commands were involved in your analysis, it's not easy to share your work with other people. Sure, you can send them all the data files you've saved, and your history and console logs, and even the little notes you wrote to yourself, but odds are pretty good that no-one else will really understand what's going on (trust me on this: I've been handed lots of random bits of output from people who've been analysing their data, and it makes very little sense unless you've got the original person who did the work sitting right next to you explaining what you're looking at)\n", "\n", "\n", "Ideally, what you'd like to be able to do is something like this... Suppose you start out with a data set `myrawdata.csv`. What you want is a single document -- let's call it `mydataanalysis.py` -- that stores all of the commands that you've used in order to do your data analysis. It would only include the commands that you want to keep for later. Then, later on, instead of typing in all those commands again, you'd just tell Python to run all of the commands that are stored in `mydataanalysis.py`. Also, in order to help you make sense of all those commands, what you'd want is the ability to add some notes or *comments* within the file, so that anyone reading the document for themselves would be able to understand what each of the commands actually does. But these comments wouldn't get in the way: when you try to get Python to run `mydataanalysis.py` it would be smart enough would recognise that these comments are for the benefit of humans, and so it would ignore them. Later on you could tweak a few of the commands inside the file (maybe in a new file called `mynewdatanalaysis.py`) so that you can adapt an old analysis to be able to handle a new problem. And you could email your friends and colleagues a copy of this file so that they can reproduce your analysis themselves. \n", "\n", "In other words, what you want is a *script*. The mechanics of exactly where you write your script, and how you run it are a bit beyond what I can cover here. You could write your script in a text file, save it with a file name that ends in .py, and then run it with a terminal command. You could use a so-called IDE (Integrated Development Environment), basically a program for writing programs. At the time I am writing this, a very popular IDE which can support many different programming languages is Visual Studio Code, but there are many other good ones out there. I'm not even going to try to list them, because I would be leaving too many good options out.\n", "\n", "Another very popular way to write scripts is to use something called Jupyter Notebooks. Jupyter Notebooks are a way to write your code in little cells, rather than in one long text document. This allows you to run individual parts of your script seperately, rather than running the whole thing at once, and this can be very useful for figuring stuff out. You can work on one cell until you get it to do what you want, then work on the next cell. Later, you may start combining cells so that you can run bigger and bigger chunks of code at once. Eventually you may find you want to simply copy all the code out of your Jupyter Notebook and paste it into a text document with a .py file extenstion, so you can just run it all at once. For learning coding, and for developing new ideas and analyses, Jupyter Notebooks are a very good option. Some programs, like Visual Studio Code, allow you to run Jupyter Notebooks inside an IDE. That's what I'm doing as I type these words.\n", "\n", "But I digress. There are 101 different ways to write, save, and run Python scripts. The key message here is to find a way that works for you, with the resources you have available to you. Let's leave the question of _where_ you will write your code behind, and turn to something more exciting: some of the basic concepts of programming." ] }, { "cell_type": "markdown", "id": "9205b9ab", "metadata": {}, "source": [ "(loops)=\n", "\n", "## Loops\n", "\n", "Oh boy, loops. For some reason, the concept of loops is often very difficult for people to grasp when they are first exposed to it. I'm not sure why, because we use loops constantly in our daily life. \n", "\n", "Imagine you are putting sugar in your tea or coffee. Let's say you like it very sweet. So you dip your spoon in the sugar bowl, and pour the sugar in the tea. Then you dip you spoon in the sugar bowl, and then you pour the sugar into the tea. And then you do it one more time. After three spoonfulls, you're done.\n", "\n", "Here's another example. You're baking a cake, and the recipe calls for four eggs. You line the eggs up on the counter, and then, one by one, you crack each one and pour the yoke and egg white into the mixing bowl, and throw away the shell. You do the same thing for each egg.\n", "\n", "Both of these are examples of loops: you perform the same action over and over, until you're done. In programming terms, the sugar-in-the-tea loop is an example of a _while_ loop. You could put as many or as few spoonfulls of sugar in, but you keep going until the tea is sweet enough for you. Put differently, _while_ the tea is _not_ sweet enough, you keep doing the sugar loop. As soon as the condition has been met (the tea is no longer not swee enough), you stop. The eggs are an example of a _for_ loop. There is a finite number of eggs (in this case four), and you keep performing the same action until you have made it through all of them. That is, _for_ each egg, you do the same thing (crack, pour, dispose) until there are no more eggs.\n", "\n", "Programming loops work exactly the same way. Let's take a look." ] }, { "cell_type": "markdown", "id": "7cb3a6ea", "metadata": {}, "source": [ "### While loops\n", "\n", "I'll stick with the tea example for demonstrating the syntax for while loops, because I'm not very imaginative. Also, I could totally go for a cup of tea right now, but I have to stay here and type this instead.\n", "\n", "Take a look at the code below. Python needs to know when to stop doing the same thing again and again, so we start by defining an end point. Here, I have declared that my tea will be sweet enough when Python has put 3 spoonfulls of sugar in: I have set the variable `sweet_enough` to 0. In the next line, I have set a starting point. Before the loop starts, there is no sugar in my tea, so I have set the variable `num_sugar_spoons` to 0. The comes the loop: as long as (while) the value of `num_sugar_spoons` is less than the stopping point `sweet_enough`, the loop continues adding 1 to `num_sugar_spoons`. Just to keep track of the progress of the loop, I have thrown a `print` statement in as well." ] }, { "cell_type": "code", "execution_count": 1, "id": "fa29bb89", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "I have added 1 spoons of sugar\n", "I have added 2 spoons of sugar\n", "I have added 3 spoons of sugar\n" ] } ], "source": [ "sweet_enough = 3\n", "num_sugar_spoons = 0\n", "while num_sugar_spoons < sweet_enough:\n", " num_sugar_spoons = num_sugar_spoons + 1\n", " print('I have added', num_sugar_spoons, 'spoons of sugar')" ] }, { "cell_type": "markdown", "id": "c20ac6df", "metadata": {}, "source": [ "Pretty easy, right?" ] }, { "cell_type": "markdown", "id": "7eaa0a78", "metadata": {}, "source": [ "### for loops\n", "\n", "`For` loops have a very similar structure. Let's crack some eggs:" ] }, { "cell_type": "code", "execution_count": 2, "id": "3261d420", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "I cracked an egg!\n", "I cracked an egg!\n", "I cracked an egg!\n" ] } ], "source": [ "eggs = ['egg', 'egg', 'egg']\n", "\n", "for egg in eggs:\n", " print('I cracked an egg!')" ] }, { "cell_type": "markdown", "id": "bc5056d5", "metadata": {}, "source": [ "Alright, I admit this was a pretty silly example. But hopefully you get the idea: I started with a finite number of items (in this case a list that contains three instances of the string 'egg'), then I had Python do something _for_ each item in the list. Let's do a few more, slightly (but not very) more interesting ones:" ] }, { "cell_type": "code", "execution_count": 4, "id": "81643512", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "I added eggs\n", "I added flour\n", "I added sugar\n", "I added cinnamon\n", "I added salt\n" ] } ], "source": [ "ingredients = ['eggs', 'flour', 'sugar', 'cinnamon', 'salt']\n", "\n", "for item in ingredients:\n", " print('I added', item)" ] }, { "cell_type": "code", "execution_count": 6, "id": "b6517a7b", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "There are 4 letters in EGGS\n", "There are 5 letters in FLOUR\n", "There are 5 letters in SUGAR\n", "There are 8 letters in CINNAMON\n", "There are 4 letters in SALT\n" ] } ], "source": [ "words = ['eggs', 'flour', 'sugar', 'cinnamon', 'salt']\n", "\n", "for word in words:\n", " print('There are', len(word), 'letters in', word.upper())" ] }, { "cell_type": "markdown", "id": "42d04957", "metadata": {}, "source": [ "As these (admittedly silly) examples illustrate, we can use `for` loops to step through a series of items, and do the same thing with each one. In the first example, we just printed the same string each time. In the second one, we took each item from the list, and printed it as the final word in a sentence. In the third example, we took each word, counted the number of letters in it, and then printed the answer in a sentence in which we also converted each word to all upper case letters. These were just very, very simple examples, but I want to underscore how extremely powerfull these loops are. _Anything_ you could do to one item in a list, you can get Python to do to _every_ item in the list. If you have a lot of actions to perform on a lot of items, this allows you to do things you would probaby never be able to do manually." ] }, { "cell_type": "markdown", "id": "036f7232", "metadata": {}, "source": [ "### Loop syntax\n", "\n", "A final thing about loops for now: if you look at both the `for` and `while` loops above, you may notice some similarities in the structure. First, there is a statement of the conditions (`while num_sugar_spoons < sweet_enough`, or `for word in words`), followed by a colon (`:`). Then, the next line or lines are indented. This syntax is _crucial_ for telling Python that you want it run a loop. It is customary in most programming languages to use indentation to show what things are \"inside\" the loop, and what things are \"outside\". In Python, the indentation is _mandatory_. Without the proper indentation, Python will not run the loop.\n", "\n", "Another thing to mention: when you define `for` loops, you also declare new variable. As an example, when we said `for word in words:`, we declared a new variable, `word`. Like all variables in Python, we could have called it anything we liked. Writing `for x in words:`, `for item in words:`, or `for flibbertigibbet in words:`, would all work the same. But the same advice that applies to other variables also applies here: don't get too cute. Give the variable a name that makes sense to you.\n", "\n", "Finally, be aware that this new variable that you declare at the beginning of your loop will inherit its type from the item in the list that it represents. This has an impact on what you can do with it in the loop. For example, suppose we have a list like this:" ] }, { "cell_type": "code", "execution_count": 7, "id": "962c5140", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "apples\n", " pears\n" ] }, { "ename": "TypeError", "evalue": "unsupported operand type(s) for +: 'int' and 'str'", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)", "Cell \u001b[0;32mIn[7], line 4\u001b[0m\n\u001b[1;32m 1\u001b[0m random_stuff \u001b[39m=\u001b[39m [\u001b[39m'\u001b[39m\u001b[39mapple\u001b[39m\u001b[39m'\u001b[39m, \u001b[39m'\u001b[39m\u001b[39m pear\u001b[39m\u001b[39m'\u001b[39m, \u001b[39m42\u001b[39m]\n\u001b[1;32m 3\u001b[0m \u001b[39mfor\u001b[39;00m thing \u001b[39min\u001b[39;00m random_stuff:\n\u001b[0;32m----> 4\u001b[0m new_thing \u001b[39m=\u001b[39m thing \u001b[39m+\u001b[39;49m \u001b[39m'\u001b[39;49m\u001b[39ms\u001b[39;49m\u001b[39m'\u001b[39;49m\n\u001b[1;32m 5\u001b[0m \u001b[39mprint\u001b[39m(new_thing)\n", "\u001b[0;31mTypeError\u001b[0m: unsupported operand type(s) for +: 'int' and 'str'" ] } ], "source": [ "random_stuff = ['apple', ' pear', 42]\n", "\n", "for thing in random_stuff:\n", " new_thing = thing + 's'\n", " print(new_thing)" ] }, { "cell_type": "markdown", "id": "41034571", "metadata": {}, "source": [ "For the first two item in the list, Python has no problem adding a an \"s\" to the end of them, because they are strngs, and so is \"s\", and Python understands what it means to add a string to a string. But the final item in the list is an integer, and Python has no way to add a string to an integer, and so it complains when it gets to that point in the loop." ] }, { "cell_type": "markdown", "id": "702f34c7", "metadata": {}, "source": [ "(if)=\n", "\n", "## Conditional statements\n", "\n", "Together with variables and loops, conditional statements are probably the third most important programming concept for you to learn. Once you master variables, loops, and conditionals, there's not much you can't do. Like loops, conditionals are also all around us in everyday life. Unlike loops, they don't seem to be as difficult for most people to grasp. This semester, I teach at 8:00 AM on Wednesdays. Yikes! That means, that _if_ it is a Wednesday, I need to leave the house earlier. Otherwise, I can leave at the normal time. It's not hard to imagine this idea re-written as code:" ] }, { "cell_type": "code", "execution_count": 8, "id": "bfb684e7", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Leave early!\n" ] } ], "source": [ "day = 'Wednesday'\n", "\n", "if day == 'Wednesday':\n", " print('Leave early!')" ] }, { "cell_type": "markdown", "id": "9ea4a39b", "metadata": {}, "source": [ "Ok, since I started by defining `day` as \"Wednesday\", it was pretty clear what was going to happen. But still, did you notice the use of `==` to make the comparison between `day` and the string `Wednesday`? \n", "\n", "To make this example ever-so-slightly more realistic, we could define a list variable that encompasses the whole week, and places the `if` statement inside a `for` loop, to do a daily check, to tell Python what to do if it is Wednesday, and an `else` statement, to tell Python what to do if it is not Wednesday." ] }, { "cell_type": "code", "execution_count": 17, "id": "4151bee8", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Today is Monday. You can leave at the normal time today.\n", "Today is Tuesday. You can leave at the normal time today.\n", "Today is Wednesday. You need to leave early!\n", "Today is Thursday. You can leave at the normal time today.\n", "Today is Friday. You can leave at the normal time today.\n", "Today is Saturday. You can leave at the normal time today.\n", "Today is Sunday. You can leave at the normal time today.\n" ] } ], "source": [ "week = ['Monday', 'Tuesday', 'Wednesday', 'Thursday', 'Friday', 'Saturday', 'Sunday']\n", "\n", "for day in week:\n", " if day == 'Wednesday':\n", " print('Today is', day + '.', 'You need to leave early!')\n", " else:\n", " print('Today is', day + '.', 'You can leave at the normal time today.')" ] }, { "cell_type": "markdown", "id": "a874d100", "metadata": {}, "source": [ "Then again, it's not just Wednesdays I need to be aware of. If it is Saturday or Sunday, then I don't want to leave at all; it's the weekend. So we can add another `if` statement and an `or` statment to create a more complicated set of choices for Python to navigate." ] }, { "cell_type": "code", "execution_count": 19, "id": "509ae446", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Today is Monday. You can leave at the normal time today.\n", "Today is Tuesday. You can leave at the normal time today.\n", "Today is Wednesday. You need to leave early!\n", "Today is Wednesday. You can leave at the normal time today.\n", "Today is Thursday. You can leave at the normal time today.\n", "Today is Friday. You can leave at the normal time today.\n", "Today is Saturday. You can sleep late!\n", "Today is Sunday. You can sleep late!\n" ] } ], "source": [ "week = ['Monday', 'Tuesday', 'Wednesday', 'Thursday', 'Friday', 'Saturday', 'Sunday']\n", "\n", "for day in week:\n", " if day == 'Wednesday':\n", " print('Today is', day + '.', 'You need to leave early!')\n", " if day == 'Saturday' or day == 'Sunday':\n", " print('Today is', day + '.', 'You can sleep late!')\n", " else:\n", " print('Today is', day + '.', 'You can leave at the normal time today.')" ] }, { "cell_type": "markdown", "id": "ee283bec", "metadata": {}, "source": [ "(functions)=\n", "\n", "## Functions\n", "\n", "Strictly speaking, you won't need to write any functions to do any of the exercises in this book, so if you're not interested, you can skip this. On the other hand, functions are pretty cool, and once you get the hang of how they work, they can really speed things up for you, make your code simpler and easier to read, and maybe even save you from making some silly mistakes.\n", "\n", "If you have been following along in the book, you've actually already been using functions, you just may not have known it. Consider the following code:" ] }, { "cell_type": "code", "execution_count": 21, "id": "078c15a2", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "perspicacious\n" ] } ], "source": [ "word = 'perspicacious'\n", "print(word)" ] }, { "cell_type": "code", "execution_count": 22, "id": "d186f70d", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "str" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "type(word)" ] }, { "cell_type": "code", "execution_count": 23, "id": "4395169b", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "13" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "len(word)" ] }, { "cell_type": "markdown", "id": "b52e293d", "metadata": {}, "source": [ "`print`, `type`, and `len` are all functions. They are little machines that accept an input, in this case we have given them the variable `word`, which contains the string \"perspicacious\", and they use that input to provide some kind of output. These are all built-in functions in Python, and thank goodness for that, because they are very useful. But we can also write our own functions, and this becomes very useful when we want to do the same sort of task again and again. \n", "\n", "Often functions are most useful when they do something fairly complicated, but just to illustrate how they work, lets look at something quite simple. Let's imagine that the `len` function didn't exist, or we didn't know about it, and we wanted a way to count the number of letters in a word. We could write our own function that would achieve this result.\n", "\n", "Let's think about how we might do this. If we couldn't just tell Python to count the number of letters in a word directly, we could still get it to do it more manually, right? One way would be to use a loop:" ] }, { "cell_type": "code", "execution_count": 29, "id": "d3c04065", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "There are 13 letters in PERSPICACIOUS\n" ] } ], "source": [ "letter_count = 0\n", "for letter in 'perspicacious':\n", " letter_count = letter_count + 1\n", "\n", "print('There are', letter_count, 'letters in PERSPICACIOUS')" ] }, { "cell_type": "markdown", "id": "13212cc3", "metadata": {}, "source": [ "This works great, but if we want to use this to count the letters in any word other than \"perspicacious\", we would need to alter our script in two places: once in the `for` loop where we tell it what word to count the letters in, and again in the `print` statement, where we would have to retype the new word with capital letters.\n", "\n", "However, if take this basic code, and put it into a function, we can re-use it again and again." ] }, { "cell_type": "code", "execution_count": 61, "id": "5ee56ae2", "metadata": {}, "outputs": [], "source": [ "def measure_word(word):\n", " letter_count = 0\n", " for letter in word:\n", " letter_count = letter_count + 1\n", " print('There are', letter_count, 'letters in', word.upper())\n" ] }, { "cell_type": "markdown", "id": "072e2e15", "metadata": {}, "source": [ "The first line in the function above tells Python we want to define a function called `measure_word`, and that this function will accept one \"argument\" (input). Then, indented, inside the function, we can just put our code from the loop above. The only change is that now, instead of hard-coding the strings \"perspicacious\" and \"PERSPICACIOUS\" into the code, we just replace these with the variable which contains the input argument. If you, like me, are using a Jupyter Notebook and you run a cell with only the function definition in it, nothing will appear to happen. But now that the function has been defined, we can use it to measure words:" ] }, { "cell_type": "code", "execution_count": 62, "id": "c3a7c67c", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "There are 3 letters in CAT\n" ] } ], "source": [ "measure_word('cat')" ] }, { "cell_type": "code", "execution_count": 63, "id": "c3378590", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "There are 13 letters in PERSPICACIOUS\n" ] } ], "source": [ "measure_word('perspicacious')" ] }, { "cell_type": "code", "execution_count": 64, "id": "cdaf8d0c", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "There are 34 letters in SUPERCALIFRAGILISTICEXPIALIDOCIOUS\n" ] } ], "source": [ "measure_word('supercalifragilisticexpialidocious')" ] }, { "cell_type": "markdown", "id": "964482a4", "metadata": {}, "source": [ "We can even use our new function inside a loop:" ] }, { "cell_type": "code", "execution_count": 65, "id": "d038dd97", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "There are 2 letters in IF\n", "There are 3 letters in YOU\n", "There are 3 letters in SAY\n", "There are 2 letters in IT\n", "There are 4 letters in LOUD\n", "There are 6 letters in ENOUGH\n", "There are 3 letters in YOU\n", "There are 4 letters in WILL\n", "There are 6 letters in ALWAYS\n", "There are 5 letters in SOUND\n", "There are 10 letters in PRECOCIOUS\n" ] } ], "source": [ "words = [ 'If', 'you', 'say', 'it', 'loud', 'enough', 'you', 'will', 'always', 'sound', 'precocious']\n", "for word in words:\n", " measure_word(word)" ] }, { "cell_type": "markdown", "id": "330479d0", "metadata": {}, "source": [ "Often we don't want our functions to print something out; instead, we want them to return a variable, which we can use later. To achieve this, we can modify our function slightly, so that instead of asking it to `print` the output, we ask it to `return` the output:" ] }, { "cell_type": "code", "execution_count": 70, "id": "49994122", "metadata": {}, "outputs": [], "source": [ "def measure_word(word):\n", " letter_count = 0\n", " for letter in word:\n", " letter_count = letter_count + 1\n", " return('There are', letter_count, 'letters in', word.upper())" ] }, { "cell_type": "markdown", "id": "d8090e2d", "metadata": {}, "source": [ "Then, when we use the function, we can assign the output to a variable, like so:" ] }, { "cell_type": "code", "execution_count": 71, "id": "8d74b517", "metadata": {}, "outputs": [], "source": [ "output = measure_word('cat')" ] }, { "cell_type": "markdown", "id": "7cb067f0", "metadata": {}, "source": [ "In this case, the output is a tuple" ] }, { "cell_type": "code", "execution_count": 72, "id": "b8371dfb", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "tuple" ] }, "execution_count": 72, "metadata": {}, "output_type": "execute_result" } ], "source": [ "type(output)" ] }, { "cell_type": "markdown", "id": "7308163f", "metadata": {}, "source": [ "And the contents look like this:" ] }, { "cell_type": "code", "execution_count": 73, "id": "80208609", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "('There are', 3, 'letters in', 'CAT')\n" ] } ], "source": [ "print(output)" ] }, { "cell_type": "markdown", "id": "eb03db2c", "metadata": {}, "source": [ "If we wanted to, we could e.g. get only the number of letters out of the result, and skip the rest. The number representing the letter count is in position 1 of our function's output, so if we only want the number of letters, and not all the surrounding text, we could write:" ] }, { "cell_type": "code", "execution_count": 74, "id": "e9be0629", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3" ] }, "execution_count": 74, "metadata": {}, "output_type": "execute_result" } ], "source": [ "output = measure_word('cat')\n", "output[1]" ] }, { "cell_type": "markdown", "id": "9c1fb167", "metadata": {}, "source": [ "Or we could even just write" ] }, { "cell_type": "code", "execution_count": 75, "id": "9b5a778e", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3" ] }, "execution_count": 75, "metadata": {}, "output_type": "execute_result" } ], "source": [ "measure_word('cat')[1]" ] }, { "cell_type": "markdown", "id": "857e6be6", "metadata": {}, "source": [ "although at that point, we might as well just use `len()`!" ] }, { "cell_type": "markdown", "id": "4ecc4309", "metadata": {}, "source": [ "Functions can easily be much more complex and useful than these examples suggest. For example, You can write your functions so that they accept more than one input, and return more than one output. It wouldn't take much to modify our code to provide the word we input, the number of letters in that word, and the first letter (in lowercase) of each input:" ] }, { "cell_type": "code", "execution_count": 83, "id": "69e32e59", "metadata": {}, "outputs": [], "source": [ "def measure_word(word):\n", " first_letter = word[0].lower()\n", " letter_count = 0\n", " for letter in word:\n", " letter_count = letter_count + 1\n", " return(word,first_letter, letter_count)" ] }, { "cell_type": "code", "execution_count": 84, "id": "db87e62e", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "('If', 'i', 2)\n", "('you', 'y', 3)\n", "('say', 's', 3)\n", "('it', 'i', 2)\n", "('loud', 'l', 4)\n", "('enough', 'e', 6)\n", "('you', 'y', 3)\n", "('will', 'w', 4)\n", "('always', 'a', 6)\n", "('sound', 's', 5)\n", "('precocious', 'p', 10)\n" ] } ], "source": [ "words = [ 'If', 'you', 'say', 'it', 'loud', 'enough', 'you', 'will', 'always', 'sound', 'precocious']\n", "\n", "for word in words:\n", " print(measure_word(word))" ] }, { "cell_type": "markdown", "id": "496d8f99", "metadata": {}, "source": [ "Or imagine if, for some reason, we only wanted to know the first letter of words that were longer than 3 letters...." ] }, { "cell_type": "code", "execution_count": 89, "id": "f0640879", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "l\n", "e\n", "w\n", "a\n", "s\n", "p\n" ] } ], "source": [ "for word in words:\n", " if measure_word(word)[2] > 3:\n", " print(measure_word(word)[1])" ] }, { "cell_type": "markdown", "id": "4327538c", "metadata": {}, "source": [ "I'll stop now. I hope you get the idea!" ] }, { "cell_type": "markdown", "id": "d574b516", "metadata": {}, "source": [ "(listcomprehensions)=\n", "\n", "## List comprehensions: A different kind of loop\n", "\n", "Python has another way of doing loops, when working with lists. I don't want to go too deep on these, but you are very likely to come across them if you spend any time looking at other people's Python code, and they can be very handy, so I will briefly mention them. List comprehensions let you take the items in a list, and then make a new list based on the old list. I think the easiest way to describe how list comprehensions work is to give some examples, starting with some very simple ones, and then some ever-so-slightly more complicated ones. Here goes:" ] }, { "cell_type": "code", "execution_count": 90, "id": "ff899946", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['If', 'you', 'say', 'it', 'loud', 'enough', 'you', 'will', 'always', 'sound', 'precocious']\n" ] } ], "source": [ "# make a new list that has all the same elements as the old list\n", "\n", "words = [ 'If', 'you', 'say', 'it', 'loud', 'enough', 'you', 'will', 'always', 'sound', 'precocious']\n", "new_words = [x for x in words]\n", "print(new_words)" ] }, { "cell_type": "code", "execution_count": 92, "id": "47f46118", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['you', 'loud', 'enough', 'you', 'sound', 'precocious']\n" ] } ], "source": [ "# make a new list that contains all the words from the old list that have an \"o\" in them\n", "\n", "words = [ 'If', 'you', 'say', 'it', 'loud', 'enough', 'you', 'will', 'always', 'sound', 'precocious']\n", "new_words = [x for x in words if 'o' in x]\n", "print(new_words)" ] }, { "cell_type": "code", "execution_count": 94, "id": "60979f96", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['loud', 'enough', 'will', 'always', 'sound', 'precocious']\n" ] } ], "source": [ "# make a new list with only the words from the old list that are longer than 3 characters\n", "\n", "words = [ 'If', 'you', 'say', 'it', 'loud', 'enough', 'you', 'will', 'always', 'sound', 'precocious']\n", "new_words = [x for x in words if len(x)>3]\n", "print(new_words)" ] }, { "cell_type": "code", "execution_count": 95, "id": "46f6525d", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['_If_', '_you_', '_say_', '_it_', '_loud_', '_enough_', '_you_', '_will_', '_always_', '_sound_', '_precocious_']\n" ] } ], "source": [ "# make a new list with underscores before and after each word in the old list\n", "\n", "words = [ 'If', 'you', 'say', 'it', 'loud', 'enough', 'you', 'will', 'always', 'sound', 'precocious']\n", "new_words = ['_' + x + '_' for x in words]\n", "print(new_words)" ] }, { "cell_type": "code", "execution_count": 118, "id": "5363b11a", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['_enough_', '_always_', '_sound_', '_precocious_']\n" ] } ], "source": [ "# make a new list with underscores before and after each word in the old list if the word is longer than 4 characters\n", "\n", "words = [ 'If', 'you', 'say', 'it', 'loud', 'enough', 'you', 'will', 'always', 'sound', 'precocious']\n", "new_words = ['_' + x + '_' for x in words if len(x) > 4]\n", "print(new_words)" ] }, { "cell_type": "code", "execution_count": 98, "id": "7f9ec176", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['you', 'enough', 'you', 'sound']\n" ] } ], "source": [ "# compare two lists and make a new list that only has words that are in both lists\n", "\n", "words = [ 'If', 'you', 'say', 'it', 'loud', 'enough', 'you', 'will', 'always', 'sound', 'precocious']\n", "keywords = ['you', 'enough', 'sound', 'dog', 'bunny', 'nihilist']\n", "new_words = [x for x in words if x in keywords]\n", "print(new_words)\n" ] }, { "cell_type": "code", "execution_count": 101, "id": "c10f2ed4", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['If', 'say', 'it', 'loud', 'will', 'always', 'precocious']\n" ] } ], "source": [ "# compare two lists and make a new list with only words from the old list that are NOT in both lists\n", "\n", "words = [ 'If', 'you', 'say', 'it', 'loud', 'enough', 'you', 'will', 'always', 'sound', 'precocious']\n", "keywords = ['you', 'enough', 'sound', 'dog', 'bunny', 'nihilist']\n", "new_words = [x for x in words if x not in keywords]\n", "print(new_words)" ] }, { "cell_type": "code", "execution_count": 103, "id": "5b23a434", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['101', '102', '103', '104', '105']\n" ] } ], "source": [ "# add 100 to each number in the old list and convert the sum to a string in the new list\n", "\n", "numbers = [1, 2, 3, 4, 5]\n", "new_numbers = [str(x + 100) for x in numbers]\n", "print(new_numbers)" ] }, { "cell_type": "code", "execution_count": 120, "id": "08836d7d", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[['1', '2', '3', '4'], ['5', '6', '7', '8']]\n" ] } ], "source": [ "# convert integers to strings within a list of lists while still maintaining the list of list structure\n", "list_of_lists = [[1,2,3,4], [5,6,7,8]]\n", "list_of_string_lists = [[str(y) for y in x] for x in list_of_lists]\n", "print(list_of_string_lists)" ] }, { "cell_type": "code", "execution_count": 116, "id": "c386480d", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[1, 2, 3, 4, 5, 6, 7, 8]]\n" ] } ], "source": [ "# \"flatten\" a list of lists into a single list\n", "list_of_lists = [[1,2,3,4], [5,6,7,8]]\n", "flattened_lists = [[y for x in list_of_lists for y in x]]\n", "print(flattened_lists)" ] }, { "cell_type": "markdown", "id": "0c1d6222", "metadata": {}, "source": [ "Ok, I put that last one there for myself, because I can never remember how to do it, and now I have a place to look it up easily! Anyway, the basic idea is that you perform some function for every item in the old list, and put the output in the new list. That function could range from nothing (simply putting all the old items in the new list) to something quite complicated. Like I say, you don't necessarily need to know about list comprehensions for the purposes of this book, but now you've seen them, so you can recognize them if you spot them in the wild, and you'll have an idea what is going on. All of these could also be done with regular old `for` loops, but list comprehensions are short and cool and make you look super pythonic.\n", "\n", "By the way, you can also use list comprehensions to overwrite a list, if you so desire:" ] }, { "cell_type": "code", "execution_count": 117, "id": "d8ee126d", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[100, 200, 300, 400]\n" ] } ], "source": [ "numbers = [1, 2, 3, 4]\n", "numbers = [x*100 for x in numbers]\n", "print(numbers)" ] }, { "cell_type": "markdown", "id": "a3436a15", "metadata": {}, "source": [ "(nesting)=\n", "\n", "## Nesting\n", "\n", "### Nested conditionals\n", "\n", "We have already talked about [loops](loops) and [conditionals](if). These are powerful on their own, but you can really take them to the next level when you start nesting them, or combining them with other types of logic. A common situation is to check whether an item meets more than one condition. Maybe I have a list of numbers like this:" ] }, { "cell_type": "code", "execution_count": 125, "id": "6c08ab74", "metadata": {}, "outputs": [], "source": [ "numbers = [4, 2, 54, 823452, 324, 2, 4435, 4, 9070878072634, 3421, 4345]" ] }, { "cell_type": "markdown", "id": "c0f8dc39", "metadata": {}, "source": [ "For reasons that are best known to myself[^ineedanexample], I want to print all of the even numbers from the list which are longer than one digit.\n", "\n", "Now, to find numbers that are longer than one digit, we first need to change them from integers to strings, so we can use `len()` on them[^nolengthonints]: \n", "\n", "[^ineedanexample]: But those reasons are pretty obviously that I need an example to illustrate nested conditionals!\n", "[^nolengthonints]: For some reason, you can't use `len()` to find the length of an integer. Dunno why, I'm sure there's a good reason. Luckily, we can just turn our ints into strs and then we can measure them all we like." ] }, { "cell_type": "code", "execution_count": 130, "id": "12240873", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "54\n", "823452\n", "324\n", "4435\n", "9070878072634\n", "3421\n", "4345\n" ] } ], "source": [ "for num in numbers:\n", " if len(str(num)) > 1:\n", " print(num)" ] }, { "cell_type": "markdown", "id": "7defe511", "metadata": {}, "source": [ "(modulo)=\n", "\n", "To figure out whether a number is even or odd, an easy way is to use `%`. This is known by the fancy name of [modulo operator](http://docs.python.org/2/reference/expressions.html#binary-arithmetic-operations), and you can follow the link if you want to learn more, but basically it just works like division, except it gives you the [remainder](https://en.wikipedia.org/wiki/Remainder) of the answer. If you divide an even number by 2, there will be nothing left over (no remainder), and so modulo 2 of an even number will be 0. So, to check if the numbers from our list are even, we could do" ] }, { "cell_type": "code", "execution_count": 131, "id": "367e30a3", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "4\n", "2\n", "54\n", "823452\n", "324\n", "2\n", "4\n", "9070878072634\n" ] } ], "source": [ "for num in numbers:\n", " if num % 2 == 0:\n", " print(num)" ] }, { "cell_type": "markdown", "id": "e9a28122", "metadata": {}, "source": [ "Great! But we want to know which numbers are _both_ even _and_ more than two digits. We can use nested loops to combine these two conditions, to give us an IF AND logic:" ] }, { "cell_type": "code", "execution_count": 132, "id": "0f337e10", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "54\n", "823452\n", "324\n", "9070878072634\n" ] } ], "source": [ "for num in numbers:\n", " if len(str(num)) > 1:\n", " if num % 2 == 0:\n", " print(num)" ] }, { "cell_type": "markdown", "id": "fc57b8e5", "metadata": {}, "source": [ "You can keep adding nested `if`s as much as you like. Say we wanted to print all the numbers from the list that are even, more than one digit, and end with a 4, why we could just do:" ] }, { "cell_type": "code", "execution_count": 136, "id": "f3d21145", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "54\n", "324\n", "9070878072634\n" ] } ], "source": [ "for num in numbers:\n", " if len(str(num)) > 1:\n", " if num % 2 == 0:\n", " if int(str(num)[-1]) == 4:\n", " print(num)" ] }, { "cell_type": "markdown", "id": "31b8d584", "metadata": {}, "source": [ "By the way, if you are wondering what's going on with all that `if int(str(num)[-1])` stuff, well... I wanted to get the last digit of the numbers, so I figured I'd use `[-1]`. But it turns out you can't do that with integers, so I used `str(num)`to turn the integers into strings so I could use `[-1]` to get the last number. But then I wanted to compare whatever the last digit was with the number 4. But by now `num` was a string, because I had changed it into one so I could find the last digit, and 4 is an integer, so it told me none of the last numbers were 4, which I knew was a baldfaced lie because I could see that two of them were, but then I realized that I could just use `int()` to turn the whole shebang back into an integer so I could compare it with 4. This the way programming works! You keep futzing around until you get it to do the thing you want." ] }, { "cell_type": "markdown", "id": "cff1ad7b", "metadata": {}, "source": [ "### Nested loops\n", "\n", "Just like we can nest conditionals within each other, we can also nest loops. Say we have a list of lists, like this:" ] }, { "cell_type": "code", "execution_count": 143, "id": "5cbb8075", "metadata": {}, "outputs": [], "source": [ "list_of_lists = [[1,2,3], [4,5,6], [7,8,9]]\n" ] }, { "cell_type": "markdown", "id": "e5a4a398", "metadata": {}, "source": [ "Maybe we want to add 10 to each number in the sublists. We can use nested loops to loop through the list of lists, pausing to loop through all the items of each sublist, and adding 10 to each one:" ] }, { "cell_type": "code", "execution_count": 144, "id": "da101a07", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "11\n", "12\n", "13\n", "14\n", "15\n", "16\n", "17\n", "18\n", "19\n" ] } ], "source": [ "\n", "for i in list_of_lists:\n", " for j in i:\n", " print(j+10)" ] }, { "cell_type": "markdown", "id": "bb430205", "metadata": {}, "source": [ "By the way, I used `i` and `j` here, insted of `list` and `sublist`, because `list` already has a meaning in Python, so you shouldn't name your list `list`. `i` and `j` are often used for this sort of thing, so I just went with that.\n", "\n", "We can of course combine nested loops with conditionals as well (and even with nested conditionals if we want), to create just about any kind of logic we want. Here is an example of a nested loop with a conditional that goes through all the numbers in each of the sublists and adds 10 to the odd ones, but leaves the even ones untouched:" ] }, { "cell_type": "code", "execution_count": 145, "id": "3d6bbbc1", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "11\n", "2\n", "13\n", "4\n", "15\n", "6\n", "17\n", "8\n", "19\n" ] } ], "source": [ "for i in list_of_lists:\n", " for j in i:\n", " if j % 2 != 0:\n", " print(j+10)\n", " else:\n", " print(j)" ] }, { "cell_type": "markdown", "id": "82bbfb35", "metadata": {}, "source": [ "(abstraction)=\n", "\n", "## Abstraction, generalization, and patterns\n", "\n", "As you become more familiar with programming, you will inevitably find yourself reusing bits of code; you will begin to see how specific programming problems you have solved are really just cases of a larger set of problems. For example, in the section on [functions](functions) we looked at this simple loop:" ] }, { "cell_type": "code", "execution_count": 121, "id": "fe95732e", "metadata": {}, "outputs": [], "source": [ "letter_count = 0\n", "for letter in 'perspicacious':\n", " letter_count = letter_count + 1" ] }, { "cell_type": "markdown", "id": "38f8d89a", "metadata": {}, "source": [ "This loop solves a very specific and fairly unusual problem: how to count the number of letters in the word \"perspicacious\". But this is a specific case of a much more general problem: how to step through items in a series while also keeping track of how many items you have stepped through. The code above is an example of one way to solve this problem: before the loop, set up an empty \"counter\" variable (in this case `letter_count`, which we have set to 0), then every time the loop goes around, add 1 to the counter variable.\n", "\n", "This is an example of a type of an even more general pattern that will be very useful to you: setting up variables \"outside\" the loop, and then modifying them from within the loop.\n" ] }, { "cell_type": "markdown", "id": "7e976c92", "metadata": {}, "source": [ "(append)=\n", "\n", "Another example of this type of pattern is the \"append to list\" pattern. You might[^mightnot] remember when we talked about the various [methods](methods) associated with different variable [types](types). One of the methods that list variables have available to them is `.append()`, the ability to stick something onto the end of a list. This comes in super handy in all sorts of situations. Here is an example.\n", "\n", "Let's say you have a list of all the different flowers in your garden. It looks like this:\n", "\n", "\n", "[^mightnot]: Or might not. If you don't, it's ok, I forgive you!" ] }, { "cell_type": "code", "execution_count": 122, "id": "82d67553", "metadata": {}, "outputs": [], "source": [ "all_flowers = ['Begonias', 'Lilacs', 'Roses', 'Pansies', 'Foxgloves', 'Buttercups', 'Sunflowers']" ] }, { "cell_type": "markdown", "id": "dd23f2d7", "metadata": {}, "source": [ "Now, say you want a list with only the flowers that start with the letter \"B\". I don't know why you want this, you just do, ok? To get achieve this, you could start by setting up an empty `b_flowers` list outside of a loop, and then loop through your `all_flowers` loop, checking each item to see if it starts with a \"B\". If it does, you append it to the `b_flowers` loop, like so:" ] }, { "cell_type": "code", "execution_count": 123, "id": "765fc630", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['Begonias', 'Buttercups']\n" ] } ], "source": [ "all_flowers = ['Begonias', 'Lilacs', 'Roses', 'Pansies', 'Foxgloves', 'Buttercups', 'Sunflowers']\n", "\n", "b_flowers = []\n", "\n", "for flower in all_flowers:\n", " if flower.startswith('B'):\n", " b_flowers.append(flower)\n", "\n", "print(b_flowers)" ] }, { "cell_type": "markdown", "id": "60f5ee71", "metadata": {}, "source": [ "Again, the specifics of the example are not important. But take a long look at the general pattern:\n", "\n", "1. Empty list before the loop\n", "2. Loop with a conditional\n", "3. Append to the list outside the loop\n", "\n", "It will serve you well!\n", "\n", "Now, [I know what you're thinking](https://www.youtube.com/watch?v=V6DCRuj2Vuc): couldn't I solve this with a list comprehension? Sure, of course you could!" ] }, { "cell_type": "code", "execution_count": 124, "id": "9475ef1d", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['Begonias', 'Buttercups']\n" ] } ], "source": [ "all_flowers = ['Begonias', 'Lilacs', 'Roses', 'Pansies', 'Foxgloves', 'Buttercups', 'Sunflowers']\n", "b_flowers = [x for x in all_flowers if x.startswith('B')]\n", "print(b_flowers)" ] }, { "cell_type": "markdown", "id": "af65a457", "metadata": {}, "source": [ "This is another great Python pattern, and it will also serve you well! The end result is exactly the same, so in this case you could use whichever you like better. The point is to learn to see past the immediate problem you are trying to solve, and begin to identify classes of problems, which will begin to match up with classes of solutions. " ] }, { "cell_type": "markdown", "id": "75f3508c", "metadata": {}, "source": [ "So that's what I mean by patterns in coding. It could be things like appending to a list outside a loop, or it could be things like using the modulo operator to check whether a number is odd or even, like we did [earlier](modulo). But what about abstraction and generalization?\n", "\n", "Functions are a good example of what I am talking about here. Often when we write code[^dilettante], we start by writing code that does something very specific, but as we reuse that code again and again, it starts to get tedious to change all the variables to match the new situation. Eventually, it becomes more practical to write a function that will accept _any_ variable of a certain kind, and do something to it.\n", "\n", "Say, for instance, that we often find ourselves checking lists like the list of flowers from before. But sometimes they aren't flowers, and sometimes we want to check for a different first letter, not just \"B\". And maybe sometimes the words start with capital letters, and sometimes they start with lower-case letters. It might be time to take our specific code from before, and generalize it to a function:\n", "\n", "\n", "[^dilettante]: Or at least when a code dilettante like me who doesn't know any better writes code" ] }, { "cell_type": "markdown", "id": "a0693173", "metadata": {}, "source": [ "Just as a reminder, this is the code from before:" ] }, { "cell_type": "code", "execution_count": null, "id": "5c902ede", "metadata": {}, "outputs": [], "source": [ "b_flowers = []\n", "\n", "for flower in all_flowers:\n", " if flower.startswith('B'):\n", " b_flowers.append(flower)" ] }, { "cell_type": "markdown", "id": "20b6d2a4", "metadata": {}, "source": [ "Now, we'll take this code, but make it more generic, and dump the whole thing into a function. Let's call our function `check_first_letter`" ] }, { "cell_type": "code", "execution_count": 154, "id": "6a068016", "metadata": {}, "outputs": [], "source": [ "def check_first_letter(input_list, target_letter):\n", " output_list = []\n", " for item in input_list:\n", " if item.lower().startswith(target_letter.lower()):\n", " output_list.append(item)\n", " print(output_list)" ] }, { "cell_type": "code", "execution_count": 156, "id": "d33fa76d", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['Begonias', 'Buttercups']\n" ] } ], "source": [ "check_first_letter(all_flowers, 'b')" ] }, { "cell_type": "code", "execution_count": 160, "id": "6c15f664", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['Pears', 'pickles', 'peppers', 'Pizza']\n" ] } ], "source": [ "foods = ['Pears', 'pickles', 'Beets', 'burgers', 'cheese', 'peppers', 'Pizza', 'bananas', 'cardamom', 'Cabbage']\n", "\n", "check_first_letter(foods, 'P')" ] }, { "cell_type": "code", "execution_count": 161, "id": "de1e6491", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['Pears', 'pickles', 'peppers', 'Pizza']\n" ] } ], "source": [ "check_first_letter(foods, 'p')" ] }, { "cell_type": "code", "execution_count": 162, "id": "a6180b4c", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['Beets', 'burgers', 'bananas']\n" ] } ], "source": [ "check_first_letter(foods, 'B')" ] }, { "cell_type": "code", "execution_count": 163, "id": "bada29d1", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "['cheese', 'cardamom', 'Cabbage']\n" ] } ], "source": [ "check_first_letter(foods, 'c')" ] }, { "cell_type": "markdown", "id": "56bd5217", "metadata": {}, "source": [ "Now we have a gerneric, generalized and abstracted function that will check the first letter of strings in a list. All we have to do is tell it which list to look at, and what letter to check for. Additionally, it doesn't matter wheter we give it the capital or lower-case version of the letter as a target, and it doesn't matter whether the item in the list starts with a capital or a lower-case letter. Our function will find them all!" ] }, { "cell_type": "markdown", "id": "79a92152", "metadata": {}, "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.3" } }, "nbformat": 4, "nbformat_minor": 5 } { "cells": [ { "cell_type": "markdown", "id": "adjusted-gross", "metadata": {}, "source": [ "(introprob)=\n", "# Statistical theory \n", "\n", "Part IV of the book is by far the most theoretical one, focusing as it does on the theory of statistical inference. Over the next three chapters my goal is to give you an introduction to [probability theory](probability), [sampling and estimation](estimation) and statistical [hypothesis testing](hypothesis-testing). Before we get started though, I want to say something about the big picture. Statistical inference is primarily about *learning from data*. The goal is no longer merely to describe our data, but to use the data to draw conclusions about the world. To motivate the discussion, I want to spend a bit of time talking about a philosophical puzzle known as the *riddle of induction*, because it speaks to an issue that will pop up over and over again throughout the book: statistical inference relies on *assumptions*. This sounds like a bad thing. In everyday life people say things like \"you should never make assumptions\", and psychology classes often talk about assumptions and biases as bad things that we should try to avoid. From bitter personal experience I have learned never to say such things around philosophers. \n", "\n", "## On the limits of logical reasoning\n", ">*The whole art of war consists in getting at what is on the other side of the hill, or, in other words, in learning what we do not know from what we do.*\n", ">\n", ">-- Arthur Wellesley, 1st Duke of Wellington\n", "\n", "I am told that quote above came about as a consequence of a carriage ride across the countryside.[^note1] He and his companion, J. W. Croker, were playing a guessing game, each trying to predict what would be on the other side of each hill. In every case it turned out that Wellesley was right and Croker was wrong. Many years later when Wellesley was asked about the game, he explained that \"the whole art of war consists in getting at what is on the other side of the hill\". Indeed, war is not special in this respect. All of life is a guessing game of one form or another, and getting by on a day to day basis requires us to make good guesses. So let's play a guessing game of our own. \n", "\n", "Suppose you and I are observing the Wellesley-Croker competition, and after every three hills you and I have to predict who will win the next one, Wellesley or Croker. Let's say that `W` refers to a Wellesley victory and `C` refers to a Croker victory. After three hills, our data set looks like this:\n", "\n", "```\n", "WWW\n", "```\n", "\n", "> You: Three in a row doesn't mean much. I suppose Wellesley might be better at this than Croker, but it might just be luck. Still, I'm a bit of a gambler. I'll bet on Wellesley.\n", "\n", "> Me: I agree that three in a row isn't informative, and I see no reason to prefer Wellesley's guesses over Croker's. I can't justify betting at this stage. Sorry. No bet for me.\n", "\n", "Your gamble paid off: three more hills go by, and Wellesley wins all three. Going into the next round of our game the score is 1-0 in favour of you, and our data set looks like this:\n", "\n", "```\n", "WWW WWW\n", "\n", "```\n", "I've organised the data into blocks of three so that you can see which batch corresponds to the observations that we had available at each step in our little side game. After seeing this new batch, our conversation continues:\n", "\n", "> You: Six wins in a row for Duke Wellesley. This is starting to feel a bit suspicious. I'm still not certain, but I reckon that he's going to win the next one too.\n", "\n", "> Me: I guess I don't see that. Sure, I agree that Wellesley has won six in a row, but I don't see any logical reason why that means he'll win the seventh one. No bet.\n", "\n", "> You: Do your really think so? Fair enough, but my bet worked out last time, and I'm okay with my choice.\n", "\n", "For a second time you were right, and for a second time I was wrong. Wellesley wins the next three hills, extending his winning record against Croker to 9-0. The data set available to us is now this: \n", "\n", "\n", "```\n", "WWW WWW WWW\n", "```\n", "\n", "And our conversation goes like this:\n", "\n", "> You: Okay, this is pretty obvious. Wellesley is way better at this game. We both agree he's going to win the next hill, right?\n", "\n", "> Me: Is there really any logical evidence for that? Before we started this game, there were lots of possibilities for the first 10 outcomes, and I had no idea which one to expect. `WWW WWW WWW W` was one possibility, but so was `WCC CWC WWC C` and `WWW WWW WWW C` or even `CCC CCC CCC C`. Because I had no idea what would happen so I'd have said they were all equally likely. I assume you would have too, right? I mean, that's what it *means* to say you have \"no idea\", isn't it?\n", "\n", "> You: I suppose so.\n", "\n", "> Me: Well then, the observations we've made logically rule out all possibilities except two: `WWW WWW WWW C` or `WWW WWW WWW W`. Both of these are perfectly consistent with the evidence we've encountered so far, aren't they? \n", "\n", "> You: Yes, of course they are. Where are you going with this?\n", "\n", "> Me: So what's changed then? At the start of our game, you'd have agreed with me that these are equally plausible, and none of the evidence that we've encountered has discriminated between these two possibilities. Therefore, both of these possibilities remain equally plausible, and I see no logical reason to prefer one over the other. So yes, while I agree with you that Wellesley's run of 9 wins in a row is remarkable, I can't think of a good reason to think he'll win the 10th hill. No bet.\n", "\n", "> You: I see your point, but I'm still willing to chance it. I'm betting on Wellesley.\n", "\n", "Wellesley's winning streak continues for the next three hills. The score in the Wellesley-Croker game is now 12-0, and the score in our game is now 3-0. As we approach the fourth round of our game, our data set is this:\n", "\n", "```\n", "WWW WWW WWW WWW\n", "```\n", "\n", "and the conversation continues:\n", "\n", "> You: Oh yeah! Three more wins for Wellesley and another victory for me. Admit it, I was right about him! I guess we're both betting on Wellesley this time around, right?\n", "\n", "> Me: I don't know what to think. I feel like we're in the same situation we were in last round, and nothing much has changed. There are only two legitimate possibilities for a sequence of 13 hills that haven't already been ruled out, `WWW WWW WWW WWW C` and `WWW WWW WWW WWW W`. It's just like I said last time: if all possible outcomes were equally sensible before the game started, shouldn't these two be equally sensible now given that our observations don't rule out either one? I agree that it feels like Wellesley is on an amazing winning streak, but where's the logical evidence that the streak will continue?\n", "\n", "> You: I think you're being unreasonable. Why not take a look at *our* scorecard, if you need evidence? You're the expert on statistics and you've been using this fancy logical analysis, but the fact is you're losing. I'm just relying on common sense and I'm winning. Maybe you should switch strategies.\n", "\n", "> Me: Hm, that is a good point and I don't want to lose the game, but I'm afraid I don't see any logical evidence that your strategy is better than mine. It seems to me that if there were someone else watching our game, what they'd have observed is a run of three wins to you. Their data would look like this: `YYY`. Logically, I don't see that this is any different to our first round of watching Wellesley and Croker. Three wins to you doesn't seem like a lot of evidence, and I see no reason to think that your strategy is working out any better than mine. If I didn't think that `WWW` was good evidence then for Wellesley being better than Croker at *their* game, surely I have no reason now to think that `YYY` is good evidence that you're better at *ours*?\n", "\n", "> You: Okay, now I think you're being a jerk.\n", "\n", "> Me: I don't see the logical evidence for that.\n", "\n", "## Learning without making assumptions is a myth\n", "There are lots of different ways in which we could dissect this dialogue, but since this is a statistics book pitched at psychologists, and not an introduction to the philosophy and psychology of reasoning, I'll keep it brief. What I've described above is sometimes referred to as the riddle of induction: it seems entirely *reasonable* to think that a 12-0 winning record by Wellesley is pretty strong evidence that he will win the 13th game, but it is not easy to provide a proper logical justification for this belief. On the contrary, despite the *obviousness* of the answer, it's not actually possible to justify betting on Wellesley without relying on some assumption that you don't have any logical justification for. \n", "\n", "The riddle of induction is most associated with the philosophical work of David Hume and more recently Nelson Goodman, but you can find examples of the problem popping up in fields as diverse literature (Lewis Carroll) and machine learning (the \"no free lunch\" theorem). There really is something weird about trying to \"learn what we do not know from what we do\". The critical point is that assumptions and biases are unavoidable if you want to learn anything about the world. There is no escape from this, and it is just as true for statistical inference as it is for human reasoning. In the dialogue, I was taking aim at your perfectly sensible inferences as a human being, but the common sense reasoning that you relied on in is no different to what a statistician would have done. Your \"common sense\" half of the dialog relied an implicit *assumption* that there exists some difference in skill between Wellesley and Croker, and what you were doing was trying to work out what that difference in skill level would be. My \"logical analysis\" rejects that assumption entirely. All I was willing to accept is that there are sequences of wins and losses, and that I did not know which sequences would be observed. Throughout the dialogue, I kept insisting that all logically possible data sets were equally plausible at the start of the Wellesely-Croker game, and the only way in which I ever revised my beliefs was to eliminate those possibilities that were factually inconsistent with the observations. \n", "\n", "That sounds perfectly sensible on its own terms. In fact, it even sounds like the hallmark of good deductive reasoning. Like Sherlock Holmes, my approach was to rule out that which is impossible, in the hope that what would be left is the truth. Yet as we saw, ruling out the impossible *never* led me to make a prediction. On its own terms, everything I said in my half of the dialogue was entirely correct. An inability to make any predictions is the logical consequence of making \"no assumptions\". In the end I lost our game, because you did make some assumptions and those assumptions turned out to be right. Skill is a real thing, and because you believed in the existence of skill you were able to learn that Wellesley had more of it than Croker. Had you relied on a less sensible assumption to drive your learning, you might not have won the game. \n", "\n", "Ultimately there are two things you should take away from this. Firstly, as I've said, you cannot avoid making assumptions if you want to learn anything from your data. But secondly, once you realise that assumptions are necessary, it becomes important to make sure you *make the right ones!* A data analysis that relies on few assumptions is not necessarily better than one that makes many assumptions: it all depends on whether those assumptions are good ones for your data. As we go through the rest of this book I'll often point out the assumptions that underpin a particular tool, and how you can check whether those assumptions are sensible. \n", "\n", "[^note1]: Source: http://www.bartleby.com/344/400.html" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.2" } }, "nbformat": 4, "nbformat_minor": 5 } ab[pw{ "cells": [ { "attachments": {}, "cell_type": "markdown", "id": "premium-incidence", "metadata": {}, "source": [ "(probability)=\n", "# Introduction to Probability\n", "\n", ">*[God] has afforded us only the twilight ... of Probability.* \n", ">\n", "> -- John Locke\n", "\n", "Up to this point in the book, we've discussed some of the key ideas in experimental design, and we've talked a little about how you can summarise a data set. To a lot of people, this is all there is to statistics: it's about calculating averages, collecting all the numbers, drawing pictures, and putting them all in a report somewhere. Kind of like stamp collecting, but with numbers. However, statistics covers much more than that. In fact, descriptive statistics is one of the smallest parts of statistics, and one of the least powerful. The bigger and more useful part of statistics is that it provides tools that let you make *inferences* about data. \n", "\n", "Once you start thinking about statistics in these terms -- that statistics is there to help us draw inferences from data -- you start seeing examples of it everywhere. For instance, here's a tiny extract from a newspaper article in the Sydney Morning Herald (30 Oct 2010):\n", "\n", "> \"I have a tough job,\" the Premier said in response to a poll which found her government is now the most unpopular Labor administration in polling history, with a primary vote of just 23 per cent.\n", "\n", "This kind of remark is entirely unremarkable in the papers or in everyday life, but let's have a think about what it entails. A polling company has conducted a survey, usually a pretty big one because they can afford it. I'm too lazy to track down the original survey, so let's just imagine that they called 1000 NSW voters at random, and 230 (23\\%) of those claimed that they intended to vote for the ALP. For the 2010 Federal election, the Australian Electoral Commission reported 4,610,795 enrolled voters in NSW; so the opinions of the remaining 4,609,795 voters (about 99.98\\% of voters) remain unknown to us. Even assuming that no-one lied to the polling company the only thing we can say with 100\\% confidence is that the true ALP primary vote is somewhere between 230/4610795 (about 0.005\\%) and 4610025/4610795 (about 99.83\\%). So, on what basis is it legitimate for the polling company, the newspaper, and the readership to conclude that the ALP primary vote is only about 23\\%?\n", "\n", "The answer to the question is pretty obvious: if I call 1000 people at random, and 230 of them say they intend to vote for the ALP, then it seems very unlikely that these are the *only* 230 people out of the entire voting public who actually intend to do so. In other words, we assume that the data collected by the polling company is pretty representative of the population at large. But how representative? Would we be surprised to discover that the true ALP primary vote is actually 24\\%? 29\\%? 37\\%? At this point everyday intuition starts to break down a bit. No-one would be surprised by 24\\%, and everybody would be surprised by 37\\%, but it's a bit hard to say whether 29\\% is plausible. We need some more powerful tools than just looking at the numbers and guessing.\n", "\n", "**_Inferential statistics_** provides the tools that we need to answer these sorts of questions, and since these kinds of questions lie at the heart of the scientific enterprise, they take up the lions share of every introductory course on statistics and research methods. However, the theory of statistical inference is built on top of **_probability theory_**. And it is to probability theory that we must now turn. This discussion of probability theory is basically background: there's not a lot of statistics per se in this chapter, and you don't need to understand this material in as much depth as the other chapters in this part of the book. Nevertheless, because probability theory does underpin so much of statistics, it's worth covering some of the basics. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "synthetic-simpson", "metadata": {}, "source": [ "(probstats)=\n", "## How are probability and statistics different?\n", "\n", "Before we start talking about probability theory, it's helpful to spend a moment thinking about the relationship between probability and statistics. The two disciplines are closely related but they're not identical. Probability theory is \"the doctrine of chances\". It's a branch of mathematics that tells you how often different kinds of events will happen. For example, all of these questions are things you can answer using probability theory:\n", "\n", "- What are the chances of a fair coin coming up heads 10 times in a row?\n", "- If I roll two six sided dice, how likely is it that I'll roll two sixes?\n", "- How likely is it that five cards drawn from a perfectly shuffled deck will all be hearts?\n", "- What are the chances that I'll win the lottery?\n", "\n", "\n", "\n", "Notice that all of these questions have something in common. In each case the \"truth of the world\" is known, and my question relates to the \"what kind of events\" will happen. In the first question I *know* that the coin is fair, so there's a 50\\% chance that any individual coin flip will come up heads. In the second question, I *know* that the chance of rolling a 6 on a single die is 1 in 6. In the third question I *know* that the deck is shuffled properly. And in the fourth question, I *know* that the lottery follows specific rules. You get the idea. The critical point is that probabilistic questions start with a known **_model_** of the world, and we use that model to do some calculations. The underlying model can be quite simple. For instance, in the coin flipping example, we can write down the model like this:\n", "\n", "$$\n", "P(\\mbox{heads}) = 0.5\n", "$$\n", "\n", "which you can read as \"the probability of heads is 0.5\". As we'll see later, in the same way that percentages are numbers that range from 0\\% to 100\\%, probabilities are just numbers that range from 0 to 1. When using this probability model to answer the first question, I don't actually know exactly what's going to happen. Maybe I'll get 10 heads, like the question says. But maybe I'll get three heads. That's the key thing: in probability theory, the *model* is known, but the *data* are not.\n", "\n", "So that's probability. What about statistics? Statistical questions work the other way around. In statistics, we *do not* know the truth about the world. All we have is the data, and it is from the data that we want to *learn* the truth about the world. Statistical questions tend to look more like these:\n", "\n", "- If my friend flips a coin 10 times and gets 10 heads, are they playing a trick on me?\n", "- If five cards off the top of the deck are all hearts, how likely is it that the deck was shuffled? \n", "- If the lottery commissioner's spouse wins the lottery, how likely is it that the lottery was rigged?\n", "\n", "\n", "\n", "This time around, the only thing we have are data. What I *know* is that I saw my friend flip the coin 10 times and it came up heads every time. And what I want to **_infer_** is whether or not I should conclude that what I just saw was actually a fair coin being flipped 10 times in a row, or whether I should suspect that my friend is playing a trick on me. The data I have look like this:\n", "```\n", "H H H H H H H H H H H\n", "```\n", "and what I'm trying to do is work out which \"model of the world\" I should put my trust in. If the coin is fair, then the model I should adopt is one that says that the probability of heads is 0.5; that is, $P(\\mbox{heads}) = 0.5$. If the coin is not fair, then I should conclude that the probability of heads is *not* 0.5, which we would write as $P(\\mbox{heads}) \\neq 0.5$. In other words, the statistical inference problem is to figure out which of these probability models is right. Clearly, the statistical question isn't the same as the probability question, but they're deeply connected to one another. Because of this, a good introduction to statistical theory will start with a discussion of what probability is and how it works.\n", "\n", "(probmeaning)=\n", "## What does probability mean?\n", "\n", "Let's start with the first of these questions. What is \"probability\"? It might seem surprising to you, but while statisticians and mathematicians (mostly) agree on what the *rules* of probability are, there's much less of a consensus on what the word really *means*. It seems weird because we're all very comfortable using words like \"chance\", \"likely\", \"possible\" and \"probable\", and it doesn't seem like it should be a very difficult question to answer. If you had to explain \"probability\" to a five year old, you could do a pretty good job. But if you've ever had that experience in real life, you might walk away from the conversation feeling like you didn't quite get it right, and that (like many everyday concepts) it turns out that you don't *really* know what it's all about. \n", "\n", "So I'll have a go at it. Let's suppose I want to bet on a soccer game between two teams of robots, *Arduino Arsenal* and *C Milan*. After thinking about it, I decide that there is an 80\\% probability that *Arduino Arsenal* winning. What do I mean by that? Here are three possibilities...\n", "\n", "- They're robot teams, so I can make them play over and over again, and if I did that, *Arduino Arsenal* would win 8 out of every 10 games on average.\n", "- For any given game, I would only agree that betting on this game is only \"fair\" if a 1 dollar bet on *C Milan* gives a 5 dollar payoff (i.e. I get my 1 dollar back plus a 4 dollar reward for being correct), as would a 4 dollar bet on *Arduino Arsenal* (i.e., my 4 dollar bet plus a 1 dollar reward). \n", "- My subjective \"belief\" or \"confidence\" in an *Arduino Arsenal* victory is four times as strong as my belief in a *C Milan* victory.\n", "\n", "Each of these seems sensible. However they're not identical, and not every statistician would endorse all of them. The reason is that there are different statistical ideologies (yes, really!) and depending on which one you subscribe to, you might say that some of those statements are meaningless or irrelevant. In this section, I give a brief introduction the two main approaches that exist in the literature. These are by no means the only approaches, but they're the two big ones. \n", "\n", "\n", "### The frequentist view\n", "\n", "The first of the two major approaches to probability, and the more dominant one in statistics, is referred to as the **_frequentist view_**, and it defines probability as a **_long-run frequency_**. Suppose we were to try flipping a fair coin, over and over again. By definition, this is a coin that has $P(H) = 0.5$. What might we observe? One possibility is that the first 20 flips might look like this:\n", "```\n", "T,H,H,H,H,T,T,H,H,H,H,T,H,H,T,T,T,T,T,H\n", "```\n", "In this case 11 of these 20 coin flips (55\\%) came up heads. Now suppose that I'd been keeping a running tally of the number of heads (which I'll call $N_H$) that I've seen, across the first $N$ flips, and calculate the proportion of heads $N_H / N$ every time. Here's what I'd get (I did literally flip coins to produce this!):" ] }, { "cell_type": "code", "execution_count": 53, "id": "shaped-skating", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "text/html": [ "
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Number_of_flipsNumber_of_headsProportion
0100.00
1210.50
2320.67
3430.75
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5640.67
6740.57
7850.63
8960.67
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" ], "text/plain": [ " Number_of_flips Number_of_heads Proportion\n", "0 1 0 0.00\n", "1 2 1 0.50\n", "2 3 2 0.67\n", "3 4 3 0.75\n", "4 5 4 0.80\n", "5 6 4 0.67\n", "6 7 4 0.57\n", "7 8 5 0.63\n", "8 9 6 0.67\n", "9 10 7 0.70\n", "10 11 8 0.73\n", "11 12 8 0.67\n", "12 13 9 0.69\n", "13 14 10 0.71\n", "14 15 10 0.67\n", "15 16 10 0.63\n", "16 17 10 0.59\n", "17 18 10 0.56\n", "18 19 10 0.53\n", "19 20 11 0.55" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "number_of_flips = [1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10, 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20]\n", "number_of_heads = [0 , 1 , 2 , 3 , 4 , 4 , 4 , 5 , 6 , 7, 8 , 8 , 9 , 10 , 10 , 10 , 10 , 10 , 10 , 11] \n", "proportion = [.00 , .50 , .67 , .75 , .80 , .67 , .57 , .63 , .67 , .70, .73 , .67 , .69 , .71 , .67 , .63 , .59 , .56 , .53 , .55]\n", "\n", "df = pd.DataFrame(\n", " {'Number_of_flips': number_of_flips,\n", " 'Number_of_heads': number_of_heads,\n", " 'Proportion': proportion\n", " })\n", "\n", "df" ] }, { "attachments": {}, "cell_type": "markdown", "id": "adjustable-nancy", "metadata": {}, "source": [ "Notice that at the start of the sequence, the *proportion* of heads fluctuates wildly, starting at .00 and rising as high as .80. Later on, one gets the impression that it dampens out a bit, with more and more of the values actually being pretty close to the \"right\" answer of .50. This is the frequentist definition of probability in a nutshell: flip a fair coin over and over again, and as $N$ grows large (approaches infinity, denoted $N\\rightarrow \\infty$), the proportion of heads will converge to 50\\%. There are some subtle technicalities that the mathematicians care about, but qualitatively speaking, that's how the frequentists define probability. Unfortunately, I don't have an infinite number of coins, or the infinite patience required to flip a coin an infinite number of times. However, I do have a computer, and computers excel at mindless repetitive tasks. So I asked my computer to simulate flipping a coin 1000 times, and then drew a picture of what happens to the proportion $N_H / N$ as $N$ increases. Actually, I did it four times, just to make sure it wasn't a fluke. The results are shown in {numref}`fig-frequentist_probability`. As you can see, the *proportion of observed heads* eventually stops fluctuating, and settles down; when it does, the number at which it finally settles is the true probability of heads." ] }, { "cell_type": "code", "execution_count": 92, "id": "dental-tours", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import random\n", "import numpy as np\n", "import pandas as pd\n", "import seaborn as sns\n", "import matplotlib.pyplot as plt\n", "\n", "\n", "\n", "def coin_flips(n):\n", " n = n\n", " heads = [random.uniform(0,1) for i in range(n)]\n", " heads = [1 if i > 0.5 else 0 for i in heads]\n", " flips = np.arange(1,n+1)\n", " proportion = (np.cumsum(heads)/flips)\n", "\n", " df = pd.DataFrame(\n", " {'flips': flips,\n", " 'proportion_heads': proportion\n", " })\n", "\n", " #ax = sns. lineplot(x=df['flips'], y=df['proportion_heads'])\n", " return(df)\n", "\n", "n = 1000\n", "\n", "run1 = coin_flips(n)\n", "run2 = coin_flips(n)\n", "run3 = coin_flips(n)\n", "run4 = coin_flips(n)\n", "\n", "\n", "df = pd.concat([run1, run2, run3, run4], axis=0)\n", "\n", "runs = ['run1']*n + ['run2']*n + ['run3']*n + ['run4']*1000\n", "\n", "df['runs'] = runs\n", "\n", "\n", "ax = sns.lineplot(data = df, x = 'flips', y = 'proportion_heads', hue = 'runs')\n", "\n", "sns.despine()\n", "\n", "#glue(\"frequentist_probability_fig\", ax, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "modern-essence", "metadata": {}, "source": [ "```{glue:figure} frequentist_probability_fig\n", ":figwidth: 600px\n", ":name: fig-frequentist_probability\n", "\n", "An illustration of how frequentist probability works. If you flip a fair coin over and over again, the proportion of heads that you’ve seen eventually settles down, and converges to the true probability of 0.5. Each line shows the result of one of four different simulated experiments: in each case, we pretend we flipped a coin 1000 times, and kept track of the proportion of flips that were heads as we went along. Although none of these sequences actually ended up with an exact value of .5, if we’d extended the experiment for an infinite number of coin flips they would have.\n", "```\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "referenced-termination", "metadata": {}, "source": [ "The frequentist definition of probability has some desirable characteristics. Firstly, it is objective: the probability of an event is *necessarily* grounded in the world. The only way that probability statements can make sense is if they refer to (a sequence of) events that occur in the physical universe. [^note1] Secondly, it is unambiguous: any two people watching the same sequence of events unfold, trying to calculate the probability of an event, must inevitably come up with the same answer. However, it also has undesirable characteristics. Firstly, infinite sequences don't exist in the physical world. Suppose you picked up a coin from your pocket and started to flip it. Every time it lands, it impacts on the ground. Each impact wears the coin down a bit; eventually, the coin will be destroyed. \n", "\n", "So, one might ask whether it really makes sense to pretend that an \"infinite\" sequence of coin flips is even a meaningful concept, or an objective one. We can't say that an \"infinite sequence\" of events is a real thing in the physical universe, because the physical universe doesn't allow infinite anything. More seriously, the frequentist definition has a narrow scope. There are lots of things out there that human beings are happy to assign probability to in everyday language, but cannot (even in theory) be mapped onto a hypothetical sequence of events. For instance, if a meteorologist comes on TV and says, \"the probability of rain in Adelaide on 2 November 2048 is 60\\%\" we humans are happy to accept this. But it's not clear how to define this in frequentist terms. There's only one city of Adelaide, and only 2 November 2048. There's no infinite sequence of events here, just a once-off thing. Frequentist probability genuinely *forbids* us from making probability statements about a single event. From the frequentist perspective, it will either rain tomorrow or it will not; there is no \"probability\" that attaches to a single non-repeatable event. \n", "\n", "Now, it should be said that there are some very clever tricks that frequentists can use to get around this. One possibility is that what the meteorologist means is something like this: \"There is a category of days for which I predict a 60\\% chance of rain; if we look only across those days for which I make this prediction, then on 60\\% of those days it will actually rain\". It's very weird and counterintuitive to think of it this way, but you do see frequentists do this sometimes. And it *will* come up [later in this book](ci).\n", "\n", "[^note1]: This doesn't mean that frequentists can't make hypothetical statements, of course; it's just that if you want to make a statement about probability, then it must be possible to redescribe that statement in terms of a sequence of potentially observable events, and the relative frequencies of different outcomes that appear within that sequence." ] }, { "attachments": {}, "cell_type": "markdown", "id": "greenhouse-roulette", "metadata": {}, "source": [ "### The Bayesian view\n", "\n", "The **_Bayesian view_** of probability is often called the subjectivist view, and it is a minority view among statisticians, but one that has been steadily gaining traction for the last several decades [^note2]. There are many flavours of Bayesianism, making hard to say exactly what \"the\" Bayesian view is. The most common way of thinking about subjective probability is to define the probability of an event as the **_degree of belief_** that an intelligent and rational agent assigns to that truth of that event. From that perspective, probabilities don't exist in the world, but rather in the thoughts and assumptions of people and other intelligent beings.\n", "\n", "However, in order for this approach to work, we need some way of operationalising \"degree of belief\". One way that you can do this is to formalise it in terms of \"rational gambling\", though there are many other ways. Suppose that I believe that there's a 60\\% probability of rain tomorrow. If someone offers me a bet: if it rains tomorrow, then I win 5 dollars, but if it doesn't rain then I lose 5 dollars. Clearly, from my perspective, this is a pretty good bet. On the other hand, if I think that the probability of rain is only 40\\%, then it's a bad bet to take. Thus, we can operationalise the notion of a \"subjective probability\" in terms of what bets I'm willing to accept. \n", "\n", "What are the advantages and disadvantages to the Bayesian approach? The main advantage is that it allows you to assign probabilities to any event you want to. You don't need to be limited to those events that are repeatable. The main disadvantage (to many people) is that we can't be purely objective -- specifying a probability requires us to specify an entity that has the relevant degree of belief. This entity might be a human, an alien, a robot, or even a statistician, but there has to be an intelligent agent out there that believes in things. To many people this is uncomfortable: it seems to make probability arbitrary. While the Bayesian approach does require that the agent in question be rational (i.e., obey the rules of probability), it does allow everyone to have their own beliefs; I can believe the coin is fair and you don't have to, even though we're both rational. The frequentist view doesn't allow any two observers to attribute different probabilities to the same event: when that happens, then at least one of them must be wrong. The Bayesian view does not prevent this from occurring. Two observers with different background knowledge can legitimately hold different beliefs about the same event. In short, where the frequentist view is sometimes considered to be too narrow (forbids lots of things that that we want to assign probabilities to), the Bayesian view is sometimes thought to be too broad (allows too many differences between observers). \n", "\n", "[^note2]: Translator's note: This is even more true today than when *Learning Statistics with R* was originally published. Bayesian methods are becoming increasingly common and less controversial in many fields." ] }, { "attachments": {}, "cell_type": "markdown", "id": "dramatic-respondent", "metadata": {}, "source": [ "### What's the difference? And who is right?\n", "\n", "Now that you've seen each of these two views independently, it's useful to make sure you can compare the two. Go back to the hypothetical robot soccer game at the start of the section. What do you think a frequentist and a Bayesian would say about these three statements? Which statement would a frequentist say is the correct definition of probability? Which one would a Bayesian do? Would some of these statements be meaningless to a frequentist or a Bayesian? If you've understood the two perspectives, you should have some sense of how to answer those questions.\n", "\n", "Okay, assuming you understand the difference, you might be wondering which of them is *right*? Honestly, I don't know that there is a right answer. As far as I can tell there's nothing mathematically incorrect about the way frequentists think about sequences of events, and there's nothing mathematically incorrect about the way that Bayesians define the beliefs of a rational agent. In fact, when you dig down into the details, Bayesians and frequentists actually agree about a lot of things. Many frequentist methods lead to decisions that Bayesians agree a rational agent would make. Many Bayesian methods have very good frequentist properties. \n", "\n", "For the most part, I'm a pragmatist so I'll use any statistical method that I trust. As it turns out, that makes me prefer Bayesian methods, for reasons I'll explain towards the end of the book, but I'm not fundamentally opposed to frequentist methods. Not everyone is quite so relaxed. For instance, consider Sir Ronald Fisher, one of the towering figures of 20th century statistics and a vehement opponent to all things Bayesian, whose paper on the mathematical foundations of statistics referred to Bayesian probability as \"an impenetrable jungle [that] arrests progress towards precision of statistical concepts\" {cite}`Fisher1922b`. Or the psychologist Paul Meehl, who suggests that relying on frequentist methods could turn you into \"a potent but sterile intellectual rake who leaves in his merry path a long train of ravished maidens but no viable scientific offspring\" {cite}`Meehl1967`. The history of statistics, as you might gather, is not devoid of entertainment.\n", "\n", "In any case, while I personally prefer the Bayesian view, the majority of statistical analyses are based on the frequentist approach. My reasoning is pragmatic: the goal of this book is to cover roughly the same territory as a typical undergraduate stats class in psychology, and if you want to understand the statistical tools used by most psychologists, you'll need a good grasp of frequentist methods. I promise you that this isn't wasted effort. Even if you end up wanting to switch to the Bayesian perspective, you really should read through at least one book on the \"orthodox\" frequentist view. Besides, I won't completely ignore the Bayesian perspective. Every now and then I'll add some commentary from a Bayesian point of view, and I'll revisit the topic in more depth in the chapter on [Bayesian Statistics](bayes).\n", "\n", "\n", " " ] }, { "attachments": {}, "cell_type": "markdown", "id": "green-shirt", "metadata": {}, "source": [ " \n", " \n", "(basicprobability)=\n", "## Basic probability theory\n", "\n", "Ideological arguments between Bayesians and frequentists notwithstanding, it turns out that people mostly agree on the rules that probabilities should obey. There are lots of different ways of arriving at these rules. The most commonly used approach is based on the work of Andrey Kolmogorov, one of the great Soviet mathematicians of the 20th century. I won't go into a lot of detail, but I'll try to give you a bit of a sense of how it works. And in order to do so, I'm going to have to talk about my pants.\n", "\n", "### Introducing probability distributions\n", "\n", "One of the disturbing truths about my life is that I only own 5 pairs of pants: three pairs of jeans, the bottom half of a suit, and a pair of tracksuit pants. Even sadder, I've given them names: I call them $X_1$, $X_2$, $X_3$, $X_4$ and $X_5$. I really do: that's why they call me Mister Imaginative. Now, on any given day, I pick out exactly one of pair of pants to wear. Not even I'm so stupid as to try to wear two pairs of pants, and thanks to years of training I never go outside without wearing pants anymore. If I were to describe this situation using the language of probability theory, I would refer to each pair of pants (i.e., each $X$) as an **_elementary event_**. The key characteristic of elementary events is that every time we make an observation (e.g., every time I put on a pair of pants), then the outcome will be one and only one of these events. Like I said, these days I always wear exactly one pair of pants, so my pants satisfy this constraint. Similarly, the set of all possible events is called a **_sample space_**. Granted, some people would call it a \"wardrobe\", but that's because they're refusing to think about my pants in probabilistic terms. Sad. \n", "\n", "Okay, now that we have a sample space (a wardrobe), which is built from lots of possible elementary events (pants), what we want to do is assign a **_probability_** of one of these elementary events. For an event $X$, the probability of that event $P(X)$ is a number that lies between 0 and 1. The bigger the value of $P(X)$, the more likely the event is to occur. So, for example, if $P(X) = 0$, it means the event $X$ is impossible (i.e., I never wear those pants). On the other hand, if $P(X) = 1$ it means that event $X$ is certain to occur (i.e., I always wear those pants). For probability values in the middle, it means that I sometimes wear those pants. For instance, if $P(X) = 0.5$ it means that I wear those pants half of the time. \n", "\n", "At this point, we're almost done. The last thing we need to recognise is that \"something always happens\". Every time I put on pants, I really do end up wearing pants (crazy, right?). What this somewhat trite statement means, in probabilistic terms, is that the probabilities of the elementary events need to add up to 1. This is known as the **_law of total probability_**, not that any of us really care. More importantly, if these requirements are satisfied, then what we have is a **_probability distribution_**. For example, this is an example of a probability distribution\n", "\n", "|Which pants |Blue jeans |Grey jeans |Black jeans |Black suit |Blue tracksuit |\n", "|:-----------|:-------------|:-------------|:-------------|:------------|:--------------|\n", "|Label |$X_1$ |$X_2$ |$X_3$ |$X_4$ |$X_5$ |\n", "|Probability |$P(X_1) = .5$ |$P(X_2) = .3$ |$P(X_3) = .1$ |$P(X_4) = 0$ |$P(X_5) = .1$ |\n", "\n", "Each of the events has a probability that lies between 0 and 1, and if we add up the probability of all events, they sum to 1. Awesome. We can even draw a nice bar graph to visualise this distribution, as shown in {numref}`fig-pants`. And at this point, we've all achieved something. You've learned what a probability distribution is, and I've finally managed to find a way to create a graph that focuses entirely on my pants. Everyone wins!" ] }, { "cell_type": "code", "execution_count": 93, "id": "editorial-metabolism", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "\n", "import pandas as pd\n", "import seaborn as sns\n", "\n", "df = pd.DataFrame(\n", " {'probabilities': [.5, .3, .1, 0, .1],\n", " 'eventNames': [\"Blue jeans\", \"Grey jeans\", \"Black jeans\", \"Black suit\", \"Blue track\"],\n", " }) \n", "\n", "fig = sns.barplot(x='eventNames', y='probabilities', data=df)\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "loving-porter", "metadata": {}, "source": [ "```{glue:figure} pants_fig\n", ":figwidth: 600px\n", ":name: fig-pants\n", "\n", "A visual depiction of the “pants” probability distribution. There are five “elementary events”, corresponding to the five pairs of pants that I own. Each event has some probability of occurring: this probability is a number between 0 to 1. The sum of these probabilities is 1.\n", "```\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "verified-silence", "metadata": {}, "source": [ "The only other thing that I need to point out is that probability theory allows you to talk about **_non elementary events_** as well as elementary ones. The easiest way to illustrate the concept is with an example. In the pants example, it's perfectly legitimate to refer to the probability that I wear jeans. In this scenario, the \"Dan wears jeans\" event said to have happened as long as the elementary event that actually did occur is one of the appropriate ones; in this case \"blue jeans\", \"black jeans\" or \"grey jeans\". In mathematical terms, we defined the \"jeans\" event $E$ to correspond to the set of elementary events $(X_1, X_2, X_3)$. If any of these elementary events occurs, then $E$ is also said to have occurred. Having decided to write down the definition of the $E$ this way, it's pretty straightforward to state what the probability $P(E)$ is: we just add everything up. In this particular case\n", "\n", "$$\n", "P(E) = P(X_1) + P(X_2) + P(X_3)\n", "$$ \n", "\n", "and, since the probabilities of blue, grey and black jeans respectively are .5, .3 and .1, the probability that I wear jeans is equal to .9. \n", "\n", "At this point you might be thinking that this is all terribly obvious and simple and you'd be right. All we've really done is wrap some basic mathematics around a few common sense intuitions. However, from these simple beginnings it's possible to construct some extremely powerful mathematical tools. I'm definitely not going to go into the details in this book, but what I will do is list some of the other rules that probabilities satisfy, in the table below. These rules can be derived from the simple assumptions that I've outlined above, and they are important if you want to understand probability theory a bit more deeply, but since we don't actually use these rules for anything in this book, I won't do so here.\n", "\n", "|English |Notation |Formula |\n", "|:-----------|:------------ |:---------------------------|\n", "|Not $A$ |$P(\\neg A)$ |$1-P(A)$ |\n", "|$A$ or $B$ |$P(A \\cup B)$ |$P(A) + P(B) - P(A \\cap B)$ |\n", "|$A$ and $B$ |$P(A \\cap B)$ |$P(A\\|B) \\cdot P(B)$ |\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "congressional-guarantee", "metadata": {}, "source": [ "(binomial)=\n", "\n", "## The binomial distribution\n", "As you might imagine, probability distributions vary enormously, and there's an enormous range of distributions out there. However, they aren't all equally important. In fact, the vast majority of the content in this book relies on one of five distributions: the binomial distribution, the normal distribution, the $t$ distribution, the $\\chi^2$ (\"chi-square\") distribution and the $F$ distribution. Given this, what I'll do over the next few sections is provide a brief introduction to all five of these, paying special attention to the binomial and the normal. I'll start with the binomial distribution, since it's the simplest of the five.\n", "\n", "### Introducing the binomial\n", "\n", "The theory of probability originated in the attempt to describe how games of chance work, so it seems fitting that our discussion of the **_binomial distribution_** should involve a discussion of rolling dice and flipping coins. Let's imagine a simple \"experiment\": in my hot little hand I'm holding 20 identical six-sided dice. On one face of each die there's a picture of a skull; the other five faces are all blank. If I proceed to roll all 20 dice, what's the probability that I'll get exactly 4 skulls? Assuming that the dice are fair, we know that the chance of any one die coming up skulls is 1 in 6; to say this another way, the skull probability for a single die is approximately $0.167$. This is enough information to answer our question, so let's have a look at how it's done. \n", "\n", "As usual, we'll want to introduce some names and some notation. We'll let $N$ denote the number of dice rolls in our experiment; which is often referred to as the **_size parameter_** of our binomial distribution. We'll also use $\\theta$ to refer to the the probability that a single die comes up skulls, a quantity that is usually called the **_success probability_** of the binomial.[^note3] Finally, we'll use $X$ to refer to the results of our experiment, namely the number of skulls I get when I roll the dice. Since the actual value of $X$ is due to chance, we refer to it as a **_random variable_**. In any case, now that we have all this terminology and notation, we can use it to state the problem a little more precisely. The quantity that we want to calculate is the probability that $X = 4$ given that we know that $\\theta = .167$ and $N=20$. The general \"form\" of the thing I'm interested in calculating could be written as \n", "\n", "$$\n", "P(X \\ | \\ \\theta, N)\n", "$$\n", "\n", "and we're interested in the special case where $X=4$, $\\theta = .167$ and $N=20$. \n", "There's only one more piece of notation I want to refer to before moving on to discuss the solution to the problem. If I want to say that $X$ is generated randomly from a binomial distribution with parameters $\\theta$ and $N$, the notation I would use is as follows:\n", "\n", "$$\n", "X \\sim \\mbox{Binomial}(\\theta, N)\n", "$$ \n", "\n", "Yeah, yeah. I know what you're thinking: notation, notation, notation. Really, who cares? Very few readers of this book are here for the notation, so I should probably move on and talk about how to use the binomial distribution. I've included the formula for the binomial distribution below, since some readers may want to play with it themselves, but since most people probably don't care that much and because we don't need the formula in this book, I won't talk about it in any detail. Instead, I just want to show you what the binomial distribution looks like. To that end, {numref}`fig-skulls` plots the binomial probabilities for all possible values of $X$ for our dice rolling experiment, from $X=0$ (no skulls) all the way up to $X=20$ (all skulls). Note that this is basically a bar chart, and is no different to the \"pants probability\" plot I drew in {numref}`fig-pants`. On the horizontal axis we have all the possible events, and on the vertical axis we can read off the probability of each of those events. So, the probability of rolling 4 skulls out of 20 times is about 0.20 (the actual answer is 0.2022036, as we'll see in a moment). In other words, you'd expect that to happen about 20\\% of the times you repeated this experiment.\n", "\n", "[^note3]: Note that the term \"success\" is pretty arbitrary, and doesn't actually imply that the outcome is something to be desired. If $\\theta$ referred to the probability that any one passenger gets injured in a bus crash, I'd still call it the success probability, but that doesn't mean I want people to get hurt in bus crashes!" ] }, { "cell_type": "code", "execution_count": 134, "id": "16efd688", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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H1D6k3+7M7cdJTh25/Vf2v7sNST6a5O4jal/R1101dLutfSbJfZNclOTb/f1eI+uf07fh9iTzXrI4R/1f9L//K5Kcm2TPkfVv7msvS3Jhkl8dWjvx2KuTVJL7jdz2G5P8YGIfOHpMfb/8j/pjwFVJ3jZy+x+b2PZ1SS4bWX9Ikotn/n+SHDay/uAkX+3/Bz+d5N7z1DePN2P2v19SVTv0je5N7u8AD6K7fPZy4KAR9UcChwIbFrn9fYBD++l7Ad8auf0A9+yn7wZ8DThiEe14FfAR4LxF1F4H3G8r/gYfBF7ST+8G7LkVf8ubgF8fUbMv8F3gHv38x4EXDax9OLAB2J3ugo6/Aw5czD4DvA04rZ8+DXjryPqH0X0Q9AvAykVs/2nArv30Wxex/XtPTL8cWD20tl++gu6ik+/Nty/Nse03Aq8e+Ddr1T+x/9v9Sj9//zH1sx5/B/CGkdu/EDiqnz4a+MLI+nXA4/vpE4E3z1PfPN6M2f8mbzvDGcT/H/Kjqm4FZobtGKSqvgT8cLEbr6obq+rr/fRPgKvpDlpD66uqftrP3q2/jbqyIMl+wDHA+8bUbQv9q50jgfcDVNWtVfXPi3y6JwPfqarvjazbFbhHkl3pDvZDP1PzMODiqvpZVd0GfBF45kJFc+wzx9EFJf39M8bUV9XVVTXfKAEL1V/Y/wwAF9N9tmhM/Y8nZvdg7g+uzvX/8t+A/zRX3YD6QeaofyndiA0/79fZvJjtJwnwe8BHR9YXMPOq/z7Ms//NUf8Q4Ev99EXAs+epn+t4M3j/m7QzBMS+wPUT85sYcYDelpLsDzyK7ixgTN2y/rR2M3BRVY2qB95F9895+8i6GQVcmOSSdEObjPEgYAvwgb6L631J9lhkO45nnn/Olqr6AfB24PvAjXSftblwYPkG4MgkeyfZne7V34oFaubygOo+40N/f/9FPs+2cCLw2bFFSf5rkuuB5wFvGFF3LPCDqrp87DYnnNJ3cZ05uHvkX/0G8LgkX0vyxSS/tcg2PA64uaq+PbLuVOAv+t/d24HXjqzfABzbTz+HgfvgrOPNova/nSEgBg/bMdVGJPcEPgGcOuvV2IKq6v9W1SF0r/oOS/LwEdt9OrC5qi4Zs81ZHlNVh9KNvvuyJEeOqN2V7pT59Kp6FPAvdKe4o6T7wOSxwP8aWbcX3aunA4BfBfZI8vwhtVV1NV13zEXA+XTdk7fNW7SdS/I6up/hb8bWVtXrqmpFX3uHcdHm2N7uwOsYESgNpwMPBg6hC/l3jKzfFdgLOAL4Y+Dj/dnAWM9l5AuU3kuBV/a/u1fSn02PcCLd/90ldN1Gty5UsDXHm0k7Q0AMGfJjqpLcje6P9TdVdc5in6fvmvkC0BxYaw6PAY5Nch1d99qTkpw1crs39PebgXPpuu2G2gRsmjjr+Vu6wBjrKODrVXXzyLqnAN+tqi1V9QvgHODfDS2uqvdX1aFVdSTdqf/YV48zbk43UjH9/ZzdHNOS5IXA04HnVd8ZvUgfYZ5ujlkeTBfOl/f74H7A15M8cOjGqurm/kXS7cB7Gbf/QbcPntN31/4D3Zn0nG+Ut/Tdk88CPjZy2wAvpNvvoHuBM6r9VfXNqnpaVT2aLqC+s0BbW8ebRe1/O0NALOmwHf0rlfcDV1fVOxdRv3zmipMk96A74H1zaH1Vvbaq9quq/el+9v9dVYNeQffb3CPJvWam6d7sHHxFV1XdBFyf5CH9oiezuGHbF/vq7fvAEUl27/8WT6brlx0kyf37+1+jO0Aspg3Q7XMv7KdfCHxqkc+zKElW0Y2UfGxV/WwR9QdOzB7LwH2wqq6sqvtX1f79PriJ7k3Um0Zse5+J2WcyYv/rfRJ4Uv9cv0F3ocTY0VGfAnyzqjaNrIPuBenj++knMfJFxsQ+uAvwn4HV86w71/FmcfvfkHey7+o3ur7jb9El7+tG1n6U7rT2F3Q794tH1j+WrkvrCuCy/nb0iPpHApf29RuY5wqKAc/1BEZexUT3HsLl/e2qsb+//jkOAdb3P8Mngb1G1u8O/CNwn0X+3G+iO6BtAD5MfzXLwNov0wXa5cCTF7vPAHsDn6M7OHwOuO/I+mf20z8HbgYuGFm/ke69uJl9sHkV0jz1n+h/f1cAnwb2Xcz/CwtcETfHtj8MXNlvew2wz8j63YCz+vZ/HXjSmPp++f8ETl7k3/6xwCX9PvQ14NEj619Bd/z6FvAW+hEw5qhvHm/G7H+TN4fakCQ17QxdTJKkRTAgJElNBoQkqcmAkCQ1GRCSpCYDQtoG+lE+B3/4anL9JD9daH1pKRgQkqQmA0Iaqf90+WfSfUfHhiT/fuKxeyQ5P8kfJnlRkr+aeOy8JE+Y53n3SfKl/nsDNiR53HR/Eml+BoQ03irghqo6uKoeTjeQH8A96T5l/JGqeu8invf36T4hfQhwMN2nYKUlY0BI410JPCXJW5M8rqp+1C//FPCBqvrQIp93HXBCkjcCj6huPH9pyRgQ0khV9S3g0XRB8edJZoay/nvgqImhpG/jl//H5v2q0+q+LOZI4AfAh5O8YJs2XBrJgJBGSvd9zD+rqrPovgBmZvjyN9ANKvjX/fx1wCFJdkmyggWGeU7y63Tf3fFeuhE5FzMsurTN7LrUDZDugh5B9w1ht9ONuvlSuu+5gO7bw85M8ja64bW/S3emMTOS6HyeAPxxkl8APwU8g9CScjRXSVKTXUySpCYDQpLUZEBIkpoMCElSkwEhSWoyICRJTQaEJKnp/wFkXjc/bf1cSQAAAABJRU5ErkJggg==", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import pandas as pd\n", "import seaborn as sns\n", "from scipy.stats import binom\n", "\n", "N = 20\n", "r = list(range(0,21,1))\n", "p = 1/6\n", "\n", "y = binom.pmf(r,N,p)\n", "\n", "df = pd.DataFrame(\n", " {'probability': y,\n", " 'skulls': np.arange(0,21,1),\n", " }) \n", "\n", "fig = sns.barplot(x='skulls', y='probability', data=df)\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "verbal-weekend", "metadata": {}, "source": [ "```{glue:figure} skulls_fig\n", ":figwidth: 600px\n", ":name: fig-skulls\n", "\n", "The binomial distribution with size parameter of N = 20 and an underlying success probability of θ = 1/6. Each vertical bar depicts the probability of one specific outcome (i.e., one possible value of X). Because this is a probability distribution, each of the probabilities must be a number between 0 and 1, and the heights of the bars must sum to 1 as well.\n", "```\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "tracked-burden", "metadata": {}, "source": [ "By the way, although it appears from the figure that it is impossible to roll more than 10 skulls in one throw of 20 dice, we intuitively know this can't be right. Surely it is _possible_, although unlikely, to roll e.g. 20 sixes in one throw of 20 dice. Indeed, it is possible, just so unlikely that the probability is too small to represent with even a single pixel in our figure, so it appears to be zero.\n", "\n", "Below, you can see the formulas for the binomial and normal distributions. We don't really use these formulas for anything in this book, but they're pretty important for more advanced work, so I thought it might be best to put them here anyway. In the equation for the binomial, $X!$ is the factorial function (i.e., multiply all whole numbers from 1 to $X$), and for the normal distribution \\\"exp\\\" refers to the exponential function. If these equations don't make a lot of sense to you, don't worry too much about them.\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "ready-jewel", "metadata": {}, "source": [ "(binomial_normal_formulas)=\n", "\n", "The formula for the binomial distribution looks like this:\n", "\n", "$$\n", "P(X | \\theta, N) = \\frac{N!}{X! (N-X)!} \\theta^X (1-\\theta)^{N-X}\n", "$$ \n", "\n", "\n", "\n", "and the formula for the normal distribution looks like this:\n", "\n", "$$\n", "p(X | \\mu, \\sigma) = \\frac{1}{\\sqrt{2\\pi}\\sigma} \\exp \\left( -\\frac{(X - \\mu)^2}{2\\sigma^2} \\right)\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "mediterranean-content", "metadata": {}, "source": [ "### Working with the binomial distribution in Python\n", "\n", "\n", "Although some people find it handy to know the above formulas, most people just want to know how to use the distributions without worrying too much about the maths. To that end, the `scipy` module has a method called `binom.pmf` that calculates binomial probabilities for us. The arguments to the function are \n", "\n", "- `k`. The number of success you are interested in.\n", "- `n`. The number of attempts\n", "- `p`. This is the success probability for any one trial in the experiment.\n", "\n", "So, in order to calculate the probability of getting `k = 4` skulls, from an experiment of `n = 20` trials, in which the probability of getting a skull on any one trial is `p = 1/6` ... well, the command I would use is simply this:\n", "\n" ] }, { "cell_type": "code", "execution_count": 140, "id": "a455ff52", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The probability of rolling 4 skulls with 20 dice is 0.20220358121717238\n" ] } ], "source": [ "from scipy.stats import binom\n", "p = binom.pmf(k=4, n=20, p=1/6)\n", "print('The probability of rolling 4 skulls with 20 dice is', p)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "d8181983", "metadata": {}, "source": [ "\n", "\n", "To give you a feel for how the binomial distribution changes when we alter the values of $\\theta$ and $N$, let's suppose that instead of rolling dice, I'm actually flipping coins. This time around, my experiment involves flipping a fair coin repeatedly, and the outcome that I'm interested in is the number of heads that I observe. In this scenario, the success probability is now $\\theta = 1/2$. Suppose I were to flip the coin $N=20$ times. In this example, I've changed the success probability, but kept the size of the experiment the same. What does this do to our binomial distribution? Well, as {numref}`binomial-fig` shows, the main effect of this is to shift the whole distribution, as you'd expect. Okay, what if we flipped a coin $N=100$ times? Well, in that case, the distribution stays roughly in the middle, but there's a bit more variability in the possible outcomes. " ] }, { "cell_type": "code", "execution_count": 143, "id": "77ab902d", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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VSVZ0le80cnJ/8/yktsZeNlySJEmSJmeYCWMv95Iay7Orah86U1qPT/Kciby4lw2XJEmSpMkZZsLYy72ktqqq7mie7wIupDPFFeDOJDsDNM93DSRaSZIkSdKDDDNh7OVeUq2SbJ/kMSPLwAuB65vNq4FjmuVjgE8NNGpJkiRJEjDE22r0ci+pJE8G1gE7AA8kOZHOFVWfCFyYZCTGj1bVJU3XJwHnJnkN8B3gpcN6D5IkSZrZDjr/A1uWLz78tQPt++BPfGLL8kUv9Sun5qeh3oexh3tJfZfOVNXR7gH22kqf3wcOGGCYkiRJkqQWw5ySKkmSJEmaxUwYJUmSJEmtTBglSZIkSa1MGCVJkiRJrUwYJUmSJEmtTBglSZIkSa1MGCVJkiRJrUwYJUmSJEmtTBglSZIkSa1MGCVJkiRJrUwYJUmSJEmtTBglSZIkSa1MGCVJkiRJrUwYJUmSJEmtTBglSZIkSa1MGCVJkiRJrUwYJUmaQZIsTXJzkg1JVrZsT5JTmu3XJtmna9uOSc5LclOSG5P89tRGL0maa0wYJUmaIZIsAE4FlgF7Akcl2XNUtWXA4uaxAjita9v7gEuq6hnAXsCNQw9akjSnmTBKkjRz7AtsqKpbquo+4Bxg+ag6y4GzquMKYMckOyfZAXgO8EGAqrqvqn40hbFLkuYgE0ZJkmaOXYDbutY3NmW91PklYDPwoSTfSPKBJNsPM1hJ0txnwihJ0syRlrLqsc42wD7AaVX1LOB/gLZzIFckWZdk3ebNmycbryRpjjNhlCRp5tgILOpa3xW4o8c6G4GNVfW1pvw8Ognkg1TV6VW1pKqWLFy4cGCBS5LmJhNGSZJmjrXA4iR7JNkWOBJYParOauDo5mqp+wN3V9WmqvoucFuSpzf1DgBumLLIJUlz0jbTHYAkSeqoqvuTnABcCiwAzqiq9UmOa7avAtYABwIbgHuBY7u6+BPgI02yecuobZIkTZgJoyRJM0hVraGTFHaXrepaLuD4rbS9GlgyzPgkSfOLU1IlSZIkSa08wihJkqRZ46Dz/33L8sWHv27KXvfF512wZfnTR7xkyl5Xmm5DPcKYZGmSm5NsSNJ2ae9nJPlqkp8meVNX+aIkn09yY5L1Sd7Qte2tSW5PcnXzOHCY70GSJEmS5quhHWFMsgA4FXgBnUt9r02yuqq6r9j2A+D1wKGjmt8PvLGqrkryGODKJJd3tX1PVZ08rNglSZIkScM9wrgvsKGqbqmq+4BzgOXdFarqrqpaC/xsVPmmqrqqWf4xcCOwyxBjlSRJkiSNMsyEcRfgtq71jfSR9CXZHXgW8LWu4hOSXJvkjCSPm1SUkiRJkqRWw0wY01JWE+ogeTRwPnBiVd3TFJ8GPA3YG9gEvGsrbVckWZdk3ebNmyfyspIkSZIkhpswbgQWda3vCtzRa+MkD6eTLH6kqrZclqqq7qyqn1fVA8D76Ux9fYiqOr2qllTVkoULF/b1BiRJkiRpPhtmwrgWWJxkjyTbAkcCq3tpmCTAB4Ebq+rdo7bt3LV6GHD9gOKVJEmSJHUZ2lVSq+r+JCcAlwILgDOqan2S45rtq5I8GVgH7AA8kOREYE/gN4BXAtclubrp8q+qag3wziR705neeiswdTfgkSRJkqR5ZGgJI0CT4K0ZVbaqa/m7dKaqjvYV2s+BpKpeOcgYJUmSJEnthjklVZIkSZI0i5kwSpIkSZJaDXVKqjTXHPjJlX21W3PoSQOOZGZaft6lfbf91BEvGmAkkiRJGgSPMEqSJEmSWpkwSpIkSZJamTBKkiRJklqZMEqSJEmSWpkwSpIkSZJamTBKkiRJklqZMEqSJEmSWpkwSpIkSZJamTBKkiRJklqZMEqSNIMkWZrk5iQbkqxs2Z4kpzTbr02yT9e2W5Ncl+TqJOumNnJJ0ly0zXQHIEmSOpIsAE4FXgBsBNYmWV1VN3RVWwYsbh77Aac1zyOeV1Xfm6KQJUlznEcYJUmaOfYFNlTVLVV1H3AOsHxUneXAWdVxBbBjkp2nOlBJ0vxgwihJ0syxC3Bb1/rGpqzXOgVcluTKJCuGFqUkad5wSqokSTNHWspqAnWeXVV3JHkScHmSm6rqSw9q3EkkVwDstttuk41XkjTHeYRRkqSZYyOwqGt9V+COXutU1cjzXcCFdKa4PkhVnV5VS6pqycKFCwcYuiRpLjJhlCRp5lgLLE6yR5JtgSOB1aPqrAaObq6Wuj9wd1VtSrJ9kscAJNkeeCFw/VQGL0mae3pKGJOcn+SgJCaYkiT1oJ+xs6ruB04ALgVuBM6tqvVJjktyXFNtDXALsAF4P/DHTflOwFeSXAN8Hbi4qi4Z0NuRJM1TvZ7DeBpwLHBKkk8A/1FVNw0vLEmSZr2+xs6qWkMnKewuW9W1XMDxLe1uAfaabNCSJHXrKWGsqs8Cn03yWOAoOifS30bnl82zq+pnQ4xRkqRZx7FTGoyDLvi3LcsXv+SPx6g5tQ4571NbllcfMfruN9Lc0fM0mSRPAF4FvBb4BvA+YB/g8qFEJknSLOfYKUma7Xo6wpjkAuAZwIeBF1fVpmbTx5OsG1ZwkiTNVo6dkqS5oNdzGD/QnFOxRZJHVNVPq2rJEOKSJGm2c+yUJM16vU5JfXtL2VcHGYgkSXOMY6ckadYb8whjkicDuwCPSvIsIM2mHYDthhybJEmzjmOnJGkuGW9K6ovonKy/K/DurvIfA381pJgkSZrNHDslSXPGmAljVZ0JnJnk8Ko6f6KdJ1lK54pwC+icy3HSqO3PAD5E54pxb6mqk8drm+TxwMeB3YFbgZdV1Q8nGpskScMw2bFTkqSZZLwpqa+oqrOB3ZP82ejtVfXulmYjbRcApwIvADYCa5Osrqobuqr9AHg9cOgE2q4EPldVJyVZ2az/5bjvVJKkKTCZsVOSpJlmvIvebN88Pxp4TMtjLPsCG6rqlqq6DzgHeNBdTavqrqpaC4y+efFYbZcDZzbLZzIq2ZQkaZpNZuyUJGlGGW9K6r83z2/ro+9dgNu61jcC+w2g7U4j97Kqqk1JntRHbJIkDcUkx05JkmaU8aaknjLW9qp6/VjN25r0EtQk23Y6SFYAKwB22223iTSVJKlvkxw7JUmaUca7SuqVk+h7I7Coa31X4I4BtL0zyc7N0cWdgbvaOqiq04HTAZYsWTKhZFOSpEmYzNgpSdKM0stVUvu1FlicZA/gduBI4A8H0HY1cAxwUvP8qUnEKEnSQE1y7JQkaUYZb0rqe6vqxCSfpmVKaFUdsrW2VXV/khOAS+ncGuOMqlqf5Lhm+6rm5sbr6NzM+IEkJwJ7VtU9bW2brk8Czk3yGuA7wEsn9pYlSRqeyYydkiTNNONNSf1w83zymLW2oqrWAGtGla3qWv4unemmPbVtyr8PHNBPPJIkTYFJjZ2SJM0k401JvbJ5/mKSbYFn0Pm19ObmdheSJKmLY6ckaS4Z7wgjAEkOAlYB36RzBdM9kryuqj4zzOAkSZqtHDslSXNBTwkj8C7geVW1ASDJ04CLAQc9SZLaOXZKkma9h/VY766RAa9xC1u5nYUkSQIcOyVJc8B4V0l9SbO4Pska4Fw652G8lM6tLyRJUhfHTknSXDLelNQXdy3fCTy3Wd4MPG4oEUmSNLs5dkqS5ozxrpJ67FQFIknSXODYKUmaS3q9SuojgdcAvwY8cqS8ql49pLgkSZrVHDslSXNBrxe9+TDwZOBFwBeBXYEfDysoSZLmgL7GziRLk9ycZEOSlS3bk+SUZvu1SfYZtX1Bkm8kuWhA70OSNI/1mjD+clX9DfA/VXUmcBDw68MLS5KkWW/CY2eSBcCpwDJgT+CoJHuOqrYMWNw8VgCnjdr+BuDGyYcvSVLvCePPmucfJXkm8Fhg96FEJEnS3NDP2LkvsKGqbqmq+4BzgOWj6iwHzqqOK4Adk+wMkGRXOonpBwb0HiRJ81yvCePpSR4H/A2wGrgBeMfQopIkafbrZ+zcBbita31jU9ZrnfcCfwE80F/IkiQ9WE8XvamqkV8qvwj80vDCkSRpbuhz7ExbV73USXIwcFdVXZnk97f6AskKOlNZ2W233XoMS5I0X/V0hDHJE5L8S5KrklyZ5L1JnjDs4CRJmq36HDs3Aou61ncF7uixzrOBQ5LcSmcq6/OTnD36Barq9KpaUlVLFi5cOMF3JUmab3qdknoOcBdwOHAE8D3g48MKSpKkOaCfsXMtsDjJHkm2BY6kM52122rg6OZqqfsDd1fVpqp6c1XtWlW7N+3+s6peMcD3I0mah3qakgo8vqr+oWv97UkOHUI8kiTNFRMeO6vq/iQnAJcCC4Azqmp9kuOa7auANcCBwAbgXuDYYQQvTaeDLviXrrUF0xbHRBxy3i/uZLP6iIOnMRJpsHpNGD+f5Ejg3Gb9CODi4YQkSdKc0NfYWVVr6CSF3WWrupYLOH6cPr4AfGFi4UqS9FBjJoxJfkznZPsAfwaMnAvxMOAnwN8NNTpJkmYZx05J0lwyZsJYVY+ZqkAkSZoLHDslSXNJr1NSSXII8Jxm9QtVddFY9SVJmu8cOyVJs12vt9U4CXgDnZsO3wC8oSmTJEktHDslSXNBr0cYDwT2rqoHAJKcCXwDWDmswCRJmuUcOyVJs16v92EE2LFr+bEDjkOSpLlox65lx05J0qzT6xHG/w18I8nn6Vz17TnAm4cWlSRJs59jpyRp1hs3YUzyMOABYH/gt+gMen9ZVd8dcmySJM1Kjp2SpLli3ISxqh5IckJVnQusnoKYJEma1Rw7JUlzRa/nMF6e5E1JFiV5/MhjqJFJkjS7OXZKkma9Xs9hfDVQwB+PKv+lwYYjSdKc4dgpSZr1ej3CuCdwKnANcDXwL8CvjdcoydIkNyfZkOQhlxFPxynN9muT7NOUPz3J1V2Pe5Kc2Gx7a5Lbu7Yd2ON7kCRpKvU1dkqSNJP0eoTxTOAe4JRm/aim7GVba5BkAZ2B8gXARmBtktVVdUNXtWXA4uaxH3AasF9V3Qzs3dXP7cCFXe3eU1Un9xi7JEnTYcJjpyRJM02vCePTq2qvrvXPJ7lmnDb7Ahuq6haAJOcAy4HuhHE5cFZVFXBFkh2T7FxVm7rqHAB8s6q+3WOskiTNBP2MnZIkzSi9Tkn9RpL9R1aS7Af8n3Ha7ALc1rW+sSmbaJ0jgY+NKjuhmcJ6RpLHjRe8JEnToJ+xU5KkGaXXhHE/4P9NcmuSW4GvAs9Ncl2Sa7fSJi1lNZE6SbYFDgE+0bX9NOBpdKasbgLe1friyYok65Ks27x581ZClCRpaPoZOyVJmlF6nZK6tI++NwKLutZ3Be6YYJ1lwFVVdedIQfdykvcDF7W9eFWdDpwOsGTJktGJquaZYy/s5yMMHzrskgFHol4cfv7avtuef/hvDTASaVL6+8MjSdIM0lPC2Of5g2uBxUn2oHPRmiOBPxxVZzWd6aXn0Pkl9u5R5y8exajpqKPOcTwMuL6P2CRJGirPvZckzQW9HmGcsKq6P8kJwKXAAuCMqlqf5Lhm+ypgDXAgsAG4Fzh2pH2S7ehcYfV1o7p+Z5K96UxdvbVluyRJkiRpAIaWMAJU1Ro6SWF32aqu5QKO30rbe4EntJS/csBhSpIkSZJa9HrRG0mSJEnSPGPCKEmSJElqZcIoSZIkSWplwihJ0gySZGmSm5NsSLKyZXuSnNJsvzbJPk35I5N8Pck1SdYnedvURy9JmmtMGCVJmiGSLABOpXMf4j2Bo5LsOaraMmBx81gBnNaU/xR4flXtBewNLE2y/1TELUmau0wYJUmaOfYFNlTVLVV1H3AOsHxUneXAWdVxBbBjc4/iqqqfNHUe3jxqyiKXJM1JJoySJM0cuwC3da1vbMp6qpNkQZKrgbuAy6vqa8MLVZI0Hwz1PoySJGlC0lI2+ijhVutU1c+BvZPsCFyY5JlVdf2DGicr6ExlZbfddpt0wNIgHHTBe7YsX/ySP53GSAZn+Xmf2bL8qSOWTWMk0uR4hFGSpJljI7Coa31X4I6J1qmqHwFfAJaOfoGqOr2qllTVkoULFw4gZEnSXGbCKEnSzLEWWJxkjyTbAkcCq0fVWQ0c3VwtdX/g7qralGRhc2SRJI8C/gC4aQpjlyTNQU5JlSRphqiq+5OcAFwKLADOqKr1SY5rtq8C1gAHAhuAe4Fjm+Y7A2c2V1p9GHBuVV001e9BkjS3mDBKkjSDVNUaOklhd9mqruUCjm9pdy3wrKEHKEmaV5ySKkmSJElqZcIoSZIkSWplwihJkiRJamXCKEmSJElqZcIoSZIkSWplwihJkiRJamXCKEmSJElqZcIoSZIkSWplwihJkiRJamXCKEmSJElqZcIoSZIkSWplwihJkiRJamXCKEmSJElqZcIoSZIkSWplwihJkiRJamXCKEmSJElqNdSEMcnSJDcn2ZBkZcv2JDml2X5tkn26tt2a5LokVydZ11X++CSXJ/nv5vlxw3wPkiRJkjRfDS1hTLIAOBVYBuwJHJVkz1HVlgGLm8cK4LRR259XVXtX1ZKuspXA56pqMfC5Zl2SJEmSNGDDPMK4L7Chqm6pqvuAc4Dlo+osB86qjiuAHZPsPE6/y4Ezm+UzgUMHGLMkSZIkqTHMhHEX4Lau9Y1NWa91CrgsyZVJVnTV2amqNgE0z09qe/EkK5KsS7Ju8+bNk3gbkiRJkjQ/bTPEvtNSVhOo8+yquiPJk4DLk9xUVV/q9cWr6nTgdIAlS5aMfl1JkiRNk4Mu/Octyxcf9ufTGMnUOPS8y7Ysf/KIF05jJNLEDfMI40ZgUdf6rsAdvdapqpHnu4AL6UxxBbhzZNpq83zXwCOXJEmSJA01YVwLLE6yR5JtgSOB1aPqrAaObq6Wuj9wd1VtSrJ9kscAJNkeeCFwfVebY5rlY4BPDfE9SJIkSdK8NbQpqVV1f5ITgEuBBcAZVbU+yXHN9lXAGuBAYANwL3Bs03wn4MIkIzF+tKouabadBJyb5DXAd4CXDus9SJIkSdJ8NsxzGKmqNXSSwu6yVV3LBRzf0u4WYK+t9Pl94IDBRipJ0syQZCnwPjo/tn6gqk4atT3N9gPp/Nj6qqq6Kski4CzgycADwOlV9b4pDV6SNOcMc0qqJEmagEnew/h+4I1V9avA/sDxLW0lSZoQE0ZJkmaOvu9hXFWbquoqgKr6MXAjD72dlSRJE2LCKEnSzDHZexgDkGR34FnA1wYfoiRpPjFhlCRp5pjsPYxJ8mjgfODEqrrnIS+QrEiyLsm6zZs3TypYSdLcZ8IoSdLMMal7GCd5OJ1k8SNVdUHbC1TV6VW1pKqWLFy4cGCBS5LmJhNGSZJmjsncwzjAB4Ebq+rdUxu2JGmuGuptNSRJUu8meQ/jZwOvBK5LcnVT9lfNLa4kSeqLCaMkSTPIJO5h/BXaz2+UJKlvTkmVJEmSJLUyYZQkSZIktTJhlCRJkiS1MmGUJEmSJLUyYZQkSZIktfIqqZqx3nHOi/pu+5dHXjrASDRbvf7C2/pue8phi8avJEmSNMd5hFGSJEmS1MqEUZIkSZLUyimpkiRJGqoDL/ynLctrDnvzNEYyMxx63n9uWf7kEc+fxkik8XmEUZIkSZLUyoRRkiRJktTKhFGSJEmS1MqEUZIkSZLUyoRRkiRJktTKhFGSJEmS1MqEUZIkSZLUyoRRkiRJktTKhFGSJEmS1MqEUZIkSZLUaqgJY5KlSW5OsiHJypbtSXJKs/3aJPs05YuSfD7JjUnWJ3lDV5u3Jrk9ydXN48BhvgdJkiRJmq+2GVbHSRYApwIvADYCa5OsrqobuqotAxY3j/2A05rn+4E3VtVVSR4DXJnk8q6276mqk4cVuyRJkiRpuEcY9wU2VNUtVXUfcA6wfFSd5cBZ1XEFsGOSnatqU1VdBVBVPwZuBHYZYqySJEmSpFGGmTDuAtzWtb6RhyZ949ZJsjvwLOBrXcUnNFNYz0jyuIFFLEmSJEnaYpgJY1rKaiJ1kjwaOB84saruaYpPA54G7A1sAt7V+uLJiiTrkqzbvHnzBEOXJGl69Hv+f7PtjCR3Jbl+aqOWJM1Vw0wYNwKLutZ3Be7otU6Sh9NJFj9SVReMVKiqO6vq51X1APB+OlNfH6KqTq+qJVW1ZOHChZN+M5IkDVvX+f/LgD2Bo5LsOapa9/n/K+j8kDriP4Clw49UkjRfDO2iN8BaYHGSPYDbgSOBPxxVZzWd6aXn0LnYzd1VtSlJgA8CN1bVu7sbjJzj2KweBvgrqiRprthy/j9AMz4uB7ovGLfl/H/giiTd5/9/qTmVQ5p2B174tq61bactjtngsPO/uGX5wsOfO42RSA81tISxqu5PcgJwKbAAOKOq1ic5rtm+ClgDHAhsAO4Fjm2aPxt4JXBdkqubsr+qqjXAO5PsTWfq6q3A64b1HiRJmmJt5/bv10OdXeicpiFJ0kAN8wgjTYK3ZlTZqq7lAo5vafcV2s9vpKpeOeAwJUmaKSZ9/v+4L5CsoDOVld122633yCRJ89Iwz2GUJEkTM6nz/3vhOf6SpIkwYZQkaebYcv5/km3pnP+/elSd1cDRzdVS96c5/3+qA5UkzQ8mjJIkzRBVdT8wcv7/jcC5I+f/j1wDgM6pHrfQOf///cAfj7RP8jHgq8DTk2xM8popfQOSpDlnqOcwSpKkien3/P9m21HDjU6SNN94hFGSJEmS1MqEUZIkSZLUyoRRkiRJktTKhFGSJEmS1MqEUZIkSZLUyqukSpIkadIO/ORbtiyvOfQfpzGS2e8l5/+fLcsXHP7saYxEMmHUEHzwrBf23fY1R182wEikwTnzgs19tTvmJQsHHIkkSdLUcUqqJEmSJKmVCaMkSZIkqZUJoyRJkiSplQmjJEmSJKmVCaMkSZIkqZUJoyRJkiSplbfVkCRJ0oQd+Mk3bVlec+jJ0xjJ3Hb4+V/bsnz+4ftNYySarzzCKEmSJElqZcIoSZIkSWplwihJkiRJamXCKEmSJElqZcIoSZIkSWrlVVIlSZI0rmWfOn7L8meWnzqNkcxvR5x/1Zbl8w7fZxoj0XzhEUZJkiRJUiuPMGqLT56xrO+2h776MwOMRJq7PvfRzX21O+APFw44EkmSpPF5hFGSJEmS1MojjJIkSXqIZZ86esvyZ5afNY2RaCwvPf+6LcufOPzXpzESzVVDPcKYZGmSm5NsSLKyZXuSnNJsvzbJPuO1TfL4JJcn+e/m+XHDfA+SJE2lYYydkiT1a2hHGJMsAE4FXgBsBNYmWV1VN3RVWwYsbh77AacB+43TdiXwuao6qRkMVwJ/Oaz3IUnSVBni2Cn1ZOnqg7YshydMYyTq18vOv2nL8rmHP2MaI9FcMcwpqfsCG6rqFoAk5wDLge6BazlwVlUVcEWSHZPsDOw+RtvlwO837c8EvsA8Txi//P6D+277e//PRQOMRNJUuf7f7+yr3TNft9OAI9GADWvslFq94pNLtyyffegl0xiJhuHIC761Zfmcl+wxjZFoNhtmwrgLcFvX+kY6v4SOV2eXcdruVFWbAKpqU5InDTJoSZKm0bDGTs1TbzzvFwnhu464hD+64Bfrp73EBHG+Oe7CX/yJWHXYIlZeePuW9ZMO22U6QtIskM4PlEPoOHkp8KKqem2z/kpg36r6k646FwP/VFVfadY/B/wF8Etba5vkR1W1Y1cfP6yqh5zHmGQFsKJZfTpw8zghPxH4Xl9v1n6mqy/7mZ/9DLIv+5l7/XyvqpaOU2fGGtbYOeo12sbHQf7/nA/cX71zX/XOfTUx7q/ejeyrvsbIYR5h3Ags6lrfFbijxzrbjtH2ziQ7N0cXdwbuanvxqjodOL3XYJOsq6olvda3n+nvy37mZz8zMSb7mV39zHDDGju3aBsf58m+HRj3V+/cV71zX02M+6t3k91Xw7xK6lpgcZI9kmwLHAmsHlVnNXB0c8W3/YG7m+mmY7VdDRzTLB8DfGqI70GSpKk0rLFTkqS+DO0IY1Xdn+QE4FJgAXBGVa1PclyzfRWwBjgQ2ADcCxw7Vtum65OAc5O8BvgO8NJhvQdJkqbSEMdOSZL6MswpqVTVGjoDW3fZqq7lAo7vtW1T/n3ggMFGCkxg+qr9zJi+7Gd+9jPIvuxnfvYzow1j7OzBvNi3A+T+6p37qnfuq4lxf/VuUvtqaBe9kSRJkiTNbsM8h1GSJEmSNIvN+4QxydIkNyfZkGTlJPo5I8ldSa6fZDyLknw+yY1J1id5Q5/9PDLJ15Nc0/TztknGtSDJN5JcNIk+bk1yXZKrk6ybRD87JjkvyU3NfvrtPvp4ehPHyOOeJCf2Gc+fNvv4+iQfS/LIPvt5Q9PH+onG0vb5S/L4JJcn+e/m+SG3n+mxn5c2MT2QpKcrbG2ln39u/s2uTXJhkh377Ocfmj6uTnJZkqf000/XtjclqSRPnMR7e2uS27s+Twf2G1OSP2n+Jq1P8s4+4/l4Vyy3Jrm6z372TnLFyP/ZJPv22c9eSb7a/P//dJIdeuin9W9hP59rjS0DGgfnIj+HEzf6+4L7auvS8n3G/dWu7buW++oXtjL2bnX/JHlz8zf/5iQvGvcFqmrePuhcFOCbdO5dtS1wDbBnn309B9gHuH6SMe0M7NMsPwb4r35iAgI8ull+OPA1YP9JxPVnwEeBiybRx63AEwfw73Ym8NpmeVtgxwF8Dr4LPLWPtrsA3wIe1ayfC7yqj36eCVwPbEfn3OLPAosn8/kD3gmsbJZXAu/os59fpXOvti8ASyYRzwuBbZrld0winh26ll8PrOqnn6Z8EZ0LhHy718/mVmJ6K/CmCf6bt/XzvObf/hHN+pP6fW9d298F/G2f8VwGLGuWDwS+0Gc/a4HnNsuvBv6hh35a/xb287n2MeZ+Htg4OBcffg772mcP+r7gvhpzXz3k+4z7q3U/tX7Xcl89aB/1/D2w+Rt2DfAIYI9mDFgwVv/z/QjjvsCGqrqlqu4DzgGW99NRVX0J+MFkA6qqTVV1VbP8Y+BGOv9RJtpPVdVPmtWHN4++TlhNsitwEPCBftoPUnNk4jnABwGq6r6q+tEkuz0A+GZVfbvP9tsAj0qyDZ2E7yH3PevBrwJXVNW9VXU/8EXgsF4bb+Xzt5zOYETzfGg//VTVjVV1c6+xjNHPZc17A7iCzj3i+unnnq7V7enhcz3G/8/30Lnhec//Nwb4f72tnz8CTqqqnzZ1Wu8z22s8SQK8DPhYn/0UMHI08LH08NneSj9PB77ULF8OHN5DP1v7Wzjhz7XGNLBxcC7yczgxW/m+4L5qMcb3GfdXu7bvWu6rxgS/By4Hzqmqn1bVt+hccXvMGUTzPWHcBbita30jfSRnw5Jkd+BZdI4O9tN+QTMV7S7g8qrqqx/gvXS+VD/QZ/sRBVyW5MokK/rs45eAzcCHmikvH0iy/STjOpIevlC3qarbgZPp3OJlE537oV3WR1fXA89J8oQk29E5mrNonDbj2ak692ajeX7SJPsbpFcDn+m3cZJ/THIb8HLgb/vs4xDg9qq6pt84RjkhnamyZ0xiWsyvAL+X5GtJvpjktyYZ0+8Bd1bVf/fZ/kTgn5t9fTLw5j77uR44pFl+KRP8bI/6WziTP9ez0YweB2cSP4c9eS8P/b7gvmq3te8z7q9Rxviu5b4a29b2z4T/7s/3hDEtZTPisrFJHg2cD5w46ohKz6rq51W1N50jOfsmeWYfcRwM3FVVV/YTwyjPrqp9gGXA8Ume00cf29A55H5aVT0L+B86h9n7ks7NrQ8BPtFn+8fR+aVmD+ApwPZJXjHRfqrqRjrTNC8HLqEzVeD+MRvNUkneQue9faTfPqrqLVW1qOnjhD5i2A54C30mmy1OA54G7E1nMHtXn/1sAzwO2B/4czr3nG37O9Wro+jzx5DGHwF/2uzrP6X5JbwPr6bzf/5KOtP67uu14SD+FmpMM3YcnEn8HI5vwN8X5oOBfp+Zywb1XUtbTPjv/nxPGDfy4F+6d6W/6YQDleThdAamj1TVBZPtr5ni8AVgaR/Nnw0ckuRWOlOVnp/k7D7juKN5vgu4kHEOf2/FRmBj19HS8+j8we3XMuCqqrqzz/Z/AHyrqjZX1c+AC4Df6aejqvpgVe1TVc+hM62g36NCI+5MsjNA8zzu9MZhS3IMcDDw8mom0k/SR+lhemOLp9EZeK5pPtu7AlcleXI/QVTVnc0PNA8A76e/zzZ0Pt8XNFPKv07nV/qeLsYzWjNt5yXAx/uMBeAYOp9p6Pyo0tf7qqqbquqFVfWbdBLYb/bSbit/C2fc53qWm5Hj4Ezi57BnW/u+4L5qt7XvM+6vh9rady331di2tn8m/Hd/vieMa4HFSfZojjQdCayezoCaowkfBG6sqndPop+Faa5CmeRRdP6z3TTRfqrqzVW1a1XtTmf//GdVTfhXnSTbJ3nMyDKdC6BM+IqyVfVd4LYkT2+KDgBumGg/XSZ7BOY7wP5Jtmv+7Q6gc47LhCV5UvO8G50v+pOJCzqf5WOa5WOAT02yv0lJshT4S+CQqrp3Ev0s7lo9hP4+19dV1ZOqavfms72RzoUtvttnTDt3rR5GH5/txieB5zd9/gqdiyB8r8++/gC4qao29tkeOgPIc5vl59Pnjxhdn+2HAX8NrBq7xZh/C2fU53oOmHHj4Ezi57B3Y3xfcF+1GOP7jPvrobb2Xct9Nbat7Z/VwJFJHpFkD2Ax8PUxe6oZcGWf6XzQOVfsv+j84v2WSfTzMTpT0X5G58vna/rs53fpHBa+Fri6eRzYRz+/AXyj6ed6erhKYg99/j59XiWVzlz9a5rH+knu672Bdc17+yTwuD772Q74PvDYSe6Xt9FJWq4HPkxzhcs++vkyncHiGuCAyX7+gCcAn6PzJf9zwOP77OewZvmnwJ3ApX32s4HOnPmRz3UvVzdt6+f8Zl9fC3wa2KWffkZtv5Xer5LaFtOHgeuamFYDO/fZz7bA2c37uwp4fr/vDfgP4LhJfoZ+F7iy+Ux+DfjNPvt5A52/s/8FnASkh35a/xb287n2Me6+Hsg4OBcffg773m+/zy+ukuq+2vp+2ptR32fcX1vdVw/5ruW+etD+mdD3QDqn5XwTuJnmauhjPdI0kiRJkiTpQeb7lFRJkiRJ0laYMEqSJEmSWpkwSpIkSZJamTBKkiRJklqZMEqSJEmSWpkwSjNckt2T9Htfv7H6vTVJXzeFlyRpujk+SlPDhFGSJEmS1MqEUZodFiR5f5L1SS5L8qgkT0tySZIrk3w5yTMAkrw4ydeSfCPJZ5Ps1JQ/oWn7jST/DqQp3z7JxUmuSXJ9kv81je9TkqSJcHyUhsyEUZodFgOnVtWvAT8CDgdOB/6kqn4TeBPwb03drwD7V9WzgHOAv2jK/w74SlO+GtitKV8K3FFVe1XVM4FLpuD9SJI0CI6P0pBtM90BSOrJt6rq6mb5SmB34HeATyQZqfOI5nlX4ONJdga2Bb7VlD8HeAlAVV2c5IdN+XXAyUneAVxUVV8e4vuQJGmQHB+lIfMIozQ7/LRr+efA44EfVdXeXY9fbbb/C/CvVfXrwOuAR3a1rdEdV9V/Ab9JZ2D8pyR/O5R3IEnS4Dk+SkNmwijNTvcA30ryUoB07NVseyxwe7N8TFebLwEvb+ovAx7XLD8FuLeqzgZOBvYZfviSJA2F46M0YCaM0uz1cuA1Sa4B1gPLm/K30pmK82Xge1313wY8J8lVwAuB7zTlvw58PcnVwFuAtw8/dEmShsbxURqgVD3kCLwkSZIkSR5hlCRJkiS1M2GUJEmSJLUyYZQkSZIktTJhlCRJkiS1MmGUJEmSJLUyYZQkSZIktTJhlCRJkiS1MmGUJEmSJLX6/wGItL9tIRXFwQAAAABJRU5ErkJggg==", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "from matplotlib import pyplot as plt\n", "\n", "\n", "from scipy.stats import binom\n", "\n", "step = 1\n", "\n", "n1 = 20\n", "n2 = 100\n", "\n", "r1 = list(range(0,21,step))\n", "r2 = list(range(0,101,step))\n", "\n", "p1 = 1/2\n", "p2 = 1/2\n", "\n", "y1 = binom.pmf(r1,n1,p1)\n", "y2 = binom.pmf(r2,n2,p2)\n", "\n", "df1 = pd.DataFrame(\n", " {'probability': y1,\n", " 'heads': r1,\n", " }) \n", "\n", "\n", "df2 = pd.DataFrame(\n", " {'probability': y2,\n", " 'heads': r2,\n", " }) \n", "\n", "\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 5))\n", "fig.suptitle('Coin Flips')\n", "\n", "\n", "axes[0].set_title('20 flips')\n", "axes[1].set_title('100 flips')\n", "\n", "ax1 = sns.barplot(x='heads', y='probability', data=df1, ax=axes[0])\n", "ax2 = sns.barplot(x='heads', y='probability', data=df2, ax=axes[1])\n", "\n", "ax2.set_xticks([0, 20, 40, 60,80, 100])\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "approved-stuart", "metadata": {}, "source": [ "```{glue:figure} fig-binomial\n", ":figwidth: 600px\n", ":name: binomial-fig\n", "\n", "The binomial distribution with size parameter of N = 20 and an underlying success probability of θ = 1/2. Compare these plots to those in {numref}`fig-skulls`, which also had a size parameter of N = 20 and an underlying success probability of θ = 1/6.\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "exposed-structure", "metadata": {}, "source": [ "(normal)=\n", "## The normal distribution\n", "\n", "While the binomial distribution is conceptually the simplest distribution to understand, it's not the most important one. That particular honour goes to the **_normal distribution_**, which is also referred to as \"the bell curve\" or a \"Gaussian distribution\". A normal distribution is described using two parameters, the mean of the distribution $\\mu$ and the standard deviation of the distribution $\\sigma$. The notation that we sometimes use to say that a variable $X$ is normally distributed is as follows:\n", "\n", "$$\n", "X \\sim \\mbox{Normal}(\\mu,\\sigma)\n", "$$\n", "\n", "Of course, that's just notation. It doesn't tell us anything interesting about the normal distribution itself. As was the case with the binomial distribution, I have included the formula for the normal distribution in this book, because I think it's important enough that everyone who learns statistics should at least look at it, but since this is an introductory text I don't want to focus on it, so I've just quietly put them in the text [above](binomial_normal_formulas).\n", "\n", "Instead of focusing on the maths, let's try to get a sense for what it means for a variable to be normally distributed. To that end, have a look at {numref}`fig-normal`, which plots a normal distribution with mean $\\mu = 0$ and standard deviation $\\sigma = 1$. You can see where the name \"bell curve\" comes from: it looks a bit like a bell. Notice that, unlike the plots that I drew to illustrate the binomial distribution, the picture of the normal distribution in {numref}`fig-normal` shows a smooth curve instead of \"histogram-like\" bars. This isn't an arbitrary choice: the normal distribution is continuous, whereas the binomial is discrete. For instance, in the die rolling example from the last section, it was possible to get 3 skulls or 4 skulls, but impossible to get 3.9 skulls. The figures that I drew in the previous section reflected this fact: in {numref}`fig-skulls`, for instance, there's a bar located at $X=3$ and another one at $X=4$, but there's nothing in between. Continuous quantities don't have this constraint. For instance, suppose we're talking about the weather. The temperature on a pleasant Spring day could be 23 degrees, 24 degrees, 23.9 degrees, or anything in between since temperature is a continuous variable, and so a normal distribution might be quite appropriate for describing Spring temperatures. In fact, the normal distribution is so handy that people tend to use it even when the variable isn't actually continuous. As long as there are enough categories (e.g., Likert scale responses to a questionnaire), it's pretty standard practice to use the normal distribution as an approximation. This works out much better in practice than you'd think." ] }, { "cell_type": "code", "execution_count": 98, "id": "tutorial-eagle", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/text/plain": "" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "normal_fig" } }, "output_type": "display_data" }, { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import scipy.stats as stats\n", "import seaborn as sns\n", "\n", "mu = 0\n", "variance = 1\n", "sigma = np.sqrt(variance)\n", "x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)\n", "\n", "fig = sns.lineplot(x = x, y = stats.norm.pdf(x, mu, sigma))\n", "plt.xlabel('Observed Value')\n", "plt.ylabel('Probability Density')\n", "\n", "sns.despine()\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "narrow-reminder", "metadata": {}, "source": [ " ```{glue:figure} normal_fig\n", ":figwidth: 600px\n", ":name: fig-normal\n", "\n", "The normal distribution with mean $mu = 0$ and standard deviation $sigma = 1$. The $x$-axis corresponds to the value of some variable, and the $y$-axis tells us something about how likely we are to observe that value. However, notice that the $y$-axis is labelled \"Probability Density\" and not \"Probability\". There is a subtle and somewhat frustrating characteristic of continuous distributions that makes the $y$ axis behave a bit oddly: the height of the curve here isn't actually the probability of observing a particular $x$ value. On the other hand, it *is* true that the heights of the curve tells you which $x$ values are more likely (the higher ones!).\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "political-ballot", "metadata": {}, "source": [ "With this in mind, let's see if we can't get an intuition for how the normal distribution works. Firstly, let's have a look at what happens when we play around with the parameters of the distribution. To that end, {numref}`fig-two-normals` plots normal distributions that have different means, but have the same standard deviation. As you might expect, all of these distributions have the same \"width\". The only difference between them is that they've been shifted to the left or to the right. In every other respect they're identical. " ] }, { "cell_type": "code", "execution_count": 144, "id": "innovative-strap", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "\n", "import numpy as np\n", "import scipy.stats as stats\n", "import pandas as pd\n", "import seaborn as sns\n", "\n", "\n", "mu = 0\n", "variance = 1\n", "sigma = np.sqrt(variance)\n", "x = np.linspace(1, 10, 100)\n", "y1 = stats.norm.pdf(x, 4, sigma)\n", "y2 = stats.norm.pdf(x, 7, sigma)\n", "\n", "\n", "fig = sns.lineplot(x = x, y = y1)\n", "ax2=fig.twinx()\n", "ax2.tick_params(left=False, labelleft=False, top=False, labeltop=False,\n", " right=False, labelright=False, bottom=False, labelbottom=False)\n", "\n", "\n", "\n", "sns.lineplot(x = x, y = y2, ax=ax2, linestyle='--')\n", "sns.despine()\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "molecular-collective", "metadata": {}, "source": [ " ```{glue:figure} two-normals_fig\n", ":figwidth: 600px\n", ":name: fig-two-normals\n", "\n", "An illustration of what happens when you change the mean of a normal distribution. The solid line depicts a normal distribution with a mean of $mu=4$. The dashed line shows a normal distribution with a mean of $mu=7$. In both cases, the standard deviation is $sigma=1$. Not surprisingly, the two distributions have the same shape, but the dashed line is shifted to the right.\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "cloudy-malta", "metadata": {}, "source": [ "In contrast, if we increase the standard deviation while keeping the mean constant, the peak of the distribution stays in the same place, but the distribution gets wider, as you can see in {numref}`fig-two-normals2`. Notice, though, that when we widen the distribution, the height of the peak shrinks. This has to happen: in the same way that the heights of the bars that we used to draw a discrete binomial distribution have to *sum* to 1, the total *area under the curve* for the normal distribution must equal 1. " ] }, { "cell_type": "code", "execution_count": 254, "id": "capable-somewhere", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/text/plain": "" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "two-normals_fig2" } }, "output_type": "display_data" }, { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "\n", "import numpy as np\n", "import scipy.stats as stats\n", "import pandas as pd\n", "import seaborn as sns\n", "\n", "\n", "mu = 5\n", "\n", "variance = 1\n", "sigma = np.sqrt(variance)\n", "x = np.linspace(1, 10, 100)\n", "y1 = stats.norm.pdf(x, mu, 1)\n", "\n", "variance = 2\n", "sigma = np.sqrt(variance)\n", "y2 = stats.norm.pdf(x, mu, 2)\n", "\n", "\n", "fig = sns.lineplot(x = x, y = y1)\n", "\n", "ax2=fig.twiny()\n", "ax2.tick_params(left=False, labelleft=False, top=False, labeltop=False,\n", " right=False, labelright=False, bottom=False, labelbottom=False)\n", "\n", "\n", "\n", "sns.lineplot(x = x, y = y2, ax=ax2, linestyle='--')\n", "sns.despine()\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "supposed-fitting", "metadata": {}, "source": [ " ```{glue:figure} two-normals_fig2\n", ":figwidth: 600px\n", ":name: fig-two-normals2\n", "\n", "An illustration of what happens when you change the the standard deviation of a normal distribution. Both distributions plotted in this figure have a mean of μ “ 5, but they have different standard deviations. The solid line plots a distribution with standard deviation σ “ 1, and the dashed line shows a distribution with standard deviation σ “ 2. As a consequence, both distributions are “centred” on the same spot, but the dashed line is wider than the solid one.\n", "```\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "ahead-evans", "metadata": {}, "source": [ "Before moving on, I want to point out one important characteristic of the normal distribution. Irrespective of what the actual mean and standard deviation are, 68.3\\% of the area falls within 1 standard deviation of the mean. Similarly, 95.4\\% of the distribution falls within 2 standard deviations of the mean, and 99.7\\% of the distribution is within 3 standard deviations. This idea is illustrated in {numref}`fig-sdnorm` and {numref}`fig-sdnorm2`." ] }, { "cell_type": "code", "execution_count": 145, "id": "progressive-insertion", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "\n", "import numpy as np\n", "import scipy.stats as stats\n", "import seaborn as sns\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 5), sharey=False)\n", "\n", "mu = 0\n", "variance = 1\n", "sigma = np.sqrt(variance)\n", "x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)\n", "sns.lineplot(x = x, y = stats.norm.pdf(x, mu, sigma), ax=axes[0])\n", "\n", "\n", "sns.lineplot(x = x, y = stats.norm.pdf(x, mu, sigma), ax=axes[1])\n", "\n", "\n", "x_fill1 = np.arange(-1, 1, 0.001)\n", "x_fill2 = np.arange(-2, 2, 0.001)\n", "\n", "y_fill1 = stats.norm.pdf(x_fill1,0,1)\n", "y_fill2 = stats.norm.pdf(x_fill2,0,1)\n", "\n", "axes[0].fill_between(x_fill1,y_fill1,0, alpha=0.2, color='blue')\n", "axes[1].fill_between(x_fill2,y_fill2,0, alpha=0.2, color='blue')\n", "\n", "axes[0].set_title(\"Shaded Area = 68.3%\")\n", "axes[1].set_title(\"Shaded Area = 95.4%\")\n", "\n", "axes[0].set(xlabel='Observed value', ylabel='Probability density')\n", "axes[1].set(xlabel='Observed value', ylabel='Probability density')\n", "\n", "sns.despine()\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "after-scanner", "metadata": {}, "source": [ " ```{glue:figure} sdnorm_fig\n", ":figwidth: 600px\n", ":name: fig-sdnorm\n", "\n", "The area under the curve tells you the probability that an observation falls within a particular range. The solid lines plot normal distributions with mean $mu=0$ and standard deviation $sigma=1$ The shaded areas illustrate \\\"areas under the curve\\\" for two important cases. On the left, we can see that there is a 68.3% chance that an observation will fall within one standard deviation of the mean. On the right, we see that there is a 95.4% chance that an observation will fall within two standard deviations of the mean.\n", "```\n" ] }, { "cell_type": "code", "execution_count": 269, "id": "systematic-peripheral", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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\n", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "sdnorm_fig2" } }, "output_type": "display_data" } ], "source": [ "\n", "import numpy as np\n", "import scipy.stats as stats\n", "import seaborn as sns\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 5), sharey=False)\n", "\n", "mu = 0\n", "variance = 1\n", "sigma = np.sqrt(variance)\n", "x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)\n", "sns.lineplot(x = x, y = stats.norm.pdf(x, mu, sigma), ax=axes[0])\n", "\n", "\n", "sns.lineplot(x = x, y = stats.norm.pdf(x, mu, sigma), ax=axes[1])\n", "\n", "\n", "x_fill1 = np.arange(-4, -1, 0.001)\n", "x_fill2 = np.arange(-1, 0, 0.001)\n", "\n", "y_fill1 = stats.norm.pdf(x_fill1,0,1)\n", "y_fill2 = stats.norm.pdf(x_fill2,0,1)\n", "\n", "axes[0].fill_between(x_fill1,y_fill1,0, alpha=0.2, color='blue')\n", "axes[1].fill_between(x_fill2,y_fill2,0, alpha=0.2, color='blue')\n", "\n", "axes[0].set_title(\"Shaded Area = 15.9%\")\n", "axes[1].set_title(\"Shaded Area = 34.1%\")\n", "\n", "axes[0].set(xlabel='Observed value', ylabel='Probability density')\n", "axes[1].set(xlabel='Observed value', ylabel='Probability density')\n", "\n", "sns.despine()\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "economic-superior", "metadata": {}, "source": [ " ```{glue:figure} sdnorm_fig2\n", ":figwidth: 600px\n", ":name: fig-sdnorm2\n", "\n", "Two more examples of the \\\"area under the curve idea\\\". There is a 15.9% chance that an observation is one standard deviation below the mean or smaller (left), and a 34.1% chance that the observation is greater than one standard deviation below the mean but still below the mean (right). Notice that if you add these two numbers together you get 15.9% + 34.1% = 50%. For normally distributed data, there is a 50% chance that an observation falls below the mean. And of course that also implies that there is a 50% chance that it falls above the mean.\n", "```\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "scenic-volleyball", "metadata": {}, "source": [ "(density)=\n", "### Probability density\n", "\n", "There's something I've been trying to hide throughout my discussion of the normal distribution, something that some introductory textbooks omit completely. They might be right to do so: this \"thing\" that I'm hiding is weird and counterintuitive even by the admittedly distorted standards that apply in statistics. Fortunately, it's not something that you need to understand at a deep level in order to do basic statistics: rather, it's something that starts to become important later on when you move beyond the basics. So, if it doesn't make complete sense, don't worry: try to make sure that you follow the gist of it.\n", "\n", "Throughout my discussion of the normal distribution, there's been one or two things that don't quite make sense. Perhaps you noticed that the $y$-axis in these figures is labelled \"Probability Density\" rather than density. Maybe you noticed that I used $p(X)$ instead of $P(X)$ when giving the formula for the normal distribution. Maybe you're wondering what the `d`in `stats.norm.pdf()` stands for. Ok, probably not. But still, you've probably guessed that `pdf` has little to do with the [Portable Document Format](https://en.wikipedia.org/wiki/PDF). And maybe, just maybe, you've been playing around with the `stats.norm.pdf()` function, and you accidentally typed in a command like this:" ] }, { "cell_type": "code", "execution_count": 192, "id": "solved-magnitude", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3.989422804014327" ] }, "execution_count": 192, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "stats.norm.pdf(1,1,0.1)\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "super-attempt", "metadata": {}, "source": [ "And if you've done the last part, you're probably very confused. I've asked Python to calculate the probability that `x = 1`, for a normally distributed variable with `mean = 1` and standard deviation `sd = 0.1`; and it tells me that the probability is 3.99. But, as we discussed earlier, probabilities *can't* be larger than 1. So either I've made a mistake, or that's not a probability. \n", "\n", "As it turns out, the second answer is correct. What we've calculated here isn't actually a probability: it's something else. To understand what that something is, you have to spend a little time thinking about what it really *means* to say that $X$ is a continuous variable. Let's say we're talking about the temperature outside. The thermometer tells me it's 23 degrees, but I know that's not really true. It's not *exactly* 23 degrees. Maybe it's 23.1 degrees, I think to myself. But I know that that's not really true either, because it might actually be 23.09 degrees. But, I know that... well, you get the idea. The tricky thing with genuinely continuous quantities is that you never really know exactly what they are.\n", "\n", "Now think about what this implies when we talk about probabilities. Suppose that tomorrow's maximum temperature is sampled from a normal distribution with mean 23 and standard deviation 1. What's the probability that the temperature will be *exactly* 23 degrees? The answer is \"zero\", or possibly, \"a number so close to zero that it might as well be zero\". Why is this? It's like trying to throw a dart at an infinitely small dart board: no matter how good your aim, you'll never hit it. In real life you'll never get a value of exactly 23. It'll always be something like 23.1 or 22.99998 or something. In other words, it's completely meaningless to talk about the probability that the temperature is exactly 23 degrees. However, in everyday language, if I told you that it was 23 degrees outside and it turned out to be 22.9998 degrees, you probably wouldn't call me a liar. Because in everyday language, \"23 degrees\" usually means something like \"somewhere between 22.5 and 23.5 degrees\". And while it doesn't feel very meaningful to ask about the probability that the temperature is exactly 23 degrees, it does seem sensible to ask about the probability that the temperature lies between 22.5 and 23.5, or between 20 and 30, or any other range of temperatures. \n", "\n", "The point of this discussion is to make clear that, when we're talking about continuous distributions, it's not meaningful to talk about the probability of a specific value. However, what we *can* talk about is the probability that the value lies within a particular range of values. To find out the probability associated with a particular range, what you need to do is calculate the \"area under the curve\". We've seen this concept already: in {numref}(fig-sdnorm), the shaded areas shown depict genuine probabilities (e.g., in the left hand panel of {numref}(fig-sdnorm) it shows the probability of observing a value that falls within 1 standard deviation of the mean). \n", "\n", "Okay, so that explains part of the story. I've explained a little bit about how continuous probability distributions should be interpreted (i.e., area under the curve is the key thing), but I haven't actually explained what the `stats.norm.pdf()` function actually calculates. Equivalently, what does the formula for $p(x)$ that I described earlier actually mean? Obviously, $p(x)$ doesn't describe a probability, but what is it? The name for this quantity $p(x)$ is a **_probability density_**, and in terms of the plots we've been drawing, it corresponds to the *height* of the curve. The densities themselves aren't meaningful in and of themselves: but they're \"rigged\" to ensure that the *area* under the curve is always interpretable as genuine probabilities. To be honest, that's about as much as you really need to know for now. [^note5]\n", "\n", "[^note5]: For those readers who know a little calculus, I'll give a slightly more precise explanation. In the same way that probabilities are non-negative numbers that must sum to 1, probability densities are non-negative numbers that must integrate to 1 (where the integral is taken across all possible values of $X$). To calculate the probability that $X$ falls between $a$ and $b$ we calculate the definite integral of the density function over the corresponding range, $\\int_a^b p(x) \\ \\mathrm{d}x$. If you don't remember or never learned calculus, don't worry about this. It's not needed for this book." ] }, { "attachments": {}, "cell_type": "markdown", "id": "critical-ability", "metadata": {}, "source": [ "(otherdists)=\n", "## Other useful distributions\n", "\n", "The normal distribution is the distribution that statistics makes most use of (for reasons to be discussed shortly), and the binomial distribution is a very useful one for lots of purposes. But the world of statistics is filled with probability distributions, some of which we'll run into in passing. In particular, the three that will appear in this book are the $t$ distribution, the $\\chi^2$ distribution and the $F$ distribution. I won't give formulas for any of these, or talk about them in too much detail, but I will show you some pictures. \n", "\n", "- The **_$t$ distribution_** is a continuous distribution that looks very similar to a normal distribution, but has heavier tails: see {numref}`fig-tdist`. This distribution tends to arise in situations where you think that the data actually follow a normal distribution, but you don't know the mean or standard deviation. To plot the probability density for a $t$ distribution, we just need `stats.t.pdf()` instead of `stats.norm.pdf()`. We'll run into this distribution again in the chapter on [Comparing Two Means](ttest). " ] }, { "cell_type": "code", "execution_count": 273, "id": "scientific-ceremony", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/text/plain": "" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "tdist-fig" } }, "output_type": "display_data" }, { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "\n", "import numpy as np\n", "import scipy.stats as stats\n", "import seaborn as sns\n", "\n", "\n", "mu = 0\n", "variance = 1\n", "sigma = np.sqrt(variance)\n", "degfree = 3\n", "x = np.linspace(-4, 4, 100)\n", "\n", "y1 = stats.norm.pdf(x, mu, sigma)\n", "\n", "y2 = stats.t.pdf(x, degfree)\n", "\n", "\n", "fig = sns.lineplot(x = x, y = y1, linestyle='--')\n", "ax2 = fig.twiny()\n", "ax2.tick_params(left=False, labelleft=False, top=False, labeltop=False,\n", " right=False, labelright=False, bottom=False, labelbottom=False)\n", "\n", "sns.lineplot(x = x, y = y2, ax=ax2)\n", "fig.set(xlabel='Observed value', ylabel='Probability density')\n", "\n", "sns.despine()\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "round-breast", "metadata": {}, "source": [ " ```{glue:figure} tdist_fig\n", ":figwidth: 600px\n", ":name: fig-tdist\n", "\n", "A $t$ distribution with 3 degrees of freedom (solid line). It looks similar to a normal distribution, but it's not quite the same. For comparison purposes, I've plotted a standard normal distribution as the dashed line. Note that the \\\"tails\\\" of the $t$ distribution are \\\"heavier\\\" (i.e., extend further outwards) than the tails of the normal distribution? That's the important difference between the two.\n", "```\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "environmental-glossary", "metadata": {}, "source": [ "- The **_$\\chi^2$ distribution_** is another distribution that turns up in lots of different places. The situation in which we'll see it is when doing [categorical data analysis](chisquare), but it's one of those things that actually pops up all over the place. When you dig into the maths (and who doesn't love doing that?), it turns out that the main reason why the $\\chi^2$ distribution turns up all over the place is that, if you have a bunch of variables that are normally distributed, square their values and then add them up (a procedure referred to as taking a \"sum of squares\"), this sum has a $\\chi^2$ distribution. You'd be amazed how often this fact turns out to be useful. Anyway, here's what a $\\chi^2$ distribution looks like: {numref}`fig-chi2`. By this point, you probably won't be surprised to find out that we can plot a chi2 distribution using `stats.chi2.pdf()`." ] }, { "cell_type": "code", "execution_count": 274, "id": "resident-passport", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/text/plain": "" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "chi2-fig" } }, "output_type": "display_data" }, { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "\n", "import numpy as np\n", "import scipy.stats as stats\n", "import seaborn as sns\n", "\n", "\n", "degfree = 3\n", "x = np.linspace(0, 10, 100)\n", "\n", "y = stats.chi2.pdf(x, degfree)\n", "\n", "fig = sns.lineplot(x = x, y = y)\n", "fig.set(xlabel='Observed value', ylabel='Probability density')\n", "\n", "sns.despine()\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "conventional-transition", "metadata": {}, "source": [ " ```{glue:figure} chi2_fig\n", ":figwidth: 600px\n", ":name: fig-chi2\n", "\n", "A $chi^2$ distribution with 3 degrees of freedom. Notice that the observed values must always be greater than zero, and that the distribution is pretty skewed. These are the key features of a chi-square distribution.\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "widespread-omaha", "metadata": {}, "source": [ "- The **_$F$ distribution_** looks a bit like a $\\chi^2$ distribution, and it arises whenever you need to compare two $\\chi^2$ distributions to one another. Admittedly, this doesn't exactly sound like something that any sane person would want to do, but it turns out to be very important in real world data analysis. Remember when I said that $\\chi^2$ turns out to be the key distribution when we're taking a \"sum of squares\"? Well, what that means is if you want to compare two different \"sums of squares\", you're probably talking about something that has an $F$ distribution. Of course, as yet I still haven't given you an example of anything that involves a sum of squares, but I will... in the chapter on [comparing several means](ANOVA). And that's where we'll run into the $F$ distribution. Oh, and here's a picture: {numref}`fig-Fdist`. Predictably, we can plot the F distribution using `stats.f.pdf()`." ] }, { "cell_type": "code", "execution_count": 275, "id": "herbal-voluntary", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/text/plain": "" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "Fdist-fig" } }, "output_type": "display_data" }, { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "\n", "import numpy as np\n", "import scipy.stats as stats\n", "import seaborn as sns\n", "\n", "\n", "degfree1 = 3\n", "degfree2 = 5\n", "x = np.linspace(0, 10, 100)\n", "\n", "y = stats.f.pdf(x, degfree1, degfree2)\n", "\n", "\n", "fig = sns.lineplot(x = x, y = y)\n", "fig.set(xlabel='Observed value', ylabel='Probability density')\n", "\n", "sns.despine()\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "inclusive-trademark", "metadata": {}, "source": [ " ```{glue:figure} Fdist_fig\n", ":figwidth: 600px\n", ":name: fig-Fdist\n", "\n", "An $F$ distribution with 3 and 5 degrees of freedom. Qualitatively speaking, it looks pretty similar to a chi-square distribution, but they're not quite the same in general.\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "ordered-hartford", "metadata": {}, "source": [ "Because these distributions are all tightly related to the normal distribution and to each other, and because they are will turn out to be the important distributions when doing inferential statistics later in this book, I think it's useful to do a little demonstration using Python, just to \"convince ourselves\" that these distributions really are related to each other in the way that they're supposed to be. First, we'll use the `random.normal()` function to generate 1000 normally-distributed observations: " ] }, { "cell_type": "code", "execution_count": 1, "id": "twelve-repair", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[-1.67904695 0.01133432 0.02261077 -0.05353278 -0.75496994 0.23851562\n", " -1.40297626 0.4028934 2.31924026 0.68549358]\n" ] } ], "source": [ "import numpy as np\n", "\n", "mu = 0 \n", "sigma = 1\n", "n = 1000\n", "dist = np.random.normal(mu, sigma, n)\n", "\n", "# inspect the first 10 samples\n", "print(dist[0:10])" ] }, { "attachments": {}, "cell_type": "markdown", "id": "excess-colorado", "metadata": {}, "source": [ "So the `dist` variable contains 1000 numbers that are normally distributed, and have mean 0 and standard deviation 1, and the actual print out of these numbers goes on for rather a long time. Next, what we can do is use the `histplot()` function from `seaborn` to draw a histogram of the data, like so:" ] }, { "cell_type": "code", "execution_count": 2, "id": "peripheral-nothing", "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "\n", "sns.histplot(dist)\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "heard-heath", "metadata": {}, "source": [ "If you do this, you should see something similar to the plot above. With a little more work we can do some formatting (see the chapter on [Drawing Graphs](DrawingGraphs)), and also plot the true distribution of the data as a solid black line (i.e., a normal distribution with mean 0 and standard deviation 1), so that you can compare the data that we just generated to the true distribution. " ] }, { "cell_type": "code", "execution_count": 303, "id": "different-dining", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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69CAhIcHtOK5Katye8MgocrJttE7jvWILv6r+DPwMtPdfHGPO7cjOLeTn54d0M0+BiOhYajVqx571S8nPzw+6KSeNbxT7f4mILHXWmSKSUWjJFJEM/0U05ky7Vy8CoFevXu4GCRB1WlyP5ueTmmq9e4x3ii38zh27qGq8qiYUWuJVNbS/XxtX7VyziIjISKpXr+52lIBwUdPrAPjggw9cTmLKC29u4LpcRKKdn28QkT/amD3GLRnpO8jc+zORUTYccYGouHgiIqNISUmx3j3GK940CH4A5InIFcAU4FJgqk9TGVOMgj7rUVb4zxAVHc2PP/7Ixo0b3Y5iygFvCn++quYCtwEvqurDQJJvYxlTtF1rvqLapY0ICw93O0pAiYyKRkSsucd4xZs7d3OcPv2DgYKraZG+i2RM0Y4fSudI2g80u30kGz95g6o1Eot9bUZmph+Tue/Y8ROEh0fw1FPjeOk/r/zq+aSkJDatX+v/YCYgeVP4fwuMAMap6nYRuRR4x7exjPm1XWsWA1C7xfVs+GhKiUM6zHygs59SBQbNz6fJbb9j7cyXuOah/1CxRu0znp87po87wUxA8mY8/u9U9Y+qOs15vF1VJ3izcxHZISIbRGStiKQ626qKyDwR2eqsq1zYr2BCxe61X1HposuIr1nH7SgBqU7z6wHYtfYrl5OYQOdNr55rnQK9RUS2ich2EdlWimN0UtXmqlow9+4oYIGqJgMLnMfGlCgr4zAHf1xP7RbXux0lYFWonkTlusnsWmOF35TMm4u7U4AXgOuANngmT29zAce8FXjL+fktoM8F7MuEiN3rl6KaTx0r/CWq0+IGDm3byMmjNpyWKZ43hf+oqn6mqvtV9VDB4uX+FZgrIqtEZLizLVFV0wGcdc2i3igiw0UkVURSc3NzvTycCVa713xFhWpJVK6T7HaUgFa7eUdQZfe6JW5HMQHMm8L/pYg8KyLtRaRlweLl/q9V1ZZAD2CkiHT0NpiqTlbV1qraOiLCm2vQJljlnDzOvu9Tqd28IyLidpyAVumiy6hYsw67bYx+UwJvKurVzrp1oW2KZ+rFEqnqHme9X0RmAW2BfSKSpKrpIpIE7C9lZhNi0jcuJz83x5p5vCAi1Gl+PT/Mn072iUyi4uLdjmQCkDe9ejoVsZyz6ItIBRGJL/gZ6ApsBD7Ec08AznrO+cc3oWDXmq+Ijq9CtcubuB2lXKjd4no0P48965e5HcUEKG969SSKyBQR+cx53FBEhnqx70RgqYisA1YAn6jq58AE4CYR2Qrc5Dw2pkh5OadI37ic2s06EBZmd+t6o1q9hsRWrs7utYvdjmIClDdNPW8CbwBjnMdbgPfw9PYplqpuA5oVsf0Q0KVUKU3I2vd9KrmnTlCnxQ1uRyk3JCyMi5p1YMfyz8jNziIiKsbtSCbAeHNxt7qqzgDyAZxxe/J8msoYx641XxEZU4GaV7VyO0q5UqfFDeRlZ7H3u2/djmICkDeF/7iIVMNzQRcRaQcc9WkqY4D8vFz2rFtKUtNrCY+w4aFKo+aVLYiKi2f3GmvuMb/mTVPPn/BckL1cRJYBNYB+Pk1lDHDwpw2cOvYLdZp73QvYOMLCI7io6XWeKRnz7D4Yc6ZzFn5VXS0i1wP1AQF+UNUcnyczIW/XmkWER0ZRq1E7t6OUS7VbdGTHN5+xf8sat6OYAFNs4ReR24t56koRQVVTfJTJGFSV3WsXk9iwLZExcW7HKZdqNbya8MhodtvYPeYsJZ3xF4y9XxO4BljoPO4ELAKs8BufycvL5cQvR2jca5jbUcqtiKgYkhq3Z/e6xUSH2x3P5r9Kmmz9t6r6WzwXdRuqal9V7Qs08ls6E7JyTp1CwsJPTyRuzk/t5h05+ctB8my8K1OIN7166hUMqubYB1zpozzGAJCdfYoayc2JrljJ7Sjl2kVNrkHCwsnOPuV2FBNAvCn8i0TkCxG5V0QGA58AX/o4lwlhmzdvJj8vz27aKgNRFRJIvKoVOadOoapuxzEBwpuxeh4AJuG5C7c5MFlV/+DjXCaEpaR4Lh/VaWHdOMtCnRY3kJ+fx4YNG9yOYgKEN2f8qOosVX3YWWb5OpQJbR988AHhERHEVq7hdpSgUNu5D6LgA9UYrwq/Mf6yfft21qxZQ5SNL1NmYhKqEhERyQcffOB2FBMgrPCbgFJwVhoZHe1ykuASGR3Nxo0b2bJli9tRTADwZljmniJiHxDGL1JSUmjRogXh4TYEc1mKivJ8kFpzjwHvzvgHAltF5B8i0sDXgUzo2rNnD19//TW3317cTePmfIWFh9OmTRsr/AbwrlfP3UAL4CfgDRFZ7kyEbnO6mTI1a5an30Dfvn1dThKc+vbty8qVK0lLS3M7inGZt716MoAPgOlAEnAbsFpErFunKTMpKSlcddVVNGhgXyx9oeCbVMEHrAld3rTx93YmSl8IRAJtVbUHnn79j3jx/nARWSMiHzuPq4rIPBHZ6qyrXODvYILAgQMH+Oqrr+xs34eSk5Np0qQJ77//vttRjMu8OePvB/xTVZuq6rOquh9AVU8AQ7x4/4PA5kKPRwELVDUZWOA8NiFu9uzZ5OXl0a+fTfXgS/369WPZsmXs2bPH7SjGRd4U/nRVPWMaHxF5BkBVF5T0RhGpA9wCvFZo863AW87PbwF9vA1rgtfMmTO54ooraNbsV9M0mzLUv39/VNUu8oY4bwr/TUVs6+Hl/l8E/oIzX68jsWDQN2dd08t9mSB18OBBFi5cSP/+/RGx4YN9qUGDBjRq1IiZM2e6HcW4qNjCLyK/E5ENwFUisr7Qsh1Yf64di0hPYL+qrjqfYE7PoVQRSc21IWWD2pw5c6yZx4/69evHkiVLSE9PP/eLTVAq6Yx/Kp7JWOY464KlldPF81yuBXqLyA48vYE6i8g7wD4RSQJw1vuLerOqTlbV1qraOiLCm6mBTXk1c+ZMLrvsMlq0aOF2lJBQ0NxjvXtCV0mFX1V1BzASyCy0ICJVz7VjVX1UVeuoaj08N4EtdD4wPgQGOy8bjOeDxYSow4cPs2DBAmvm8aNGjRrRoEEDa+4JYec64wdYBaQ661WFHp+vCcBNIrIVz/WDCRewL1POzZkzh9zcXPr37+92lJDSv39/Fi9ezL59+9yOYlxQ0tSLPZ31pap6mbMuWC4rzUFUdVGh/R1S1S6qmuysD1/Yr2DKs5kzZ1KvXj1atmzpdpSQ0q9fP/Lz8613T4gq6eJuy5IWf4Y0wenIkSPMnz/fmnlc0LhxY+rXr2/NPSGqpKumz5fwnAKdyziLCTGzZs0iJyeHAQMGuB0l5IgIAwYMYOzYsaSnp5OUlOR2JONHJTX1dCphsaJvLtj06dO54oorrJnHJQMGDEBVbQiHEFTsGb+IdFbVhSJS5Bi5qmqNg+a87d+/nwULFvDoo49aM49LGjZsSNOmTZk+fTqT/ndKsf36k5KS2LR+rX/DGZ8qqannejwDs/Uq4jkFrPCb8/b++++Tn5/PwIED3Y4S0gYOHMjo0aOpVKUa3Sd8VORr5o7p499QxueKLfyq+piz/q3/4phQMX36dBo1akTjxo3djhISMjKPUbVG4q+25+XleZ7POOrvSMZF57wlVkSqAY8B1+E5018KPKmqh3yczQSpXbt2sWTJEsaOHet2lJCh+fl0HTe7yOfmjR/K4Z+/928g4ypvBmmbDhwA+uIZovkA8J4vQ5ngNmPGDADrzRMgLm59I6iSuW+n21GMn3hT+Kuq6lhV3e4sTwGVfZzLBLHp06fTsmVLkpOT3Y5igLqtuwCQljrf5STGX7wp/F+KyEARCXOWO4BPfB3MBKeffvqJlStX2tl+AImrUhNESFs5H1V1O47xg5Lu3M0UkQzgfjzj9mQ7y3TgYf/EM8Hm3XffRUS488473Y5iChEJJyN9O7/s2up2FOMHJd3AFa+qCc46TFUjnCVMVRP8GdIEB1XlnXfe4YYbbqBu3bpuxzGFhYUTFh7Bz99+4XYS4wfeNPUgIlVEpK2IdCxYfB3MBJ+VK1eydetW7r7bm+kcjD+JCEmN2/Pzinnk5+e5Hcf42DkLv4gMAxYDXwBPOOvHfRvLBKN33nmH6Oho+vbt63YUU4RLru5G1tGD7P9htdtRjI95c8b/INAG+FlVOwEt8HTpNMZrOTk5TJ8+nd69e1OpUiW345giXNT0WiJjK/LzN5+7HcX4mDeFP0tVswBEJFpVvwfq+zaWCTZz587lwIED1swTwMIjo6nbqhO71nxF7qmTbscxPuTNZLa7RKQyMBuYJyJHgD2+DGWCQ6OmzU8P/HUs4ygiwuAhw04PymaDfwWeS67uzralH7F73RIuadvV7TjGR85Z+FX1NufHx0XkS6ASYN8FzTmlp6fTddxscrKOM+eRnlzWsQ+t7/rz6edt8K/AU+OKZsRVSeTnb7+wwh/EvO3V01JE/gg0BXaparYX74kRkRUisk5ENonIE872qiIyT0S2OusqF/YrmEC3c9WX5OWcot7V3d2OYs5BwsK45Oqu7P1uBSePHnQ7jvERb3r1/B14C6gGVAfeEJG/ebHvU0BnVW0GNAe6i0g7YBSwQFWTgQXOYxPEti/7mPjEi6l2mY3EWR5ces0taH4eO+wib9Dy5oz/TqCNqj7mDNXcDhh0rjepxzHnYaSzKHArng8SnHWf0oY25UfmvjQO/rSeS6/taROulBPxiRdT/fKmbP/6ExvCIUh5U/h3ADGFHkcDP3mzcxEJF5G1wH5gnqp+CySqajqAs65ZzHuHi0iqiKTm5uZ6czgTgLZ//SkSFk69dtbMU55ces0tZO79mUPbNrodxfhASWP1/EtEXsLTZLNJRN4UkTeAjcCx4t5XmKrmqWpzoA7QVkS8/q6vqpNVtbWqto6I8KbzkQk0qsqObz4lqVE7YitVdzuOKYW6rTsTHhXD9q9tPMZgVFJFTXXWq4BZhbYvKu1BVPUXEVkEdAf2iUiSqqaLSBKebwMmCOXmZHMy4ygtB/zJ7SimlCJjKlC3VWfSUudTsUIFt+OYMlbS1IsF7fCISBRwpfPwB1XNOdeORaQGkOMU/VjgRuAZ4ENgMDDBWc85//gmkJ3KyiK6YmWSml7rdhRzHi695hZ2LP+U7Ihwt6OYMubN1Is34LkIuwMQoK6IDFbVxed4axLwloiE42lSmqGqH4vIcmCGiAwF0oD+5x/fBKpDhw6Rk32KKzv0ITwi0u045jzUSG5OxRq1yfrFvpQHG28az58HuqrqDwAiciUwDWhV0ptUdT2ecX3O3n4I6FL6qKY8+b//+z8ALr22p8tJzPkSES695hY2zJnM1q1bbca0IOJNr57IgqIPoKpb8HTNNKZIqsqkSZMIj4ikcu3L3Y5jLkDBB/err77qchJTlrwp/KtEZIqI3OAs/4vngq8xRVq0aBFbtmwhOibm3C82AS22UnUio6J54403yMrKcjuOKSPeNPWMAEYCf8TTxr8YeNmXoUz5UXggtgIFA7JlZZfcByAj8xhVayQW81xmmWU0FyY6JpbDhw/z/vvv2+iqQaLEwi8iYcAqVW0MvOCfSKY8KRiIrcDJo4f4aFQfkrsM4MevZhf7PgDNzz/jvYXNfKBz2YU0FyQiMpLk5GQmTZpkhT9IlNjUo6r5wDoRudhPeUw5t33Zx2h+Hpd3uNXtKKaMiAj3338/y5YtY8OGDW7HMWXAmzb+JDx37i4QkQ8LFl8HM+VPfn4ePy2ZTc36rUiodYnbcUwZuvfee4mOjmbSpEluRzFlwJs2/id8nsIEhb0bv+HE4X007/cHt6OYMlatWjXuuOMO3n77bSZMmEB8fLzbkcwFKGmsnhgReQjPDVZXActU9auCxV8BTfmxZeEMYitXp3bzjm5HMT4wcuRIMjMzefPNN92OYi5QSU09bwGtgQ1ADzw3chlTpF92/8S+zSu54oZ+hIXboHrB6Oqrr6Zdu3ZMnDiRvLw8t+OYC1BS4W+oqner6qtAP6CDnzKZcmjLghmER0bbRd0g9/DDD/PTTz/xySc2amd5VlLhP90JW1VtQHxTrKzMI/z87RfUa9+D6IqV3I5jfOj222+nbt26/POf/3Q7irkAJRX+ZiKS4SyZQNOCn0Ukw18BTeD7afFs8nOzubLzHW5HMT4WERHBH/7wBxYtWsTatWvdjmPOU7GFX1XDVTXBWeJVNaLQzwn+DGkCl6ry46IUajVqR0JSPbfjGD8YNmwYcXFxvPjii25HMefJm378xhQr+1QWWRmHuLKLne2HiipVqnDvvfcybdq0Xw3XYcoHK/zmvOXn55N18gSVal9OrYZXux3H+NHDDz9Mbm4uL7xgI7mUR1b4zXmbPXs2+Xl5NOzxG0TE7TjGj6644goGDBjAK6+8wuHDh92OY0rJCr85L6rK008/TVhYOHVa2YBqoejRRx/l+PHj/Otf/3I7iiklK/zmvMydO5dVq1YRExdHWJjNyRqKmjRpQu/evZk4cSKZNox2ueKzwi8idUXkSxHZLCKbRORBZ3tVEZknIluddRVfZTC+8/TTT1OnTh2iom2ylVA2evRojhw5YjN0lTO+POPPBf5HVRsA7YCRItIQGAUsUNVkYIHz2JQjS5cuZfHixfz5z3+2tv0Qd/XVV9OlSxeef/55m6GrHPFZ4VfVdFVd7fycCWwGagO34hkHCGfdx1cZjG889thj1KxZk2HDhrkdxQSAMWPGsHfvXhuyuRzxSxu/iNQDWgDfAomqmg6eDwegZjHvGS4iqSKSmptrI0YEivnz57Nw4ULGjBlDXFyc23FMAOjUqRNdunTh6aeftrb+csLnhV9EKgIfAA+pqtdDPajqZFVtraqtIyJstMdAoKqMHj2aiy++mPvvv9/tOCaAjBs3jgMHDtjdvOWETyuqiETiKfrvqmqKs3mfiCSparqIJAH7fZnBlJ3Zs2ezcuVKXn/9daKjo92OYwJEo6bNSU9PJzIqir8/9hjPv/gSYWH/PadMSkpi0/q17gU0v+Kzwi+eq35TgM2qWvj2vg+BwcAEZz3HVxlM2cnLy+Nvf/sbV111Fffcc4/bcUwASU9Pp+u42fyy+ye+GPsbklp3o1nfkaefnzumj3vhTJF82dRzLXAP0FlE1jrLzXgK/k0ishW4yXlsAty7777Ld999x9ixY7GmN1OUyrUv55Kru7H1y5mcOHLA7TimBL7s1bNUVUVVm6pqc2f5VFUPqWoXVU121na/d4A7duwYjz76KK1bt+b22293O44JYI17DUNVWT/7FbejmBLYnbvmnCZMmMCePXuYOHHiGW23xpytYvWLqH/jQH7+5nMObtvodhxTDPsrNiXatm0bzz33HIMGDeKaa65xO44pBxr0+A0xlaqz5r1/ovn5bscxRbDGWlOiRx55hIiICJ555hm3oxiXZGQeo2qNxBKeP7PvfmRMBZrd/ju+fWMs25d/6ut45jxY4TfFWrBgAbNmzWLcuHHUrl3b7TjGJZqfT9dxs4t9fuYDvx6d9ZK23fhxUQobZk8iJsIaFgKN/RcxRTp58iS///3viYyM4tkXXqRqjcQil7PP9owBkLAwWgx8mKzMI5w8cdztOOYsdsZvivTkk0+yZcsWKiZUptvTxd9qUdTZnjEA1eo1JLlTP7YunMmSJUvo0KGD25GMw874za+sXr2aZ599liFDhhAZFeV2HFOONbn1fsLCwhg2bJiN3hlArPCbM+Tk5DBkyBBq1qzJ888/73YcU85FxsQRVzGBLVu28OSTT7odxzis8JszPPPMM6xbt46XX36ZypUrux3HBIHIqCh++9vf8o9//INVq1a5Hcdghd8U8s033/DEE08wcOBA+vTp43YcE0Sef/55atWqxV133cWxY8fcjhPy7OJuiCsYWTE/P5/MXzyjZ3w2d/7pftvWa8dcqIzMY1x+5VXk5GSze/duqlWvQYX4BMBG7nSLFf4Ql56ezk1PzWL5//6djDWL6PznV6h+WePTz1uvHXOhCt8HsPHD19j0yeu0uOsv1GvX3UbudIk19Ri2LfuInasW0OTW4WcUfWPKWsNb7qXGFc1YNfU5MvftdDtOyLLCH+Jyc3JYPe15Ehu04aqug9yOY4JcWHgE7YY+TlhEBEtf+auN5eMSK/whLC0tjWMZvxBXpSbthz2J2Mibxg/iqiZyzf3jyNy3k2OZGeTl5bkdKeTYX3qIOnbsGL1790aBDiOfJbpiJbcjmRCSWL8VLe/8H3JzsvnLX/7idpyQY4U/BOXm5nL33XezYcMGKsYnkJBUz+1IJgRd0bEP0TGxvPDCC0yePNntOCHFZ4VfRF4Xkf0isrHQtqoiMk9EtjrrKr46vilafn4+Q4YMYc6cObz44otERtmk6cY9sRUqcvPNNzNixAimTZvmdpyQ4csz/jeB7mdtGwUsUNVkYIHz2PhYo6bNqVojkSrVaxJboSJvv/02MXEVeOzJp6yfvnGViDBz5kw6duzIPffcw5w5xQ8IaMqOL+fcXQycPZ/urcBbzs9vAX18dXzzXwV99RNb3kR21kmu6nY3vV+YS9dxs9F8dTueCXFxcXF89NFHtGrVijvuuIMvvvjC7UhBz99t/Imqmg7grGsW90IRGS4iqSKSmpub67eAwUhVSX33H/wwbypX3NCPprf9DhFxO5Yxp8XHx/PZZ5/RoEEDevXqxYwZM9yOFNQC9uKuqk5W1daq2joiwm4wPl9ZWVkcz8xg25I5NOjxG1oOfNiKvglIVatW5csvv+Tqq69m4MCBvPzyy25HClr+Lvz7RCQJwFnv9/PxQ8qBAwfo3r07OdmnaDHgIZr2GWFF3wS0KlWqMHfuXHr27MnIkSMZPXq09fP3AX8X/g+Bwc7PgwG7kuMjK1asoFWrVnzzzTdUiE/gys53uB3JGK/ExsaSkpLCfffdx/jx4+nZsyeHD599udBcCF9255wGLAfqi8guERkKTABuEpGtwE3OY1OGVJVXX32VDh06EB4eztdff01UdIzbsYwplYiICF599VUmT57MwoULadWqlY3lX4Z82avnTlVNUtVIVa2jqlNU9ZCqdlHVZGdtH+NlaM+ePfTu3ZsRI0bQqVMnUlNTadmypduxjDkvIsJ9993HkiVLyMvLo127djzxxBPk5OS4Ha3cC9iLu8Z7qsrbb79No0aNmD9/Pi+88AKffPIJ1apVczuaMSXKyDxG1RqJxS6Nmjanbdu2rF27lgEDBvD444/Ttm1b1q1b53b0cs26y5RzK1eu5E9/+hNLly7lmmuu4Y033uDKK690O5YxXik8Vn9RCsbrr1q1Ku+88w79+vVjxIgRtGzZkvvuu48nn3ySmjWL7RVuimFn/OXUjz/+yG9+8xvatm3Lli1bePXVV1m8eLEVfRPU+vTpw3fffccDDzzAlClTSE5OZvz48WTaHeilYoW/nFm/fj133XUX9evXZ8aMGYwaNYqtW7cyfPhwwsPD3Y5njM9VrVqViRMnsnHjRjp27Mjo0aO5+OKLeeyxxzh06JDb8coFK/zlQFZWFlOnTqVTp040a9aMjz76iEceeYQdO3Ywfvx4EhIS3I5ojN/Vr1+fjz76iBUrVtCpUyeefPJJ6tSpw+DBg1m2bBmqNhxJcazwB6icnBzmzZvH/fffT+3atRk0aBBpaWmMHz+etLQ0nnnmGWrVquV2TGNc16ZNG1JSUti0aRP33nsvs2bN4rrrrqNhw4Y8/vjjbNy40T4EzmKFP4Ds3r2bN998k7vuuotatWrRtWtXpk6dSrdu3Zg/fz5bt25l1KhRVKlio1kbc7aGDRvyyiuvsGfPHqZMmULNmjV58sknadKkCVdddRUPPvggH3/8MceOHXM7quusV49LMjMz2bhxI6tWrWL58uV88803bNu2DYDExERuueUW+vbtS9euXYmNjXU5rTHlR8WKFRkyZAhDhgxh7969zJ49m1mzZjF58mReeuklIiIiaNasGe3ataNdu3Y0b96c+vXrExkZ6XZ0v7HC70MZGRmkpaWxc+dOtm3bxtatW9m6dSubN29m+/btp1+XlJRE+/btGTlyJDfeeCNNmjSxMXWM4b/9/IuSlJTEpvVrS3x/rVq1GDFiBCNGjCArK4tly5axcOFCli9fzptvvsl//vMfACIjI2nQoAH169cnOTmZ5ORk6tWrR926dalTpw7R0cE1YZEV/rOoKtnZ2WRnZ3Pq1CmysrI4efKkZ5TL48dPLxkZGWRkZHD06FGOHDnC4cOHOXToEPv372ffvn3s27fvV18p4+LiSE5Opk2bNgwdOpSmTZvSvHlz6tSpY4XemCKU1M+/oI+/t2JiYujSpQtdunQBIC8vj++++47169ezfv16NmzYwOrVq0lJSfnVwHDVqlUjMTGRxMREatSoQdWqValWrRqVK1cmISGBhIQE4uPjqVChAhUqVCAuLo7Y2FhiYmKIiYkhOjqaqKgoIiIiAuJvPagL/9ixY5k6dSr5+fmnl7y8vDOW3NxccnJyzliXVnR0NNWqVaNKlSrUrFmTNm3akJiYSO3atbn44oupW7cu9w69jwMHDpC2O5203enMW7jojH14c/ZSnEZNm5Oenl7s8zbLljG/Fh4eTpMmTWjSpAmDBg06vT0nJ4cdO3ac/raelpbG3r172bt3L/v27WPdunUcPnyYw4cPn9fIoZGRkaeXiIgIIiIiCA8PP72EhYWdXoeFhZ0ee6ssBXXhv+iii2jatOnpf0AROeMfuPA/euH/GAWfztHR0cTExBAbG0tsbCxxcXGnP9ETEhKoVKkS8fHxXrXBHzx4kG5PFz8YaWnPXgpLT08v8e7HmQ90Pu99GxNqIiMjTzf3lERVOX78OEePHuXo0aMcO3bsdIvAiRMnTrcUnDx58nQrQnZ2Njk5OaeXgpPP3Nxc8vLyTp+cFj5ZjY+PL/PfMagL/9ChQxk6dKjbMYwxQUhEqFixIhUrVqR27dpuxymVoC78xpjgVdKFX7iw5tNgZ4XfGFMueTvAm/k1K/wB4lxnLydOZhEXW/SEKnbx1hhTGlb4A8S5zl5mPtCZri98XuxzxhjjLRuywRhjQowrZ/wi0h2YCIQDr6mqzb1rjPGrku5/KalpFUq+cHyu+2oC4aKz3wu/iIQD/8Ez2fouYKWIfKiq3/k7izEmdJV0/0tJTatQ8oXjc91XEwgXnd1o6mkL/Kiq21Q1G5gO3OpCDmOMCUni73GqRaQf0F1VhzmP7wGuVtUHznrdcGC487AlcPI8DxkBlH4cBt+zXKVjuUrHcpVOoOaCC8t2QlVrFLVDfytqhKJfffqo6mRg8gUfTCRVVVtf6H7KmuUqHctVOpardAI1F/gmmxtNPbuAuoUe1wH2uJDDGGNCkhuFfyWQLCKXikgUMBD40IUcxhgTkvze1KOquSLyAPAFnu6cr6vqJh8e8oKbi3zEcpWO5Sody1U6gZoLfJDN7xd3jTHGuMvu3DXGmBBjhd8YY0JMSBV+EXlERFREqrudBUBExorIehFZKyJzReQitzMBiMizIvK9k22WiFR2OxOAiPQXkU0iki8irne9E5HuIvKDiPwoIqPczgMgIq+LyH4R2eh2lsJEpK6IfCkim53/hg+6nQlARGJEZIWIrHNyPeF2psJEJFxE1ojIx2W535Ap/CJSF88wEWluZynkWVVtqqrNgY+Bv7ucp8A8oLGqNgW2AI+6nKfARuB2YLHbQQoNPdIDaAjcKSIN3U0FwJtAd7dDFCEX+B9VbQC0A0YGyL/XKaCzqjYDmgPdRaSdu5HO8CCwuax3GjKFH/gn8BeKuFnMLaqaUehhBQIkm6rOVdWCOwW/wXOvhetUdbOq/uB2DkdADj2iqouBw27nOJuqpqvqaufnTDzFzPX5CtXjmPMw0lkC4u9QROoAtwCvlfW+Q6Lwi0hvYLeqrnM7y9lEZJyI7AQGEThn/IUNAT5zO0QAqg3sLPR4FwFQyMoDEakHtAC+dTkKcLo5ZS2wH5inqgGRC3gRz8lqflnvOGgmYhGR+UCtIp4aA4wGuvo3kUdJuVR1jqqOAcaIyKPAA8BjgZDLec0YPF/R3/VHJm9zBQivhh4xZxKRisAHwENnfeN1jarmAc2da1mzRKSxqrp6jUREegL7VXWViNxQ1vsPmsKvqjcWtV1EmgCXAutEBDzNFqtFpK2q7nUrVxGmAp/gp8J/rlwiMhjoCXRRP97sUYp/L7fZ0COlJCKReIr+u6qa4naes6nqLyKyCM81Ercvjl8L9BaRm4EYIEFE3lHVu8ti50Hf1KOqG1S1pqrWU9V6eP5gW/qj6J+LiCQXetgb+N6tLIU5E+X8FeitqifczhOgbOiRUhDPWdcUYLOqvuB2ngIiUqOg15qIxAI3EgB/h6r6qKrWcWrWQGBhWRV9CIHCH+AmiMhGEVmPpykqILq4Af8G4oF5TlfTSW4HAhCR20RkF9Ae+EREvnAri3Pxu2Dokc3ADB8PPeIVEZkGLAfqi8guERnqdibHtcA9QGfn/6m1ztms25KAL52/wZV42vjLtOtkILIhG4wxJsTYGb8xxoQYK/zGGBNirPAbY0yIscJvjDEhxgq/McaEGCv8plwQkToiMkdEtorITyIy0ek/j4jcKyL/djvj2UTk2Llf5b/9GFPACr8JeM7NPynAbFVNBq4EKgLjfHjMoLmr3ZizWeE35UFnIEtV34DTY6s8DAwRkTjnNXVF5HNnfPzHAESkgoh84oy1vlFEBjjbW4nIVyKySkS+EJEkZ/siEXlaRL7CM37SDhEJc56LE5GdIhIpIpc7x1olIktE5CrnNZeKyHIRWSkiY4v6RUTkGRH5faHHj4vI/4hIRRFZICKrRWSDiPxqpE8RuaHwuOwi8m8Rubek38mYoljhN+VBI2BV4Q3OAF9pwBXOprZ4RjhtDvQXz0Qt3YE9qtpMVRsDnzvjxfwL6KeqrYDXOfObQ2VVvV5VnwDWAdc723sBX6hqDp7Jr//gvP8R4GXnNROBV1S1DVDckCDTgQGFHt8BzASygNtUtSXQCXje+aZzTl78Tsacwb7OmvJAKHrky8Lb56nqIQARSQGuAz4FnhORZ4CPVXWJiDQGGuMZjgIgHEgvtM/3zvp5APAlnvFSXnZGl7wGmFmoLkc762uBvs7PbwPPnB1YVdeISE3xzLZWAziiqmlO8X5aRDriGYa3NpBI8R8ghdU/x+9kzBms8JvyYBP/LagAiEgCntExfwJa8esPBlXVLSLSCrgZGC8ic4FZwCZVbV/MsY4X+vlD531VnWMsxDNhzi/OrGlF8WYMlPeBfniGn57ubBuE54OglarmiMgOPKMyFpbLmd/SC54XSv6djDmDNfWY8mABECciv4HT0x4+D7xZaPTQm0SkqjPCYh9gmXNWfUJV3wGeA1oCPwA1RKS9s69IEWlU1EGdmZlW4GnC+VhV85wmpu0i0t95v4hIM+cty/B8MwBPIS/OdOd1/fB8CABUwjP+eo6IdAIuKeJ9PwMNRSRaRCoBXZztXv9OxoAVflMOOPMB3Ian7X4rnnmAs/BMsFNgKZ7mlbXAB6qaCjQBVohndqUxwFPONIn9gGdEZJ3z+mtKOPx7wN2c2QQ0CBjqvH8T/51y8UE8c8muxFPIi/t9NuEZ/XS3qhY0ybwLtBaRVGf/vxoaWFV3AjOA9c7r1zjbS/s7mRBno3MaY0yIsTN+Y4wJMVb4jTEmxFjhN8aYEGOF3xhjQowVfmOMCTFW+I0xJsRY4TfGmBDz/0UrEu8OJma3AAAAAElFTkSuQmCC", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import seaborn as sns\n", "from scipy import stats\n", "\n", "x = np.linspace(-4,4,100)\n", "\n", "mu = 0 \n", "sigma = 1\n", "n = 1000\n", "dist = np.random.normal(mu, sigma, n)\n", "\n", "y = stats.norm.pdf(x, mu, sigma)\n", "\n", "fig = sns.histplot(dist, binwidth=0.2)\n", "ax2 = fig.twinx()\n", "ax2.tick_params(left=False, labelleft=False, top=False, labeltop=False,\n", " right=False, labelright=False, bottom=False, labelbottom=False)\n", "\n", "fig.set(xlabel='Observed value', ylabel='Probability density')\n", "\n", "sns.lineplot(x=x,y=y, ax=ax2, color='black')\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "loaded-uzbekistan", "metadata": {}, "source": [ "In the previous example all I did was generate lots of normally distributed observations using `random.normal()` and then compared those to the true probability distribution in the figure (using `stats.norm.pdf()` to generate the black line in the figure. Now let's try something trickier. We'll try to generate some observations that follow a chi-square distribution with 3 degrees of freedom, but instead of using `stats.chi2.pdf()`, we'll start with variables that are normally distributed, and see if we can exploit the known relationships between normal and chi-square distributions to do the work. As I mentioned earlier, a chi-square distribution with $k$ degrees of freedom is what you get when you take $k$ normally-distributed variables (with mean 0 and standard deviation 1), square them, and add them up. Since we want a chi-square distribution with 3 degrees of freedom, we'll need to create three sets of normally-distributed data. Let's call them `normal_a`, `normal_b`, and `normal_c`.[^note6]\n", "\n", "[^note6]: Of course, you can give variables any name you want, and we could just as well call these Larry, Moe, and Curly, or Huey, Dewey, and Louie, but in programming, the most boringly obvious names are usually the best ones. When writing code, we are looking for clarity, not dramatic effect!" ] }, { "cell_type": "code", "execution_count": 3, "id": "bottom-luxembourg", "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "\n", "normal_a = np.random.normal(0, 1, 1000) # a set of normally-distributed data\n", "normal_b = np.random.normal(0, 1, 1000) # another set of normally-distributed data\n", "normal_c = np.random.normal(0, 1, 1000) # and another!" ] }, { "attachments": {}, "cell_type": "markdown", "id": "worldwide-runner", "metadata": {}, "source": [ "Now that we've done that, the theory says we should square these and add them together, like this" ] }, { "cell_type": "code", "execution_count": 4, "id": "structural-macintosh", "metadata": {}, "outputs": [], "source": [ "chi_square_data = np.square(normal_a) + np.square(normal_b) + np.square(normal_c)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "neural-andorra", "metadata": {}, "source": [ "and the resulting `chi_square_data` variable should contain 1000 observations that follow a chi-square distribution with 3 degrees of freedom. You can use the `sns.histplot()` function to have a look at these observations yourself, using a command like this," ] }, { "cell_type": "code", "execution_count": 5, "id": "technical-guitar", "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "\n", "sns.histplot(chi_square_data)\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "special-savings", "metadata": {}, "source": [ "and you should obtain a result that looks pretty similar to the chi-square plot in {numref}`fig-chi2`. Once again, with a bit more code, we can plot the actual chi-square distribution with 3 degrees of freedom over our histogram generated from random samples and compare:" ] }, { "cell_type": "code", "execution_count": 36, "id": "express-plumbing", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import seaborn as sns\n", "from scipy import stats\n", "\n", "x = np.linspace(0, 15, 100)\n", "y = stats.chi2.pdf(x, 3)\n", "\n", "fig = sns.histplot(chi_square_data)\n", "ax2 = fig.twinx()\n", "\n", "fig.set(xlabel='Observed value', ylabel='Probability density')\n", "\n", "sns.lineplot(x=x, y=y, ax=ax2, color='black')\n", "\n", "ax2.tick_params(left=False, labelleft=False, top=False, labeltop=False,\n", " right=False, labelright=False, bottom=False, labelbottom=False)\n", "\n", "sns.despine()\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "tribal-cabinet", "metadata": {}, "source": [ "It's pretty clear that -- even though I used `random.norm()` to do all the work rather than `stats.chi2.pdf` -- the observations stored in the `chi_square_data` variable really do follow a chi-square distribution. Admittedly, this probably doesn't seem all that interesting right now, but later on when we start encountering the chi-square distribution in the chapter on [Categorical data analysis](chisquare), it will be useful to understand the fact that these distributions are related to one another. \n", "\n", "We can extend this demonstration to the $t$ distribution and the $F$ distribution. Earlier, I implied that the $t$ distribution is related to the normal distribution when the standard deviation is unknown. That's certainly true, and that's the what we'll see later on in the chapter on [Comparing two means](ttest), but there's a somewhat more precise relationship between the normal, chi-square and $t$ distributions. Suppose we \"scale\" our chi-square data by dividing it by the degrees of freedom, like so:" ] }, { "cell_type": "code", "execution_count": 37, "id": "wrong-privacy", "metadata": {}, "outputs": [], "source": [ "scaled_chi_square_data = chi_square_data / 3" ] }, { "attachments": {}, "cell_type": "markdown", "id": "statutory-blood", "metadata": {}, "source": [ "We then take a set of normally distributed variables and divide them by (the square root of) our scaled chi-square variable which had $df=3$, and the result is a $t$ distribution with 3 degrees of freedom. If we plot the histogram of `t_3`, we end up with something that looks very similar to the t distribution." ] }, { "cell_type": "code", "execution_count": 39, "id": "fallen-andrew", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "normal_d = np.random.normal(0, 1, 1000) # yet another set of normally distributed data\n", "t_3 = normal_d / np.sqrt(scaled_chi_square_data) # divide by square root of scaled chi-square to get t\n", "\n", "sns.histplot(t_3)\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "welsh-dutch", "metadata": {}, "source": [ "Adding the usual formatting plus the true t-distribution with three degrees of freedom, we get something like this:" ] }, { "cell_type": "code", "execution_count": 41, "id": "decent-computer", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import seaborn as sns\n", "from scipy import stats\n", "\n", "x = np.linspace(-10, 10, 100)\n", "y = stats.t.pdf(x, 3)\n", "\n", "fig = sns.histplot(t_3)\n", "ax2 = fig.twinx()\n", "\n", "fig.set(xlabel='Observed value', ylabel='Probability density')\n", "\n", "sns.lineplot(x=x, y=y, ax=ax2, color='black')\n", "\n", "ax2.tick_params(left=False, labelleft=False, top=False, labeltop=False,\n", " right=False, labelright=False, bottom=False, labelbottom=False)\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "designed-story", "metadata": {}, "source": [ "Similarly, we can obtain an $F$ distribution by taking the ratio between two scaled chi-square distributions. Suppose, for instance, we wanted to generate data from an $F$ distribution with 3 and 20 degrees of freedom. We could do this using `numpy.random.f()`, but we could also do the same thing by generating two chi-square variables, one with 3 degrees of freedom, and the other with 20 degrees of freedom. As the example with `chi_square_data` illustrates, we can actually do this using `numpy.random.normal()` if we really want to, but this time I'll take a short cut:" ] }, { "cell_type": "code", "execution_count": 47, "id": "dutch-advocacy", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 47, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "chi_square_3 = np.random.chisquare(3, 1000) # generate chi square data with df = 3...\n", "chi_square_20 = np.random.chisquare(20, 1000) # generate chi square data with df = 20...\n", "\n", "scaled_chi_square_3 = chi_square_3 / 3 # scale first the chi square variable...\n", "scaled_chi_square_20 = chi_square_20 / 20 # and scale the other one...\n", "\n", "f_3_20 = scaled_chi_square_3 / scaled_chi_square_20 # take the ratio of the two chi squares...\n", "\n", "sns.histplot(f_3_20) # draw a picture\n", "\n", "sns.despine() # ... and remove that unsightly box around the plot" ] }, { "attachments": {}, "cell_type": "markdown", "id": "threatened-librarian", "metadata": {}, "source": [ "The resulting `f_3_20` variable does in fact store variables that follow an $F$ distribution with 3 and 20 degrees of freedom, which we can check by overlaying the real $F$ distribution with $df_1 = 3$ and $df_2 = 20$. Again, they match. I'll hide the code this time, but you can click the button to take a look." ] }, { "cell_type": "code", "execution_count": 48, "id": "stupid-waste", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import seaborn as sns\n", "from scipy import stats\n", "\n", "x = np.linspace(0, 9, 100)\n", "y = stats.f.pdf(x, 3,20)\n", "\n", "fig = sns.histplot(f_3_20)\n", "ax2 = fig.twinx()\n", "\n", "fig.set(xlabel='Observed value', ylabel='Probability density')\n", "\n", "sns.lineplot(x=x, y=y, ax=ax2, color='black')\n", "\n", "ax2.tick_params(left=False, labelleft=False, top=False, labeltop=False,\n", " right=False, labelright=False, bottom=False, labelbottom=False)\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "protecting-explanation", "metadata": {}, "source": [ "Okay, time to wrap this section up. We've seen three new distributions: $\\chi^2$, $t$ and $F$. They're all continuous distributions, and they're all closely related to the normal distribution. I've talked a little bit about the precise nature of this relationship, and shown you some Python commands that illustrate this relationship. The key thing for our purposes, however, is not that you have a deep understanding of all these different distributions, nor that you remember the precise relationships between them. The main thing is that you grasp the basic idea that these distributions are all deeply related to one another, and to the normal distribution. Later on in this book, we're going to run into data that are normally distributed, or at least assumed to be normally distributed. What I want you to understand right now is that, if you make the assumption that your data are normally distributed, you shouldn't be surprised to see $\\chi^2$, $t$ and $F$ distributions popping up all over the place when you start trying to do your data analysis. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "sticky-driver", "metadata": {}, "source": [ "## Summary\n", "\n", "In this chapter we've talked about probability. We've talked what probability means, and why statisticians can't agree on what it means. We talked about the rules that probabilities have to obey. And we introduced the idea of a probability distribution, and spent a good chunk of the chapter talking about some of the more important probability distributions that statisticians work with. The section by section breakdown looks like this:\n", "\n", "- [Probability theory versus statistics](probstats)\n", "- Frequentist versus Bayesian [views of probability](probmeaning)\n", "- Basics of [probability theory](basicprobability)\n", "- [Binomial](binomial) distribution, [normal](normal) distribution, and [others](otherdists)\n", "\n", "\n", "As you'd expect, my coverage is by no means exhaustive. Probability theory is a large branch of mathematics in its own right, entirely separate from its application to statistics and data analysis. As such, there are thousands of books written on the subject and universities generally offer multiple classes devoted entirely to probability theory. Even the \"simpler\" task of documenting standard probability distributions is a big topic. I've described five standard probability distributions in this chapter, but sitting on my bookshelf I have a 45-chapter book called \"Statistical Distributions\" {cite}`Evans2000` that lists a *lot* more than that. Fortunately for you, very little of this is necessary. You're unlikely to need to know dozens of statistical distributions when you go out and do real world data analysis, and you definitely won't need them for this book, but it never hurts to know that there are other possibilities out there.\n", "\n", "Picking up on that last point, there's a sense in which this whole chapter is something of a digression. Many undergraduate psychology classes on statistics skim over this content very quickly (I know mine did), and even the more advanced classes will often \"forget\" to revisit the basic foundations of the field. Most academic psychologists would not know the difference between probability and density, and until recently very few would have been aware of the difference between Bayesian and frequentist probability. However, I think it's important to understand these things before moving onto the applications. For example, there are a lot of rules about what you're \"allowed\" to say when doing statistical inference, and many of these can seem arbitrary and weird. However, they start to make sense if you understand that there is this Bayesian/frequentist distinction. Similarly, in the chapter on [comparing two means](ttest) we're going to talk about something called the $t$-test, and if you really want to have a grasp of the mechanics of the $t$-test, it really helps to have a sense of what a $t$-distribution actually looks like. You get the idea, I hope." ] } ], "metadata": { "celltoolbar": "Tags", "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.5" } }, "nbformat": 4, "nbformat_minor": 5 } { "cells": [ { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(estimation)=\n", "# Estimating unknown quantities from a sample\n", "\n", " \n", "At the start of the last chapter I highlighted the critical distinction between *descriptive statistics* and *inferential statistics*. As discussed in the section on [descriptive statistics](descriptives), the role of descriptive statistics is to concisely summarise what we *do* know. In contrast, the purpose of inferential statistics is to \"learn what we do not know from what we do\". Now that we have a foundation in probability theory, we are in a good position to think about the problem of statistical inference. What kinds of things would we like to learn about? And how do we learn them? These are the questions that lie at the heart of inferential statistics, and they are traditionally divided into two \"big ideas\": estimation and hypothesis testing. The goal in this chapter is to introduce the first of these big ideas, estimation theory, but I'm going to witter on about sampling theory first because estimation theory doesn't make sense until you understand sampling. As a consequence, this chapter divides naturally into two parts. The sections on [Defining a population](srs) through [](samplingdists) are focused on sampling theory, and [estimating population parameters](pointestimates) and [confidence intervals](ci) make use of sampling theory to discuss how statisticians think about estimation.\n", "\n", "\n", "## Samples, populations and sampling\n", "(srs)=\n", "\n", "In the prelude to Part I discussed the riddle of induction, and highlighted the fact that *all* learning requires you to make assumptions. Accepting that this is true, our first task to come up with some fairly general assumptions about data that make sense. This is where **_sampling theory_** comes in. If probability theory is the foundation upon which all statistical theory builds, sampling theory is the frame around which you can build the rest of the house. Sampling theory plays a huge role in specifying the assumptions upon which your statistical inferences rely. And in order to talk about \"making inferences\" the way statisticians think about it, we need to be a bit more explicit about what it is that we're drawing inferences *from* (the sample) and what it is that we're drawing inferences *about* (the population). \n", "\n", "In almost every situation of interest, what we have available to us as researchers is a **_sample_** of data. We might have run an experiment with some number of participants; a polling company might have phoned some number of people to ask questions about voting intentions; etc. Regardless: the data set available to us is finite, and incomplete. We can't possibly get every person in the world to do our experiment; a polling company doesn't have the time or the money to ring up every voter in the country etc. In our earlier discussion of [descriptive statistics](descriptives), this sample was the only thing we were interested in. Our only goal was to find ways of describing, summarising and graphing that sample. This is about to change.\n", "\n", "### Defining a population\n", "\n", "A sample is a concrete thing. You can open up a data file, and there's the data from your sample. A **_population_**, on the other hand, is a more abstract idea. It refers to the set of all possible people, or all possible observations, that you want to draw conclusions about, and is generally *much* bigger than the sample. In an ideal world, the researcher would begin the study with a clear idea of what the population of interest is, since the process of designing a study and testing hypotheses about the data that it produces does depend on the population about which you want to make statements. However, that doesn't always happen in practice: usually the researcher has a fairly vague idea of what the population is and designs the study as best he/she can on that basis. \n", "\n", "Sometimes it's easy to state the population of interest. For instance, in the \"polling company\" example that opened the chapter, the population consisted of all voters enrolled at the a time of the study -- millions of people. The sample was a set of 1000 people who all belong to that population. In most situations the situation is much less simple. In a typical a psychological experiment, determining the population of interest is a bit more complicated. Suppose I run an experiment using 100 undergraduate students as my participants. My goal, as a cognitive scientist, is to try to learn something about how the mind works. So, which of the following would count as \"the population\":\n", "\n", "- All of the undergraduate psychology students at the University of Adelaide?\n", "- Undergraduate psychology students in general, anywhere in the world?\n", "- Australians currently living?\n", "- Australians of similar ages to my sample?\n", "- Anyone currently alive?\n", "- Any human being, past, present or future?\n", "- Any biological organism with a sufficient degree of intelligence operating in a terrestrial environment?\n", "- Any intelligent being?\n", "\n", "Each of these defines a real group of mind-possessing entities, all of which might be of interest to me as a cognitive scientist, and it's not at all clear which one ought to be the true population of interest. As another example, consider the Wellesley-Croker game that we discussed in the prelude. The sample here is a specific sequence of 12 wins and 0 losses for Wellesley. What is the population?\n", "\n", "- All outcomes until Wellesley and Croker arrived at their destination?\n", "- All outcomes if Wellesley and Croker had played the game for the rest of their lives?\n", "- All outcomes if Wellseley and Croker lived forever and played the game until the world ran out of hills?\n", "- All outcomes if we created an infinite set of parallel universes and the Wellesely/Croker pair made guesses about the same 12 hills in each universe?\n", "\n", "Again, it's not obvious what the population is.\n", "\n", "### Simple random samples" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "\n", "```{figure} ../img/estimation/srs1.png\n", "---\n", "width: 600px\n", "align: center\n", "name: srs1-fig\n", "---\n", "Simple random sampling without replacement from a finite population\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Irrespective of how I define the population, the critical point is that the sample is a subset of the population, and our goal is to use our knowledge of the sample to draw inferences about the properties of the population. The relationship between the two depends on the *procedure* by which the sample was selected. This procedure is referred to as a **_sampling method_**, and it is important to understand why it matters.\n", "\n", "To keep things simple, let's imagine that we have a bag containing 10 chips. Each chip has a unique letter printed on it, so we can distinguish between the 10 chips. The chips come in two colours, black and white. This set of chips is the population of interest, and it is depicted graphically on the left of {numref}`srs1-fig`. As you can see from looking at the picture, there are 4 black chips and 6 white chips, but of course in real life we wouldn't know that unless we looked in the bag. Now imagine you run the following \"experiment\": you shake up the bag, close your eyes, and pull out 4 chips without putting any of them back into the bag. First out comes the $a$ chip (black), then the $c$ chip (white), then $j$ (white) and then finally $b$ (black). If you wanted, you could then put all the chips back in the bag and repeat the experiment, as depicted on the right hand side of {numref}`srs1-fig`. Each time you get different results, but the procedure is identical in each case. The fact that the same procedure can lead to different results each time, we refer to it as a *random* process. [^note1] However, because we shook the bag before pulling any chips out, it seems reasonable to think that every chip has the same chance of being selected. A procedure in which every member of the population has the same chance of being selected is called a **_simple random sample_**. The fact that we did *not* put the chips back in the bag after pulling them out means that you can't observe the same thing twice, and in such cases the observations are said to have been sampled **_without replacement_**. \n", "\n", "To help make sure you understand the importance of the sampling procedure, consider an alternative way in which the experiment could have been run. Suppose that my 5-year old son had opened the bag, and decided to pull out four black chips without putting any of them back in the bag. This *biased* sampling scheme is depicted in {numref}`brs-fig`. Now consider the evidentiary value of seeing 4 black chips and 0 white chips. Clearly, it depends on the sampling scheme, does it not? If you know that the sampling scheme is biased to select only black chips, then a sample that consists of only black chips doesn't tell you very much about the population! For this reason, statisticians really like it when a data set can be considered a simple random sample, because it makes the data analysis *much* easier. \n", "\n", "[^note1]: The proper mathematical definition of randomness is extraordinarily technical, and way beyond the scope of this book. We'll be non-technical here and say that a process has an element of randomness to it whenever it is possible to repeat the process and get different answers each time." ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "\n", "```{image} ../img/estimation/brs.png\n", ":alt: biased-random-sample\n", ":width: 600px\n", ":align: center\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "\n", "A third procedure is worth mentioning. This time around we close our eyes, shake the bag, and pull out a chip. This time, however, we record the observation and then put the chip back in the bag. Again we close our eyes, shake the bag, and pull out a chip. We then repeat this procedure until we have 4 chips. Data sets generated in this way are still simple random samples, but because we put the chips back in the bag immediately after drawing them it is referred to as a sample **_with replacement_**. The difference between this situation and the first one is that it is possible to observe the same population member multiple times, as illustrated below. " ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "```{image} ../img/estimation/srs2.png\n", ":alt: biased-random-sample\n", ":width: 600px\n", ":align: center\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "In my experience, most psychology experiments tend to be sampling without replacement, because the same person is not allowed to participate in the experiment twice. However, most statistical theory is based on the assumption that the data arise from a simple random sample *with* replacement. In real life, this very rarely matters. If the population of interest is large (e.g., has more than 10 entities!) the difference between sampling with- and without- replacement is too small to be concerned with. The difference between simple random samples and biased samples, on the other hand, is not such an easy thing to dismiss.\n", "\n", "\n", "### Most samples are not simple random samples\n", "\n", "As you can see from looking at the list of possible populations that I showed above, it is almost impossible to obtain a simple random sample from most populations of interest. When I run experiments, I'd consider it a minor miracle if my participants turned out to be a random sampling of the undergraduate psychology students at Adelaide university, even though this is by far the narrowest population that I might want to generalise to. A thorough discussion of other types of sampling schemes is beyond the scope of this book, but to give you a sense of what's out there I'll list a few of the more important ones:\n", "\n", " \n", "- *Stratified sampling*. Suppose your population is (or can be) divided into several different subpopulations, or *strata*. Perhaps you're running a study at several different sites, for example. Instead of trying to sample randomly from the population as a whole, you instead try to collect a separate random sample from each of the strata. Stratified sampling is sometimes easier to do than simple random sampling, especially when the population is already divided into the distinct strata. It can also be more efficient that simple random sampling, especially when some of the subpopulations are rare. For instance, when studying schizophrenia it would be much better to divide the population into two^[Nothing in life is that simple: there's not an obvious division of people into binary categories like \"schizophrenic\" and \"not schizophrenic\". But this isn't a clinical psychology text, so please forgive me a few simplifications here and there.] strata (schizophrenic and not-schizophrenic), and then sample an equal number of people from each group. If you selected people randomly, you would get so few schizophrenic people in the sample that your study would be useless. This specific kind of of stratified sampling is referred to as *oversampling* because it makes a deliberate attempt to over-represent rare groups. \n", "- *Snowball sampling* is a technique that is especially useful when sampling from a \"hidden\" or hard to access population, and is especially common in social sciences. For instance, suppose the researchers want to conduct an opinion poll among transgender people. The research team might only have contact details for a few trans folks, so the survey starts by asking them to participate (stage 1). At the end of the survey, the participants are asked to provide contact details for other people who might want to participate. In stage 2, those new contacts are surveyed. The process continues until the researchers have sufficient data. The big advantage to snowball sampling is that it gets you data in situations that might otherwise be impossible to get any. On the statistical side, the main disadvantage is that the sample is highly non-random, and non-random in ways that are difficult to address. On the real life side, the disadvantage is that the procedure can be unethical if not handled well, because hidden populations are often hidden for a reason. I chose transgender people as an example here to highlight this: if you weren't careful you might end up outing people who don't want to be outed (very, very bad form), and even if you don't make that mistake it can still be intrusive to use people's social networks to study them. It's certainly very hard to get people's informed consent *before* contacting them, yet in many cases the simple act of contacting them and saying \"hey we want to study you\" can be hurtful. Social networks are complex things, and just because you can use them to get data doesn't always mean you should.\n", "- *Convenience sampling* is more or less what it sounds like. The samples are chosen in a way that is convenient to the researcher, and not selected at random from the population of interest. Snowball sampling is one type of convenience sampling, but there are many others. A common example in psychology are studies that rely on undergraduate psychology students. These samples are generally non-random in two respects: firstly, reliance on undergraduate psychology students automatically means that your data are restricted to a single subpopulation. Secondly, the students usually get to pick which studies they participate in, so the sample is a self selected subset of psychology students not a randomly selected subset. In real life, most studies are convenience samples of one form or another. This is sometimes a severe limitation, but not always.\n", "\n", "\n", "### How much does it matter if you don't have a simple random sample?\n", "\n", "Okay, so real world data collection tends not to involve nice simple random samples. Does that matter? A little thought should make it clear to you that it *can* matter if your data are not a simple random sample: just think about the difference between Figures {numref}`srs1-fig` and {numref}`brs-fig`. However, it's not quite as bad as it sounds. Some types of biased samples are entirely unproblematic. For instance, when using a stratified sampling technique you actually *know* what the bias is because you created it deliberately, often to *increase* the effectiveness of your study, and there are statistical techniques that you can use to adjust for the biases you've introduced (not covered in this book!). So in those situations it's not a problem. \n", "\n", "More generally though, it's important to remember that random sampling is a means to an end, not the end in itself. Let's assume you've relied on a convenience sample, and as such you can assume it's biased. A bias in your sampling method is only a problem if it causes you to draw the wrong conclusions. When viewed from that perspective, I'd argue that we don't need the sample to be randomly generated in *every* respect: we only need it to be random with respect to the psychologically-relevant phenomenon of interest. Suppose I'm doing a study looking at working memory capacity. In study 1, I actually have the ability to sample randomly from all human beings currently alive, with one exception: I can only sample people born on a Monday. In study 2, I am able to sample randomly from the Australian population. I want to generalise my results to the population of all living humans. Which study is better? The answer, obviously, is study 1. Why? Because we have no reason to think that being \"born on a Monday\" has any interesting relationship to working memory capacity. In contrast, I can think of several reasons why \"being Australian\" might matter. Australia is a wealthy, industrialised country with a very well-developed education system. People growing up in that system will have had life experiences much more similar to the experiences of the people who designed the tests for working memory capacity. This shared experience might easily translate into similar beliefs about how to \"take a test\", a shared assumption about how psychological experimentation works, and so on. These things might actually matter. For instance, \"test taking\" style might have taught the Australian participants how to direct their attention exclusively on fairly abstract test materials relative to people that haven't grown up in a similar environment; leading to a misleading picture of what working memory capacity is. \n", "\n", "There are two points hidden in this discussion. Firstly, when designing your own studies, it's important to think about what population you care about, and try hard to sample in a way that is appropriate to that population. In practice, you're usually forced to put up with a \"sample of convenience\" (e.g., psychology lecturers sample psychology students because that's the least expensive way to collect data, and our coffers aren't exactly overflowing with gold), but if so you should at least spend some time thinking about what the dangers of this practice might be.\n", "\n", "Secondly, if you're going to criticise someone else's study because they've used a sample of convenience rather than laboriously sampling randomly from the entire human population, at least have the courtesy to offer a specific theory as to *how* this might have distorted the results. Remember, everyone in science is aware of this issue, and does what they can to alleviate it. Merely pointing out that \"the study only included people from group BLAH\" is entirely unhelpful, and borders on being insulting to the researchers, who are *of course* aware of the issue. They just don't happen to be in possession of the infinite supply of time and money required to construct the perfect sample. In short, if you want to offer a responsible critique of the sampling process, then be *helpful*. Rehashing the blindingly obvious truisms that I've been rambling on about in this section isn't helpful.\n", "\n", "### Population parameters and sample statistics\n", "\n", "Okay. Setting aside the thorny methodological issues associated with obtaining a random sample and my rather unfortunate tendency to rant about lazy methodological criticism, let's consider a slightly different issue. Up to this point we have been talking about populations the way a scientist might. To a psychologist, a population might be a group of people. To an ecologist, a population might be a group of bears. In most cases the populations that scientists care about are concrete things that actually exist in the real world. Statisticians, however, are a funny lot. On the one hand, they *are* interested in real world data and real science in the same way that scientists are. On the other hand, they also operate in the realm of pure abstraction in the way that mathematicians do. As a consequence, statistical theory tends to be a bit abstract in how a population is defined. In much the same way that psychological researchers operationalise our abstract theoretical ideas in terms of [concrete measurements](studydesign), statisticians operationalise the concept of a \"population\" in terms of mathematical objects that they know how to work with. You've already come across these objects in the chapter on [probability](probability)): they're called probability distributions.\n", "\n", "The idea is quite simple. Let's say we're talking about IQ scores. To a psychologist, the population of interest is a group of actual humans who have IQ scores. A statistician \"simplifies\" this by operationally defining the population as the probability distribution depicted in {numref}`fig-IQ`. IQ tests are designed so that the average IQ is 100, the standard deviation of IQ scores is 15, and the distribution of IQ scores is normal. These values are referred to as the **_population parameters_** because they are characteristics of the entire population. That is, we say that the population mean $\\mu$ is 100, and the population standard deviation $\\sigma$ is 15." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "import seaborn as sns\n", "import scipy.stats as stats\n", "import math\n", "\n", "# arrange a grid of three plots\n", "fig, axes = plt.subplots(1, 3, figsize=(15, 5))\n", "\n", "# plot normal distribution\n", "mu = 100\n", "sigma = 15\n", "x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)\n", "y = stats.norm.pdf(x, mu, sigma)\n", "ax0 = sns.lineplot(x=x,y=y, ax=axes[0])\n", "\n", "# plot histogram of 100 samples from normal distribution\n", "IQ = np.random.normal(loc=100,scale=15,size=100)\n", "ax1 = sns.histplot(IQ, ax=axes[1])\n", "\n", "# plot histogram of 10000 samples from normal distribution\n", "IQ = np.random.normal(loc=100,scale=15,size=10000)\n", "ax2 = sns.histplot(IQ,ax=axes[2])\n", "\n", "# add titles, labels, and formatting\n", "labels = ['A', 'B', 'C']\n", "titles = ['Normal Distribution', '100 samples', '10,000 samples']\n", "for i,ax in enumerate(axes):\n", " ax.set_title(titles[i])\n", " ax.text(-0.1, 1.15, labels[i], transform=ax.transAxes,\n", " fontsize=16, fontweight='bold', va='top', ha='right')\n", " ax.set(xticklabels=[])\n", " ax.set(yticklabels=[])\n", " ax.set(ylabel=None)\n", " ax.tick_params(axis='both', \n", " which='both',\n", " bottom=False,\n", " left=False)\n", "\n", "# remove top and bottom spines\n", "sns.despine()\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} IQ_fig\n", ":figwidth: 600px\n", ":name: fig-IQ\n", "\n", "The population distribution of IQ scores (panel A) and two samples drawn randomly from it. In panel B we have a sample of 100 observations, and panel C we have a sample of 10,000 observations.\n", "```\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Now suppose I run an experiment. I select 100 people at random and administer an IQ test, giving me a simple random sample from the population. My sample would consist of a collection of numbers like this:\n", "```\n", " 106 101 98 80 74 ... 107 72 100\n", "```\n", "Each of these IQ scores is sampled from a normal distribution with mean 100 and standard deviation 15. So if I plot a histogram of the sample, I get something like the one shown in {numref}`fig-IQ` panel B. As you can see, the histogram is *roughly* the right shape, but it's a very crude approximation to the true population distribution shown in {numref}`fig-IQ` panel A. When I calculate the mean of my sample, I get a number that is fairly close to the population mean 100 but not identical. In this case, it turns out that the people in my sample have a mean IQ of 98.5, and the standard deviation of their IQ scores is 15.9. These **_sample statistics_** are properties of my data set, and although they are fairly similar to the true population values, they are not the same. In general, sample statistics are the things you can calculate from your data set, and the population parameters are the things you want to learn about. Later on in this chapter I'll talk about how you can [estimate population parameters](pointestimates) using your sample statistics and how to work out [how confident you are in your estimates](ci) but before we get to that there's a few more ideas in sampling theory that you need to know about. \n", "\n", "\n", "## The law of large numbers\n", "(lawlargenumbers)=\n", "\n", "In the previous section I showed you the results of one fictitious IQ experiment with a sample size of $N=100$. The results were somewhat encouraging: the true population mean is 100, and the sample mean of 98.5 is a pretty reasonable approximation to it. In many scientific studies that level of precision is perfectly acceptable, but in other situations you need to be a lot more precise. If we want our sample statistics to be much closer to the population parameters, what can we do about it?\n", "\n", "The obvious answer is to collect more data. Suppose that we ran a much larger experiment, this time measuring the IQs of 10,000 people. We can simulate the results of this experiment using Python. To create the simulated data in plotted in {numref}`fig-IQ` panel B, I used the ``numpy`` package to sample from a normal distribution with a mean of 100 and a standard deviation of 15. By altering the size of the sample drawn and plotting histograms of the frequencies of the values in the samples, it is easy to create simulated data that illustrate the effect of increasing the sampling size. The histogram of this much larger sample is shown in {numref}`fig-IQ` panel C. Even a moment's inspections makes clear that the larger sample is a much better approximation to the true population distribution than the smaller one. \n", "\n", "Let's take a look at the how close our estimates of the true population distribution are, if we simply change the sampling size. Below, I have created three samples from a normal distribution with a mean of 100, and a standard deviation of 15. The only thing that changes is the number of samples drawn. Because these are random samples from the normal distribution, the exact values of the estimated mean and standard deviation may change slightly each time the code is run, but overall tendency will always be for larger samples to give a better estimate of the true parameters of the poulation distribution:" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "10 samples. Mean: 102.84232415781497 Standard deviation: 12.828447346156201\n", "100 samples. Mean: 101.23645948381213 Standard deviation: 16.117663878002418\n", "10000 samples. Mean: 99.92258949264254 Standard deviation: 15.004958779974102\n" ] } ], "source": [ "import numpy as np\n", "import statistics\n", "\n", "IQ_10 = np.random.normal(loc=100,scale=15,size=10)\n", "IQ_100 = np.random.normal(loc=100,scale=15,size=100)\n", "IQ_10000 = np.random.normal(loc=100,scale=15,size=10000)\n", "\n", "print(\"10 samples. Mean: \", statistics.mean(IQ_10), \" Standard deviation: \", statistics.stdev(IQ_10))\n", "print(\"100 samples. Mean: \", statistics.mean(IQ_100), \" Standard deviation: \", statistics.stdev(IQ_100))\n", "print(\"10000 samples. Mean: \", statistics.mean(IQ_10000), \" Standard deviation: \", statistics.stdev(IQ_10000))" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "I feel a bit silly saying this, but the thing I want you to take away from this is that large samples generally give you better information. I feel silly saying it because it's so bloody obvious that it shouldn't need to be said. In fact, it's such an obvious point that when Jacob Bernoulli -- one of the founders of probability theory -- formalised this idea back in 1713, he was kind of a jerk about it. Here's how he described the fact that we all share this intuition {cite}`Stigler1986`:\n", "\n", "> *For even the most stupid of men, by some instinct of nature, by himself and without any instruction (which is a remarkable thing), is convinced that the more observations have been made, the less danger there is of wandering from one's goal* \n", "\n", "Okay, so the passage comes across as a bit condescending, but his main point is correct: it really does feel obvious that more data will give you better answers. The question is, why is this so? Not surprisingly, this intuition that we all share turns out to be correct, and statisticians refer to it as the **_law of large numbers_**. The law of large numbers is a mathematical law that applies to many different sample statistics, but the simplest way to think about it is as a law about averages. The sample mean is the most obvious example of a statistic that relies on averaging (because that's what the mean is... an average), so let's look at that. When applied to the sample mean, what the law of large numbers states is that as the sample gets larger, the sample mean tends to get closer to the true population mean. Or, to say it a little bit more precisely, as the sample size \"approaches\" infinity (written as $N \\rightarrow \\infty$) the sample mean approaches the population mean ($\\bar{X} \\rightarrow \\mu$).[^notelawlargenumbers]\n", "\n", "I don't intend to subject you to a proof that the law of large numbers is true, but it's one of the most important tools for statistical theory. The law of large numbers is the thing we can use to justify our belief that collecting more and more data will eventually lead us to the truth. For any particular data set, the sample statistics that we calculate from it **_will be wrong_**, but the law of large numbers tells us that if we keep collecting more data those sample statistics will tend to get closer and closer to the true population parameters.\n", "\n", "[^notenotelawlargenumbers]: Technically, the law of large numbers pertains to any sample statistic that can be described as an average of independent quantities. That's certainly true for the sample mean. However, it's also possible to write many other sample statistics as averages of one form or another. The variance of a sample, for instance, can be rewritten as a kind of average and so is subject to the law of large numbers. The minimum value of a sample, however, cannot be written as an average of anything and is therefore not governed by the law of large numbers." ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(samplingdists)=\n", "## Sampling distributions and the central limit theorem\n", "\n", "The law of large numbers is a very powerful tool, but it's not going to be good enough to answer all our questions. Among other things, all it gives us is a \"long run guarantee\". In the long run, if we were somehow able to collect an infinite amount of data, then the law of large numbers guarantees that our sample statistics will be correct. But as John Maynard Keynes famously argued in economics, a long run guarantee is of little use in real life:\n", "\n", "> *[The] long run is a misleading guide to current affairs. In the long run we are all dead. Economists set themselves too easy, too useless a task, if in tempestuous seasons they can only tell us, that when the storm is long past, the ocean is flat again.* {cite}`Keynes1923`\n", "\n", "As in economics, so too in psychology and statistics. It is not enough to know that we will *eventually* arrive at the right answer when calculating the sample mean. Knowing that an infinitely large data set will tell me the exact value of the population mean is cold comfort when my *actual* data set has a sample size of $N=100$. In real life, then, we must know something about the behaviour of the sample mean when it is calculated from a more modest data set!\n", "\n", "### Sampling distribution of the mean\n", "\n", "\n", "With this in mind, let's abandon the idea that our studies will have sample sizes of 10000, and consider a very modest experiment indeed. This time around we'll sample $N=5$ people and measure their IQ scores. As before, I can simulate this experiment in Python using numpy's `random.normal()` function. We can convert these to integers with `astype(int)`" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Simulated data: [140 116 96 119 122]\n", "Mean of simulated data: 118\n" ] } ], "source": [ "import numpy as np\n", "import statistics\n", "\n", "IQ_1 = np.random.normal(loc=100,scale=15,size=5).astype(int)\n", "print(\"Simulated data: \", IQ_1)\n", "print(\"Mean of simulated data: \", statistics.mean(IQ_1))" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Because we are sampling at random from the normal distribution, the mean of the simulated data will change every time we run the code. In fact, this number will likely change every time I updat this book, since the code will run again, and pick five new values from the normal distribution. Depending on the luck of the draw, the mean may be closer or further from the true mean of 100. Now imagine that I decided to **_replicate_** the experiment. That is, I repeat the procedure as closely as possible: I randomly sample 5 new people and measure their IQ. Again, Python allows me to simulate the results of this procedure:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Simulated data: [ 97 95 73 111 123]\n", "Mean of simulated data: 99\n" ] } ], "source": [ "IQ_2 = np.random.normal(loc=100,scale=15,size=5).astype(int)\n", "print(\"Simulated data: \", IQ_2)\n", "print(\"Mean of simulated data: \", statistics.mean(IQ_2))" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "If I repeat the experiment 10 times, I can a whole series of simulated experiments, each showing the results of five \"people\" participating in my \"experiment\", like so:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Person 1Person 2Person 3Person 4Person 5Sample Mean
Replication 1101104129117100110.2
Replication 2108105111113100107.4
Replication 38199100838689.8
Replication 410887809310394.2
Replication 5102106117888098.6
Replication 610791101114107104.0
Replication 71069398106108102.2
Replication 89810710294126105.4
Replication 910485747511590.6
Replication 108111479959492.6
\n", "
" ], "text/plain": [ " Person 1 Person 2 Person 3 Person 4 Person 5 Sample Mean\n", "Replication 1 101 104 129 117 100 110.2\n", "Replication 2 108 105 111 113 100 107.4\n", "Replication 3 81 99 100 83 86 89.8\n", "Replication 4 108 87 80 93 103 94.2\n", "Replication 5 102 106 117 88 80 98.6\n", "Replication 6 107 91 101 114 107 104.0\n", "Replication 7 106 93 98 106 108 102.2\n", "Replication 8 98 107 102 94 126 105.4\n", "Replication 9 104 85 74 75 115 90.6\n", "Replication 10 81 114 79 95 94 92.6" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "df = pd.DataFrame(\n", " {'Person 1': np.random.normal(loc=100,scale=15,size=10).astype(int),\n", " 'Person 2': np.random.normal(loc=100,scale=15,size=10).astype(int),\n", " 'Person 3': np.random.normal(loc=100,scale=15,size=10).astype(int),\n", " 'Person 4': np.random.normal(loc=100,scale=15,size=10).astype(int),\n", " 'Person 5': np.random.normal(loc=100,scale=15,size=10).astype(int),\n", " }) \n", "\n", "df['Sample Mean']=df.mean(axis=1)\n", "df.index=['Replication '+str(i+1) for i in range(df.shape[0])]\n", "df" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Now suppose that I decided to keep going in this fashion, replicating this \"five IQ scores\" experiment over and over again. Every time I replicate the experiment I write down the sample mean. Over time, I'd be amassing a new data set, in which every experiment generates a single data point. The first 10 observations from my data set are the sample means listed in the table below, so my data set starts out like this:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[110.2, 107.4, 89.8, 94.2, 98.6, 104.0, 102.2, 105.4, 90.6, 92.6]" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "list(df['Sample Mean'])" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "What if I continued like this for 10,000 replications, and then drew a histogram? Using the magical powers of Python that's exactly what I did, and you can see the results in {numref}`fig-IQ_samp_dist`. As this picture illustrates, the average of 5 IQ scores is usually between 90 and 110. But more importantly, what it highlights is that if we replicate an experiment over and over again, what we end up with is a *distribution* of sample means! This distribution has a special name in statistics: it's called the **_sampling distribution of the mean_**. " ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import scipy.stats as stats\n", "import seaborn as sns\n", "import numpy as np\n", "import statistics\n", "import math\n", "\n", "# define a normal distribution with a mean of 100 and a standard deviation of 15\n", "mu = 100\n", "sigma = 15\n", "x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)\n", "y = stats.norm.pdf(x, mu, sigma)\n", "\n", "# run 10000 simulated experiments with 5 subjects each, and calculate the sample mean for each experiment\n", "sample_means = []\n", "for i in range(1,10000):\n", " sample_mean = statistics.mean(np.random.normal(loc=100,scale=15,size=5).astype(int))\n", " sample_means.append(sample_mean)\n", "\n", "\n", "# plot a histogram of the distribution of sample means, together with the population distribution\n", "fig, ax = plt.subplots()\n", "sns.histplot(sample_means, ax=ax)\n", "ax2 = ax.twinx()\n", "sns.lineplot(x=x,y=y, ax=ax2, color='black')\n", "\n", "# format the figure\n", "axes=[ax, ax2]\n", "for ax in axes:\n", " ax.set(yticklabels=[])\n", " ax.set(ylabel=None)\n", " ax.set(xlabel='IQ Score')\n", " ax.tick_params(axis='both', \n", " which='both',\n", " left=False,\n", " right=False)\n", " ax.spines[['right', 'top']].set_visible(False)\n", " \n", "\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} IQ_samp_dist\n", ":figwidth: 600px\n", ":name: fig-IQ_samp_dist\n", "\n", "The sampling distribution of the mean for the \\\"five IQ scores experiment\\\". If you sample 5 people at random and calculate their *average* IQ, you'll almost certainly get a number between 80 and 120, even though there are quite a lot of individuals who have IQs above 120 or below 80. For comparison, the black line plots the population distribution of IQ scores.\n", "```\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "Sampling distributions are another important theoretical idea in statistics, and they're crucial for understanding the behaviour of small samples. For instance, when I ran the very first \"five IQ scores\" experiment, the sample mean turned out to be 95. What the sampling distribution in {numref}`fig-IQ_samp_dist` tells us, though, is that the \"five IQ scores\" experiment is not very accurate. If I repeat the experiment, the sampling distribution tells me that I can expect to see a sample mean anywhere between 80 and 120. \n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "\n", "### Sampling distributions exist for any sample statistic!\n", "\n", "One thing to keep in mind when thinking about sampling distributions is that *any* sample statistic you might care to calculate has a sampling distribution. For example, suppose that each time I replicated the \"five IQ scores\" experiment I wrote down the largest IQ score in the experiment. This might give me a data set that started out like this:\n", "```\n", " 110 117 122 119 113 ... \n", "```\n", "Doing this over and over again would give me a very different sampling distribution, namely the *sampling distribution of the maximum*. The sampling distribution of the maximum of 5 IQ scores is shown in {numref}`fig-IQ_max`. Not surprisingly, if you pick 5 people at random and then find the person with the highest IQ score, they're going to have an above average IQ. Most of the time you'll end up with someone whose IQ is measured in the 100 to 140 range. " ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "import scipy.stats as stats\n", "import math\n", "\n", "# define a normal distribution with a mean of 100 and a standard deviation of 15\n", "mu = 100\n", "sigma = 15\n", "x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)\n", "y = stats.norm.pdf(x, mu, sigma)\n", "\n", "# run 10000 simulated experiments with 5 subjects each, and find the maximum score for each experiment\n", "sample_maxes = []\n", "for i in range(1,10000):\n", " sample_max = max(np.random.normal(loc=100,scale=15,size=5).astype(int))\n", " sample_maxes.append(sample_max)\n", "\n", "\n", "# plot a histogram of the distribution of sample maximums, together with the population distribution\n", "fig, ax = plt.subplots()\n", "sns.histplot(sample_maxes, ax=ax)\n", "ax2 = ax.twinx()\n", "sns.lineplot(x=x,y=y, ax=ax2, color='black')\n", "\n", "# format the figure\n", "axes=[ax, ax2]\n", "for ax in axes:\n", " ax.set(yticklabels=[])\n", " ax.set(ylabel=None)\n", " ax.set(xlabel='IQ Score')\n", " ax.tick_params(axis='both', \n", " which='both',\n", " left=False,\n", " right=False)\n", " ax.spines[['right', 'top']].set_visible(False)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} IQ_max_fig\n", ":figwidth: 600px\n", ":name: fig-IQ_max\n", "\n", "The sampling distribution of the *maximum* for the \\\"five IQ scores experiment\\\". If you sample 5 people at random and select the one with the highest IQ score, you'll probably see someone with an IQ between 100 and 140.\n", "```\n", "\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(clt)=\n", "### The central limit theorem\n", "\n", "At this point I hope you have a pretty good sense of what sampling distributions are, and in particular what the sampling distribution of the mean is. In this section I want to talk about how the sampling distribution of the mean changes as a function of sample size. Intuitively, you already know part of the answer: if you only have a few observations, the sample mean is likely to be quite inaccurate: if you replicate a small experiment and recalculate the mean you'll get a very different answer. In other words, the sampling distribution is quite wide. If you replicate a large experiment and recalculate the sample mean you'll probably get the same answer you got last time, so the sampling distribution will be very narrow. You can see this visually in {numref}`fig-IQ-clm`: the bigger the sample size, the narrower the sampling distribution gets. We can quantify this effect by calculating the standard deviation of the sampling distribution, which is referred to as the **_standard error_**. The standard error of a statistic is often denoted SE, and since we're usually interested in the standard error of the sample *mean*, we often use the acronym SEM. As you can see just by looking at the picture, as the sample size $N$ increases, the SEM decreases." ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "import scipy.stats as stats\n", "import seaborn as sns\n", "import statistics\n", "import math\n", "\n", "# define a normal distribution with a mean of 100 and a standard deviation of 15\n", "mu = 100\n", "sigma = 15\n", "x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)\n", "y = stats.norm.pdf(x, mu, sigma)\n", "\n", "# arrange a grid of three plots\n", "fig, axes = plt.subplots(1, 3, figsize=(15, 5))\n", "\n", "\n", "labels = ['A', 'B', 'C']\n", "titles = ['N = 1', 'N = 2', 'N = 10']\n", "\n", "# run 10000 simulated experiments with either 1, 2, or 10 subjects each, and calculate the sample mean for each experiment\n", "for s,n in enumerate([1, 2, 10]):\n", " sample_means = []\n", " for i in range(1,10000):\n", " sample_mean = statistics.mean(np.random.normal(loc=100,scale=15,size=n).astype(int))\n", " sample_means.append(sample_mean)\n", "\n", " # plot a histogram of the distribution of sample means, together with the population distribution\n", " ax1 = sns.histplot(sample_means, ax=axes[s], binwidth=4)\n", " ax1.set_title(titles[s])\n", " ax1.text(-0.1, 1.15, labels[s], transform=ax1.transAxes,fontsize=16, fontweight='bold', va='top', ha='right')\n", " ax2 = ax1.twinx()\n", " sns.lineplot(x=x,y=y, color='black')\n", " \n", "\n", " # format the figures\n", " subaxes=[ax1, ax2]\n", " for ax in subaxes:\n", " ax.set(yticklabels=[])\n", " ax.set(ylabel=None)\n", " ax.set(xlabel='IQ Score')\n", " ax.tick_params(axis='both', \n", " which='both',\n", " left=False,\n", " right=False)\n", " ax.spines[['right', 'top']].set_visible(False)\n", " \n", "\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "```{glue:figure} IQ_clm-fig\n", ":figwidth: 600px\n", ":name: fig-IQ-clm\n", "\n", "An illustration of the how sampling distribution of the mean depends on sample size. In each panel, I generated 10,000 samples of IQ data, and calculated the mean IQ observed within each of these data sets. The histograms in these plots show the distribution of these means (i.e., the sampling distribution of the mean). Each individual IQ score was drawn from a normal distribution with mean 100 and standard deviation 15, which is shown as the solid black line). In panel A, each data set contained only a single observation, so the mean of each sample is just one person’s IQ score. As a consequence, the sampling distribution of the mean is of course identical to the population distribution of IQ scores. However, when we raise the sample size to 2, the mean of any one sample tends to be closer to the population mean than a one person’s IQ score, and so the histogram (i.e., the sampling distribution) is a bit narrower than the population distribution (panel B). By the time we raise the sample size to 10 (panel C), we can see that the distribution of sample means tend to be fairly tightly clustered around the true population mean.\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "\n", "Okay, so that's one part of the story. However, there's something I've been glossing over so far. All my examples up to this point have been based on the \"IQ scores\" experiments, and because IQ scores are roughly normally distributed, I've assumed that the population distribution is normal. What if it isn't normal? What happens to the sampling distribution of the mean? The remarkable thing is this: no matter what shape your population distribution is, as $N$ increases the sampling distribution of the mean starts to look more like a normal distribution. To give you a sense of this, I ran some simulations using Python. To do this, I wrote a function called `plotSamples` that produces the \"ramped\" beta distribution shown in the first histogram below when $N=1$. You can use the \"click to show\" button to take a look at the code, if you want to see how it works. The important thing for our purposes is that, as you can see by comparing the triangular shaped histogram to the bell curve plotted by the black line, the population distribution doesn't look very much like a normal distribution at all. Then I provided the function with increasingly larger numbers of \"participants\" for each simulated experiment. As the size of $N$ increases, the sampling distribution of the mean looks increasingly normal, and by the time we reach a sample size of $N=8$ it's almost perfectly normal. In other words, as long as your sample size isn't tiny, the sampling distribution of the mean will be approximately normal no matter what your population distribution looks like!" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "tags": [ "hide-input" ] }, "outputs": [], "source": [ "import math\n", "import numpy as np\n", "import pandas as pd\n", "import seaborn as sns\n", "import matplotlib.pyplot as plt\n", "\n", "# parameters of the beta\n", "a=2\n", "b=1\n", "\n", "def plotSamples(n):\n", " # create normal distribution with mean and standard deviation of the beta\n", " mu = a / (a+b)\n", " sigma = math.sqrt( a*b / (a+b)**2 / (a+b+1) )\n", " x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)\n", " y = stats.norm.pdf(x, mu, sigma/math.sqrt(n))\n", "\n", " # find sample means from samples of \"ramped\" beta distribution\n", "\n", " values = []\n", " for i in range(n):\n", " v = []\n", " for j in range(50000):\n", " v.append(np.random.beta(a,b))\n", " values.append(v)\n", " df = pd.DataFrame(values)\n", " sample_means = df.mean(axis=0)\n", "\n", " # plot a histogram of the distribution of sample means, together with the population distribution\n", " fig, ax = plt.subplots(sharex=True)\n", " sns.histplot(sample_means)\n", " ax2 = ax.twinx()\n", " sns.lineplot(x=x,y=y, ax=ax2, color='black')\n", " \n", " # format the plots\n", " axes = [ax, ax2]\n", " for ax in axes:\n", " ax.set(yticklabels=[])\n", " ax.set(ylabel=None)\n", " ax.tick_params(left=False) \n", " ax.spines[['top', 'right']].set_visible(False)\n", " ax.tick_params(axis='both', \n", " which='both',\n", " left=False,\n", " right=False)\n", " ax.set_title(\"Sample size = \" + str(n))\n" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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", 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", 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plotSamples(8)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "On the basis of these figures, it seems like we have evidence for all of the following claims about the sampling distribution of the mean:\n", "\n", "- The mean of the sampling distribution is the same as the mean of the population\n", "- The standard deviation of the sampling distribution (i.e., the standard error) gets smaller as the sample size increases\n", "- The shape of the sampling distribution becomes normal as the sample size increases\n", "\n", "As it happens, not only are all of these statements true, there is a very famous theorem in statistics that proves all three of them, known as the **_central limit theorem_**. Among other things, the central limit theorem tells us that if the population distribution has mean $\\mu$ and standard deviation $\\sigma$, then the sampling distribution of the mean also has mean $\\mu$, and the standard error of the mean is \n", "\n", "$$\n", "\\mbox{SEM} = \\frac{\\sigma}{ \\sqrt{N} }\n", "$$\n", "\n", "Because we divide the population standard devation $\\sigma$ by the square root of the sample size $N$, the SEM gets smaller as the sample size increases. It also tells us that the shape of the sampling distribution becomes normal. [^note_clm] \n", "\n", "This result is useful for all sorts of things. It tells us why large experiments are more reliable than small ones, and because it gives us an explicit formula for the standard error it tells us *how much* more reliable a large experiment is. It tells us why the normal distribution is, well, *normal*. In real experiments, many of the things that we want to measure are actually averages of lots of different quantities (e.g., arguably, \"general\" intelligence as measured by IQ is an average of a large number of \"specific\" skills and abilities), and when that happens, the averaged quantity should follow a normal distribution. Because of this mathematical law, the normal distribution pops up over and over again in real data.\n", "\n", "[^note_clm]: As usual, I'm being a bit sloppy here. The central limit theorem is a bit more general than this section implies. Like most introductory stats texts, I've discussed one situation where the central limit theorem holds: when you're taking an average across lots of independent events drawn from the same distribution. However, the central limit theorem is much broader than this. There's a whole class of things called \"$U$-statistics\" for instance, all of which satisfy the central limit theorem and therefore become normally distributed for large sample sizes. The mean is one such statistic, but it's not the only one." ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(pointestimates)=\n", "## Estimating population parameters\n", "\n", "In all the IQ examples in the previous sections, we actually knew the population parameters ahead of time. As every undergraduate gets taught in their very first lecture on the measurement of intelligence, IQ scores are *defined* to have mean 100 and standard deviation 15. However, this is a bit of a lie. How do we know that IQ scores have a true population mean of 100? Well, we know this because the people who designed the tests have administered them to very large samples, and have then \"rigged\" the scoring rules so that their sample has mean 100. That's not a bad thing of course: it's an important part of designing a psychological measurement. However, it's important to keep in mind that this theoretical mean of 100 only attaches to the population that the test designers used to design the tests. Good test designers will actually go to some lengths to provide \"test norms\" that can apply to lots of different populations (e.g., different age groups, nationalities etc). \n", "\n", "This is very handy, but of course almost every research project of interest involves looking at a different population of people to those used in the test norms. For instance, suppose you wanted to measure the effect of low level lead poisoning on cognitive functioning in Port Pirie, a South Australian industrial town with a lead smelter. Perhaps you decide that you want to compare IQ scores among people in Port Pirie to a comparable sample in Whyalla, a South Australian industrial town with a steel refinery. [^note_hard] Regardless of which town you're thinking about, it doesn't make a lot of sense simply to *assume* that the true population mean IQ is 100. No-one has, to my knowledge, produced sensible norming data that can automatically be applied to South Australian industrial towns. We're going to have to **_estimate_** the population parameters from a sample of data. So how do we do this?\n", "\n", "[^note_hard]: Please note that if you were *actually* interested in this question, you would need to be a *lot* more careful than I'm being here. You *can't* just compare IQ scores in Whyalla to Port Pirie and assume that any differences are due to lead poisoning. Even if it were true that the only differences between the two towns corresponded to the different refineries (and it isn't, not by a long shot), you need to account for the fact that people already *believe* that lead pollution causes cognitive deficits: if you recall back to the chapter on [study design](studydesign), this means that there are different demand effects for the Port Pirie sample than for the Whyalla sample. In other words, you might end up with an illusory group difference in your data, caused by the fact that people *think* that there is a real difference. I find it pretty implausible to think that the locals wouldn't be well aware of what you were trying to do if a bunch of researchers turned up in Port Pirie with lab coats and IQ tests, and even less plausible to think that a lot of people would be pretty resentful of you for doing it. Those people won't be as co-operative in the tests. Other people in Port Pirie might be *more* motivated to do well because they don't want their home town to look bad. The motivational effects that would apply in Whyalla are likely to be weaker, because people don't have any concept of \"iron ore poisoning\" in the same way that they have a concept for \"lead poisoning\". Psychology is *hard*.\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "\n", "### Estimating the population mean\n", "\n", "Suppose we go to Port Pirie and 100 of the locals are kind enough to sit through an IQ test. The average IQ score among these people turns out to be $\\bar{X}=98.5$. So what is the true mean IQ for the entire population of Port Pirie? Obviously, we don't know the answer to that question. It could be $97.2$, but if could also be $103.5$. Our sampling isn't exhaustive so we cannot give a definitive answer. Nevertheless if I was forced at gunpoint to give a \"best guess\" I'd have to say $98.5$. That's the essence of statistical estimation: giving a best guess. \n", "\n", "In this example, estimating the unknown poulation parameter is straightforward. I calculate the sample mean, and I use that as my **_estimate of the population mean_**. It's pretty simple, and in the next section I'll explain the statistical justification for this intuitive answer. However, for the moment what I want to do is make sure you recognise that the sample statistic and the estimate of the population parameter are conceptually different things. A sample statistic is a description of your data, whereas the estimate is a guess about the population. With that in mind, statisticians often use different notation to refer to them. For instance, if true population mean is denoted $\\mu$, then we would use $\\hat\\mu$ to refer to our estimate of the population mean. In contrast, the sample mean is denoted $\\bar{X}$ or sometimes $m$. However, in simple random samples, the estimate of the population mean is identical to the sample mean: if I observe a sample mean of $\\bar{X} = 98.5$, then my estimate of the population mean is also $\\hat\\mu = 98.5$. To help keep the notation clear, here's a handy table:" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "|Symbol |What it's called |Do we know what it is? |\n", "|:-----------|:-------------------------------|:---------------------------------|\n", "|$\\bar{X}$ |Sample mean |Yes - Calculated from the raw data |\n", "|$\\mu$ |True population mean |Almost never known for sure |\n", "|$\\hat{\\mu}$ |Estimate of the population mean |Yes - Identical to the sample mean |" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "### Estimating the population standard deviation\n", "\n", "So far, estimation seems pretty simple, and you might be wondering why I forced you to read through all that stuff about sampling theory. In the case of the mean, our estimate of the population parameter (i.e. $\\hat\\mu$) turned out to identical to the corresponding sample statistic (i.e. $\\bar{X}$). However, that's not always true. To see this, let's have a think about how to construct an **_estimate of the population standard deviation_**, which we'll denote $\\hat\\sigma$. What shall we use as our estimate in this case? Your first thought might be that we could do the same thing we did when estimating the mean, and just use the sample statistic as our estimate. That's almost the right thing to do, but not quite. \n", "\n", "Here's why. Suppose I have a sample that contains a single observation. For this example, it helps to consider a sample where you have no intutions at all about what the true population values might be, so let's use something completely fictitious. Suppose the observation in question measures the *cromulence* of my shoes. It turns out that my shoes have a cromulence of 20. So here's my sample:\n", "\n", "```\n", "20\n", "```\n", "\n", "This is a perfectly legitimate sample, even if it does have a sample size of $N=1$. It has a sample mean of 20, and because every observation in this sample is equal to the sample mean (obviously!) It has a sample standard deviation of 0. As a description of the *sample* this seems quite right: the sample contains a single observation and therefore there is no variation observed within the sample. A sample standard deviation of $s = 0$ is the right answer here. But as an estimate of the *population* standard deviation, it feels completely insane, right? Admittedly, you and I don't know anything at all about what \"cromulence\" is, but we know something about data: the only reason that we don't see any variability in the *sample* is that the sample is too small to display any variation! So, if you have a sample size of $N=1$, it *feels* like the right answer is just to say \"no idea at all\". \n", "\n", "Notice that you *don't* have the same intuition when it comes to the sample mean and the population mean. If forced to make a best guess about the population mean, it doesn't feel completely insane to guess that the population mean is 20. Sure, you probably wouldn't feel very confident in that guess, because you have only the one observation to work with, but it's still the best guess you can make. \n", "\n", "Let's extend this example a little. Suppose I now make a second observation. My data set now has $N=2$ observations of the cromulence of shoes, and the complete sample now looks like this:\n", "```\n", "20, 22\n", "```\n", "This time around, our sample is *just* large enough for us to be able to observe some variability: two observations is the bare minimum number needed for any variability to be observed! For our new data set, the sample mean is $\\bar{X}=21$, and the sample standard deviation is $s=1$. What intuitions do we have about the population? Again, as far as the population mean goes, the best guess we can possibly make is the sample mean: if forced to guess, we'd probably guess that the population mean cromulence is 21. What about the standard deviation? This is a little more complicated. The sample standard deviation is only based on two observations, and if you're at all like me you probably have the intuition that, with only two observations, we haven't given the population \"enough of a chance\" to reveal its true variability to us. It's not just that we suspect that the estimate is *wrong*: after all, with only two observations we expect it to be wrong to some degree. The worry is that the error is *systematic*. Specifically, we suspect that the sample standard deviation is likely to be smaller than the population standard deviation. " ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import statistics\n", "import numpy as np\n", "import seaborn as sns\n", "\n", "# generate data from 10000 \"IQ\" studies, where each study consists of two scores\n", "n = 2\n", "sample_sds = []\n", "for i in range(1,10000):\n", " sample_sd = statistics.stdev(np.random.normal(loc=100,scale=15,size=n))\n", " sample_sds.append(sample_sd)\n", "\n", "\n", "# plot a histogram of the distribution of sample standard deviations, together with dashed line indicating \n", "# population standard deviation\n", "#fig, ax = plt.subplots()\n", "ax=sns.histplot(sample_sds, binwidth=4)\n", "plt.axvline(15, color = 'black', linestyle = \"dashed\")\n", "\n", "ax.set(yticklabels=[])\n", "ax.set(ylabel=None)\n", "ax.set(xlabel='Sample Standard Deviation')\n", "ax.set_title(\"Population Standard Deviation\")\n", "ax.tick_params(left=False) \n", "ax.spines[['top', 'right']].set_visible(False)\n", "ax.tick_params(axis='both', \n", " which='both',\n", " left=False,\n", " right=False)\n", "\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "\n", " ```{glue:figure} sampdistsd-fig\n", ":figwidth: 600px\n", ":name: fig-sampdistsd\n", "\n", "The sampling distribution of the sample standard deviation for a \\\"two IQ scores\\\" experiment. The true population standard deviation is 15 (dashed line), but as you can see from the histogram, the vast majority of experiments will produce a much smaller sample standard deviation than this. On average, this experiment would produce a sample standard deviation of only 8.5, well below the true value! In other words, the sample standard deviation is a *biased* estimate of the population standard deviation.\n", "\n", "```\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "\n", "This intuition feels right, but it would be nice to demonstrate this somehow. There are in fact mathematical proofs that confirm this intuition, but unless you have the right mathematical background they don't help very much. Instead, what I'll do is use Python to simulate the results of some experiments. With that in mind, let's return to our IQ studies. Suppose the true population mean IQ is 100 and the standard deviation is 15. I can use the `rnorm()` function to generate the the results of an experiment in which I measure $N=2$ IQ scores, and calculate the sample standard deviation. If I do this over and over again, and plot a histogram of these sample standard deviations, what I have is the *sampling distribution of the standard deviation*. I've plotted this distribution in {numref}`fig-sampdistsd`. Even though the true population standard deviation is 15, the average of the *sample* standard deviations is only 8.5. Notice that this is a very different result to what we found in {numref}`fig-IQ-clm` when we plotted the sampling distribution of the mean. If you look at that sampling distribution, what you see is that the population mean is 100, and the average of the sample means is also 100. " ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Now let's extend the simulation. Instead of restricting ourselves to the situation where we have a sample size of $N=2$, let's repeat the exercise for sample sizes from 1 to 10. If we plot the average sample mean and average sample standard deviation as a function of sample size, you get the results shown in {numref}`fig-estimatorbias`. On the left hand side (panel a), I've plotted the average sample mean and on the right hand side (panel b), I've plotted the average standard deviation. The two plots are quite different: *on average*, the average sample mean is equal to the population mean. It is an **_unbiased estimator_**, which is essentially the reason why your best estimate for the population mean is the sample mean. [^notebias] The plot on the right is quite different: on average, the sample standard deviation $s$ is *smaller* than the population standard deviation $\\sigma$. It is a **_biased estimator_**. In other words, if we want to make a \"best guess\" $\\hat\\sigma$ about the value of the population standard deviation $\\sigma$, we should make sure our guess is a little bit larger than the sample standard deviation $s$. \n", "\n", "[^notebias]: I should note that I'm hiding something here. Unbiasedness is a desirable characteristic for an estimator, but there are other things that matter besides bias. However, it's beyond the scope of this book to discuss this in any detail. I just want to draw your attention to the fact that there's some hidden complexity here." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import statistics\n", "import numpy as np\n", "import seaborn as sns\n", "import pandas as pd\n", "from matplotlib import pyplot as plt\n", "\n", "\n", "\n", "ns = range(2,11)\n", "\n", "\n", "averageSampleSds = []\n", "averageSampleMeans = []\n", "\n", "# Simulate data for N = 2 to 10\n", "for n in ns:\n", " sample_sds = []\n", " sample_means = []\n", " for i in range(1,10000):\n", " sample_sd = statistics.stdev(np.random.normal(loc=100,scale=15,size=n))#.astype(int))\n", " sample_sds.append(sample_sd)\n", " sample_mean = statistics.mean(np.random.normal(loc=100,scale=15,size=n))#.astype(int))\n", " sample_means.append(sample_mean)\n", " averageSampleSds.append(statistics.mean(sample_sds))\n", " averageSampleMeans.append(statistics.mean(sample_means))\n", "\n", "# Simulate data for N = 1. This is not possible in the loop above, because Python can't calculate a SD\n", "# from only one observation\n", "n = 1\n", "sample_mean_1 = []\n", "for i in range(1,10000):\n", " sample_mean = statistics.mean(np.random.normal(loc=100,scale=15,size=n).astype(int))\n", " sample_mean_1.append(sample_mean)\n", "\n", "# Add in sample mean and SD for N=1 at the beginning of the lists\n", "# For N = 1, the sample SD is simply 0\n", "averageSampleSds.insert(0,0)\n", "averageSampleMeans.insert(0,statistics.mean(sample_mean_1))\n", "\n", "# Collect simulated data in a dataframe, together with a vector from 1 to 10 representing N\n", "df = pd.DataFrame(\n", " {'N': range(1,11),\n", " 'SampleMeans': averageSampleMeans,\n", " 'SampleSDs': averageSampleSds\n", " })\n", "\n", "# Plot the data\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 5), sharey=False)\n", "fig.suptitle('Simulated IQ Data')\n", "\n", "# Format the figure\n", "sns.lineplot(data=df, x='N', y='SampleMeans',ax=axes[0], linestyle = \"dashdot\")\n", "sns.lineplot(data=df, x='N', y='SampleSDs',ax=axes[1], linestyle = \"dashdot\")\n", "axes[0].set(ylim=(0,120))\n", "axes[1].set(ylim=(0,17))\n", "axes[0].axhline(100, color = 'black', linestyle = \"dashed\")\n", "axes[1].axhline(15, color = 'black', linestyle = \"dashed\")\n", "axes[0].set_title(\"Sample Means\")\n", "axes[1].set_title(\"Sample Standard Deviations\")\n", "axes[0].spines[['top', 'right']].set_visible(False)\n", "axes[1].spines[['top', 'right']].set_visible(False)\n", "\n", "labels = ['A', 'B']\n", "for s, ax in enumerate(axes):\n", " axes[s].text(-0.1, 1, labels[s], transform=axes[s].transAxes,fontsize=16, fontweight='bold', va='top', ha='right')\n", "\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "\n", " ```{glue:figure} estimatorbias-fig\n", ":figwidth: 600px\n", ":name: fig-estimatorbias\n", "\n", "An illustration of the fact that the sample mean is an unbiased estimator of the population mean (panel a), but the sample standard deviation is a biased estimator of the population standard deviation (panel b). To generate the figure, I generated 10,000 simulated data sets with 1 observation each, 10,000 more with 2 observations, and so on up to a sample size of 10. Each data set consisted of fake IQ data: that is, the data were normally distributed with a true population mean of 100 and standard deviation 15. *On average*, the sample means turn out to be 100, regardless of sample size (panel a). However, the sample standard deviations turn out to be systematically too small (panel b), especially for small sample sizes.\n", "\n", "```\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "\n", "The fix to this systematic bias turns out to be very simple. Here's how it works. Before tackling the standard deviation, let's look at the variance. If you recall from the section on [measures of variability](variability), the sample variance is defined to be the average of the squared deviations from the sample mean. That is:\n", "\n", "$$\n", "s^2 = \\frac{1}{N} \\sum_{i=1}^N (X_i - \\bar{X})^2\n", "$$ \n", "\n", "The sample variance $s^2$ is a biased estimator of the population variance $\\sigma^2$. But as it turns out, we only need to make a tiny tweak to transform this into an unbiased estimator. All we have to do is divide by $N-1$ rather than by $N$. If we do that, we obtain the following formula:\n", "\n", "$$\n", "\\hat\\sigma^2 = \\frac{1}{N-1} \\sum_{i=1}^N (X_i - \\bar{X})^2 \n", "$$\n", "\n", "This is an unbiased estimator of the population variance $\\sigma$. Moreover, this finally answers the question we raised [before](variability). Why did Python give us slightly different answers when we used the `statistics.variance()` function? Because `statistics.variance()` calculates $\\hat\\sigma^2$ not $s^2$, that's why. A similar story applies for the standard deviation. If we divide by $N-1$ rather than $N$, our estimate of the population standard deviation becomes:\n", "\n", "$$\n", "\\hat\\sigma = \\sqrt{\\frac{1}{N-1} \\sum_{i=1}^N (X_i - \\bar{X})^2} \n", "$$\n", "\n", "\n", "One final point: in practice, a lot of people tend to refer to $\\hat{\\sigma}$ (i.e., the formula where we divide by $N-1$) as the *sample* standard deviation. Technically, this is incorrect: the *sample* standard deviation should be equal to $s$ (i.e., the formula where we divide by $N$). These aren't the same thing, either conceptually or numerically. One is a property of the sample, the other is an estimated characteristic of the population. However, in almost every real life application, what we actually care about is the estimate of the population parameter, and so people always report $\\hat\\sigma$ rather than $s$. This is the right number to report, of course, it's that people tend to get a little bit imprecise about terminology when they write it up, because \"sample standard deviation\" is shorter than \"estimated population standard deviation\". It's no big deal, and in practice I do the same thing everyone else does. Nevertheless, I think it's important to keep the two *concepts* separate: it's never a good idea to confuse \"known properties of your sample\" with \"guesses about the population from which it came\". The moment you start thinking that $s$ and $\\hat\\sigma$ are the same thing, you start doing exactly that. \n", "\n", "To finish this section off, here's another couple of tables to help keep things clear:" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "|Symbol |What it's called |Do we know what it is? |\n", "|:----------------|:---------------------------------------------|:-------------------------------------------------------|\n", "|$s$ |Sample standard deviation |Yes - calculated from the raw data |\n", "|$\\sigma$ |Population standard deviation |Almost never known for sure |\n", "|$\\hat{\\sigma}$ |Estimate of the population standard deviation |Yes - but not the same as the sample standard deviation |\n", "|$s^2$ |Sample variance |Yes - calculated from the raw data |\n", "|$\\sigma^2$ |Population variance |Almost never known for sure |\n", "|$\\hat{\\sigma}^2$ |Estimate of the population variance |Yes - but not the same as the sample variance |\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "(ci)=\n", "## Estimating a confidence interval\n", "\n", "> *Statistics means never having to say you're certain* -- Unknown origin [^notestatsquote]\n", "\n", "Up to this point in this chapter, I've outlined the basics of sampling theory which statisticians rely on to make guesses about population parameters on the basis of a sample of data. As this discussion illustrates, one of the reasons we need all this sampling theory is that every data set leaves us with some degree of uncertainty, so our estimates are never going to be perfectly accurate. The thing that has been missing from this discussion is an attempt to *quantify* the amount of uncertainty that attaches to our estimate. It's not enough to be able guess that, say, the mean IQ of undergraduate psychology students is 115 (yes, I just made that number up). We also want to be able to say something that expresses the degree of certainty that we have in our guess. For example, it would be nice to be able to say that there is a 95\\% chance that the true mean lies between 109 and 121. The name for this is a **_confidence interval_** for the mean.\n", "\n", "Armed with an understanding of sampling distributions, constructing a confidence interval for the mean is actually pretty easy. Here's how it works. Suppose the true population mean is $\\mu$ and the standard deviation is $\\sigma$. I've just finished running my study that has $N$ participants, and the mean IQ among those participants is $\\bar{X}$. We know from our discussion of the [central limit theorem](clt) that the sampling distribution of the mean is approximately normal. We also know from our [discussion of the normal distribution](normal) that there is a 95\\% chance that a normally-distributed quantity will fall within two standard deviations of the true mean. To be more precise, we can use the `norm.ppf()` function from `scipy.stats` to compute the 2.5th and 97.5th percentiles of the normal distribution.\n", "\n", "[^notestatsquote]: This quote appears on a great many t-shirts and websites, and even gets a mention in a few academic papers (e.g., \\url{http://www.amstat.org/publications/jse/v10n3/friedman.html] but I've never found the original source." ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[-1.9599639845400545, 1.959963984540054]" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from scipy.stats import norm\n", "list(norm.ppf([.025, 0.975]))" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Okay, so I lied earlier on. The more correct answer is that there is a 95\\% chance that a normally-distributed quantity will fall within 1.96 standard deviations of the true mean. Next, recall that the standard deviation of the sampling distribution is referred to as the standard error, and the standard error of the mean is written as SEM. When we put all these pieces together, we learn that there is a 95\\% probability that the sample mean $\\bar{X}$ that we have actually observed lies within 1.96 standard errors of the population mean. Mathematically, we write this as:\n", "\n", "$$\n", "\\mu - \\left( 1.96 \\times \\mbox{SEM} \\right) \\ \\leq \\ \\bar{X}\\ \\leq \\ \\mu + \\left( 1.96 \\times \\mbox{SEM} \\right) \n", "$$\n", "\n", "where the SEM is equal to $\\sigma / \\sqrt{N}$, and we can be 95\\% confident that this is true. However, that's not answering the question that we're actually interested in. The equation above tells us what we should expect about the sample mean, given that we know what the population parameters are. What we *want* is to have this work the other way around: we want to know what we should believe about the population parameters, given that we have observed a particular sample. However, it's not too difficult to do this. Using a little high school algebra, a sneaky way to rewrite our equation is like this:\n", "\n", "$$\n", "\\bar{X} - \\left( 1.96 \\times \\mbox{SEM} \\right) \\ \\leq \\ \\mu \\ \\leq \\ \\bar{X} + \\left( 1.96 \\times \\mbox{SEM}\\right)\n", "$$\n", "\n", "What this is telling is is that the range of values has a 95\\% probability of containing the population mean $\\mu$. We refer to this range as a **_95\\% confidence interval_**, denoted $\\mbox{CI}_{95}$. In short, as long as $N$ is sufficiently large -- large enough for us to believe that the sampling distribution of the mean is normal -- then we can write this as our formula for the 95\\% confidence interval:\n", "\n", "$$\n", "\\mbox{CI}_{95} = \\bar{X} \\pm \\left( 1.96 \\times \\frac{\\sigma}{\\sqrt{N}} \\right)\n", "$$\n", "\n", "Of course, there's nothing special about the number 1.96: it just happens to be the multiplier you need to use if you want a 95\\% confidence interval. If I'd wanted a 70\\% confidence interval, I could have used the `norm.ppf` function to calculate the 15th and 85th quantiles:" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[-1.0364333894937898, 1.0364333894937898]" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "list(norm.ppf([.15, .85]))" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "and so the formula for $\\mbox{CI}_{70}$ would be the same as the formula for $\\mbox{CI}_{95}$ except that we'd use 1.04 as our magic number rather than 1.96.\n", "\n", "### A slight mistake in the formula\n", "\n", "As usual, I lied. The formula that I've given above for the 95\\% confidence interval is approximately correct, but I glossed over an important detail in the discussion. Notice my formula requires you to use the standard error of the mean, SEM, which in turn requires you to use the true population standard deviation $\\sigma$. Yet, in the section on [estimating population parameters](pointestimates) I stressed the fact that we don't actually *know* the true population parameters. Because we don't know the true value of $\\sigma$, we have to use an estimate of the population standard deviation $\\hat{\\sigma}$ instead. This is pretty straightforward to do, but this has the consequence that we need to use the quantiles of the $t$-distribution rather than the normal distribution to calculate our magic number; and the answer depends on the sample size. When $N$ is very large, we get pretty much the same value using `t.ppf()` that we would if we used `norm.ppf()`..." ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[-1.960201263621358, 1.9602012636213575]" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from scipy.stats import t\n", "N = 10000 # suppose our sample size is 10,000\n", "list(t.ppf([.025, 0.975], df = N-1))" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "But when $N$ is small, we get a much bigger number when we use the $t$ distribution:" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[-2.262157162740992, 2.2621571627409915]" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from scipy.stats import t\n", "N = 10 # suppose our sample size is 10\n", "list(t.ppf([.025, 0.975], df = N-1))" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "There's nothing too mysterious about what's happening here. Bigger values mean that the confidence interval is wider, indicating that we're more uncertain about what the true value of $\\mu$ actually is. When we use the $t$ distribution instead of the normal distribution, we get bigger numbers, indicating that we have more uncertainty. And why do we have that extra uncertainty? Well, because our estimate of the population standard deviation $\\hat\\sigma$ might be wrong! If it's wrong, it implies that we're a bit less sure about what our sampling distribution of the mean actually looks like... and this uncertainty ends up getting reflected in a wider confidence interval. \n", "\n", "\n", "### Interpreting a confidence interval\n", "\n", "The hardest thing about confidence intervals is understanding what they *mean*. Whenever people first encounter confidence intervals, the first instinct is almost always to say that \"there is a 95\\% probabaility that the true mean lies inside the confidence interval\". It's simple, and it seems to capture the common sense idea of what it means to say that I am \"95\\% confident\". Unfortunately, it's not quite right. The intuitive definition relies very heavily on your own personal *beliefs* about the value of the population mean. I say that I am 95\\% confident because those are my beliefs. In everyday life that's perfectly okay, but if you remember back to the section on [the meaning of probability](probmeaning), you'll notice that talking about personal belief and confidence is a Bayesian idea. Personally (speaking as a Bayesian) I have no problem with the idea that the phrase \"95\\% probability\" is allowed to refer to a personal belief. However, confidence intervals are *not* Bayesian tools. Like everything else in this chapter, confidence intervals are *frequentist* tools, and if you are going to use frequentist methods then it's not appropriate to attach a Bayesian interpretation to them. If you use frequentist methods, you must adopt frequentist interpretations!\n", "\n", "Okay, so if that's not the right answer, what is? Remember what we said about frequentist probability: the only way we are allowed to make \"probability statements\" is to talk about a sequence of events, and to count up the *frequencies* of different kinds of events. From that perspective, the interpretation of a 95\\% confidence interval must have something to do with replication. Specifically: if we replicated the experiment over and over again and computed a 95\\% confidence interval for each replication, then 95\\% of those *intervals* would contain the true mean. More generally, 95\\% of all confidence intervals constructed using this procedure should contain the true population mean. This idea is illustrated in {numref}`fig-cirep`, which shows 50 confidence intervals constructed for a \"measure 10 IQ scores\" experiment Panel A and another 50 confidence intervals for a \"measure 25 IQ scores\" experiment Panel B. " ] }, { "cell_type": "code", "execution_count": 106, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "from scipy.stats import t, sem\n", "import numpy as np\n", "import seaborn as sns\n", "import pandas as pd\n", "import matplotlib.pyplot as plt\n", "import random\n", "\n", "#set a random number generator for reproducibility\n", "rng = np.random.default_rng(42)\n", "\n", "#define simulation parameters\n", "n_experiments = 50\n", "ns = [10, 25]\n", "\n", "#prepare labels for the figure panels\n", "labels = ['A', 'B']\n", "\n", "# prepare the figure, with two subplots\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 5), sharey=False, sharex=False)\n", "fig.suptitle('Simulated IQ Data')\n", "\n", "# simulate the two sets of experiments, and make the figures.\n", "for f, ax in enumerate(axes):\n", "\n", " # collect the upper and lower CI bounds, and the sample means for each experiment\n", " uppers = []\n", " lowers = []\n", " sample_means = []\n", " \n", " # for the x-axis, make a list of incrementing numbers\n", " x = list(range(0,n_experiments))\n", "\n", " # generate simulated data for each experiment by sampling from a normal distribution\n", " # with the parameters defined above\n", " # store the sample means and CI bounds in the lists above\n", " for i in x:\n", " simdata = rng.normal(loc=100,scale=15,size=ns[f]).astype(int)\n", " sample_means.append(statistics.mean(simdata))\n", " ci = t.interval(alpha=.95, df=len(simdata)-1, loc=np.mean(simdata), scale=sem(simdata))\n", " uppers.append(ci[1])\n", " lowers.append(ci[0])\n", "\n", " \n", " # find the experiments where the CI did not capture the population mean\n", " # and mark these with red. Color the others blue. \n", " too_high = [x[s] for s, val in enumerate(lowers) if val > 100]\n", " too_low = [x[s] for s, val in enumerate(uppers) if val < 100]\n", " no_mean = too_high + too_low\n", " highlight = ['blue']*n_experiments\n", " for s, val in enumerate(no_mean):\n", " highlight[val] = 'red'\n", "\n", " # plot the data and format the plots\n", " sns.pointplot(x=x, y=sample_means, join=False, ax=ax)\n", " ax.set_ylim([70,130])\n", " ax.vlines(x=x, ymin=lowers, ymax=uppers, color = highlight)\n", " ax.axhline(y=100, linestyle = \"dashed\")\n", " ax.set_title(\"Sample Size = \" + str(ns[f]))\n", " ax.set(ylabel=None)\n", " ax.text(-0.1, 1, labels[f], transform=ax.transAxes,fontsize=16, fontweight='bold', va='top', ha='right')\n", " ax.tick_params(left=False)\n", " ax.spines[['top', 'right']].set_visible(False)\n", " ax.tick_params(axis='both', \n", " which='both',\n", " left=False,\n", " right=False,\n", " bottom=False,\n", " labelbottom=False)\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ " ```{glue:figure} cirep-fig\n", ":figwidth: 600px\n", ":name: fig-cirep\n", "\n", "95% confidence intervals. Panel A shows 50 simulated replications of an experiment in which we measure the IQs of 10 people. The blue points mark the location of the sample mean, and the lines shows the 95% confidence interval around each mean. The dashed line in the middle shows the true population mean, which is 100. Panel B shows a similar simulation, but this time we simulate replications of an experiment that measures the IQs of 25 people. Confidence intervals which are colored red indicate experiments in which the confidence interval around the sample mean did not capture the population mean.\n", "\n", "```\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "The critical difference here is that the Bayesian claim makes a probability statement about the population mean (i.e., it refers to our uncertainty about the population mean), which is not allowed under the frequentist interpretation of probability because you can't \"replicate\" a population! In the frequentist claim, the population mean is fixed and no probabilistic claims can be made about it. Confidence intervals, however, are repeatable so we can replicate experiments. Therefore a frequentist is allowed to talk about the probability that the *confidence interval* (a random variable) contains the true mean; but is not allowed to talk about the probability that the *true population mean* (not a repeatable event) falls within the confidence interval. \n", "\n", "I know that this seems a little pedantic, but it does matter. It matters because the difference in interpretation leads to a difference in the mathematics. There is a Bayesian alternative to confidence intervals, known as *credible intervals*. In most situations credible intervals are quite similar to confidence intervals, but in other cases they are drastically different. As promised, though, I'll talk more about the Bayesian perspective in a [later chapter](bayes).\n", "\n", "\n", "### Calculating confidence intervals in Python\n", "\n", "\n", "To produce the confidence intervals for the plots of simulated IQ data above, I used the ``t``, ``sem``, and ``mean``functions available in the ``scipy.stats``package. Another option is to use the ``tconfint_mean `` function from the ``statsmodels`` package. As you can see, both methods give nearly identical results. Method 1 is good insofar is at requires you to explicitly specify the desired confidence interval, the degrees of freedom, and the standard error of the mean. Method 2 takes care of all of this for us, which makes it easier, but is a bit more of a black box." ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Method 1: (9.514149929408497, 15.485850070591503)\n", "Method 2: (9.514149929408497, 15.485850070591503)\n" ] } ], "source": [ "# Sample data:\n", "data = range(1,25)\n", "\n", "# Method 1:\n", "import numpy as np\n", "import scipy.stats as st\n", "ci_1 = t.interval(alpha=0.95, \n", " df=len(data)-1, \n", " loc=np.mean(data), \n", " scale=sem(data))\n", "\n", "\n", "# Method 2:\n", "import statsmodels.stats.api as sms\n", "ci_2 = sms.DescrStatsW(data).tconfint_mean()\n", "\n", "# Compare the methods\n", "print(\"Method 1: \", ci_1)\n", "print(\"Method 2: \", ci_2)" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "### Plotting confidence intervals in Python\n", "\n", "There are many different ways you can draw graphs that show confidence intervals as error bars, and the method you select will depend on what you are trying to achieve. However, ``seaborn`` offers some good, off-the-shelf methods for plotting confidence intervals, which should cover most of the common cases. More in-depth information about these can be found in the seaborn documentation, but here are a few common cases, using seaborn's built-in \"tips\" dataset." ] }, { "cell_type": "code", "execution_count": 341, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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total_billtipsexsmokerdaytimesize
016.991.01FemaleNoSunDinner2
110.341.66MaleNoSunDinner3
221.013.50MaleNoSunDinner3
323.683.31MaleNoSunDinner2
424.593.61FemaleNoSunDinner4
\n", "
" ], "text/plain": [ " total_bill tip sex smoker day time size\n", "0 16.99 1.01 Female No Sun Dinner 2\n", "1 10.34 1.66 Male No Sun Dinner 3\n", "2 21.01 3.50 Male No Sun Dinner 3\n", "3 23.68 3.31 Male No Sun Dinner 2\n", "4 24.59 3.61 Female No Sun Dinner 4" ] }, "execution_count": 341, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import seaborn as sns\n", "tips = sns.load_dataset(\"tips\")\n", "tips.head()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "To compare the mean total bill for lunches and dinners for smokers and non-smokers, we can use ``sns.pointplot``. Notice that you can specifiy the size of desired confidence interval. By convention, people tend to use a 95% confidence interval, and this is the default in seaborn, but it is possible to specify a different one. Just make sure you report what size confidence interval you are showing! In the figure to the right, below, I have used a 40% confidence interval, but I probably wouldn't do that in a paper, because it is likely to confuse or mislead readers who expect a 95% CI." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "import matplotlib.pyplot as plt\n", "tips = sns.load_dataset(\"tips\")\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 5))\n", "\n", "\n", "sns.pointplot(x=\"time\", y=\"total_bill\", hue=\"smoker\", data=tips, ax=axes[0])\n", "sns.pointplot(x=\"time\", y=\"total_bill\", hue=\"smoker\", ci = 40, data=tips, ax=axes[1])\n", "axes[0].set_title(\"95% Confidence Interval\")\n", "axes[1].set_title(\"40% Confidence Interval\")\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "For regression plots, seaborn computes a confidence interval for regression line by default. This can be turned off with ``ci=None``, but I think it is good practice to include it, because it gives a nice visual indication of the strength of the model." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "tips = sns.load_dataset(\"tips\")\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 5))\n", "\n", "sns.regplot(x=\"total_bill\", y=\"tip\", data=tips, ax=axes[0])\n", "sns.regplot(x=\"total_bill\", y=\"tip\", data=tips, ci = None, ax=axes[1])\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "For regression plots with discrete variables on the x-axis, seaborn has options for either showing all datapoints, or showing only the mean with error-bars indicating the confidence interval. There are many more details and options to be found in the seaborn documentation. For more complex or custom figures, like the one in {numref}`fig-cirep` showing confidence intervals for simulated IQ data, you will need to dive into ``matplotlib``, which allows much more customization than is available simply using ``seaborn``." ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "tips = sns.load_dataset(\"tips\")\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 5))\n", "\n", "sns.regplot(x=\"size\", y=\"tip\", data=tips, ax=axes[0])\n", "sns.regplot(x=\"size\", y=\"tip\", data=tips, x_estimator=np.mean, ax=axes[1])\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "## Summary\n", "\n", "In this chapter I've covered two main topics. The first half of the chapter talks about sampling theory, and the second half talks about how we can use sampling theory to construct estimates of the population parameters. The section breakdown looks like this:\n", "\n", "- [Basic ideas](srs) about samples, sampling and populations \n", "- Statistical theory of sampling: the [law of large numbers](lawlargenumbers), [sampling distributions and the central limit theorem](samplingdists).\n", "- [Estimating means and standard deviations](pointestimates)\n", "- [Estimating a confidence interval](ci)\n", "\n", "As always, there's a lot of topics related to sampling and estimation that aren't covered in this chapter, but for an introductory psychology class this is fairly comprehensive I think. For most applied research, you won't need much more theory than this. One big question that I haven't touched on in this chapter is what you do when you don't have a simple random sample. There is a lot of statistical theory you can draw on to handle this situation, but it's well beyond the scope of this book." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.3" } }, "nbformat": 4, "nbformat_minor": 4 } { "cells": [ { "attachments": {}, "cell_type": "markdown", "id": "valued-people", "metadata": {}, "source": [ "(hypothesis-testing)=\n", "# Hypothesis Testing" ] }, { "attachments": {}, "cell_type": "markdown", "id": "sought-miniature", "metadata": {}, "source": [ "\n", ">*The process of induction is the process of assuming the simplest law that can be made to harmonize with our experience. This process, however, has no logical foundation but only a psychological one. It is clear that there are no grounds for believing that the simplest course of events will really happen. It is an hypothesis that the sun will rise tomorrow: and this means that we do not know whether it will rise.*\n", ">\n", ">-- Ludwig Wittgenstein [^note1] \n", "\n", "In the last chapter, I discussed the ideas behind estimation, which is one of the two \"big ideas\" in inferential statistics. It's now time to turn out attention to the other big idea, which is *hypothesis testing*. In its most abstract form, hypothesis testing really a very simple idea: the researcher has some theory about the world, and wants to determine whether or not the data actually support that theory. However, the details are messy, and most people find the theory of hypothesis testing to be the most frustrating part of statistics. The structure of the chapter is as follows. Firstly, I'll describe how hypothesis testing works, in a fair amount of detail, using a simple running example to show you how a hypothesis test is \"built\". I'll try to avoid being too dogmatic while doing so, and focus instead on the underlying logic of the testing procedure.[^note2] Afterwards, I'll spend a bit of time talking about the various dogmas, rules and heresies that surround the theory of hypothesis testing.\n", "\n", "[^note1]: The quote comes from Wittgenstein's (1922) text, *Tractatus Logico-Philosphicus*.\n", "[^note2]: A technical note. The description below differs subtly from the standard description given in a lot of introductory texts. The orthodox theory of null hypothesis testing emerged from the work of Sir Ronald Fisher and Jerzy Neyman in the early 20th century; but Fisher and Neyman actually had very different views about how it should work. The standard treatment of hypothesis testing that most texts use is a hybrid of the two approaches. The treatment here is a little more Neyman-style than the orthodox view, especially as regards the meaning of the $p$ value." ] }, { "attachments": {}, "cell_type": "markdown", "id": "american-courage", "metadata": {}, "source": [ "(hypotheses)=\n", "## A menagerie of hypotheses\n", "\n", "\n", "Eventually we all succumb to madness. For me, that day will arrive once I'm finally promoted to full professor. Safely ensconced in my ivory tower, happily protected by tenure, I will finally be able to take leave of my senses (so to speak), and indulge in that most thoroughly unproductive line of psychological research: the search for extrasensory perception (ESP).[^note3]\n", "\n", "Let's suppose that this glorious day has come. My first study is a simple one, in which I seek to test whether clairvoyance exists. Each participant sits down at a table, and is shown a card by an experimenter. The card is black on one side and white on the other. The experimenter takes the card away, and places it on a table in an adjacent room. The card is placed black side up or white side up completely at random, with the randomisation occurring only after the experimenter has left the room with the participant. A second experimenter comes in and asks the participant which side of the card is now facing upwards. It's purely a one-shot experiment. Each person sees only one card, and gives only one answer; and at no stage is the participant actually in contact with someone who knows the right answer. My data set, therefore, is very simple. I have asked the question of $N$ people, and some number $X$ of these people have given the correct response. To make things concrete, let's suppose that I have tested $N = 100$ people, and $X = 62$ of these got the answer right... a surprisingly large number, sure, but is it large enough for me to feel safe in claiming I've found evidence for ESP? This is the situation where hypothesis testing comes in useful. However, before we talk about how to *test* hypotheses, we need to be clear about what we mean by hypotheses.\n", "\n", "### Research hypotheses versus statistical hypotheses\n", "\n", "The first distinction that you need to keep clear in your mind is between research hypotheses and statistical hypotheses. In my ESP study, my overall scientific goal is to demonstrate that clairvoyance exists. In this situation, I have a clear research goal: I am hoping to discover evidence for ESP. In other situations I might actually be a lot more neutral than that, so I might say that my research goal is to determine whether or not clairvoyance exists. Regardless of how I want to portray myself, the basic point that I'm trying to convey here is that a research hypothesis involves making a substantive, testable scientific claim... if you are a psychologist, then your research hypotheses are fundamentally *about* psychological constructs. Any of the following would count as **_research hypotheses_**:\n", "\n", "- *Listening to music reduces your ability to pay attention to other things.* This is a claim about the causal relationship between two psychologically meaningful concepts (listening to music and paying attention to things), so it's a perfectly reasonable research hypothesis.\n", "- *Intelligence is related to personality*. Like the last one, this is a relational claim about two psychological constructs (intelligence and personality), but the claim is weaker: correlational not causal.\n", "- *Intelligence *is* speed of information processing*. This hypothesis has a quite different character: it's not actually a relational claim at all. It's an ontological claim about the fundamental character of intelligence (and I'm pretty sure it's wrong). It's worth expanding on this one actually: It's usually easier to think about how to construct experiments to test research hypotheses of the form \"does X affect Y?\" than it is to address claims like \"what is X?\" And in practice, what usually happens is that you find ways of testing relational claims that follow from your ontological ones. For instance, if I believe that intelligence *is* speed of information processing in the brain, my experiments will often involve looking for relationships between measures of intelligence and measures of speed. As a consequence, most everyday research questions do tend to be relational in nature, but they're almost always motivated by deeper ontological questions about the state of nature. \n", "\n", "Notice that in practice, my research hypotheses could overlap a lot. My ultimate goal in the ESP experiment might be to test an ontological claim like \"ESP exists\", but I might operationally restrict myself to a narrower hypothesis like \"Some people can `see' objects in a clairvoyant fashion\". That said, there are some things that really don't count as proper research hypotheses in any meaningful sense: \n", "\n", "- *Love is a battlefield*. This is too vague to be testable. While it's okay for a research hypothesis to have a degree of vagueness to it, it has to be possible to operationalise your theoretical ideas. Maybe I'm just not creative enough to see it, but I can't see how this can be converted into any concrete research design. If that's true, then this isn't a scientific research hypothesis, it's a pop song. That doesn't mean it's not interesting -- a lot of deep questions that humans have fall into this category. Maybe one day science will be able to construct testable theories of love, or to test to see if God exists, and so on; but right now we can't, and I wouldn't bet on ever seeing a satisfying scientific approach to either. \n", "- *The first rule of tautology club is the first rule of tautology club*. This is not a substantive claim of any kind. It's true by definition. No conceivable state of nature could possibly be inconsistent with this claim. As such, we say that this is an unfalsifiable hypothesis, and as such it is outside the domain of science. Whatever else you do in science, your claims must have the possibility of being wrong. \n", "- *More people in my experiment will say \"yes\" than \"no\"*. This one fails as a research hypothesis because it's a claim about the data set, not about the psychology (unless of course your actual research question is whether people have some kind of \"yes\" bias!). As we'll see shortly, this hypothesis is starting to sound more like a statistical hypothesis than a research hypothesis. \n", "\n", "\n", "As you can see, research hypotheses can be somewhat messy at times; and ultimately they are *scientific* claims. **_Statistical hypotheses_** are neither of these two things. Statistical hypotheses must be mathematically precise, and they must correspond to specific claims about the characteristics of the data-generating mechanism (i.e., the \"population\"). Even so, the intent is that statistical hypotheses bear a clear relationship to the substantive research hypotheses that you care about! For instance, in my ESP study my research hypothesis is that some people are able to see through walls or whatever. What I want to do is to \"map\" this onto a statement about how the data were generated. So let's think about what that statement would be. The quantity that I'm interested in within the experiment is $P(\\mbox{\"correct\"})$, the true-but-unknown probability with which the participants in my experiment answer the question correctly. Let's use the Greek letter $\\theta$ (theta) to refer to this probability. Here are four different statistical hypotheses:\n", "\n", "\n", "- If ESP doesn't exist and if my experiment is well designed, then my participants are just guessing. So I should expect them to get it right half of the time and so my statistical hypothesis is that the true probability of choosing correctly is $\\theta = 0.5$. \n", "- Alternatively, suppose ESP does exist and participants can see the card. If that's true, people will perform better than chance. The statistical hypotheis would be that $\\theta > 0.5$. \n", "- A third possibility is that ESP does exist, but the colours are all reversed and people don't realise it (okay, that's wacky, but you never know...). If that's how it works then you'd expect people's performance to be *below* chance. This would correspond to a statistical hypothesis that $\\theta < 0.5$. \n", "- Finally, suppose ESP exists, but I have no idea whether people are seeing the right colour or the wrong one. In that case, the only claim I could make about the data would be that the probability of making the correct answer is *not* equal to 50. This corresponds to the statistical hypothesis that $\\theta \\neq 0.5$. \n", "\n", "All of these are legitimate examples of a statistical hypothesis because they are statements about a population parameter and are meaningfully related to my experiment.\n", "\n", "What this discussion makes clear, I hope, is that when attempting to construct a statistical hypothesis test, the researcher actually has two quite distinct hypotheses to consider. First, he or she has a research hypothesis (a claim about psychology), and this corresponds to a statistical hypothesis (a claim about the data generating population). In my ESP example, these might be\n", "\n", "[^note3]: My apologies to anyone who actually believes in this stuff, but on my reading of the literature on ESP, it's just not reasonable to think this is real. To be fair, though, some of the studies are rigorously designed; so it's actually an interesting area for thinking about psychological research design. And of course it's a free country, so you can spend your own time and effort proving me wrong if you like, but I wouldn't think that's a terribly practical use of your intellect." ] }, { "attachments": {}, "cell_type": "markdown", "id": "buried-beijing", "metadata": {}, "source": [ "\n", "My research hypothesis: “ESP exists” \n", "\n", "My statistical hypothesis: $\\theta \\neq 0.5$\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "optional-electronics", "metadata": {}, "source": [ "And the key thing to recognise is this: *a statistical hypothesis test is a test of the statistical hypothesis, not the research hypothesis*. If your study is badly designed, then the link between your research hypothesis and your statistical hypothesis is broken. To give a silly example, suppose that my ESP study was conducted in a situation where the participant can actually see the card reflected in a window; if that happens, I would be able to find very strong evidence that $\\theta \\neq 0.5$, but this would tell us nothing about whether \"ESP exists\". \n", "\n", "\n", "### Null hypotheses and alternative hypotheses\n", "\n", "So far, so good. I have a research hypothesis that corresponds to what I want to believe about the world, and I can map it onto a statistical hypothesis that corresponds to what I want to believe about how the data were generated. It's at this point that things get somewhat counterintuitive for a lot of people. Because what I'm about to do is invent a new statistical hypothesis (the \"null\" hypothesis, $H_0$) that corresponds to the exact opposite of what I want to believe, and then focus exclusively on that, almost to the neglect of the thing I'm actually interested in (which is now called the \"alternative\" hypothesis, $H_1$). In our ESP example, the null hypothesis is that $\\theta = 0.5$, since that's what we'd expect if ESP *didn't* exist. My hope, of course, is that ESP is totally real, and so the *alternative* to this null hypothesis is $\\theta \\neq 0.5$. In essence, what we're doing here is dividing up the possible values of $\\theta$ into two groups: those values that I really hope aren't true (the null), and those values that I'd be happy with if they turn out to be right (the alternative). Having done so, the important thing to recognise is that the goal of a hypothesis test is *not* to show that the alternative hypothesis is (probably) true; the goal is to show that the null hypothesis is (probably) false. Most people find this pretty weird. \n", "\n", "The best way to think about it, in my experience, is to imagine that a hypothesis test is a criminal trial [^note4] ... *the trial of the null hypothesis*. The null hypothesis is the defendant, the researcher is the prosecutor, and the statistical test itself is the judge. Just like a criminal trial, there is a presumption of innocence: the null hypothesis is *deemed* to be true unless you, the researcher, can prove beyond a reasonable doubt that it is false. You are free to design your experiment however you like (within reason, obviously!), and your goal when doing so is to maximise the chance that the data will yield a conviction... for the crime of being false. The catch is that the statistical test sets the rules of the trial, and those rules are designed to protect the null hypothesis -- specifically to ensure that if the null hypothesis is actually true, the chances of a false conviction are guaranteed to be low. This is pretty important: after all, the null hypothesis doesn't get a lawyer. And given that the researcher is trying desperately to prove it to be false, *someone* has to protect it. \n", "\n", "[^note4]: This analogy only works if you're from an adversarial legal system like UK/US/Australia. As I understand these things, the French inquisitorial system is quite different." ] }, { "attachments": {}, "cell_type": "markdown", "id": "modular-savannah", "metadata": {}, "source": [ "(errortypes)=\n", "## Two types of errors\n", "\n", "Before going into details about how a statistical test is constructed, it's useful to understand the philosophy behind it. I hinted at it when pointing out the similarity between a null hypothesis test and a criminal trial, but I should now be explicit. Ideally, we would like to construct our test so that we never make any errors. Unfortunately, since the world is messy, this is never possible. Sometimes you're just really unlucky: for instance, suppose you flip a coin 10 times in a row and it comes up heads all 10 times. That feels like very strong evidence that the coin is biased (and it is!), but of course there's a 1 in 1024 chance that this would happen even if the coin was totally fair. In other words, in real life we *always* have to accept that there's a chance that we did the wrong thing. As a consequence, the goal behind statistical hypothesis testing is not to *eliminate* errors, but to *minimise* them.\n", "\n", "At this point, we need to be a bit more precise about what we mean by \"errors\". Firstly, let's state the obvious: it is either the case that the null hypothesis is true, or it is false; and our test will either reject the null hypothesis or retain it.[^note5] So, as the table below illustrates, after we run the test and make our choice, one of four things might have happened:\n", "\n", "[^note5]: An aside regarding the language you use to talk about hypothesis testing. Firstly, one thing you really want to avoid is the word \"prove\": a statistical test really doesn't *prove* that a hypothesis is true or false. Proof implies certainty, and as the saying goes, statistics means never having to say you're certain. On that point almost everyone would agree. However, beyond that there's a fair amount of confusion. Some people argue that you're only allowed to make statements like \"rejected the null\", \"failed to reject the null\", or possibly \"retained the null\". According to this line of thinking, you can't say things like \"accept the alternative\" or \"accept the null\". Personally I think this is too strong: in my opinion, this conflates null hypothesis testing with Karl Popper's falsificationist view of the scientific process. While there are similarities between falsificationism and null hypothesis testing, they aren't equivalent. However, while I personally think it's fine to talk about accepting a hypothesis (on the proviso that \"acceptance\" doesn't actually mean that it's necessarily true, especially in the case of the null hypothesis), many people will disagree. And more to the point, you should be aware that this particular weirdness exists, so that you're not caught unawares by it when writing up your own results." ] }, { "attachments": {}, "cell_type": "markdown", "id": "differential-conversion", "metadata": {}, "source": [ "| | retain H0 | reject H0 |\n", "|-------------|------------------|------------------|\n", "| H0 is true | correct decision | error (type I) |\n", "| H0 is false | error (type II) | correct decision |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "preliminary-convenience", "metadata": {}, "source": [ "As a consequence there are actually *two* different types of error here. If we reject a null hypothesis that is actually true, then we have made a **_type I error_**. On the other hand, if we retain the null hypothesis when it is in fact false, then we have made a **_type II error_**. \n", "\n", "Remember how I said that statistical testing was kind of like a criminal trial? Well, I meant it. A criminal trial requires that you establish \"beyond a reasonable doubt\" that the defendant did it. All of the evidentiary rules are (in theory, at least) designed to ensure that there's (almost) no chance of wrongfully convicting an innocent defendant. The trial is designed to protect the rights of a defendant: as the English jurist William Blackstone famously said, it is \"better that ten guilty persons escape than that one innocent suffer.\" In other words, a criminal trial doesn't treat the two types of error in the same way... punishing the innocent is deemed to be much worse than letting the guilty go free. A statistical test is pretty much the same: the single most important design principle of the test is to *control* the probability of a type I error, to keep it below some fixed probability. This probability, which is denoted $\\alpha$, is called the **_significance level_** of the test (or sometimes, the *size* of the test). And I'll say it again, because it is so central to the whole set-up... a hypothesis test is said to have significance level $\\alpha$ if the type I error rate is no larger than $\\alpha$. \n", "\n", "So, what about the type II error rate? Well, we'd also like to keep those under control too, and we denote this probability by $\\beta$. However, it's much more common to refer to the **_power_** of the test, which is the probability with which we reject a null hypothesis when it really is false, which is $1-\\beta$. To help keep this straight, here's the same table again, but with the relevant numbers added:" ] }, { "attachments": {}, "cell_type": "markdown", "id": "mature-portal", "metadata": {}, "source": [ "| | retain H0 | reject H0 |\n", "|-------------|-----------------------------------------------|-------------------------------|\n", "| H0 is true | $1-\\alpha$ (probability of correct retention) | $\\alpha$ (type I error rate) |\n", "| H0 is false | $\\beta$ (type II error rate) | $1-\\beta$ (power of the test)|" ] }, { "attachments": {}, "cell_type": "markdown", "id": "undefined-remains", "metadata": {}, "source": [ "A \"powerful\" hypothesis test is one that has a small value of $\\beta$, while still keeping $\\alpha$ fixed at some (small) desired level. By convention, scientists make use of three different $\\alpha$ levels: $.05$, $.01$ and $.001$. Notice the asymmetry here~... the tests are designed to *ensure* that the $\\alpha$ level is kept small, but there's no corresponding guarantee regarding $\\beta$. We'd certainly *like* the type II error rate to be small, and we try to design tests that keep it small, but this is very much secondary to the overwhelming need to control the type I error rate. As Blackstone might have said if he were a statistician, it is \"better to retain 10 false null hypotheses than to reject a single true one\". To be honest, I don't know that I agree with this philosophy -- there are situations where I think it makes sense, and situations where I think it doesn't -- but that's neither here nor there. It's how the tests are built." ] }, { "attachments": {}, "cell_type": "markdown", "id": "associate-position", "metadata": {}, "source": [ "(teststatistics)=\n", "## Test statistics and sampling distributions\n", "\n", "At this point we need to start talking specifics about how a hypothesis test is constructed. To that end, let's return to the ESP example. Let's ignore the actual data that we obtained, for the moment, and think about the structure of the experiment. Regardless of what the actual numbers are, the *form* of the data is that $X$ out of $N$ people correctly identified the colour of the hidden card. Moreover, let's suppose for the moment that the null hypothesis really is true: ESP doesn't exist, and the true probability that anyone picks the correct colour is exactly $\\theta = 0.5$. What would we *expect* the data to look like? Well, obviously, we'd expect the proportion of people who make the correct response to be pretty close to 50\\%. Or, to phrase this in more mathematical terms, we'd say that $X/N$ is approximately $0.5$. Of course, we wouldn't expect this fraction to be *exactly* 0.5: if, for example we tested $N=100$ people, and $X = 53$ of them got the question right, we'd probably be forced to concede that the data are quite consistent with the null hypothesis. On the other hand, if $X = 99$ of our participants got the question right, then we'd feel pretty confident that the null hypothesis is wrong. Similarly, if only $X=3$ people got the answer right, we'd be similarly confident that the null was wrong. Let's be a little more technical about this: we have a quantity $X$ that we can calculate by looking at our data; after looking at the value of $X$, we make a decision about whether to believe that the null hypothesis is correct, or to reject the null hypothesis in favour of the alternative. The name for this thing that we calculate to guide our choices is a **_test statistic_**. \n", "\n", "\n", "Having chosen a test statistic, the next step is to state precisely which values of the test statistic would cause us to reject the null hypothesis, and which values would cause us to keep it. In order to do so, we need to determine what the **_sampling distribution of the test statistic_** would be if the null hypothesis were actually true (we talked about [sampling distributions](samplingdists) earlier). Why do we need this? Because this distribution tells us exactly what values of $X$ our null hypothesis would lead us to expect. And therefore, we can use this distribution as a tool for assessing how closely the null hypothesis agrees with our data. Using ``random.binomial`` from ``numpy``, we can estimate a binomial distribution with a $\\theta = 0.5$, e.g. estimating from 10,000 trials:" ] }, { "cell_type": "code", "execution_count": 7, "id": "alone-legislation", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "from myst_nb import glue\n", "from numpy import random\n", "import matplotlib.pyplot as plt\n", "import seaborn as sns\n", "\n", "# sample from a binomial distribution\n", "data = random.binomial(n=100, p=.5, size=10000)\n", "\n", "\n", "esp = sns.histplot(data, bins=20,binwidth=0.5)\n", "esp.set(xlim=(0,100))\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "hundred-religious", "metadata": {}, "source": [ "```{glue:figure} estimation-fig\n", ":figwidth: 600px\n", ":name: fig-esp-estimation\n", "\n", "The sampling distribution for our test statistic X when the null hypothesis is true. For our ESP scenario, this is a binomial distribution. Not surprisingly, since the null hypothesis says that the probability of a correct response is θ = .5, the sampling distribution says that the most likely value is 50 (out of 100) correct responses. Most of the probability mass lies between 40 and 60.\n", "```\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "statutory-valuation", "metadata": {}, "source": [ "How do we actually determine the sampling distribution of the test statistic? For a lot of hypothesis tests this step is actually quite complicated, and later on in the book you'll see me being slightly evasive about it for some of the tests (some of them I don't even understand myself). However, sometimes it's very easy. And, fortunately for us, our ESP example provides us with one of the easiest cases. Our population parameter $\\theta$ is just the overall probability that people respond correctly when asked the question, and our test statistic $X$ is the *count* of the number of people who did so, out of a sample size of $N$. We've seen a distribution like this before, in the section on [the binomial distribution]((binomial)): that's exactly what the binomial distribution describes! So, to use the notation and terminology that I introduced in that section, we would say that the null hypothesis predicts that $X$ is binomially distributed, which is written\n", "\n", "$$\n", "X \\sim \\mbox{Binomial}(\\theta,N)\n", "$$\n", "\n", "Since the null hypothesis states that $\\theta = 0.5$ and our experiment has $N=100$ people, we have the sampling distribution we need. This sampling distribution is plotted in Figure {numref}`fig-esp-estimation`. No surprises really: the null hypothesis says that $X=50$ is the most likely outcome, and it says that we're almost certain to see somewhere between 40 and 60 correct responses. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "preceding-bermuda", "metadata": {}, "source": [ "(decisionmaking)=\n", "## Making decisions\n", "\n", "\n", "Okay, we're very close to being finished. We've constructed a test statistic ($X$), and we chose this test statistic in such a way that we're pretty confident that if $X$ is close to $N/2$ then we should retain the null, and if not we should reject it. The question that remains is this: exactly which values of the test statistic should we associate with the null hypothesis, and which exactly values go with the alternative hypothesis? In my ESP study, for example, I've observed a value of $X=62$. What decision should I make? Should I choose to believe the null hypothesis, or the alternative hypothesis?\n", "\n", "### Critical regions and critical values\n", "\n", "To answer this question, we need to introduce the concept of a **_critical region_** for the test statistic $X$. The critical region of the test corresponds to those values of $X$ that would lead us to reject the null hypothesis (which is why the critical region is also sometimes called the rejection region). How do we find this critical region? Well, let's consider what we know: \n", "\n", "- $X$ should be very big or very small in order to reject the null hypothesis.\n", "- If the null hypothesis is true, the sampling distribution of $X$ is Binomial $(0.5, N)$.\n", "- If $\\alpha =.05$, the critical region must cover 5\\% of this sampling distribution. \n", "\n", "It's important to make sure you understand this last point: the critical region corresponds to those values of $X$ for which we would reject the null hypothesis, and the sampling distribution in question describes the probability that we would obtain a particular value of $X$ if the null hypothesis were actually true. Now, let's suppose that we chose a critical region that covers 20\\% of the sampling distribution, and suppose that the null hypothesis is actually true. What would be the probability of incorrectly rejecting the null? The answer is of course 20\\%. And therefore, we would have built a test that had an $\\alpha$ level of $0.2$. If we want $\\alpha = .05$, the critical region is only *allowed* to cover 5\\% of the sampling distribution of our test statistic." ] }, { "cell_type": "code", "execution_count": 28, "id": "interested-depth", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "from myst_nb import glue\n", "from numpy import random\n", "import matplotlib.pyplot as plt\n", "import seaborn as sns\n", "\n", "# sample from a binomial distribution\n", "data = random.binomial(n=100, p=.5, size=10000)\n", "\n", "# plot distribution and color critical region\n", "ax = sns.histplot(data, bins=20,binwidth=.5, color=\"black\")\n", "ax.set_title(\"Critical regions for a two-sided test\")\n", "ax.annotate(\"\", xy=(40, 500), xytext=(30, 500), arrowprops=dict(arrowstyle=\"<-\"))\n", "ax.annotate(\"lower critical region \\n (2.5% of the distribution)\", xy=(40, 600), xytext=(5, 580))\n", "ax.annotate(\"\", xy=(70, 500), xytext=(60, 500), arrowprops=dict(arrowstyle=\"->\"))\n", "ax.annotate(\"upper critical region \\n (2.5% of the distribution)\", xy=(70, 500), xytext=(60, 580))\n", "ax.set(xlim=(0,100))\n", "for p in ax.patches:\n", " if p.get_x() >= 40:\n", " if p.get_x() <= 60:\n", " p.set_color(\"lightgrey\")\n", "\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "directed-assembly", "metadata": {}, "source": [ "```{glue:figure} espcritcal-fig\n", ":figwidth: 600px\n", ":name: fig-esp-critical\n", "\n", "The critical region associated with the hypothesis test for the ESP study, for a hypothesis test with a significance level of $\\alpha = .05$. The plot itself shows the sampling distribution of $X$ under the null hypothesis: the grey bars correspond to those values of $X$ for which we would retain the null hypothesis. The black bars show the critical region: those values of $X$ for which we would reject the null. Because the alternative hypothesis is two sided (i.e., allows both $\\theta <.5$ and $\\theta >.5$), the critical region covers both tails of the distribution. To ensure an $\\alpha$ level of $.05$, we need to ensure that each of the two regions encompasses 2.5% of the sampling distribution.\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "daily-coast", "metadata": {}, "source": [ "As it turns out, those three things uniquely solve the problem: our critical region consists of the most *extreme values*, known as the **_tails_** of the distribution. This is illustrated in {numref}`fig-esp-critical`. As it turns out, if we want $\\alpha = .05$, then our critical regions correspond to $X \\leq 40$ and $X \\geq 60$.[^note6] That is, if the number of people saying \"true\" is between 41 and 59, then we should retain the null hypothesis. If the number is between 0 to 40 or between 60 to 100, then we should reject the null hypothesis. The numbers 40 and 60 are often referred to as the **_critical values_**, since they define the edges of the critical region.\n", "\n", "\n", "At this point, our hypothesis test is essentially complete: (1) we choose an $\\alpha$ level (e.g., $\\alpha = .05$, (2) come up with some test statistic (e.g., $X$) that does a good job (in some meaningful sense) of comparing $H_0$ to $H_1$, (3) figure out the sampling distribution of the test statistic on the assumption that the null hypothesis is true (in this case, binomial) and then (4) calculate the critical region that produces an appropriate $\\alpha$ level (0-40 and 60-100). All that we have to do now is calculate the value of the test statistic for the real data (e.g., $X = 62$) and then compare it to the critical values to make our decision. Since 62 is greater than the critical value of 60, we would reject the null hypothesis. Or, to phrase it slightly differently, we say that the test has produced a **_significant_** result. \n", "\n", "[^note6]: Strictly speaking, the test I just constructed has $\\alpha = .057$, which is a bit too generous. However, if I'd chosen 39 and 61 to be the boundaries for the critical region, then the critical region only covers 3.5\\% of the distribution. I figured that it makes more sense to use 40 and 60 as my critical values, and be willing to tolerate a 5.7\\% type I error rate, since that's as close as I can get to a value of $\\alpha = .05$." ] }, { "attachments": {}, "cell_type": "markdown", "id": "fifty-toilet", "metadata": {}, "source": [ "### A note on statistical \"significance\"\n", "\n", ">*Like other occult techniques of divination, the statistical method has a private jargon deliberately contrived to obscure its methods from non-practitioners.*\n", ">\n", ">-- Attributed to G. O. Ashley[^note7]\n", "\n", "A very brief digression is in order at this point, regarding the word \"significant\". The concept of statistical significance is actually a very simple one, but has a very unfortunate name. If the data allow us to reject the null hypothesis, we say that \"the result is *statistically significant*\", which is often shortened to \"the result is significant\". This terminology is rather old, and dates back to a time when \"significant\" just meant something like \"indicated\", rather than its modern meaning, which is much closer to \"important\". As a result, a lot of modern readers get very confused when they start learning statistics, because they think that a \"significant result\" must be an important one. It doesn't mean that at all. All that \"statistically significant\" means is that the data allowed us to reject a null hypothesis. Whether or not the result is actually important in the real world is a very different question, and depends on all sorts of other things. \n", "\n", "[^note7]:(The internet seems fairly convinced that Ashley said this, though I can't for the life of me find anyone willing to give a source for the claim.)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "activated-hammer", "metadata": {}, "source": [ "(one-two-sided)=\n", "### The difference between one sided and two sided tests\n", "\n", "\n", "There's one more thing I want to point out about the hypothesis test that I've just constructed. If we take a moment to think about the statistical hypotheses I've been using, \n", "\n", "$$\n", "\\begin{array}{cc}\n", "H_0 : & \\theta = .5 \\\\\n", "H_1 : & \\theta \\neq .5 \n", "\\end{array}\n", "$$\n", "\n", "we notice that the alternative hypothesis covers *both* the possibility that $\\theta < .5$ and the possibility that $\\theta > .5$. This makes sense if I really think that ESP could produce better-than-chance performance *or* worse-than-chance performance (and there are some people who think that). In statistical language, this is an example of a **_two-sided test_**. It's called this because the alternative hypothesis covers the area on both \"sides\" of the null hypothesis, and as a consequence the critical region of the test covers both tails of the sampling distribution (2.5\\% on either side if $\\alpha =.05$), as illustrated earlier in {numref}`fig-esp-critical`. \n", "\n", "However, that's not the only possibility. It might be the case, for example, that I'm only willing to believe in ESP if it produces better than chance performance. If so, then my alternative hypothesis would only cover the possibility that $\\theta > .5$, and as a consequence the null hypothesis now becomes $\\theta \\leq .5$:\n", "\n", "$$\n", "\\begin{array}{cc}\n", "H_0 : & \\theta \\leq .5 \\\\\n", "H_1 : & \\theta > .5 \n", "\\end{array}\n", "$$\n", "\n", "When this happens, we have what's called a **_one-sided test_**, and when this happens the critical region only covers one tail of the sampling distribution. This is illustrated in {numref}`fig-esp-critical-onesided`.\n", "\n" ] }, { "cell_type": "code", "execution_count": 14, "id": "binary-shoot", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "from myst_nb import glue\n", "from numpy import random\n", "import matplotlib.pyplot as plt\n", "import seaborn as sns\n", "\n", "# sample from a binomial distribution\n", "data = random.binomial(n=100, p=.5, size=10000)\n", "\n", "# plot distribution and color critical region\n", "ax = sns.histplot(data, bins=20,binwidth=.5, color=\"black\")\n", "ax.set_title(\"Critical region for a one-sided test\")\n", "\n", "#ax.annotate(\"\", xy=(40, 500), xytext=(30, 500), arrowprops=dict(arrowstyle=\"<-\"))\n", "#ax.annotate(\"lower critical region \\n (2.5% of the distribution)\", xy=(40, 600), xytext=(22, 580))\n", "ax.annotate(\"\", xy=(70, 500), xytext=(60, 500), arrowprops=dict(arrowstyle=\"->\"))\n", "ax.annotate(\"upper critical region \\n (5% of the distribution)\", xy=(70, 500), xytext=(55, 580))\n", "ax.set(xlim=(0,100))\n", "for p in ax.patches:\n", " if p.get_x() <= 58:\n", " p.set_color(\"lightgrey\")\n", "\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "false-wallace", "metadata": {}, "source": [ "```{glue:figure} espcritical-onesided-fig\n", ":figwidth: 600px\n", ":name: fig-esp-critical-onesided\n", "\n", "The critical region for a one sided test. In this case, the alternative hypothesis is that θ ą .05, so we would only reject the null hypothesis for large values of X. As a consequence, the critical region only covers the upper tail of the sampling distribution; specifically the upper 5% of the distribution. Contrast this to the two-sided version in {numref}`fig-esp-critical`\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "selected-amazon", "metadata": {}, "source": [ "(pvalue)=\n", "## The $p$ value of a test\n", "\n", "In one sense, our hypothesis test is complete; we've constructed a test statistic, figured out its sampling distribution if the null hypothesis is true, and then constructed the critical region for the test. Nevertheless, I've actually omitted the most important number of all: **_the $p$ value_**. It is to this topic that we now turn. There are two somewhat different ways of interpreting a $p$ value, one proposed by Sir Ronald Fisher and the other by Jerzy Neyman. Both versions are legitimate, though they reflect very different ways of thinking about hypothesis tests. Most introductory textbooks tend to give Fisher's version only, but I think that's a bit of a shame. To my mind, Neyman's version is cleaner, and actually better reflects the logic of the null hypothesis test. You might disagree though, so I've included both. I'll start with Neyman's version...\n", "\n", "\n", "### A softer view of decision making\n", "\n", "One problem with the hypothesis testing procedure that I've described is that it makes no distinction at all between a result this \"barely significant\" and those that are \"highly significant\". For instance, in my ESP study the data I obtained only just fell inside the critical region - so I did get a significant effect, but was a pretty near thing. In contrast, suppose that I'd run a study in which $X=97$ out of my $N=100$ participants got the answer right. This would obviously be significant too, but by a much larger margin; there's really no ambiguity about this at all. The procedure that I described makes no distinction between the two. If I adopt the standard convention of allowing $\\alpha = .05$ as my acceptable Type I error rate, then both of these are significant results. \n", "\n", "This is where the $p$ value comes in handy. To understand how it works, let's suppose that we ran lots of hypothesis tests on the same data set: but with a different value of $\\alpha$ in each case. When we do that for my original ESP data, what we'd get is something like this" ] }, { "attachments": {}, "cell_type": "markdown", "id": "civil-score", "metadata": {}, "source": [ "| Value of $\\alpha$ | 0.05 | 0.04 | 0.03 | 0.02 | 0.01 |\n", "|--------------------|------|------|------|------|------|\n", "| Reject the null? | Yes | Yes | Yes | No | No |\n", "\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "correct-commissioner", "metadata": {}, "source": [ "When we test ESP data ($X=62$ successes out of $N=100$ observations) using $\\alpha$ levels of .03 and above, we'd always find ourselves rejecting the null hypothesis. For $\\alpha$ levels of .02 and below, we always end up retaining the null hypothesis. Therefore, somewhere between .02 and .03 there must be a smallest value of $\\alpha$ that would allow us to reject the null hypothesis for this data. This is the $p$ value; as it turns out the ESP data has $p = .021$. In short:\n", "\n", "> $p$ is defined to be the smallest Type I error rate ($\\alpha$) that you have to be willing to tolerate if you want to reject the null hypothesis. \n", "\n", "If it turns out that $p$ describes an error rate that you find intolerable, then you must retain the null. If you're comfortable with an error rate equal to $p$, then it's okay to reject the null hypothesis in favour of your preferred alternative. \n", "\n", "In effect, $p$ is a summary of all the possible hypothesis tests that you could have run, taken across all possible $\\alpha$ values. And as a consequence it has the effect of \"softening\" our decision process. For those tests in which $p \\leq \\alpha$ you would have rejected the null hypothesis, whereas for those tests in which $p > \\alpha$ you would have retained the null. In my ESP study I obtained $X=62$, and as a consequence I've ended up with $p = .021$. So the error rate I have to tolerate is 2.1\\%. In contrast, suppose my experiment had yielded $X=97$. What happens to my $p$ value now? This time it's shrunk to $p = 1.36 \\times 10^{-25}$, which is a tiny, tiny [^note8] Type I error rate. For this second case I would be able to reject the null hypothesis with a lot more confidence, because I only have to be \"willing\" to tolerate a type I error rate of about 1 in 10 trillion trillion in order to justify my decision to reject.\n", "\n", "\n", "### The probability of extreme data\n", "\n", "The second definition of the $p$-value comes from Sir Ronald Fisher, and it's actually this one that you tend to see in most introductory statistics textbooks. Notice how, when I constructed the critical region, it corresponded to the *tails* (i.e., extreme values) of the sampling distribution? That's not a coincidence: almost all \"good\" tests have this characteristic (good in the sense of minimising our type II error rate, $\\beta$). The reason for that is that a good critical region almost always corresponds to those values of the test statistic that are least likely to be observed if the null hypothesis is true. If this rule is true, then we can define the $p$-value as the probability that we would have observed a test statistic that is at least as extreme as the one we actually did get. In other words, if the data are extremely implausible according to the null hypothesis, then the null hypothesis is probably wrong.\n", "\n", "\n", "### A common mistake\n", "\n", "Okay, so you can see that there are two rather different but legitimate ways to interpret the $p$ value, one based on Neyman's approach to hypothesis testing and the other based on Fisher's. Unfortunately, there is a third explanation that people sometimes give, especially when they're first learning statistics, and it is *absolutely and completely wrong*. This mistaken approach is to refer to the $p$ value as \"the probability that the null hypothesis is true\". It's an intuitively appealing way to think, but it's wrong in two key respects: (1) null hypothesis testing is a frequentist tool, and the frequentist approach to probability does *not* allow you to assign probabilities to the null hypothesis... according to this view of probability, the null hypothesis is either true or it is not; it cannot have a \"5\\% chance\" of being true. (2) even within the Bayesian approach, which does let you assign probabilities to hypotheses, the $p$ value would not correspond to the probability that the null is true; this interpretation is entirely inconsistent with the mathematics of how the $p$ value is calculated. Put bluntly, despite the intuitive appeal of thinking this way, there is *no* justification for interpreting a $p$ value this way. Never do it.\n", "\n", "[^note8]: That's $p = .000000000000000000000000136$ for folks that don't like scientific notation!" ] }, { "attachments": {}, "cell_type": "markdown", "id": "parallel-issue", "metadata": {}, "source": [ "(writeup)=\n", "## Reporting the results of a hypothesis test\n", "\n", "When writing up the results of a hypothesis test, there's usually several pieces of information that you need to report, but it varies a fair bit from test to test. Throughout the rest of the book I'll spend a little time talking about how to report the results of different tests (see Section \\@ref(chisqreport) for a particularly detailed example), so that you can get a feel for how it's usually done. However, regardless of what test you're doing, the one thing that you always have to do is say something about the $p$ value, and whether or not the outcome was significant. \n", "\n", "The fact that you have to do this is unsurprising; it's the whole point of doing the test. What might be surprising is the fact that there is some contention over exactly how you're supposed to do it. Leaving aside those people who completely disagree with the entire framework underpinning null hypothesis testing, there's a certain amount of tension that exists regarding whether or not to report the exact $p$ value that you obtained, or if you should state only that $p < \\alpha$ for a significance level that you chose in advance (e.g., $p<.05$). \n", "\n", "### The issue\n", "\n", "To see why this is an issue, the key thing to recognise is that $p$ values are *terribly* convenient. In practice, the fact that we can compute a $p$ value means that we don't actually have to specify any $\\alpha$ level at all in order to run the test. Instead, what you can do is calculate your $p$ value and interpret it directly: if you get $p = .062$, then it means that you'd have to be willing to tolerate a Type I error rate of 6.2\\% to justify rejecting the null. If you personally find 6.2\\% intolerable, then you retain the null. Therefore, the argument goes, why don't we just report the actual $p$ value and let the reader make up their own minds about what an acceptable Type I error rate is? This approach has the big advantage of \"softening\" the decision making process -- in fact, if you accept the Neyman definition of the $p$ value, that's the whole point of the $p$ value. We no longer have a fixed significance level of $\\alpha = .05$ as a bright line separating \"accept\" from \"reject\" decisions; and this removes the rather pathological problem of being forced to treat $p = .051$ in a fundamentally different way to $p = .049$. \n", "\n", "This flexibility is both the advantage and the disadvantage to the $p$ value. The reason why a lot of people don't like the idea of reporting an exact $p$ value is that it gives the researcher a bit *too much* freedom. In particular, it lets you change your mind about what error tolerance you're willing to put up with *after* you look at the data. For instance, consider my ESP experiment. Suppose I ran my test, and ended up with a $p$ value of .09. Should I accept or reject? Now, to be honest, I haven't yet bothered to think about what level of Type I error I'm \"really\" willing to accept. I don't have an opinion on that topic. But I *do* have an opinion about whether or not ESP exists, and I *definitely* have an opinion about whether my research should be published in a reputable scientific journal. And amazingly, now that I've looked at the data I'm starting to think that a 9\\% error rate isn't so bad, especially when compared to how annoying it would be to have to admit to the world that my experiment has failed. So, to avoid looking like I just made it up after the fact, I now say that my $\\alpha$ is .1: a 10\\% type I error rate isn't too bad, and at that level my test is significant! I win.\n", "\n", "In other words, the worry here is that I might have the best of intentions, and be the most honest of people, but the temptation to just \"shade\" things a little bit here and there is really, really strong. As anyone who has ever run an experiment can attest, it's a long and difficult process, and you often get *very* attached to your hypotheses. It's hard to let go and admit the experiment didn't find what you wanted it to find. And that's the danger here. If we use the \"raw\" $p$-value, people will start interpreting the data in terms of what they *want* to believe, not what the data are actually saying... and if we allow that, well, why are we bothering to do science at all? Why not let everyone believe whatever they like about anything, regardless of what the facts are? Okay, that's a bit extreme, but that's where the worry comes from. According to this view, you really *must* specify your $\\alpha$ value in advance, and then only report whether the test was significant or not. It's the only way to keep ourselves honest. \n", "\n", "\n", "### Two proposed solutions\n", "\n", "In practice, it's pretty rare for a researcher to specify a single $\\alpha$ level ahead of time. Instead, the convention is that scientists rely on three standard significance levels: .05, .01 and .001. When reporting your results, you indicate which (if any) of these significance levels allow you to reject the null hypothesis. This is summarised in the table below. This allows us to soften the decision rule a little bit, since $p<.01$ implies that the data meet a stronger evidentiary standard than $p<.05$ would. Nevertheless, since these levels are fixed in advance by convention, it does prevent people choosing their $\\alpha$ level after looking at the data. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "certain-myrtle", "metadata": {}, "source": [ "| Usual notation | Sig. stars | English translation | The null is... |\n", "|----------------|------------|----------------------------------------------------------------------------|----------------|\n", "| p > 0.05 | | The test wasn't significant | Retained |\n", "| p < 0.05 | * | The test was significant at $\\alpha$ = 0.05 but not at $\\alpha$ = 0.01 or $\\alpha$ = 0.001. | Rejected |\n", "| p < 0.01 | ** | The test was significant at $\\alpha$ = 0.05 and at $\\alpha$ = 0.01 but not at $\\alpha$ = 0.001. | Rejected |\n", "| p < 0.001 | *** | The test was significant at all levels | Rejected |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "original-addiction", "metadata": {}, "source": [ "Nevertheless, quite a lot of people still prefer to report exact $p$ values. To many people, the advantage of allowing the reader to make up their own mind about how to interpret $p = .06$ outweighs any disadvantages. In practice, however, even among those researchers who prefer exact $p$ values it is quite common to just write $p<.001$ instead of reporting an exact value for small $p$. This is in part because a lot of software doesn't actually print out the $p$ value when it's that small (e.g., SPSS just writes $p = .000$ whenever $p<.001$), and in part because a very small $p$ value can be kind of misleading. The human mind sees a number like .0000000001 and it's hard to suppress the gut feeling that the evidence in favour of the alternative hypothesis is a near certainty. In practice however, this is usually wrong. Life is a big, messy, complicated thing: and every statistical test ever invented relies on simplifications, approximations and assumptions. As a consequence, it's probably not reasonable to walk away from *any* statistical analysis with a feeling of confidence stronger than $p<.001$ implies. In other words, $p<.001$ is really code for \"as far as *this test* is concerned, the evidence is overwhelming.\" \n", "\n", "In light of all this, you might be wondering exactly what you should do. There's a fair bit of contradictory advice on the topic, with some people arguing that you should report the exact $p$ value, and other people arguing that you should use the tiered approach illustrated in the table above. As a result, the best advice I can give is to suggest that you look at papers/reports written in your field and see what the convention seems to be. If there doesn't seem to be any consistent pattern, then use whichever method you prefer." ] }, { "attachments": {}, "cell_type": "markdown", "id": "professional-trademark", "metadata": {}, "source": [ "## Running the hypothesis test in practice\n", "\n", "At this point some of you might be wondering if this is a \"real\" hypothesis test, or just a toy example that I made up. It's real. In the previous discussion I built the test from first principles, thinking that it was the simplest possible problem that you might ever encounter in real life. However, this test already exists: it's called the *binomial test*, and it's implemented in a function called `binom_test()` from the `scipy.stats` package. To test the null hypothesis that the response probability is one-half `p = .5`, [^note9] using data in which `x = 62` of `n = 100` people made the correct response, here's how to do it in Python:\n", "\n", "[^note9]: Note that the `p` here has nothing to do with a $p$ value. The `p` argument in the `binom_test()` function corresponds to the probability of making a correct response, according to the null hypothesis. In other words, it's the $\\theta$ value." ] }, { "cell_type": "code", "execution_count": 4, "id": "beginning-guitar", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.02097873567785172" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from scipy.stats import binom_test\n", "binom_test(x = 62, n = 100, p = 0.5, alternative = 'two-sided')" ] }, { "attachments": {}, "cell_type": "markdown", "id": "neutral-reference", "metadata": {}, "source": [ "Well. There's a number, but what does it mean? Sometimes the output of these Python functions can be fairly terse. But here `binom_test()` is giving us the $p$-value for the test we specified. In this case, the $p$-value of 0.02 is less than the usual choice of $\\alpha = .05$, so we can reject the null. Usually we will want to know more than just the $p$-value for a test, and Python has ways of giving us this information, but for now, however, I just wanted to make the point that Python packages contain a whole lot of functions corresponding to different kinds of hypothesis test. And while I'll usually spend quite a lot of time explaining the logic behind how the tests are built, every time I discuss a hypothesis test the discussion will end with me showing you a fairly simple Python command that you can use to run the test in practice." ] }, { "attachments": {}, "cell_type": "markdown", "id": "shared-pension", "metadata": {}, "source": [ "(effectsize)=\n", "## Effect size, sample size and power\n", "\n", "In previous sections I've emphasised the fact that the major design principle behind statistical hypothesis testing is that we try to control our Type I error rate. When we fix $\\alpha = .05$ we are attempting to ensure that only 5\\% of true null hypotheses are incorrectly rejected. However, this doesn't mean that we don't care about Type II errors. In fact, from the researcher's perspective, the error of failing to reject the null when it is actually false is an extremely annoying one. With that in mind, a secondary goal of hypothesis testing is to try to minimise $\\beta$, the Type II error rate, although we don't usually *talk* in terms of minimising Type II errors. Instead, we talk about maximising the *power* of the test. Since power is defined as $1-\\beta$, this is the same thing. " ] }, { "cell_type": "code", "execution_count": 17, "id": "agricultural-zimbabwe", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "from myst_nb import glue\n", "from numpy import random\n", "import matplotlib.pyplot as plt\n", "import seaborn as sns\n", "\n", "# sample from a binomial distribution\n", "data = random.binomial(n=100, p=.55, size=10000)\n", "\n", "# plot distribution and color critical region\n", "ax = sns.histplot(data, bins=20,binwidth=.5, color=\"black\")\n", "ax.set_title(\"Sampling distribution for X if $\\\\theta = 0.55$\")\n", "ax.annotate(\"\", xy=(40, 500), xytext=(30, 500), arrowprops=dict(arrowstyle=\"<-\"))\n", "ax.annotate(\"lower critical region \\n (2.5% of the distribution)\", xy=(40, 600), xytext=(10, 580))\n", "ax.annotate(\"\", xy=(70, 500), xytext=(60, 500), arrowprops=dict(arrowstyle=\"->\"))\n", "ax.annotate(\"upper critical region \\n (2.5% of the distribution)\", xy=(70, 500), xytext=(60, 580))\n", "ax.set(xlim=(0,100))\n", "for p in ax.patches:\n", " if p.get_x() >= 40:\n", " if p.get_x() <= 60:\n", " p.set_color(\"lightgrey\")\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "civil-panel", "metadata": {}, "source": [ "```{glue:figure} esp-alternative-fig\n", ":figwidth: 600px\n", ":name: fig-esp-alternative\n", "\n", "Sampling distribution under the *alternative* hypothesis, for a population parameter value of $\\\\theta$ = 0.55. A reasonable proportion of the distribution lies in the rejection region.\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "active-coating", "metadata": {}, "source": [ "### The power function\n", "\n", "Let's take a moment to think about what a Type II error actually is. A Type II error occurs when the alternative hypothesis is true, but we are nevertheless unable to reject the null hypothesis. Ideally, we'd be able to calculate a single number $\\beta$ that tells us the Type II error rate, in the same way that we can set $\\alpha = .05$ for the Type I error rate. Unfortunately, this is a lot trickier to do. To see this, notice that in my ESP study the alternative hypothesis actually corresponds to lots of possible values of $\\theta$. In fact, the alternative hypothesis corresponds to every value of $\\theta$ *except* 0.5. Let's suppose that the true probability of someone choosing the correct response is 55\\% (i.e., $\\theta = .55$). If so, then the *true* sampling distribution for $X$ is not the same one that the null hypothesis predicts: the most likely value for $X$ is now 55 out of 100. Not only that, the whole sampling distribution has now shifted, as shown in {numref}`fig-esp-alternative`. The critical regions, of course, do not change: by definition, the critical regions are based on what the null hypothesis predicts. What we're seeing in this figure is the fact that when the null hypothesis is wrong, a much larger proportion of the sampling distribution distribution falls in the critical region. And of course that's what should happen: the probability of rejecting the null hypothesis is larger when the null hypothesis is actually false! However $\\theta = .55$ is not the only possibility consistent with the alternative hypothesis. Let's instead suppose that the true value of $\\theta$ is actually 0.7. What happens to the sampling distribution when this occurs? The answer, shown in {numref}`fig-esp-alternative2`, is that almost the entirety of the sampling distribution has now moved into the critical region. Therefore, if $\\theta = 0.7$ the probability of us correctly rejecting the null hypothesis (i.e., the power of the test) is much larger than if $\\theta = 0.55$. In short, while $\\theta = .55$ and $\\theta = .70$ are both part of the alternative hypothesis, the Type II error rate is different." ] }, { "cell_type": "code", "execution_count": 25, "id": "manufactured-annex", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "from myst_nb import glue\n", "from numpy import random\n", "import matplotlib.pyplot as plt\n", "import seaborn as sns\n", "\n", "# sample from a binomial distribution\n", "data = random.binomial(n=100, p=.7, size=10000)\n", "\n", "\n", "# plot distribution and color critical region\n", "ax = sns.histplot(data, bins=20,binwidth=.5, color=\"black\")\n", "ax.set_title(\"Sampling distribution for X if $\\\\theta = 0.7$\")\n", "ax.annotate(\"\", xy=(40, 500), xytext=(30, 500), arrowprops=dict(arrowstyle=\"<-\"))\n", "ax.annotate(\"lower critical region \\n (2.5% of the distribution)\", xy=(40, 600), xytext=(5, 580))\n", "ax.annotate(\"\", xy=(80, 500), xytext=(60, 500), arrowprops=dict(arrowstyle=\"->\"))\n", "ax.annotate(\"upper critical region \\n (2.5% of the distribution)\", xy=(70, 500), xytext=(80, 580))\n", "ax.set(xlim=(0,100))\n", "for p in ax.patches:\n", " if p.get_x() >= 40:\n", " if p.get_x() <= 60:\n", " p.set_color(\"lightgrey\")\n", "\n", "\n", "sns.despine()\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "improved-malaysia", "metadata": {}, "source": [ "```{glue:figure} esp-alternative-fig2\n", ":figwidth: 600px\n", ":name: fig-esp-alternative2\n", "\n", "Sampling distribution under the *alternative* hypothesis, for a population parameter value of $\\\\theta$ = 0.7. Almost all of the distribution lies in the rejection region.\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "alien-drunk", "metadata": {}, "source": [ "What all this means is that the power of a test (i.e., $1-\\beta$) depends on the true value of $\\theta$. To illustrate this, I've calculated the expected probability of rejecting the null hypothesis for all values of $\\theta$, and plotted it in {numref}`fig-powerfunction`. This plot describes what is usually called the **_power function_** of the test. It's a nice summary of how good the test is, because it actually tells you the power ($1-\\beta$) for all possible values of $\\theta$. As you can see, when the true value of $\\theta$ is very close to 0.5, the power of the test drops very sharply, but when it is further away, the power is large. " ] }, { "cell_type": "code", "execution_count": 20, "id": "stainless-campus", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "from myst_nb import glue\n", "import numpy as np\n", "from scipy.stats import binom\n", "import seaborn as sns\n", "theta = np.arange(0.01,.99,0.01)\n", "\n", "n = 100 \n", "\n", "prob = []\n", "for k in theta:\n", " prob.append(binom.cdf(40,n, k) + 1 - binom.cdf(59,n, k))\n", "\n", "\n", "#sns.lineplot(theta, prob_lower)\n", "ax = sns.lineplot(x = theta, y = prob)\n", "ax.set_title(\"Power Function for the Test (N=100)\")\n", "ax.set(xlabel='True value of $\\\\theta$', ylabel='Probablility of rejecting the Null')\n", "\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "constitutional-script", "metadata": {}, "source": [ "```{glue:figure} powerfunction-fig\n", ":figwidth: 600px\n", ":name: fig-powerfunction\n", "\n", "The probability that we will reject the null hypothesis, plotted as a function of the true value of $\\theta$. Obviously, the test is more powerful (greater chance of correct rejection) if the true value of $\\theta$ is very different from the value that the null hypothesis specifies (i.e., $\\theta=.5$). Notice that when $\\theta$ actually is equal to .5 (plotted as a black dot), the null hypothesis is in fact true: rejecting the null hypothesis in this instance would be a Type I error.\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "funded-intensity", "metadata": {}, "source": [ "### Effect size\n", "\n", ">*Since all models are wrong the scientist must be alert to what is importantly wrong. It is inappropriate to be concerned with mice when there are tigers abroad*\n", ">\n", ">-- George Box 1976\n", "\n", "The plot shown in {numref}`fig-powerfunction` captures a fairly basic point about hypothesis testing. If the true state of the world is very different from what the null hypothesis predicts, then your power will be very high; but if the true state of the world is similar to the null (but not identical) then the power of the test is going to be very low. Therefore, it's useful to be able to have some way of quantifying how \"similar\" the true state of the world is to the null hypothesis. A statistic that does this is called a measure of **_effect size_** (e.g. {cite}`Cohen1988` or {cite}`Ellis2010`). Effect size is defined slightly differently in different contexts (and so this section just talks in general terms) but the qualitative idea that it tries to capture is always the same: how big is the difference between the *true* population parameters, and the parameter values that are assumed by the null hypothesis? In our ESP example, if we let $\\theta_0 = 0.5$ denote the value assumed by the null hypothesis, and let $\\theta$ denote the true value, then a simple measure of effect size could be something like the difference between the true value and null (i.e., $\\theta - \\theta_0$), or possibly just the magnitude of this difference, $\\mbox{abs}(\\theta - \\theta_0)$." ] }, { "attachments": {}, "cell_type": "markdown", "id": "handled-sarah", "metadata": {}, "source": [ "| | big effect size | small effect size |\n", "| :--------------------- | :---------------------- | :----------------------- |\n", "| significant result | difference is real, and of practical importance | difference is real, but might not be interesting |\n", "| non-significant result | no effect observed | no effect observed |\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "noble-carnival", "metadata": {}, "source": [ "Why calculate effect size? Let's assume that you've run your experiment, collected the data, and gotten a significant effect when you ran your hypothesis test. Isn't it enough just to say that you've gotten a significant effect? Surely that's the *point* of hypothesis testing? Well, sort of. Yes, the point of doing a hypothesis test is to try to demonstrate that the null hypothesis is wrong, but that's hardly the only thing we're interested in. If the null hypothesis claimed that $\\theta = .5$, and we show that it's wrong, we've only really told half of the story. Rejecting the null hypothesis implies that we believe that $\\theta \\neq .5$, but there's a big difference between $\\theta = .51$ and $\\theta = .8$. If we find that $\\theta = .8$, then not only have we found that the null hypothesis is wrong, it appears to be *very* wrong. On the other hand, suppose we've successfully rejected the null hypothesis, but it looks like the true value of $\\theta$ is only .51 (this would only be possible with a large study). Sure, the null hypothesis is wrong, but it's not at all clear that we actually *care*, because the effect size is so small. In the context of my ESP study we might still care, since any demonstration of real psychic powers would actually be pretty cool [^note10], but in other contexts a 1\\% difference isn't very interesting, even if it is a real difference. For instance, suppose we're looking at differences in high school exam scores between males and females, and it turns out that the female scores are 1\\% higher on average than the males. If I've got data from thousands of students, then this difference will almost certainly be *statistically significant*, but regardless of how small the $p$ value is it's just not very interesting. You'd hardly want to go around proclaiming a crisis in boys education on the basis of such a tiny difference would you? It's for this reason that it is becoming more standard (slowly, but surely) to report some kind of standard measure of effect size along with the the results of the hypothesis test. The hypothesis test itself tells you whether you should believe that the effect you have observed is real (i.e., not just due to chance); the effect size tells you whether or not you should care.\n", "\n", "[^note10]: Although in practice a very small effect size is worrying, because even very minor methodological flaws might be responsible for the effect; and in practice no experiment is perfect, so there are always methodological issues to worry about." ] }, { "attachments": {}, "cell_type": "markdown", "id": "environmental-legend", "metadata": {}, "source": [ "(power)=\n", "### Increasing the power of your study\n", "\n", "Not surprisingly, scientists are fairly obsessed with maximising the power of their experiments. We want our experiments to work, and so we want to maximise the chance of rejecting the null hypothesis if it is false (and of course we usually want to believe that it is false!) As we've seen, one factor that influences power is the effect size. So the first thing you can do to increase your power is to increase the effect size. In practice, what this means is that you want to design your study in such a way that the effect size gets magnified. For instance, in my ESP study I might believe that psychic powers work best in a quiet, darkened room; with fewer distractions to cloud the mind. Therefore I would try to conduct my experiments in just such an environment: if I can strengthen people's ESP abilities somehow, then the true value of $\\theta$ will go up [^note11] and therefore my effect size will be larger. In short, clever experimental design is one way to boost power; because it can alter the effect size.\n", "\n", "Unfortunately, it's often the case that even with the best of experimental designs you may have only a small effect. Perhaps, for example, ESP really does exist, but even under the best of conditions it's very very weak. Under those circumstances, your best bet for increasing power is to increase the sample size. In general, the more observations that you have available, the more likely it is that you can discriminate between two hypotheses. If I ran my ESP experiment with 10 participants, and 7 of them correctly guessed the colour of the hidden card, you wouldn't be terribly impressed. But if I ran it with 10,000 participants and 7,000 of them got the answer right, you would be much more likely to think I had discovered something. In other words, power increases with the sample size. This is illustrated in {numref}`fig-powerfunctionsample`, which shows the power of the test for a true parameter of $\\theta = 0.7$, for all sample sizes $N$ from 1 to 100, where I'm assuming that the null hypothesis predicts that $\\theta_0 = 0.5$. \n", "\n", "[^note11]: Notice that the true population parameter $\\theta$ doesn't necessarily correspond to an immutable fact of nature. In this context $\\theta$ is just the true probability that people would correctly guess the colour of the card in the other room. As such the population parameter can be influenced by all sorts of things. Of course, this is all on the assumption that ESP actually exists!" ] }, { "cell_type": "code", "execution_count": 21, "id": "presidential-translation", "metadata": {}, "outputs": [ { "data": { "image/png": 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WAef78d7zgCxV3e/e2J+nc5OSAnHiNIrGAkeBphO6AmN6yCf7SwB4e1sBGw4epaKukY2HSlk6MblduR+dO4GYiFDufXMHqsqqXYXMGTmQhOjwtjKRYaHER4VjTE/oMhGo6t9aH8Am4ALgFuA+wJ8By2nAYY/nOe4xT78HJgF5OEnmh6ra0vGNROQmEckUkcyiIuunNj2vuUVZv7+ES2YMIzU+kl+9tZOP9hbT3KIsnZjSruygmAjuOGcCH+4t5pn12ezMr+DsjJQu3tmY7uezj0BEJonIX4E3cOYOTFbVR/xsuvFWx+3Y1HQ+sBkYBswEfi8i8Z1epPqYqs5V1bnJyckdTxvT7XbkVVBR18SySSnced5Evjhcxn+u3ElcVBizRyR2Kn/DwpGMSY7hZ69vA2CZJQLTi/hadO4lYCXwCbAUWAHEi8ggEfFnW6McYLjH83Scb/6ebgReUUcWcADI8D98Y3rGJ/uLAVg4JokrZqczaWg8OaW1nDl+cNvS0Z7CQ0P46Vcm06KQPnBA29LRxvQGvmoErZPG7gTWA5nARveR6cd7bwDGi8hoEYkArsZJJp6ycfocEJFUYCLOEhbGdIvteeXsOVLps8yn+0u49ZnPqW9qbjv2yb4SxibHkBIfRWiIcM9XJgFw7uSuZ/eelZHCtxaN4p+WjLW5AqZX8bX66Kgv88aq2iQitwF/B0KBP6vqdhG52T3/KHAv8Bd3NzQB7lLV4i/zucaciNuf20R9Uwur71xKuJdv8gAvb8zhra35LBybxPULRtLY3MJnB45y+exjXV6njxvM+/+8hFHHWfTNdgozvVFAN69X1ZU4zUuexx71+D0POC+QMRjTlbKaBvYVVQOwYnMeV3axhPPGQ6WAs2HMV+emsz2vguqGZhaOab/O/5hka+4xfZNtXm+C1qbDZQDERobx0Josmls6T5spqarnQHE1Z44fTH55HS9sOMwn+5xhowvG+NNVZkzvZ4nABK1N2WWECPzs4snsL6rmnW0Fncq01gZuXzae00YNdLaR3F1ExpA4kmIjuztkYwLCr0QgIme0Lv8gIskiMjqwYRkTeJuyS8kYEs+Vs9MZkxzDg+/v7bRS6MZDpUSEhjAtLYEfnTuBIxX1fHbwKAvGJPVQ1Macev7sUPYfwF3AT9xD4cBfAxmUMYHW0qJszi5j1ghnY/dbl45jV0Elq3YWtiuXeaiUqWnxRIWHsmjsYOaPdpqDFo21RGD6D39qBJcDlwDV0NbBa/sZmz7l8+xSKuqOrQC6r6iKyvomZrn7AF8ycxjDBw3g96uz2moF9U3NbM0pZ87IY3sF3/OVyZw5fjCLxtmG8Kb/8CcRNKjzP0MBRMQ2RTV9SnV9E1//wyf8fMX2tmOfZztt/62zgMNDQ7hp8Vg2Hy7jswNHAdiWW05DcwtzRh7rFJ6WnsDT35lPbGRAB9wZ0638SQQvisgfgEQR+R7wHvB4YMMy5tTZnldBY7Pyxhd5FFY4K6hvyi4jYUA4owcf+17z1TnpJMVE8OgH+wDIPOgkC88agTH9kT/7EfwP8DLwN5yZvz9T1QcDHZgxp8q23HIAGpuVpz89BDg1glkjEtvN8I0KD+Vbi0axencRuwoq2HiolFFJ0STH2egg07/5NWpIVd9V1X9R1TtV9d1AB2XMqbQtt5yUuEjOmZTKM+uzKaqsZ29hFbNHdP6mf8PCkQwID+WxD/az8VBpu2YhY/orf0YNXSEie0WkXEQqRKRSRCq6IzhjToWtueVMS0vgO2eM5mh1A794YzuqMMvLKqGJ0RFcPW84r27OpaS6wZqFTFDwp0bw/4BLVDVBVeNVNU5VOy0VbUxvVNPQxL6iKqamJbBgzCAmD43nzS35iMCMLraF/O6ZYwhxm4zmjrJEYPo/fxLBEVXdGfBIjAmAHXkVtChMS0tARPj2Gc5cyHHJsV3uCJaWOIDLZ6WREhfJOFs/yAQBX3sWX+H+mikiLwCvAfWt51X1lcCGZsyJUVXW7Cli8Xhn43hwmoXAGfYJcPGModz/7h5OP848gF9dNpXKuibbQ9gEBV+DoS/2+L2G9quEKmCJwHSbspoGDpXUdNmcA84s4Buf2MC9l03lhgUjAScRJMdFtm0mHxkWyjt3nElkWKjPz4sKDyUq3HcZY/oLX/sRtK4tdLqqrvM8JyKnBzowYzw9tDqLJ9Yd5OO7zybFval31DpM9JlPD3H9/BGICNvcjmJPcbZJvDHt+NNH4G3OgM0jMN1qR34FTS3Ky5/ndFlmV76z09iugko+zy6jpqGJrEKno9gY0zVffQQLgUVAsoj82ONUPM6OY8Z0m90Fzk3+hQ2HuaWLrR53FlQwc3giWYVVPLP+ENfNH9HWUWyM6ZqvGkEEEIuTLOI8HhXAVYEPzRhHcVU9xVUNTEtL4FBJDZ/uP9qpTHOLsrugkjkjB3LZrGG8uSWfj/Y6G8hYIjDGN199BB8AH4jIX1T1UDfGZEw7e9zawO3LxvPjFzfzwoZsFnZYBvpgSTX1TS1MGhrP5KHx/PXTbP6wdh+DYyNJjbclIozxxZ+1hiwJmB61+4iTCGYMT+DyWWms3FZAeU1juzI7853J7hlD4pg8LJ7ZIxKpaWhmalq812YkY8wxtlWl6fX2HKlkYHQ4ybGRfP204TQ0tfDqpvadxrvyKwkNEcalOBPArneHj1qzkDHHZ4nA9Hq7CiqZkBqHiDBlWALT0xN4fsPhdttK7syvYGxyTNvY/wunDeVrc9O5dOawngrbmD7juLtriMgDXg6XA5mq+vqpD8kEM1WlRWmbGayq7Cmo5Ko56W1lvjZ3OPe8to2tueVMT08EnGThuUBcVHgo/++qGd0auzF9lT81gihgJrDXfUwHBgHfEZH/C1hkJijd8cJmbnoqs+15blkt1Q3NTBhybHfUS2YOIzIshJcyneah8tpGcstqmTTU1kI05mT4kwjGAWer6oPuhjTnAJNw9jI+z+crjTlBGw4cZdWuQnLLaoFj8wcmph5LBPFR4SyfOoTXN+dS19jMrtaO4qG2lbYxJ8OfRJAGeO5THAMMU9VmPBahM+bLqq5vIq/c2UrytU25wLERQ541AoCvzhlORV0T7+44wi43WUwaYjUCY06Gv/sRbBaRJ0TkL8Am4H/cTezfC2RwJrjsL6oGICI0hFc35aLqTBIblhDVacnoRWOTSEscwEsbc9iZX8HA6HCbL2DMSfJnHsGfcJaaeM19nKGqf1TValX9l8CGZ4LJvqIqAK6dP4Kswiq251Wwu6CyU20AICREuHJ2Gh/uLeKjrGIyhth8AWNOlr/DR0OAIuAoME5EFgcuJBOssgqrCA0Rvn/WWMJDhZc35rC/qJqJXhIBwFVzhqMKOaW11j9gzJfgz/DR/wa+DmwHWtzDCqwNYFwmCGUVVjFyUDQpcVGcNTGF5z7LpqG5pV1HsacRSdHMHz2I9QeO2oghY76E4yYC4DJgoqpax7AJqH1FVYx1ZwZfPiuNf+w4AsCELhIBOM1I6w8cZYY7n8AYc+L8aRraD5zUTh4islxEdotIlojc3UWZpSKyWUS2i8gHJ/M5pu9ram7hYEk1Y909gs/KSCE+KowQoW3ZCG8umTGM1Xcu7bL5yBhzfP7UCGpwRg2tov2exbf7epGIhAIPAecCOcAGEVmhqjs8yiQCDwPLVTVbRFJO/BJMf5B9tIbGZm276UeFh3Lt/JFszS3zuWWkiDB6cEyX540xx+dPIljhPk7UPCBLVfcDiMjzwKXADo8y1wKvqGo2gKoWnsTnmD7o2fXZzBiewJRhzqJwWYXOiCHPb/93X5DRI7EZE2yOmwhU9cmTfO804LDH8xxgfocyE4BwEVmDs+nN71T1qY5vJCI3ATcBjBgx4iTDMb1FfVMz97y2lTPHJ/Pkt+cBsM+dQzAm2b7dG9PduuwjEJEX3Z9bRWRLx4cf7+1tULd2eB4GzAG+ApwP/FREJnR6kepjqjpXVecmJyf78dGmpzS3KG9vzaelpeNf9TGHj9bQorAuq5ij1Q2AUyNIjY/sNHHMGBN4vjqLf+j+vAi42MvjeHKA4R7P04E8L2XecSenFeMMSbUlI/uwtXuKuOWZz1m5Lb/LMq0ziJtalHe2FQCQVVTV1lFsjOleXSYCVW39n/x9VT3k+QC+78d7bwDGi8hoEYkArqZzX8PrwJkiEiYi0ThNRztP/DJMb9E6O/jtrQVdljlQ7CSCIfFRvLklD1Vlf2GVz9FBxpjA8Wf46Llejl1wvBepahNwG/B3nJv7i6q6XURuFpGb3TI7gXeALcBnwB9VdZu/wZve51BJDQDv7yqktqHZa5mDJdUkxUTw1bnpfLq/hO15FVTWN1mNwJge0mVnsYjcgvPNf2yHPoE44GN/3lxVVwIrOxx7tMPz3wC/8Tdg07sdLKkmMiyE2sZm1uwu5IJpQzuV2V9UzejBMVw8YxgPvp/F79/PAnzPFzDGBI6vGsGzOH0Br9O+b2COql7XDbGZPuhQSQ3LJqWQFBPBW1u99xMcKHYSwYTUOCakxvLOdqcZyWoExvQMX30E5ap6EPgdcNSjf6BRRDoOAzWGhqYWckprGJscy/lTh/D+rkLqGts3D1XVN1FYWc8odxLYRdOdPYVjI8NsGWljeog/fQSPAFUez6vdY8a0k1PqDAsdmRTDhVOHUtPQzJrdRe3KHHQ7ise0JQKn6WhsSqwtI21MD/EnEYiqtg0KV9UW/JuRbIJMa0fxqKRoFowZxMDocFZ2aB5qHTE02p04NiY5lqUTk1k8fnD3BmuMaePPDX2/iNzOsVrA93EWojOmnYMlzk1+1OAYwkJDOH/KEN74Io+6xua29YJaE8GopGMziP9y47zuD9YY08afGsHNODuU5XJsmYibAhmU6ZsOldQQGxlGUkwEABdOG0p1QzMf7DnWPHSguJphCVE+F5IzxnQvf9YaKsSZDGaMTwdLqhmZFN3W1r9wbBIJA8J5Z1sB508ZArgjhmw9IWN6lePWCERkgoisEpFt7vPpInJP4EMzvZmqdhoRdKikpl2TT3hoCOdNTuW9HUeob2p2ZhAXVdmy0cb0Mv40DT0O/ARoBFDVLVgNIeg9v+EwC/9rFZV1jYCzsczhozWMTIpuV+7CaUOprG/i46wSSmsaqahrYvRgmy9gTG/iTyKIVtXPOhxrCkQwpu/YklNGaU1j2/DQvLI6mlq0XY0AYNG4JOKiwli5NZ8Dxc4o5NGDozu9nzGm5/iTCIpFZCzuEtIichXQ9dKSps+raWjqcp2gVtlHnaGirbOCD7gjhjrWCCLDQjl3Uir/2HGEvUdaE4HVCIzpTfxJBLcCfwAyRCQXuAO4JZBBmZ71vacyueOFTT7LtCaCNe7s4UNuIvDW/n/BtKGU1zby7GfZhIUI6QMHnPqgjTEnzZ9RQ/uBc0QkBghR1crAh2V60s78SqrqmqiubyImsvM/kcbmFvLK6piRnsAXOeWsyyrmYHENA8JDSY7rvEzEmeMHExMRypaccsYMjiE81J/vH8aY7uJr9dHrVfWvIvLjDsfBaSY6CqxQ1dLAhmi6U3V9U9uuYeuyijnPHfbpKa+sluYW5eunjWB/8U7e2VbA0eqGdkNHPUWFh7JsUiorvshrW2PIGNN7+Ppq1vo/Ns7LIx5ni8m3Axqd6Xa5ZbVtv6/eXei1TGuz0NjkGM6ZlMq7O4+wr6iqU0expwunOQnFho4a0/t0WSNQ1T+4P3/RVRkR+WUggjI9J6fUucmnJQ7g/V2FqGqnb/mtiWBEUjTnT0nl1U25lNU0cv7UzrWHVksmpDBjeCKLJ9ie08b0Nl9qQpmq/izQAZrulVPq1AiuWzCCIxX1bM+r6FQmu6SGiLAQUuOiWDwhmahw55+RrxrBgIhQXr/1dJZYIjCm17EJZaad3NJaIsJC+Oqc4Yg4W052lH20huEDBxASIkRHhLXd3DsOHTXG9A02ocy0k1NaS3riAJLjIpmRnthlIhgx6NhN/6o5w4mJCCVjSHx3hmqMOUVsQplpJ6e0hjR3nP/ZGSl8kVNGcVV923lVJbukfSI4d3IqW35+PoPcVUeNMX3LyU4ouzmQQZmek1Na2zbh6+yMFFRpt8tYWU0jlfVNDB/UvhkoNMR2FzOmr/KZCEQkFLhFVc8BkoEMVT3D3bvY9DM1DU2UVDeQPtC5yU8ZFk9qfCSrdh5pK9M6Ymikj45hY0zf4jMRqGozznwBVLXaZhX3b7nuiKHWGoGIcHZGKmv3FLUtOd02dHSQdQwb01/40zS0SURWiMgNInJF6yPgkZlul9MhEQCcPyWV6oZm1mUVA8cSwfBBtl6QMf2FP3sWDwJKgLM9jinwSkAiMj0mp6w1ERz7tr9o7GDiIsP4+/YClk1KJbukhsGxkURH+PNPxxjTF/iz6NyN3RGI6V6qyj2vbePSmWnMGz0IcEYMRYSGkBx7bOG4iLAQzp6Uwrs7jtDU3OIOHbXagDH9iS0DGaRKaxp5Zn02f/pof9uxnNJa0tyJYp6WTxlCaU0jGw6Wkn20xjqKjelnLBEEqdaO4Q/3FlPf5HQEew4d9bRkYjKRYSG8uSWP/PLaTkNHjTF9W5eJQER+6P48vfvCMd0lt8zp9K1paOazA0edY6U1pCV2TgTREWGcOT6Zv32eQ4vaiCFj+htfNYLWvoEHuyMQ071aRwiFhQjv7yqktqGZ4qqGLncPWz51CHWNLYAlAmP6G1+JYKeIHAQmisgWj8dWEdnSTfGZk7C/qIqv/+GTtg1mvMktqyUmIpQzxg/m/V2FbTUEzxFDns6ZlNI2e9gWlzOmf+kyEajqNcACIAu42ONxkfvzuERkuYjsFpEsEbnbR7nTRKTZXcfIfEmf7C9h/YGj/MPdWN6bXLdjeFlGCodKali7x5kn0FWNIDE6ggVjBhEZ1n5UkTGm7zvezOICVZ2Bs8hc6+5kef4sMeEuT/EQcAEwGbhGRCZ3Ue6/gb+fePjGm/yyOgBWeVk5tFVuWS1piQM4KyMFgL9+6vyVdlUjALh7+SR+ffm0TqOKjDF9mz8b0ywB9uLc1B8G9ojIYj/eex6Qpar7VbUBeB641Eu5HwB/A7q+a5kTkudODFuXdWxEUEe5ZU6NIH1gNBNT49hfXE14qJDiZfP5VtPSE7hqTnpAYjbG9Bx/ho/+L3Ceqi5R1cXA+cD9frwuDTjs8TzHPdZGRNKAy4FHfb2RiNwkIpkikllUVOSrqMG5yUeEhlDT0Mz6/Uc7na+ub6KsppFh7gih1lpBWmLnOQTGmP7Pn0QQrqq7W5+o6h4g3I/XebujaIfn/wfc5S5u1yVVfUxV56rq3OTk4N7q8IM9RTz58UGfZfLKazkrwxn7721jmdYN6luHii6b5CQCX81Cxpj+y59EkCkifxKRpe7jcWCjH6/LAYZ7PE8H8jqUmQs8745Ougp4WEQu8+O9g9ZTHx/kV2/toLre+yZxzS1KQXkdY5JjWTQ2idW7nQ3oPXVcZXTW8EQGx0YyLiU2sMEbY3olf1YOuwVnc5rbcb7lr8XpKzieDcB4ERkN5OLsc3ytZwFVHd36u4j8BXhTVV/zJ/BglV9eR2Ozsi6rmPOmDOl0vriqnsZmZVjiAIYlDmD1a9vYX1zN2ORjN/mcthqBUwMICw1hxW2nExdlC8kZE4yOWyNQ1XpV/V9VvUJVL1fV+1W13o/XNQG34YwG2gm8qKrbReRmEbEdzk5SfrlzE1+923vfel7bTT6Ks922//d3ti+bW1rbqWN4WOIA4qL8afEzxvQ3Af0KqKorgZUdjnntGFbVbwUylv6grrGZ0ppGAFbvKkJVEWnfFZPnDh0dljiAtMQBTEyN4/1dhXxv8Zi2MrlltQxNsI5hY4zDFp3rQwrKnZv8vNGDKKioY1dB5w3jWmsEQxPcfYcnpbDh4FEq6hrbynS1ppAxJjj5M49gancEYo4vz20Wumae0wfvrXkot6yW2Mgw4t32/rMzUmhqUdbuKWpXJq2LGcTGmODjT43gURH5TES+LyKJgQ7IdK21RjAjPZGpafGs9jI0NK+slmGJUW1NRrNHDGRgdDir3H6ChqYWCivrrUZgjGnjT2fxGcB1OENBM0XkWRE5N+CRmU7y3UQwNGEAZ01MYeOhUsprGtuVySuvbZsoBhAa4mxA//6uQpqaW8gvr0UVqxEYY9r41UegqnuBe4C7gCXAAyKyyzax71755bUkRoczICKUpRNTaFFYu7f9TOv8srp2iQDg3MkplNc2knmo9NgcAqsRGGNc/vQRTBeR+3GGgJ4NXKyqk9zf/VlqwpwiBeV1DImPAmDm8EQGRoe3ax6qa2ympLqBYQlR7V535vhkIkJDeG/HkWNzCKxGYIxx+TN89PfA48C/qWpt60FVzRORewIWmekkz+PbfmiIsGRCMmv2FNHSooSESNuIoY41gpjIMBaNS+LdnUeIiQxD5NioImOM8adp6BVVfdozCbRuY6mqTwcsMtNJQUUdQzy+7S+dmMLR6ga25JYD7ecQdHTOpFQOldTwwZ4iUuIiiQizkcPGGIc/d4NveDn2rVMchzmOusZmjnZo9lk8IRkRWOMOI83rsJicp9aF5TYfLrMRQ8aYdnxtXn+NiLwBjBaRFR6P1UBJ94Vo4NjQ0SEeTTqDYiKYOTyR1budDuO88lpEIDU+qtPrhyYMYFpaAgBptsqoMcaDrz6Cj3F2JhsM/NbjeCVgexZ3s2NDR9vf5JdOSOH/Vu2hpKqevLJakmO7bvY5Z1IqW3PLrUZgjGnH157Fh1R1jaouVNUPPB6fuwvKmW7Uuthcx0RwVkYy6g4jzfMydNTTuZNTARhlm88bYzx0WSMQkY9U9QwRqaT9hjICqKrGBzw608ZzMpmnqcMSGBwbwZrdReSV1TJpaNd/LZOHxfPCTQuYMTwxkKEaY/qYLhOBO6MYVY3rvnBMq5zSGmIjw0iMjgDaTybzFBIiLJ6QzPu7CqltaG7rFO7K/DFJAYvZGNM3+eosHuTr0Z1BBqMbn9jAnS8d64rxnEzW0VkTUyiraaS+qcVn05Axxnjjq7N4I06TUFd7D4/xctycAs0tysGSag6WVFNR10h8VDj55V23/585fjAhAi1qE8WMMSfOV2fxaFUd4/7s+LAkEECt2002Nmvb7mL55e0nk3lKjI5g9oiBgPc5BMYY44uvpqEM9+dsb4/uCzH45Ja1TeLm7W35XieTdXTO5FTCQqRtQ3pjjPGXr6ahfwa+R/s5BK0UZ9E5EwCtM4TnjR7EB3uK2F9UDbSfTNbRt08fzZIJyQyMieiWGI0x/YevUUPfc3+e1X3hGDiWCL5zxmj+6emNPL8hG8BnjSAiLMTn0FFjjOmKr3kEPvcaUNVXTn04BpzF4+Iiw1iWkcKgmAhe3pgD0GUfgTHGfBm+moYu9nFOAUsEJ6GpuYWHVu/jGwtHdtmM07qncFhoCOdPSeW5zw4DNiLIGBMYvpqGbuzOQILFtrwK7n9vD5HhIdy8ZKzXMs6+w85Nf/nUoTz32WGvk8mMMeZU8GeHsiQReUBEPheRjSLyOxGx6aknqcBdM2jtnqIuy7RuQA+wcEwS8VFhXU4mM8aYL8ufHcqeB9YCV7rPrwNeAM4JVFD9Wety0pkHS6lpaCI6ov1fQU1DE6U1jW01goiwEP51eQYh4m1enzHGfHn+JIJBqnqvx/NfichlAYqn38uvcBJBQ3ML6/cf5ayM9msDte4y5jkx7PoFI7svQGNM0PFnh7LVInK1iIS4j68BbwU6sP7qSHkdKXGRRIWH8IGX5qGu9h02xphA8TV8tHX5aQF+DPzVPRUCVAH/EfDo+qGCijpGDIomJjKMtXstERhjep6vtYbiVDXe/RmiqmHuI8T2Ijh5RyrqSU2I4szxg9lfVE1OaU2783lltYQIpMZF9lCExphg40/TECIyUETmicji1kegA+uPVJX88lqGxkexZEIyAB/uLW5XJrfMWW46LNSvvxpjjPnS/Bk++l2cUUN/B37h/vx5YMPqnypqm6hrbGFIQhTjUmIZmhDVaRip5xwCY4zpDv587fwhcBpwyF13aBbQ9SB406UCd8RQanwUIsLi8cl8lFVMU3NLW5m8cksExpju5U8iqFPVOgARiVTVXcBEf95cRJaLyG4RyRKRu72cv05EtriPj0VkxomF37e0JoLWNYMWT0imsq6JL3LKAGhpUfKPswG9Mcacav4kghwRSQReA94VkdeBvOO9SERCgYeAC4DJwDUiMrlDsQPAElWdDtwLPOZ/6L1PS4tS09DU5fnWWcWts4TPGDeYsBBh5dYCAIqr62lobiEt0WYRG2O6z3ETgaperqplqvpz4KfAn4DL/HjveUCWqu5X1QacGcqXdnjvj1W11H36KZB+ArH3Os98ls2i+96nvLbR6/mC8nrAaRoCSIgO5/ypQ3h5Yw61Dc1tk8msRmCM6U7+jhqaLSK3A9OBHPfGfjxpwGGP5znusa58B3i7i8+/SUQyRSSzqKj3dk/syKugrKaRN7d4rzAVVNSRFBNBRNixP/YbFoykvLaRN7bk2RwCY0yP8GfU0M+AJ4EkYDDwhIjc48d7d7XpvbfPOAsnEdzl7byqPqaqc1V1bnJysh8f3TNab+QvZuZ4PX+koq6tNtBq/uhBjE+J5a+fHrJEYIzpEf6sNXQNMMujw/g+4HPgV8d5XQ4w3ON5Ol76FkRkOvBH4AJVLfEn6N4qr6yW0BDhi8Nl7C6oZOKQuHbn88vrOu0yJiLcsHAkP3t9O+GhIcRGhhEf5c9fizHGnBr+NA0dBDzvXpHAPj9etwEYLyKjRSQCuBpY4VlAREbgbHBzg6ru8SviXkpVySur5eLpQwkLEV7KPNypzJGKOlK97DJ2+aw0oiNC2XiolGGJztBSY4zpLl0mAhF5UEQeAOqB7SLyFxF5AtiGs9aQT6raBNyGMwFtJ/Ciqm4XkZtF5Ga32M9wmpweFpHNIpL5Ja+nx1TUNlHd0MzUtATOmZTKq5tyaWg6Nj+gvqmZo9UNXvcViIsK5/JZTveJNQsZY7qbrzaI1pvyRuBVj+Nr/H1zVV0JrOxw7FGP378LfNff9+vNct32/bTEAYw5LYZ3thfw/q5Clk8dAkBhhTNiqKsNZq5fMJJn1mfbdpTGmG7na6vKJ1t/d5t2JrhPd6uq9/GRQcyzo3fKsHhS4iJ5KfNwWyLIL28/mayjSUPj+dlFk5k3elD3BGyMMS5/Rg0tBfbiTA57GNhji851lld+LBGEhYZw5Zx01uwpotCdTdxxVrE33z5jNFPTEgIfrDHGePCns/i3wHmqukRVFwPnA/cHNqy+J7esloiwEJJiIgC4cnY6zS3Kii+cgVJHyo+tM2SMMb2JP4kgXFV3tz5xR/eEBy6kvim3tJZhCVGEhDgjfsalxDI9PYFXN+UCTo1gQHioDQ01xvQ6/iSCjSLyJxFZ6j4ex+lANh68LR992cw0tudVsOdIJQXldQxJsKGhxpjex59EcDOwHbgdZ0nqHe4x4yHPy6qhl8wcRmiI8OqmXAoq6rocMWSMMT3JZzuFiIQAG1V1KvC/3RNS39PY3MKRys6JYHBsJIvHD+Z1t3lo/pikngjPGGN88lkjUNUW4At3BnBQe/rTQ7ywIdvruYLyOlTxunz0ZbPSyCuvI6+88zpDxhjTG/jTczkUZ2bxZ0B160FVvSRgUfVCD6/OIr+8jsiwUC6b1X4RVV+LxZ03eQixkWFU1TcxJN42pDfG9D7+JIJfBDyKXq6qvslNAiH8y8tfkBwXyenjBredb51DkOYlEQyICGW5u+fAEJs1bIzphXytNRQlIncAXwUygHWq+kHro7sC7A32FzlLK9172VTGJsdy89Mb2Zlf0Xb+eBvKXDt/BNERoUwaGuf1vDHG9CRffQRPAnOBrTjbTf62WyLqhfa5iWD2iESeuPE0YqPCuOP5zag62yvkltWSFBNBVHio19fPHjGQ7b84n5FJMd0WszHG+MtXIpisqter6h+Aq4AzuymmXiersIrQEGHEoBiGJgzgB2ePZ/eRSrblOrUCb3MIOrL5A8aY3spXImhbWM5dUjpo7SusZmRSdNsWk1+ZNpSI0BBe2eTsROYkAhsRZIzpm3wlghkiUuE+KoHprb+LSIWP1/U7WUVVjE2ObXueEB3OOZNTWLE5j8bmFmd5CdtHwBjTR3WZCFQ1VFXj3UecqoZ5/B7fnUH2pMbmFg6VVLdLBACXz0qnpLqBN7fkUd3Q7HXEkDHG9AX+LDER1A4fraGxWRmX0j4RLJmQzKCYCB5a7ezaaTUCY0xfZYngOLIKnRFDY5Pbj/iJCAvh4ulD285bIjDG9FWWCI5jX5EzmXpshxoBwBWz09t+t85iY0xfZYngOLIKq0iJiyQ+qvMWDNPTExiTHENEaAiDY2z5CGNM32S7pBzHvqKqTv0DrUSEfzlvIpsPl7VtSGOMMX2NJQIfVJV9hVWdFpnzdMG0oVwwbWg3RmWMMaeWNQ35UFRZT2V9U6eOYmOM6U8sEfiQ5a4xNC7FFoszxvRflgh82Nc6dDTFagTGmP7LEoEP+4qqiYkItb2GjTH9miUCH7IKqxibEmsrhxpj+jVLBEB2SQ3NLdruWFV9E9vzyhmX7H3oqDHG9BdBnwg+3V/C4t+s5ievbGnbaAbg12/tpKy2kesWjOjB6IwxJvCCOhE0tyi/fGMH4aHCi5k5PPnxQQDW7C7kuc+yuWnxGOaMHNSzQRpjTIAF9YSylzceZkd+Bb+7eiZvbsnn3rd2khofxc/f2M6E1Fh+dM6Eng7RGGMCLqA1AhFZLiK7RSRLRO72cl5E5AH3/BYRmR3IeDxV1jXym7/vZu7IgVwyYxj3f30mY5NjuOWZzympauC3X53Z5R7ExhjTnwQsEYhIKPAQzsb3k4FrRGRyh2IXAOPdx03AI4GKp6OHVu+juKqBn108GREhNjKMx78xl7TEAfzL+ROZlp7QXaEYY0yPCmTT0DwgS1X3A4jI88ClwA6PMpcCT6nTS/upiCSKyFBVzT/VwXywp4hfvXnsow8UV3Pl7HSmpye2HRuZFMNHd51lw0WNMUElkIkgDTjs8TwHmO9HmTSgXSIQkZtwagyMGHFyo3hiI8MYn3psKOickQO58/yJncpZEjDGBJtAJgJvd1Q9iTKo6mPAYwBz587tdN4fc0YOZM7IOSfzUmOM6dcC2VmcAwz3eJ4O5J1EGWOMMQEUyESwARgvIqNFJAK4GljRocwK4Bvu6KEFQHkg+geMMcZ0LWBNQ6raJCK3AX8HQoE/q+p2EbnZPf8osBK4EMgCaoAbAxWPMcYY78RzWYW+YO7cuZqZmdnTYRhjTF/T5UiYoF5iwhhjjCUCY4wJepYIjDEmyFkiMMaYINfnOotFpAg4dAIvGQwUByic3ixYrxuC99rtuoPLiV53saou93aizyWCEyUimao6t6fj6G7Bet0QvNdu1x1cTuV1W9OQMcYEOUsExhgT5IIhETzW0wH0kGC9bgjea7frDi6n7Lr7fR+BMcYY34KhRmCMMcYHSwTGGBPk+nUiEJHlIrJbRLJE5O6ejidQRGS4iKwWkZ0isl1EfugeHyQi74rIXvfnwJ6ONRBEJFRENonIm+7zfn/d7rauL4vILvfvfWGQXPeP3H/j20TkORGJ6o/XLSJ/FpFCEdnmcazL6xSRn7j3ud0icv6Jfl6/TQQiEgo8BFwATAauEZHJPRtVwDQB/6yqk4AFwK3utd4NrFLV8cAq93l/9ENgp8fzYLju3wHvqGoGMAPn+vv1dYtIGnA7MFdVp+Isb381/fO6/wJ0nPzl9Trd/+tXA1Pc1zzs3v/81m8TATAPyFLV/araADwPXNrDMQWEquar6ufu75U4N4U0nOt90i32JHBZjwQYQCKSDnwF+KPH4X593SISDywG/gSgqg2qWkY/v25XGDBARMKAaJwdDfvddavqWuBoh8NdXeelwPOqWq+qB3D2d5l3Ip/XnxNBGnDY43mOe6xfE5FRwCxgPZDauuOb+zOlB0MLlP8D/hVo8TjW3697DFAEPOE2if1RRGLo59etqrnA/wDZQD7Ojob/oJ9ft4eurvNL3+v6cyLwtglDvx4rKyKxwN+AO1S1oqfjCTQRuQgoVNWNPR1LNwsDZgOPqOosoJr+0Rzik9smfikwGhgGxIjI9T0bVa/wpe91/TkR5ADDPZ6n41Qj+yURCcdJAs+o6ivu4SMiMtQ9PxQo7Kn4AuR04BIROYjT9He2iPyV/n/dOUCOqq53n7+Mkxj6+3WfAxxQ1SJVbQReARbR/6+7VVfX+aXvdf05EWwAxovIaBGJwOlMWdHDMQWEiAhOe/FOVf1fj1MrgG+6v38TeL27YwskVf2Jqqar6iicv9/3VfV6+v91FwCHRWSie2gZsIN+ft04TUILRCTa/Te/DKc/rL9fd6uurnMFcLWIRIrIaGA88NkJvbOq9tsHcCGwB9gH/HtPxxPA6zwDpyq4BdjsPi4EknBGF+x1fw7q6VgD+GewFHjT/b3fXzcwE8h0/85fAwYGyXX/AtgFbAOeBiL743UDz+H0gzTifOP/jq/rBP7dvc/tBi440c+zJSaMMSbI9eemIWOMMX6wRGCMMUHOEoExxgQ5SwTGGBPkLBEYY0yQs0Rg+iwR+Xd3JcotIrJZROYH+PPWiIjfm4WLyAIRWe/GtlNEfu4ev+RUr4YrIioiv/V4fmfr5xlzPGE9HYAxJ0NEFgIXAbNVtV5EBgMRPRxWR08CX1PVL9zVICcCqOoKTv3kxnrgChH5L1UtPsXvbfo5qxGYvmooUKyq9QCqWqyqeQAi8jMR2eCuWf+YOwu19Rv9/SKy1v2GfpqIvOKu7/4rt8wod43/J92axssiEt3xw0XkPBH5REQ+F5GX3HWeOkrBmRSEqjar6g73td8Skd+7v2/2eNSKyBIRiXHXo9/gLirnz6q5TTh72P7oRP8gjbFEYPqqfwDDRWSPiDwsIks8zv1eVU9TZ836ATg1h1YNqroYeBRniv6twFTgWyKS5Ja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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "from scipy.stats import binom\n", "size = list(range(1,100))\n", "theta = 0.7\n", "\n", "# qbinom(p, size, prob, lower.tail = TRUE, log.p = FALSE)\n", "power = []\n", "for n in size:\n", " critlo = binom.ppf(0.25,n,.5)-1\n", " crithi = binom.ppf(0.975,n,.5)\n", " power.append(binom.cdf(critlo,n,theta) + 1-binom.cdf(crithi,n,theta))\n", "\n", "ax = sns.lineplot(x = size, y = power)\n", "ax.set(xlabel = 'Sample Size, N', ylabel = 'Probablility of rejecting the Null')\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "metallic-characterization", "metadata": {}, "source": [ "```\n", "{glue:figure} powerfunctionsample-fig\n", ":figwidth: 600px\n", ":name: fig-powerfunctionsample\n", "\n", "The power of our test, plotted as a function of the sample size $N$. In this case, the true value of $\\\\theta$ is 0.7, but the null hypothesis is that $\\\\theta = 0.5$. Overall, larger $N$ means greater power. (The small zig-zags in this function occur because of some odd interactions between $\\\\theta$, $\\\\alpha$ and the fact that the binomial distribution is discrete; it doesn't matter for any serious purpose)\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "filled-november", "metadata": {}, "source": [ "Because power is important, whenever you're contemplating running an experiment it would be pretty useful to know how much power you're likely to have. It's never possible to know for sure, since you can't possibly know what your effect size is. However, it's often (well, sometimes) possible to guess how big it should be. If so, you can guess what sample size you need! This idea is called **_power analysis_**, and if it's feasible to do it, then it's very helpful, since it can tell you something about whether you have enough time or money to be able to run the experiment successfully. It's increasingly common to see people arguing that power analysis should be a required part of experimental design, so it's worth knowing about. I don't discuss power analysis in this book, however. This is partly for a boring reason and partly for a substantive one. The boring reason is that I haven't had time to write about power analysis yet. The substantive one is that I'm still a little suspicious of power analysis. Speaking as a researcher, I have very rarely found myself in a position to be able to do one -- it's either the case that (a) my experiment is a bit non-standard and I don't know how to define effect size properly, or (b) I literally have so little idea about what the effect size will be that I wouldn't know how to interpret the answers. Not only that, after extensive conversations with someone who does stats consulting for a living (my wife, as it happens), I can't help but notice that in practice the *only* time anyone ever asks her for a power analysis is when she's helping someone write a grant application. In other words, the only time any scientist ever seems to want a power analysis in real life is when they're being forced to do it by bureaucratic process. It's not part of anyone's day to day work. In short, I've always been of the view that while power is an important concept, power *analysis* is not as useful as people make it sound, except in the rare cases where (a) someone has figured out how to calculate power for your actual experimental design and (b) you have a pretty good idea what the effect size is likely to be. Maybe other people have had better experiences than me, but I've personally never been in a situation where both (a) and (b) were true. Maybe I'll be convinced otherwise in the future, and probably a future version of this book would include a more detailed discussion of power analysis, but for now this is about as much as I'm comfortable saying about the topic." ] }, { "attachments": {}, "cell_type": "markdown", "id": "motivated-mason", "metadata": {}, "source": [ "(nhstmess)=\n", "## Some issues to consider\n", "\n", "What I've described to you in this chapter is the orthodox framework for null hypothesis significance testing (NHST). Understanding how NHST works is an absolute necessity, since it has been the dominant approach to inferential statistics ever since it came to prominence in the early 20th century. It's what the vast majority of working scientists rely on for their data analysis, so even if you hate it you need to know it. However, the approach is not without problems. There are a number of quirks in the framework, historical oddities in how it came to be, theoretical disputes over whether or not the framework is right, and a lot of practical traps for the unwary. I'm not going to go into a lot of detail on this topic, but I think it's worth briefly discussing a few of these issues.\n", "\n", "### Neyman versus Fisher\n", "\n", "The first thing you should be aware of is that orthodox NHST is actually a mash-up of two rather different approaches to hypothesis testing, one proposed by Sir Ronald Fisher and the other proposed by Jerzy Neyman (for a historical summary see {cite}`Lehmann2011`. The history is messy because Fisher and Neyman were real people whose opinions changed over time, and at no point did either of them offer \"the definitive statement\" of how we should interpret their work many decades later. That said, here's a quick summary of what I take these two approaches to be. \n", "\n", "First, let's talk about Fisher's approach. As far as I can tell, Fisher assumed that you only had the one hypothesis (the null), and what you want to do is find out if the null hypothesis is inconsistent with the data. From his perspective, what you should do is check to see if the data are \"sufficiently unlikely\" according to the null. In fact, if you remember back to our earlier discussion, that's how Fisher defines the $p$-value. According to Fisher, if the null hypothesis provided a very poor account of the data, you could safely reject it. But, since you don't have any other hypotheses to compare it to, there's no way of \"accepting the alternative\" because you don't necessarily have an explicitly stated alternative. That's more or less all that there was to it. \n", "\n", "In contrast, Neyman thought that the point of hypothesis testing was as a guide to action, and his approach was somewhat more formal than Fisher's. His view was that there are multiple things that you could *do* (accept the null or accept the alternative) and the point of the test was to tell you which one the data support. From this perspective, it is critical to specify your alternative hypothesis properly. If you don't know what the alternative hypothesis is, then you don't know how powerful the test is, or even which action makes sense. His framework genuinely requires a competition between different hypotheses. For Neyman, the $p$ value didn't directly measure the probability of the data (or data more extreme) under the null, it was more of an abstract description about which \"possible tests\" were telling you to accept the null, and which \"possible tests\" were telling you to accept the alternative.\n", "\n", "As you can see, what we have today is an odd mishmash of the two. We talk about having both a null hypothesis and an alternative (Neyman), but usually [^note12] define the $p$ value in terms of exreme data (Fisher), but we still have $\\alpha$ values (Neyman). Some of the statistical tests have explicitly specified alternatives (Neyman) but others are quite vague about it (Fisher). And, according to some people at least, we're not allowed to talk about accepting the alternative (Fisher). It's a mess: but I hope this at least explains why it's a mess.\n", "\n", "[^note12]: Although this book describes both Neyman's and Fisher's definition of the $p$ value, most don't. Most introductory textbooks will only give you the Fisher version." ] }, { "attachments": {}, "cell_type": "markdown", "id": "marine-virgin", "metadata": {}, "source": [ "### Bayesians versus frequentists\n", "\n", "Earlier on in this chapter I was quite emphatic about the fact that you *cannot* interpret the $p$ value as the probability that the null hypothesis is true. NHST is fundamentally a frequentist tool (see the chapter on [probability](probability)) and as such it does not allow you to assign probabilities to hypotheses: the null hypothesis is either true or it is not. The Bayesian approach to statistics interprets probability as a degree of belief, so it's totally okay to say that there is a 10\\% chance that the null hypothesis is true: that's just a reflection of the degree of confidence that you have in this hypothesis. You aren't allowed to do this within the frequentist approach. Remember, if you're a frequentist, a probability can only be defined in terms of what happens after a large number of independent replications (i.e., a long run frequency). If this is your interpretation of probability, talking about the \"probability\" that the null hypothesis is true is complete gibberish: a null hypothesis is either true or it is false. There's no way you can talk about a long run frequency for this statement. To talk about \"the probability of the null hypothesis\" is as meaningless as \"the colour of freedom\". It doesn't have one!\n", "\n", "Most importantly, this *isn't* a purely ideological matter. If you decide that you are a Bayesian and that you're okay with making probability statements about hypotheses, you have to follow the Bayesian rules for calculating those probabilities. I'll talk more about this in the chapter on [Bayesian statistics](bayes), but for now what I want to point out to you is the $p$ value is a *terrible* approximation to the probability that $H_0$ is true. If what you want to know is the probability of the null, then the $p$ value is not what you're looking for!" ] }, { "attachments": {}, "cell_type": "markdown", "id": "japanese-preparation", "metadata": {}, "source": [ "### Traps\n", "\n", "\n", "As you can see, the theory behind hypothesis testing is a mess, and even now there are arguments in statistics about how it \"should\" work. However, disagreements among statisticians are not our real concern here. Our real concern is practical data analysis. And while the \"orthodox\" approach to null hypothesis significance testing has many drawbacks, even an unrepentant Bayesian like myself would agree that they can be useful if used responsibly. Most of the time they give sensible answers, and you can use them to learn interesting things. Setting aside the various ideologies and historical confusions that we've discussed, the fact remains that the biggest danger in all of statistics is *thoughtlessness*. I don't mean stupidity, here: I literally mean thoughtlessness. The rush to interpret a result without spending time thinking through what each test actually says about the data, and checking whether that's consistent with how you've interpreted it. That's where the biggest trap lies. \n", "\n", "To give an example of this, consider the following example see {cite}`Gelman2006`. Suppose I'm running my ESP study, and I've decided to analyse the data separately for the male participants and the female participants. Of the male participants, 33 out of 50 guessed the colour of the card correctly. This is a significant effect ($p = .03$). Of the female participants, 29 out of 50 guessed correctly. This is not a significant effect ($p = .32$). Upon observing this, it is extremely tempting for people to start wondering why there is a difference between males and females in terms of their psychic abilities. However, this is wrong. If you think about it, we haven't *actually* run a test that explicitly compares males to females. All we have done is compare males to chance (binomial test was significant) and compared females to chance (binomial test was non significant). If we want to argue that there is a real difference between the males and the females, we should probably run a test of the null hypothesis that there is no difference! We can do that using a different hypothesis test,[^note13] but when we do that it turns out that we have no evidence that males and females are significantly different ($p = .54$). *Now* do you think that there's anything fundamentally different between the two groups? Of course not. What's happened here is that the data from both groups (male and female) are pretty borderline: by pure chance, one of them happened to end up on the magic side of the $p = .05$ line, and the other one didn't. That doesn't actually imply that males and females are different. This mistake is so common that you should always be wary of it: **the difference between significant and not-significant is *not* evidence of a real difference** -- if you want to say that there's a difference between two groups, then you have to test for that difference! \n", "\n", "The example above is just that: an example. I've singled it out because it's such a common one, but the bigger picture is that data analysis can be tricky to get right. Think about *what* it is you want to test, *why* you want to test it, and whether or not the answers that your test gives could possibly make any sense in the real world. \n", "\n", "[^note13]: In this case, the Pearson [chi-square test of independence](chisquare) is what we use." ] }, { "attachments": {}, "cell_type": "markdown", "id": "structured-percentage", "metadata": {}, "source": [ "## Summary\n", "\n", "Null hypothesis testing is one of the most ubiquitous elements to statistical theory. The vast majority of scientific papers report the results of some hypothesis test or another. As a consequence it is almost impossible to get by in science without having at least a cursory understanding of what a $p$-value means, making this one of the most important chapters in the book. As usual, I'll end the chapter with a quick recap of the key ideas that we've talked about:\n", "\n", "\n", "- Research hypotheses and statistical [hypotheses](hypotheses). Null and alternative [hypotheses](hypotheses).\n", "- Type 1 and Type 2 [errors](errortypes)\n", "- [Test statistics](teststatistics) and sampling distributions\n", "- Hypothesis testing as a [decision making](decisionmaking) process\n", "- [$p$-values](pvalue) as \"soft\" decisions\n", "- [Writing up](writeup) the results of a hypothesis test\n", "- [Effect size and power](effectsize)\n", "- A few issues to consider regarding [hypothesis testing](nhstmess)\n", "\n", "\n", "\n", "Later in the book, in the section on [Bayesian statistics](bayes), I'll revisit the theory of null hypothesis tests from a Bayesian perspective, and introduce a number of new tools that you can use if you aren't particularly fond of the orthodox approach. But for now, though, we're done with the abstract statistical theory, and we can start discussing specific data analysis tools." ] }, { "cell_type": "code", "execution_count": null, "id": "nutritional-approval", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "celltoolbar": "Tags", "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.12" } }, "nbformat": 4, "nbformat_minor": 5 } { "cells": [ { "attachments": {}, "cell_type": "markdown", "id": "backed-military", "metadata": {}, "source": [ "(chisquare)=\n", "# Categorical data analysis" ] }, { "attachments": {}, "cell_type": "markdown", "id": "medieval-japanese", "metadata": {}, "source": [ "Now that we've got the basic theory behind hypothesis testing, it's time to start looking at specific tests that are commonly used in psychology. So where should we start? Not every textbook agrees on where to start, but I'm going to start with \"$\\chi^2$ tests\" (this chapter) and \"$t$-tests\" [next chapter](ttest). Both of these tools are very frequently used in scientific practice, and while they're not as powerful as [\"analysis of variance\"](ANOVA) and [\"regression\"](regression) they're much easier to understand.\n", "\n", "The term \"categorical data\" is just another name for \"nominal scale data\". It's nothing that we haven't already discussed, it's just that in the context of data analysis people tend to use the term \"categorical data\" rather than \"nominal scale data\". I don't know why. In any case, **_categorical data analysis_** refers to a collection of tools that you can use when your data are nominal scale. However, there are a lot of different tools that can be used for categorical data analysis, and this chapter only covers a few of the more common ones." ] }, { "attachments": {}, "cell_type": "markdown", "id": "latest-disorder", "metadata": {}, "source": [ "(goftest)=\n", "\n", "## The $\\chi^2$ goodness-of-fit test\n", "\n", "The $\\chi^2$ goodness-of-fit test is one of the oldest hypothesis tests around: it was invented by Karl Pearson around the turn of the century {cite:p}`Pearson1900`, with some corrections made later by Sir Ronald Fisher {cite}`Fisher1922`. To introduce the statistical problem that it addresses, let's start with some psychology... \n", "\n", "### The cards data\n", "\n", "Over the years, there have been a lot of studies showing that humans have a lot of difficulties in simulating randomness. Try as we might to \"act\" random, we *think* in terms of patterns and structure, and so when asked to \"do something at random\", what people actually do is anything but random. As a consequence, the study of human randomness (or non-randomness, as the case may be) opens up a lot of deep psychological questions about how we think about the world. With this in mind, let's consider a very simple study. Suppose I asked people to imagine a shuffled deck of cards, and mentally pick one card from this imaginary deck \"at random\". After they've chosen one card, I ask them to mentally select a second one. For both choices, what we're going to look at is the suit (hearts, clubs, spades or diamonds) that people chose. After asking, say, $N=200$ people to do this, I'd like to look at the data and figure out whether or not the cards that people pretended to select were really random. The data are contained in the file called `cards.csv`. Let's take a look:" ] }, { "cell_type": "code", "execution_count": 2, "id": "urban-calvin", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
idchoice_1choice_2
0subj1spadesclubs
1subj2diamondsclubs
2subj3heartsclubs
3subj4spadesclubs
4subj5heartsspades
............
195subj196spadeshearts
196subj197heartsspades
197subj198clubsclubs
198subj199spadeshearts
199subj200heartshearts
\n", "

200 rows × 3 columns

\n", "
" ], "text/plain": [ " id choice_1 choice_2\n", "0 subj1 spades clubs\n", "1 subj2 diamonds clubs\n", "2 subj3 hearts clubs\n", "3 subj4 spades clubs\n", "4 subj5 hearts spades\n", ".. ... ... ...\n", "195 subj196 spades hearts\n", "196 subj197 hearts spades\n", "197 subj198 clubs clubs\n", "198 subj199 spades hearts\n", "199 subj200 hearts hearts\n", "\n", "[200 rows x 3 columns]" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "df = pd.read_csv('https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/cards.csv')\n", "df\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "prepared-mixer", "metadata": {}, "source": [ "As you can see, the `cards` data frame contains three variables, an `id` variable that assigns a unique identifier to each participant, and the two variables `choice_1` and `choice_2` that indicate the card suits that people chose. Here's the first few entries in the data frame:" ] }, { "cell_type": "code", "execution_count": 3, "id": "abstract-pepper", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
idchoice_1choice_2
0subj1spadesclubs
1subj2diamondsclubs
2subj3heartsclubs
3subj4spadesclubs
4subj5heartsspades
\n", "
" ], "text/plain": [ " id choice_1 choice_2\n", "0 subj1 spades clubs\n", "1 subj2 diamonds clubs\n", "2 subj3 hearts clubs\n", "3 subj4 spades clubs\n", "4 subj5 hearts spades" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.head()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "threatened-grade", "metadata": {}, "source": [ "For the moment, let's just focus on the first choice that people made. We'll use the `value_counts()` function to count the number of times that we observed people choosing each suit. I'll save the table to a variable called `observed`, for reasons that will become clear very soon:" ] }, { "cell_type": "code", "execution_count": 4, "id": "modular-running", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "hearts 64\n", "diamonds 51\n", "spades 50\n", "clubs 35\n", "Name: choice_1, dtype: int64" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "observed = df['choice_1'].value_counts()\n", "observed" ] }, { "attachments": {}, "cell_type": "markdown", "id": "reverse-kingdom", "metadata": {}, "source": [ "That little frequency table is quite helpful. Looking at it, there's a bit of a hint that people *might* be more likely to select hearts than clubs, but it's not completely obvious just from looking at it whether that's really true, or if this is just due to chance. So we'll probably have to do some kind of statistical analysis to find out, which is what I'm going to talk about in the next section. \n", " \n", "\n", "Excellent. From this point on, we'll treat this table as the data that we're looking to analyse. However, since I'm going to have to talk about this data in mathematical terms (sorry!) it might be a good idea to be clear about what the notation is. In Python, if I wanted to pull out the number of people that selected diamonds, I can type `observed[1]` (diamonds is the second element in the list, but remember Python is zero-indexed, so clubs is 0, diamonds is 1, etc.). The mathematical notation for this is pretty similar, except that we shorten the human-readable word \"observed\" to the letter $O$, and we use subscripts rather than brackets. Also, since we are trying to make this more or less human-readable, let's start our subscripts at 1 rather than zero. So the second observation in our table is written as `observed[1]` in Python, and is written as $O_2$ in maths. The relationship between the English descriptions, the Python commands, and the mathematical symbols are illustrated below: " ] }, { "attachments": {}, "cell_type": "markdown", "id": "iraqi-bosnia", "metadata": {}, "source": [ "|label | index $i$ | math. symbol | Python command | the value |\n", "|:-----------------------|:---------:|:------------:|:-------------:|:---------:|\n", "|hearts $\\heartsuit$ | 0 | $O_1$ | `observed[0]` | 64 |\n", "|diamonds $\\diamondsuit$ | 1 | $O_2$ | `observed[1]` | 51 |\n", "|spades $\\spadesuit$ | 2 | $O_3$ | `observed[2]` | 50 |\n", "|clubs $\\clubsuit$ | 3 | $O_4$ | `observed[3]` | 35 |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "dental-county", "metadata": {}, "source": [ "Hopefully that's pretty clear. It's also worth nothing that mathematicians prefer to talk about things in general rather than specific things, so you'll also see the notation $O_i$, which refers to the number of observations that fall within the $i$-th category (where $i$ could be 1, 2, 3 or 4). Finally, if we want to refer to the set of all observed frequencies, statisticians group all of observed values into a vector, which I'll refer to as $O$. \n", "\n", "$$\n", "O = (O_1, O_2, O_3, O_4)\n", "$$\n", "\n", "Again, there's nothing new or interesting here: it's just notation. If I say that $O~=~(64, 51, 50, 35)$ all I'm doing is describing the table of observed frequencies (i.e., `observed`), but I'm referring to it using mathematical notation, rather than by referring to an Python variable. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "higher-conversion", "metadata": {}, "source": [ "### The null hypothesis and the alternative hypothesis\n", "\n", "As the last section indicated, our research hypothesis is that \"people don't choose cards randomly\". What we're going to want to do now is translate this into some statistical hypotheses, and construct a statistical test of those hypotheses. The test that I'm going to describe to you is **_Pearson's $\\chi^2$ goodness of fit test_**, and as is so often the case, we have to begin by carefully constructing our null hypothesis. In this case, it's pretty easy. First, let's state the null hypothesis in words:\n", "\n", "\n", "$H_0$: All four suits are chosen with equal probability\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "killing-arnold", "metadata": {}, "source": [ "Now, because this is statistics, we have to be able to say the same thing in a mathematical way. To do this, let's use the notation $P_j$ to refer to the true probability that the $j$-th suit is chosen. If the null hypothesis is true, then each of the four suits has a 25\\% chance of being selected: in other words, our null hypothesis claims that $P_1 = .25$, $P_2 = .25$, $P_3 = .25$ and finally that $P_4 = .25$. However, in the same way that we can group our observed frequencies into a vector $O$ that summarises the entire data set, we can use $P$ to refer to the probabilities that correspond to our null hypothesis. So if I let the vector $P = (P_1, P_2, P_3, P_4)$ refer to the collection of probabilities that describe our null hypothesis, then we have\n", "\n", "\n", "$H_0: {P} = (.25, .25, .25, .25)$\n", "\n", "In this particular instance, our null hypothesis corresponds to a vector of probabilities $P$ in which all of the probabilities are equal to one another. But this doesn't have to be the case. For instance, if the experimental task was for people to imagine they were drawing from a deck that had twice as many clubs as any other suit, then the null hypothesis would correspond to something like $P = (.4, .2, .2, .2)$. As long as the probabilities are all positive numbers, and they all sum to 1, them it's a perfectly legitimate choice for the null hypothesis. However, the most common use of the goodness of fit test is to test a null hypothesis that all of the categories are equally likely, so we'll stick to that for our example. \n", "\n", "What about our alternative hypothesis, $H_1$? All we're really interested in is demonstrating that the probabilities involved aren't all identical (that is, people's choices weren't completely random). As a consequence, the \"human friendly\" versions of our hypotheses look like this:\n", "\n", "\n", "$H_0$: All four suits are chosen with equal probability\n", "\n", "$H_1$: At least one of the suit-choice probabilities *isn't* .25\n", "\n", "\n", "\n", "and the \"mathematician friendly\" version is\n", "\n", "$H_0$: $P = (.25, .25, .25, .25)$\n", "\n", "$H_1$: $P \\neq (.25,.25,.25,.25)$\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "interstate-rainbow", "metadata": {}, "source": [ "Maybe what I should do is store the $P$ list in Python as well, since we're almost certainly going to need it later. And because I'm so imaginative, I'll call this panads series `probabilities`:" ] }, { "cell_type": "code", "execution_count": 5, "id": "external-anger", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "hearts 0.25\n", "diamonds 0.25\n", "spades 0.25\n", "clubs 0.25\n", "dtype: float64" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# make dictionary of values\n", "dict = {'hearts' : .25,\n", " 'diamonds' : .25,\n", " 'spades' : .25,\n", " 'clubs': .25}\n", " \n", "# create series from dictionary\n", "probabilities = pd.Series(dict)\n", "probabilities" ] }, { "attachments": {}, "cell_type": "markdown", "id": "reliable-plasma", "metadata": {}, "source": [ "### The \"goodness of fit\" test statistic\n", "\n", "\n", "At this point, we have our observed frequencies $O$ and a collection of probabilities $P$ corresponding the null hypothesis that we want to test. We've stored these in Python as the corresponding variables `observed` and `probabilities`. What we want to do now is construct a test of the null hypothesis. As always, if we want to test $H_0$ against $H_1$, we're going to need a test statistic. The basic trick that a goodness of fit test uses is to construct a test statistic that measures how \"close\" the data are to the null hypothesis. If the data don't resemble what you'd \"expect\" to see if the null hypothesis were true, then it probably isn't true. Okay, if the null hypothesis were true, what would we expect to see? Or, to use the correct terminology, what are the **_expected frequencies_*?. There are $N=200$ observations, and (if the null is true) the probability of any one of them choosing a heart is $P_3 = .25$, so I guess we're expecting $200 \\times .25 = 50$ hearts, right? Or, more specifically, if we let $E_i$ refer to \"the number of category $i$ responses that we're expecting if the null is true\", then\n", "\n", "$$\n", "E_i = N \\times P_i\n", "$$\n", "\n", "This is pretty easy to calculate in Python:" ] }, { "cell_type": "code", "execution_count": 6, "id": "warming-quilt", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "hearts 50.0\n", "diamonds 50.0\n", "spades 50.0\n", "clubs 50.0\n", "dtype: float64" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N = 200 #sample size\n", "expected = N * probabilities\n", "expected" ] }, { "attachments": {}, "cell_type": "markdown", "id": "embedded-salmon", "metadata": {}, "source": [ "None of which is very surprising: if there are 200 observation that can fall into four categories, and we think that all four categories are equally likely, then on average we'd expect to see 50 observations in each category, right?\n", "\n", "Now, how do we translate this into a test statistic? Clearly, what we want to do is compare the *expected* number of observations in each category ($E_i$) with the *observed* number of observations in that category ($O_i$). And on the basis of this comparison, we ought to be able to come up with a good test statistic. To start with, let's calculate the difference between what the null hypothesis expected us to find and what we actually did find. That is, we calculate the \"observed minus expected\" difference score, $O_i - E_i$. This is illustrated in the following table. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "solved-coordinator", "metadata": {}, "source": [ "| | | $\\clubsuit$| $\\diamondsuit$| $\\heartsuit$| $\\spadesuit$|\n", "|:------------------|:-----------|-----------:|--------------:|------------:|------------:|\n", "|expected frequency |$E_i$ | 50| 50| 50| 50|\n", "|observed frequency |$O_i$ | 35| 51| 64| 50|\n", "|difference score |$O_i - E_i$ | -15| 1| 14| 0|" ] }, { "attachments": {}, "cell_type": "markdown", "id": "grateful-jenny", "metadata": {}, "source": [ "The same calculations can be done in Python, using our `expected` and `observed` variables:" ] }, { "cell_type": "code", "execution_count": 7, "id": "stupid-albert", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "hearts 14.0\n", "diamonds 1.0\n", "spades 0.0\n", "clubs -15.0\n", "dtype: float64" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "observed-expected" ] }, { "attachments": {}, "cell_type": "markdown", "id": "saving-rider", "metadata": {}, "source": [ "Regardless of whether we do the calculations by hand or whether we do them in Python, it's clear that people chose more hearts and fewer clubs than the null hypothesis predicted. However, a moment's thought suggests that these raw differences aren't quite what we're looking for. Intuitively, it feels like it's just as bad when the null hypothesis predicts too few observations (which is what happened with hearts) as it is when it predicts too many (which is what happened with clubs). So it's a bit weird that we have a negative number for clubs and a positive number for hearts. One easy way to fix this is to square everything, so that we now calculate the squared differences, $(E_i - O_i)^2$. As before, we could do this by hand, but it's easier to do it in Python..." ] }, { "cell_type": "code", "execution_count": 8, "id": "reflected-rough", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "hearts 196.0\n", "diamonds 1.0\n", "spades 0.0\n", "clubs 225.0\n", "dtype: float64" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(observed - expected)**2" ] }, { "attachments": {}, "cell_type": "markdown", "id": "framed-documentary", "metadata": {}, "source": [ "Now we're making progress. What we've got now is a collection of numbers that are big whenever the null hypothesis makes a bad prediction (clubs and hearts), but are small whenever it makes a good one (diamonds and spades). Next, for some technical reasons that I'll explain in a moment, let's also divide all these numbers by the expected frequency $E_i$, so we're actually calculating $\\frac{(E_i-O_i)^2}{E_i}$. Since $E_i = 50$ for all categories in our example, it's not a very interesting calculation, but let's do it anyway. The Python command becomes:" ] }, { "cell_type": "code", "execution_count": 9, "id": "apparent-remainder", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "hearts 3.92\n", "diamonds 0.02\n", "spades 0.00\n", "clubs 4.50\n", "dtype: float64" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "(observed - expected)**2 / expected\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "pediatric-history", "metadata": {}, "source": [ "In effect, what we've got here are four different \"error\" scores, each one telling us how big a \"mistake\" the null hypothesis made when we tried to use it to predict our observed frequencies. So, in order to convert this into a useful test statistic, one thing we could do is just add these numbers up. The result is called the **_goodness of fit_** statistic, conventionally referred to either as $X^2$ or GOF. We can calculate it using this command in Python:" ] }, { "cell_type": "code", "execution_count": 10, "id": "figured-exercise", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "8.44" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sum((observed - expected)**2/expected)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "unexpected-mounting", "metadata": {}, "source": [ "The formula for this statistic looks remarkably similar to the Python command. If we let $k$ refer to the total number of categories (i.e., $k=4$ for our cards data), then the $X^2$ statistic is given by:\n", "\n", "$$\n", "X^2 = \\sum_{i=1}^k \\frac{(O_i - E_i)^2}{E_i}\n", "$$ \n", "\n", "Intuitively, it's clear that if $X^2$ is small, then the observed data $O_i$ are very close to what the null hypothesis predicted $E_i$, so we're going to need a large $X^2$ statistic in order to reject the null. As we've seen from our calculations, in our cards data set we've got a value of $X^2 = 8.44$. So now the question becomes, is this a big enough value to reject the null?" ] }, { "attachments": {}, "cell_type": "markdown", "id": "understanding-suite", "metadata": {}, "source": [ "### The sampling distribution of the GOF statistic (advanced)\n", "\n", "To determine whether or not a particular value of $X^2$ is large enough to justify rejecting the null hypothesis, we're going to need to figure out what the sampling distribution for $X^2$ would be if the null hypothesis were true. So that's what I'm going to do in this section. I'll show you in a fair amount of detail how this sampling distribution is constructed, and then -- in the next section -- use it to build up a hypothesis test. If you want to cut to the chase and are willing to take it on faith that the sampling distribution is a **_chi-squared ($\\chi^2$) distribution_** with $k-1$ degrees of freedom, you can skip the rest of this section. However, if you want to understand *why* the goodness of fit test works the way it does, read on...\n", "\n", "Okay, let's suppose that the null hypothesis is actually true. If so, then the true probability that an observation falls in the $i$-th category is $P_i$ -- after all, that's pretty much the definition of our null hypothesis. Let's think about what this actually means. If you think about it, this is kind of like saying that \"nature\" makes the decision about whether or not the observation ends up in category $i$ by flipping a weighted coin (i.e., one where the probability of getting a head is $P_j$). And therefore, we can think of our observed frequency $O_i$ by imagining that nature flipped $N$ of these coins (one for each observation in the data set)... and exactly $O_i$ of them came up heads. Obviously, this is a pretty weird way to think about the experiment. But what it does (I hope) is remind you that we've actually seen this scenario before. It's exactly the same set up that gave rise to the [binomial distribution](binomial). In other words, if the null hypothesis is true, then it follows that our observed frequencies were generated by sampling from a binomial distribution:\n", "\n", "$$\n", "O_i \\sim \\mbox{Binomial}(P_i, N)\n", "$$\n", "\n", "Now, if you remember from our discussion of the [central limit theorem](clt), the binomial distribution starts to look pretty much identical to the normal distribution, especially when $N$ is large and when $P_i$ isn't *too* close to 0 or 1. In other words as long as $N \\times P_i$ is large enough -- or, to put it another way, when the expected frequency $E_i$ is large enough -- the theoretical distribution of $O_i$ is approximately normal. Better yet, if $O_i$ is normally distributed, then so is $(O_i - E_i)/\\sqrt{E_i}$ ... since $E_i$ is a fixed value, subtracting off $E_i$ and dividing by $\\sqrt{E_i}$ changes the mean and standard deviation of the normal distribution; but that's all it does. Okay, so now let's have a look at what our goodness of fit statistic actually *is*. What we're doing is taking a bunch of things that are normally-distributed, squaring them, and adding them up. Wait. We've seen that before too! As we discussed in the section on the [central limit theorem](clt), when you take a bunch of things that have a standard normal distribution (i.e., mean 0 and standard deviation 1), square them, then add them up, then the resulting quantity has a chi-square distribution. So now we know that the null hypothesis predicts that the sampling distribution of the goodness of fit statistic is a chi-square distribution. Cool. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "acting-kidney", "metadata": {}, "source": [ "There's one last detail to talk about, namely the degrees of freedom. If you remember back to the section on [other useful distributions](otherdists), I said that if the number of things you're adding up is $k$, then the degrees of freedom for the resulting chi-square distribution is $k$. Yet, what I said at the start of this section is that the actual degrees of freedom for the chi-square goodness of fit test is $k-1$. What's up with that? The answer here is that what we're supposed to be looking at is the number of genuinely *independent* things that are getting added together. And, as I'll go on to talk about in the next section, even though there's $k$ things that we're adding, only $k-1$ of them are truly independent; and so the degrees of freedom is actually only $k-1$. That's the topic of the next section.[^note1]\n", "\n", "[^note1]: I should point out that this issue does complicate the story somewhat: I'm not going to cover it in this book, but there's a sneaky trick that you can do to rewrite the equation for the goodness of fit statistic as a sum over $k-1$ independent things. When we do so we get the \"proper\" sampling distribution, which is chi-square with $k-1$ degrees of freedom. In fact, in order to get the maths to work out properly, you actually have to rewrite things that way. But it's beyond the scope of an introductory book to show the maths in that much detail: all I wanted to do is give you a sense of why the goodness of fit statistic is associated with the chi-squared distribution.\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "supposed-feedback", "metadata": {}, "source": [ "When I introduced the chi-square distribution in [other useful distributions](otherdists), I was a bit vague about what \"**_degrees of freedom_**\" actually *means*. Obviously, it matters: looking {numref}`fig-manychi` you can see that if we change the degrees of freedom, then the chi-square distribution changes shape quite substantially. But what exactly *is* it? Again, when I introduced the distribution and explained its relationship to the normal distribution, I did offer an answer... it's the number of \"normally distributed variables\" that I'm squaring and adding together. But, for most people, that's kind of abstract, and not entirely helpful. What we really need to do is try to understand degrees of freedom in terms of our data. So here goes." ] }, { "cell_type": "code", "execution_count": 1, "id": "further-concord", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "\n", "from myst_nb import glue\n", "import seaborn as sns\n", "import pandas as pd\n", "import numpy as np\n", "from scipy.stats import chi2\n", "\n", "\n", "\n", "\n", "x = np.linspace(0, 10, 100)\n", "\n", "y3 = chi2.pdf(x, df=3)\n", "y4 = chi2.pdf(x, df=4)\n", "y5 = chi2.pdf(x, df=5)\n", "labels = ['df = 3']*len(x) + ['df = 4']*len(x) + ['df = 5']*len(x)\n", "y = list(y3)+list(y4)+list(y5)\n", "x = list(x)*3\n", "\n", "df = pd.DataFrame(\n", " {'y': y,\n", " 'x': x,\n", " 'df': labels\n", " }) \n", "\n", "fig = sns.lineplot(x= \"x\", y= \"y\",\n", " style=\"df\",\n", " data=df)\n", "\n", "sns.despine()\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "nonprofit-eagle", "metadata": {}, "source": [ " ```{glue:figure} manychi_fig\n", ":figwidth: 600px\n", ":name: fig-manychi\n", "\n", "Chi-square distributions with different values for the \\\"degrees of freedom\\\".\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "numeric-maker", "metadata": {}, "source": [ "The basic idea behind degrees of freedom is quite simple: you calculate it by counting up the number of distinct \"quantities\" that are used to describe your data; and then subtracting off all of the \"constraints\" that those data must satisfy.[^note2] This is a bit vague, so let's use our cards data as a concrete example. We describe out data using four numbers, $O_1$, $O_2$, $O_3$ and $O_4$ corresponding to the observed frequencies of the four different categories (hearts, clubs, diamonds, spades). These four numbers are the *random outcomes* of our experiment. But, my experiment actually has a fixed constraint built into it: the sample size $N$.[^note3] That is, if we know how many people chose hearts, how many chose diamonds and how many chose clubs; then we'd be able to figure out exactly how many chose spades. In other words, although our data are described using four numbers, they only actually correspond to $4-1 = 3$ degrees of freedom. A slightly different way of thinking about it is to notice that there are four *probabilities* that we're interested in (again, corresponding to the four different categories), but these probabilities must sum to one, which imposes a constraint. Therefore, the degrees of freedom is $4-1 = 3$. Regardless of whether you want to think about it in terms of the observed frequencies or in terms of the probabilities, the answer is the same. In general, when running the chi-square goodness of fit test for an experiment involving $k$ groups, then the degrees of freedom will be $k-1$.\n", "\n", "[^note2]: I feel obliged to point out that this is an over-simplification. It works nicely for quite a few situations; but every now and then we'll come across degrees of freedom values that aren't whole numbers. Don't let this worry you too much -- when you come across this, just remind yourself that \"degrees of freedom\" is actually a bit of a messy concept, and that the nice simple story that I'm telling you here isn't the whole story. For an introductory class, it's usually best to stick to the simple story: but I figure it's best to warn you to expect this simple story to fall apart. If I didn't give you this warning, you might start getting confused when you see $df = 3.4$ or something; and (incorrectly) thinking that you had misunderstood something that I've taught you, rather than (correctly) realising that there's something that I haven't told you.\n", "\n", "[^note3]: [In practice, the sample size isn't always fixed... e.g., we might run the experiment over a fixed period of time, and the number of people participating depends on how many people show up. That doesn't matter for the current purposes.]" ] }, { "attachments": {}, "cell_type": "markdown", "id": "governmental-ebony", "metadata": {}, "source": [ "### Testing the null hypothesis \n", "\n", "The final step in the process of constructing our hypothesis test is to figure out what the rejection region is. That is, what values of $X^2$ would lead us to reject the null hypothesis. As we saw earlier, large values of $X^2$ imply that the null hypothesis has done a poor job of predicting the data from our experiment, whereas small values of $X^2$ imply that it's actually done pretty well. Therefore, a pretty sensible strategy would be to say there is some critical value, such that if $X^2$ is bigger than the critical value we reject the null; but if $X^2$ is smaller than this value we retain the null. In other words, to use the language we introduced in the chapter on [hypothesis testing](hypothesis-testing), the chi-squared goodness of fit test is always a **_one-sided test_**. Right, so all we have to do is figure out what this critical value is. And it's pretty straightforward. If we want our test to have significance level of $\\alpha = .05$ (that is, we are willing to tolerate a Type I error rate of 5\\%), then we have to choose our critical value so that there is only a 5\\% chance that $X^2$ could get to be that big if the null hypothesis is true. That is to say, we want the 95th percentile of the sampling distribution. This is illustrated in {numref}`fig-goftest`." ] }, { "cell_type": "code", "execution_count": 13, "id": "verbal-disposition", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/text/plain": "" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "goftest-fig" } }, "output_type": "display_data" }, { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "\n", "\n", "import seaborn as sns\n", "import pandas as pd\n", "import numpy as np\n", "from scipy.stats import chi2\n", "\n", "\n", "x = np.linspace(0, 14, 100)\n", "\n", "y = chi2.pdf(x, df=3)\n", "\n", "df = pd.DataFrame(\n", " {'y': y,\n", " 'x': x\n", " }) \n", "\n", "fig = sns.lineplot(x= \"x\", y= \"y\",\n", " data=df)\n", "critical_value = chi2.ppf(0.95, 3)\n", "fig.fill_between(x, 0, y, where = x > critical_value-0.1, color = 'blue', alpha = 0.3)\n", "\n", "fig.annotate(\"The observed GOF value is 8.44\", xy=(8.44, 0), xytext=(10, 0.12), arrowprops={\"arrowstyle\":\"->\", \"color\":\"black\"})\n", "fig.annotate(\"The critical value is 7.81\", xy=(critical_value, 0), xytext=(6, 0.15), arrowprops={\"arrowstyle\":\"->\", \"color\":\"black\"})\n", "fig.set(xlabel = 'Value of the GOF Statistic', ylabel='')\n", "\n", "sns.despine()\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "alive-march", "metadata": {}, "source": [ "```{glue:figure} goftest-fig\n", ":figwidth: 600px\n", ":name: fig-goftest\n", "\n", "Illustration of how the hypothesis testing works for the chi-square goodness of fit test.\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "prostate-preservation", "metadata": {}, "source": [ "Ah, but -- I hear you ask -- how do I calculate the 95th percentile of a chi-squared distribution with $k-1$ degrees of freedom? If only Python had some function, called... oh, I don't know, chi2 percent, or chi2 percent point ... that would let you calculate this percentile. What about the \"chi2 percent point function\", `chi2.ppf()`, hmm? Doesn't that sound like it might do the trick? Like this..." ] }, { "cell_type": "code", "execution_count": 14, "id": "inside-property", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "7.81" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from scipy.stats import chi2\n", "round(chi2.ppf(0.95, 3),2)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "3da7303a", "metadata": {}, "source": [ "So if our $X^2$ statistic is bigger than 7.81 or so, then we can reject the null hypothesis. Since we actually calculated that before (i.e., $X^2 = 8.44$), we know that we can reject the null. If we want an exact $p$-value, we can calculate it using the `chisquare()` function from `scipy.stats`:" ] }, { "cell_type": "code", "execution_count": 15, "id": "a891c30c", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Power_divergenceResult(statistic=8.44, pvalue=0.0377418520240214)" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from scipy.stats import chisquare\n", "\n", "chisquare(f_obs = observed, f_exp = expected)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "47c51b9b", "metadata": {}, "source": [ "So, in this case we would reject the null hypothesis, since $p < .05$. And that's it, basically. You now know \"Pearson's $\\chi^2$ test for the goodness of fit\". Lucky you." ] }, { "attachments": {}, "cell_type": "markdown", "id": "f6a7fdc1", "metadata": {}, "source": [ "(gofTestInPython)=\n", "### Doing the test in Python\n", "\n", "Gosh darn it. Although we did manage to do everything in Python as we were going through that little example, it does rather feel as if we're typing too many things into the magic computing box. And I *hate* typing. Unfortunately, to my knowledge, there is no ready-made function in Python to go directly from the raw data to a finished chi2 goodness-of-fit analysis, but we can get pretty close with `chisquare` from `scipy.stats`. If we don't supply any expected values, then the function assumes that we expect equal counts for all items, so all we need to do is feed the frequency table directly into the `chisquare` function, like so:" ] }, { "cell_type": "code", "execution_count": 16, "id": "add3ca96", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "x2 = 8.44\n", "p = 0.038\n" ] } ], "source": [ "import pandas as pd\n", "from scipy.stats import chisquare\n", "\n", "df = pd.read_csv('https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/cards.csv')\n", "\n", "ans = chisquare(f_obs = df['choice_1'].value_counts())\n", "\n", "print(\"x2 = \", ans[0])\n", "print(\"p = \", round(ans[1],3))" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a73cf98f", "metadata": {}, "source": [ "### Specifying a different null hypothesis\n", "\n", "At this point you might be wondering what to do if you want to run a goodness of fit test, but your null hypothesis is *not* that all categories are equally likely. For instance, let's suppose that someone had made the theoretical prediction that people should choose red cards 60\\% of the time, and black cards 40\\% of the time (I've no idea why you'd predict that), but had no other preferences. If that were the case, the null hypothesis would be to expect 30\\% of the choices to be hearts, 30\\% to be diamonds, 20\\% to be spades and 20\\% to be clubs. This seems like a silly theory to me, and it's pretty easy to test it using our data. All we need to do is specify the probabilities associated with the null hypothesis. We create a vector like this:" ] }, { "cell_type": "code", "execution_count": 17, "id": "bc33db93", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "hearts 0.3\n", "diamonds 0.3\n", "spades 0.2\n", "clubs 0.2\n", "dtype: float64" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "\n", "# make dictionary of values\n", "dict = {'hearts' : .3,\n", " 'diamonds' : .3,\n", " 'spades' : .2,\n", " 'clubs': .2}\n", " \n", "# create series from dictionary\n", "nullProbs = pd.Series(dict)\n", "nullProbs\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "ba9af8fa", "metadata": {}, "source": [ "Now we have explicitly specified our null hypothesis in terms of probabilities. There is one more step, though, because `chisquare` expects the counts in the observed and \"expected\" data to add up to the same thing; otherwise there is no way for it to compare them. So, to convert our null hypothesis probabilities to count values, we need to multiply them by the total number of items counted in our observed data (which in the case of the `cards` data was 200, remember?) Once that is taken care of, we can go ahead and run the test:" ] }, { "cell_type": "code", "execution_count": 18, "id": "24d32583", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "x2 = 4.741666666666667\n", "p = 0.192\n" ] } ], "source": [ "import pandas as pd\n", "from scipy.stats import chisquare\n", "\n", "df = pd.read_csv('https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/cards.csv')\n", "\n", "# make a frequency table for the observed data\n", "observed = df['choice_1'].value_counts()\n", "\n", "# make a frequency table for our expected data, using the probabilities we specified above\n", "expected = nullProbs * sum(observed)\n", "\n", "# run the test\n", "ans = chisquare(f_obs = observed, f_exp = expected)\n", "\n", "# make the output easier to read\n", "print(\"x2 = \", ans[0])\n", "print(\"p = \", round(ans[1],3))" ] }, { "attachments": {}, "cell_type": "markdown", "id": "2a4154d3", "metadata": {}, "source": [ "The null hypothesis and the expected frequencies are different to what they were last time. As a consequence our $X^2$ test statistic is different, and our $p$-value is different too. Annoyingly, the $p$-value is .192, so we can't reject the null hypothesis. Sadly, despite the fact that the null hypothesis corresponds to a very silly theory, these data don't provide enough evidence against it." ] }, { "attachments": {}, "cell_type": "markdown", "id": "141bbe90", "metadata": {}, "source": [ "(chisqreport})=\n", "### How to report the results of the test\n", "\n", "So now you know how the test works, and you know how to do the test using a wonderful magic computing box. The next thing you need to know is how to write up the results. After all, there's no point in designing and running an experiment and then analysing the data if you don't tell anyone about it! So let's now talk about what you need to do when reporting your analysis. Let's stick with our card-suits example. If I wanted to write this result up for a paper or something, the conventional way to report this would be to write something like this:\n", "\n", "> Of the 200 participants in the experiment, 64 selected hearts for their first choice, 51 selected diamonds, 50 selected spades, and 35 selected clubs. A chi-square goodness of fit test was conducted to test whether the choice probabilities were identical for all four suits. The results were significant ($\\chi^2(3) = 8.44, p<.05$), suggesting that people did not select suits purely at random.\n", "\n", "This is pretty straightforward, and hopefully it seems pretty unremarkable. That said, there's a few things that you should note about this description:\n", "\n", "- *The statistical test is preceded by the descriptive statistics*. That is, I told the reader something about what the data look like before going on to do the test. In general, this is good practice: always remember that your reader doesn't know your data anywhere near as well as you do. So unless you describe it to them properly, the statistical tests won't make any sense to them, and they'll get frustrated and cry.\n", "- *The description tells you what the null hypothesis being tested is*. To be honest, writers don't always do this, but it's often a good idea in those situations where some ambiguity exists; or when you can't rely on your readership being intimately familiar with the statistical tools that you're using. Quite often the reader might not know (or remember) all the details of the test that your using, so it's a kind of politeness to \"remind\" them! As far as the goodness of fit test goes, you can usually rely on a scientific audience knowing how it works (since it's covered in most intro stats classes). However, it's still a good idea to be explicit about stating the null hypothesis (briefly!) because the null hypothesis can be different depending on what you're using the test for. For instance, in the cards example my null hypothesis was that all the four suit probabilities were identical (i.e., $P_1 = P_2 = P_3 = P_4 = 0.25$), but there's nothing special about that hypothesis. I could just as easily have tested the null hypothesis that $P_1 = 0.7$ and $P_2 = P_3 = P_4 = 0.1$ using a goodness of fit test. So it's helpful to the reader if you explain to them what your null hypothesis was. Also, notice that I described the null hypothesis in words, not in maths. That's perfectly acceptable. You can describe it in maths if you like, but since most readers find words easier to read than symbols, most writers tend to describe the null using words if they can.\n", "- *A \"stat block\" is included*. When reporting the results of the test itself, I didn't just say that the result was significant, I included a \"stat block\" (i.e., the dense mathematical-looking part in the parentheses), which reports all the \"raw\" statistical data. For the chi-square goodness of fit test, the information that gets reported is the test statistic (that the goodness of fit statistic was 8.44), the information about the distribution used in the test ($\\chi^2$ with 3 degrees of freedom, which is usually shortened to $\\chi^2(3)$), and then the information about whether the result was significant (in this case $p<.05$). The particular information that needs to go into the stat block is different for every test, and so each time I introduce a new test I'll show you what the stat block should look like.[^note4] However the general principle is that you should always provide enough information so that the reader could check the test results themselves if they really wanted to. \n", "- *The results are interpreted*. In addition to indicating that the result was significant, I provided an interpretation of the result (i.e., that people didn't choose randomly). This is also a kindness to the reader, because it tells them something about what they should believe about what's going on in your data. If you don't include something like this, it's really hard for your reader to understand what's going on.[^note5]\n", "\n", "As with everything else, your overriding concern should be that you *explain* things to your reader. Always remember that the point of reporting your results is to communicate to another human being. I cannot tell you just how many times I've seen the results section of a report or a thesis or even a scientific article that is just gibberish, because the writer has focused solely on making sure they've included all the numbers, and forgotten to actually communicate with the human reader. \n", "\n", "[^note4]: Well, sort of. The conventions for how statistics should be reported tend to differ somewhat from discipline to discipline; I've tended to stick with how things are done in psychology, since that's what I do. But the general principle of providing enough information to the reader to allow them to check your results is pretty universal, I think.\n", "\n", "[^note5]: To some people, this advice might sound odd, or at least in conflict with the \"usual\" advice on how to write a technical report. Very typically, students are told that the \"results\" section of a report is for describing the data and reporting statistical analysis; and the \"discussion\" section is for providing interpretation. That's true as far as it goes, but I think people often interpret it way too literally. The way I usually approach it is to provide a quick and simple interpretation of the data in the results section, so that my reader understands what the data are telling us. Then, in the discussion, I try to tell a bigger story; about how my results fit with the rest of the scientific literature. In short; don't let the \"interpretation goes in the discussion\" advice turn your results section into incomprehensible garbage. Being understood by your reader is *much* more important." ] }, { "attachments": {}, "cell_type": "markdown", "id": "f70ad7b0", "metadata": {}, "source": [ "### A comment on statistical notation (advanced)\n", "\n", ">*Satan delights equally in statistics and in quoting scripture*\n", ">\n", "> -- H.G. Wells\n", "\n", "\n", "If you've been reading very closely, and are as much of a mathematical pedant as I am, there is one thing about the way I wrote up the chi-square test in the last section that might be bugging you a little bit. There's something that feels a bit wrong with writing \"$\\chi^2(3) = 8.44$\", you might be thinking. After all, it's the goodness of fit statistic that is equal to 8.44, so shouldn't I have written $X^2 = 8.44$ or maybe GOF$=8.44$? This seems to be conflating the *sampling distribution* (i.e., $\\chi^2$ with $df = 3$) with the *test statistic* (i.e., $X^2$). Odds are you figured it was a typo, since $\\chi$ and $X$ look pretty similar. Oddly, it's not. Writing $\\chi^2(3) = 8.44$ is essentially a highly condensed way of writing \"the sampling distribution of the test statistic is $\\chi^2(3)$, and the value of the test statistic is 8.44\". \n", "\n", "In one sense, this is kind of stupid. There are *lots* of different test statistics out there that turn out to have a chi-square sampling distribution: the $X^2$ statistic that we've used for our goodness of fit test is only one of many (albeit one of the most commonly encountered ones). In a sensible, perfectly organised world, we'd *always* have a separate name for the test statistic and the sampling distribution: that way, the stat block itself would tell you exactly what it was that the researcher had calculated. Sometimes this happens. For instance, the test statistic used in the Pearson goodness of fit test is written $X^2$; but there's a closely related test known as the $G$-test[^note6] {cite}`Sokal1994`, in which the test statistic is written as $G$. As it happens, the Pearson goodness of fit test and the $G$-test both test the same null hypothesis; and the sampling distribution is exactly the same (i.e., chi-square with $k-1$ degrees of freedom). If I'd done a $G$-test for the cards data rather than a goodness of fit test, then I'd have ended up with a test statistic of $G = 8.65$, which is slightly different from the $X^2 = 8.44$ value that I got earlier; and produces a slightly smaller $p$-value of $p = .034$. Suppose that the convention was to report the test statistic, then the sampling distribution, and then the $p$-value. If that were true, then these two situations would produce different stat blocks: my original result would be written $X^2 = 8.44, \\chi^2(3), p = .038$, whereas the new version using the $G$-test would be written as $G = 8.65, \\chi^2(3), p = .034$. However, using the condensed reporting standard, the original result is written $\\chi^2(3) = 8.44, p = .038$, and the new one is written $\\chi^2(3) = 8.65, p = .034$, and so it's actually unclear which test I actually ran. \n", "\n", "\n", "So why don't we live in a world in which the contents of the stat block uniquely specifies what tests were ran? The deep reason is that life is messy. We (as users of statistical tools) want it to be nice and neat and organised... we want it to be *designed*, as if it were a product. But that's not how life works: statistics is an intellectual discipline just as much as any other one, and as such it's a massively distributed, partly-collaborative and partly-competitive project that no-one really understands completely. The things that you and I use as data analysis tools weren't created by an Act of the Gods of Statistics; they were invented by lots of different people, published as papers in academic journals, implemented, corrected and modified by lots of other people, and then explained to students in textbooks by someone else. As a consequence, there's a *lot* of test statistics that don't even have names; and as a consequence they're just given the same name as the corresponding sampling distribution. As we'll see later, any test statistic that follows a $\\chi^2$ distribution is commonly called a \"chi-square statistic\"; anything that follows a $t$-distribution is called a \"$t$-statistic\" and so on. But, as the $X^2$ versus $G$ example illustrates, two different things with the same sampling distribution are still, well, different. \n", "\n", "As a consequence, it's sometimes a good idea to be clear about what the actual test was that you ran, especially if you're doing something unusual. If you just say \"chi-square test\", it's not actually clear what test you're talking about. Although, since the two most common chi-square tests are the goodness of fit test and the [independence test](chisqindependence)), most readers with stats training can probably guess. Nevertheless, it's something to be aware of.\n", "\n", "[^note6]: [Complicating matters, the $G$-test is a special case of a whole class of tests that are known as *likelihood ratio tests*. I don't cover LRTs in this book, but they are quite handy things to know about.] " ] }, { "attachments": {}, "cell_type": "markdown", "id": "d897d044", "metadata": {}, "source": [ "(chisqindependence)=\n", "## The $\\chi^2$ test of independence (or association)\n", "\n", "\n", "| | |\n", "|:----------|:--------------------------------------------------------------------------------------------------------------------------------------------|\n", "|GUARDBOT1: |Halt! |\n", "|GUARDBOT2: |Be you robot or human? |\n", "|LEELA: |Robot...we be. |\n", "|FRY: |Uh, yup! Just two robots out roboting it up! Eh? |\n", "|GUARDBOT1: |Administer the test. |\n", "|GUARDBOT2: |Which of the following would you most prefer? A: A puppy, B: A pretty flower from your sweetie, or C: A large properly-formatted data file? |\n", "|GUARDBOT1: |Choose! |\n", "\n", "-- Futurama, \"Fear of a Bot Planet" ] }, { "attachments": {}, "cell_type": "markdown", "id": "fb5f0110", "metadata": {}, "source": [ "The other day I was watching an animated documentary examining the quaint customs of the natives of the planet *Chapek 9*. Apparently, in order to gain access to their capital city, a visitor must prove that they're a robot, not a human. In order to determine whether or not visitor is human, they ask whether the visitor prefers puppies, flowers or large, properly formatted data files. \"Pretty clever,\" I thought to myself \"but what if humans and robots have the same preferences? That probably wouldn't be a very good test then, would it?\" As it happens, I got my hands on the testing data that the civil authorities of *Chapek 9* used to check this. It turns out that what they did was very simple... they found a bunch of robots and a bunch of humans and asked them what they preferred. I saved their data in a file called `chapek9.csv`, which I can now load and have a quick look at:" ] }, { "cell_type": "code", "execution_count": 1, "id": "9aca4fac", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
specieschoice
0robotflower
1humandata
2humandata
3humandata
4robotdata
\n", "
" ], "text/plain": [ " species choice\n", "0 robot flower\n", "1 human data\n", "2 human data\n", "3 human data\n", "4 robot data" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "df = pd.read_csv('https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/chapek9.csv')\n", "\n", "df.head()" ] }, { "cell_type": "code", "execution_count": 5, "id": "7b2e3ccc", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
specieschoice
count180180
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" ], "text/plain": [ " species choice\n", "count 180 180\n", "unique 2 3\n", "top human data\n", "freq 93 109" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.describe()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "3142690a", "metadata": {}, "source": [ "In total there are 180 entries in the data frame, one for each person (counting both robots and humans as \"people\") who was asked to make a choice. Specifically, there's 93 humans and 87 robots; and overwhelmingly the preferred choice is the data file. However, these summaries don't address the question we're interested in. To do that, we need a more detailed description of the data. What we want to do is look at the `choices` broken down *by* `species`. That is, we need to [cross-tabulate]freqtables) the data. There's quite a few ways to do this, as we've seen, but since our data are stored in a data frame, it's convenient to use the `crosstab()` method from `pandas`." ] }, { "cell_type": "code", "execution_count": 13, "id": "4a11f221", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
specieshumanrobot
choice
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flower1330
puppy1513
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" ], "text/plain": [ "species human robot\n", "choice \n", "data 65 44\n", "flower 13 30\n", "puppy 15 13" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pd.crosstab(index=df[\"choice\"], columns=df[\"species\"],margins=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "ace00c05", "metadata": {}, "source": [ "That's more or less what we're after. If set `margins=` to `True`, then we can get the row and column totals as well (which is convenient for the purposes of explaining the statistical tests):" ] }, { "cell_type": "code", "execution_count": 10, "id": "1cecf38c", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
specieshumanrobotAll
choice
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flower133043
puppy151328
All9387180
\n", "
" ], "text/plain": [ "species human robot All\n", "choice \n", "data 65 44 109\n", "flower 13 30 43\n", "puppy 15 13 28\n", "All 93 87 180" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pd.crosstab(index=df[\"choice\"], columns=df[\"species\"],margins=True)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "588290b1", "metadata": {}, "source": [ "With maybe a teeny bit of tidying up, like so:" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a427dc0e", "metadata": {}, "source": [ "| | Robot | Human | Total |\n", "|:---------|:-----:|:-----:|:-----:|\n", "|Puppy | 13 | 15 | 28 |\n", "|Flower | 30 | 13 | 43 |\n", "|Data file | 44 | 65 | 109 |\n", "|Total | 87 | 93 | 180 |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "99519b7d", "metadata": {}, "source": [ "we would actually have a nice way to report the descriptive statistics for this data set. In any case, it's quite clear that the vast majority of the humans chose the data file, whereas the robots tended to be a lot more even in their preferences. Leaving aside the question of *why* the humans might be more likely to choose the data file for the moment (which does seem quite odd, admittedly), our first order of business is to determine if the discrepancy between human choices and robot choices in the data set is statistically significant." ] }, { "attachments": {}, "cell_type": "markdown", "id": "24235b9b", "metadata": {}, "source": [ "### Constructing our hypothesis test\n", "\n", "How do we analyse this data? Specifically, since my *research* hypothesis is that \"humans and robots answer the question in different ways\", how can I construct a test of the *null* hypothesis that \"humans and robots answer the question the same way\"? As before, we begin by establishing some notation to describe the data:\n", "\n", "| |Robot |Human |Total |\n", "|:---------|:--------|:--------|:-------|\n", "|Puppy |$O_{11}$ |$O_{12}$ |$R_{1}$ |\n", "|Flower |$O_{21}$ |$O_{22}$ |$R_{2}$ |\n", "|Data file |$O_{31}$ |$O_{32}$ |$R_{3}$ |\n", "|Total |$C_{1}$ |$C_{2}$ |$N$ |\n", "\n", "In this notation we say that $O_{ij}$ is a count (observed frequency) of the number of respondents that are of species $j$ (robots or human) who gave answer $i$ (puppy, flower or data) when asked to make a choice. The total number of observations is written $N$, as usual. Finally, I've used $R_i$ to denote the row totals (e.g., $R_1$ is the total number of people who chose the flower), and $C_j$ to denote the column totals (e.g., $C_1$ is the total number of robots).[^note7]\n", "\n", "So now let's think about what the null hypothesis says. If robots and humans are responding in the same way to the question, it means that the probability that \"a robot says puppy\" is the same as the probability that \"a human says puppy\", and so on for the other two possibilities. So, if we use $P_{ij}$ to denote \"the probability that a member of species $j$ gives response $i$\" then our null hypothesis is that:\n", "\n", "| | |\n", "|:------|:---------------------------------------------------------|\n", "|$H_0$: |All of the following are true: |\n", "| |$P_{11} = P_{12}$ (same probability of saying puppy) |\n", "| |$P_{21} = P_{22}$ (same probability of saying flower) and |\n", "| |$P_{31} = P_{32}$ (same probability of saying data). |\n", "\n", "And actually, since the null hypothesis is claiming that the true choice probabilities don't depend on the species of the person making the choice, we can let $P_i$ refer to this probability: e.g., $P_1$ is the true probability of choosing the puppy.\n", "\n", "Next, in much the same way that we did with the goodness of fit test, what we need to do is calculate the expected frequencies. That is, for each of the observed counts $O_{ij}$, we need to figure out what the null hypothesis would tell us to expect. Let's denote this expected frequency by $E_{ij}$. This time, it's a little bit trickier. If there are a total of $C_j$ people that belong to species $j$, and the true probability of anyone (regardless of species) choosing option $i$ is $P_i$, then the expected frequency is just: \n", "\n", "$$\n", "E_{ij} = C_j \\times P_i\n", "$$\n", "\n", "Now, this is all very well and good, but we have a problem. Unlike the situation we had with the goodness of fit test, the null hypothesis doesn't actually specify a particular value for $P_i$. It's something we have to [estimate](estimation) from the data! Fortunately, this is pretty easy to do. If 28 out of 180 people selected the flowers, then a natural estimate for the probability of choosing flowers is $28/180$, which is approximately $.16$. If we phrase this in mathematical terms, what we're saying is that our estimate for the probability of choosing option $i$ is just the row total divided by the total sample size:\n", "\n", "$$\n", "\\hat{P}_i = \\frac{R_i}{N}\n", "$$ \n", "\n", "Therefore, our expected frequency can be written as the product (i.e. multiplication) of the row total and the column total, divided by the total number of observations:[^note8]\n", "\n", "$$\n", "E_{ij} = \\frac{R_i \\times C_j}{N}\n", "$$\n", "\n", "Now that we've figured out how to calculate the expected frequencies, it's straightforward to define a test statistic; following the exact same strategy that we used in the goodness of fit test. In fact, it's pretty much the *same* statistic. For a contingency table with $r$ rows and $c$ columns, the equation that defines our $X^2$ statistic is \n", "\n", "$$ \n", "X^2 = \\sum_{i=1}^r \\sum_{j=1}^c \\frac{({E}_{ij} - O_{ij})^2}{{E}_{ij}}\n", "$$\n", "\n", "The only difference is that I have to include two summation sign (i.e., $\\sum$) to indicate that we're summing over both rows and columns. As before, large values of $X^2$ indicate that the null hypothesis provides a poor description of the data, whereas small values of $X^2$ suggest that it does a good job of accounting for the data. Therefore, just like last time, we want to reject the null hypothesis if $X^2$ is too large.\n", "\n", "Not surprisingly, this statistic is $\\chi^2$ distributed. All we need to do is figure out how many degrees of freedom are involved, which actually isn't too hard. As I mentioned before, you can (usually) think of the degrees of freedom as being equal to the number of data points that you're analysing, minus the number of constraints. A contingency table with $r$ rows and $c$ columns contains a total of $r \\times c$ observed frequencies, so that's the total number of observations. What about the constraints? Here, it's slightly trickier. The answer is always the same\n", "\n", "$$\n", "df = (r-1)(c-1)\n", "$$\n", "\n", "but the explanation for *why* the degrees of freedom takes this value is different depending on the experimental design. For the sake of argument, let's suppose that we had honestly intended to survey exactly 87 robots and 93 humans (column totals fixed by the experimenter), but left the row totals free to vary (row totals are random variables). Let's think about the constraints that apply here. Well, since we deliberately fixed the column totals by Act of Experimenter, we have $c$ constraints right there. But, there's actually more to it than that. Remember how our null hypothesis had some free parameters (i.e., we had to estimate the $P_i$ values)? Those matter too. I won't explain why in this book, but every free parameter in the null hypothesis is rather like an additional constraint. So, how many of those are there? Well, since these probabilities have to sum to 1, there's only $r-1$ of these. So our total degrees of freedom is:\n", "\n", "$$\n", "\\begin{array}{rcl}\n", "df &=& \\mbox{(number of observations)} - \\mbox{(number of constraints)} \\\\\n", "&=& (rc) - (c + (r-1)) \\\\\n", "&=& rc - c - r + 1 \\\\\n", "&=& (r - 1)(c - 1)\n", "\\end{array}\n", "$$\n", "\n", "Alternatively, suppose that the only thing that the experimenter fixed was the total sample size $N$. That is, we quizzed the first 180 people that we saw, and it just turned out that 87 were robots and 93 were humans. This time around our reasoning would be slightly different, but would still lead is to the same answer. Our null hypothesis still has $r-1$ free parameters corresponding to the choice probabilities, but it now *also* has $c-1$ free parameters corresponding to the species probabilities, because we'd also have to estimate the probability that a randomly sampled person turns out to be a robot.[^note9] Finally, since we did actually fix the total number of observations $N$, that's one more constraint. So now we have, $rc$ observations, and $(c-1) + (r-1) + 1$ constraints. What does that give?\n", "\n", "$$\n", "\\begin{array}{rcl}\n", "df &=& \\mbox{(number of observations)} - \\mbox{(number of constraints)} \\\\\n", "&=& rc - ( (c-1) + (r-1) + 1) \\\\\n", "&=& rc - c - r + 1 \\\\\n", "&=& (r - 1)(c - 1)\n", "\\end{array}\n", "$$\n", "\n", "Amazing. \n", "\n", "\n", "[^note7]: A technical note. The way I've described the test pretends that the column totals are fixed (i.e., the researcher intended to survey 87 robots and 93 humans) and the row totals are random (i.e., it just turned out that 28 people chose the puppy). To use the terminology from my mathematical statistics textbook {cite:ps}`Hogg2005` I should technically refer to this situation as a chi-square test of homogeneity; and reserve the term chi-square test of independence for the situation where both the row and column totals are random outcomes of the experiment. In the initial drafts of this book that's exactly what I did. However, it turns out that these two tests are identical; and so I've collapsed them together.\n", "\n", "[^note8]:Technically, $E_{ij}$ here is an estimate, so I should probably write it $\\hat{E}_{ij}$. But since no-one else does, I won't either.\n", "\n", "[^note9]: A problem many of us worry about in real life." ] }, { "attachments": {}, "cell_type": "markdown", "id": "a4161a9b", "metadata": {}, "source": [ "(AssocTestInPython)=\n", "### Doing the test in Python\n", "\n", "\n", "Okay, now that we know how the test works, let's have a look at how it's done in Python. As tempting as it is to lead you through the tedious calculations so that you're forced to learn it the long way, I figure there's no point. I already showed you how to do it the long way for the goodness of fit test in the last section, and since the test of independence isn't conceptually any different, you won't learn anything new by doing it the long way. So instead, I'll go straight to showing you the easy way. As always, Python lets you do it multiple ways. We've already used `scipy.stats` for calculating the $\\chi^2$ goodness-of-fit test above, so I'll start with that. Then, I want to introduce you to the `pingouin` package, which bundles up a bunch of statistical tests and makes calculating and reporting them a good deal easier." ] }, { "attachments": {}, "cell_type": "markdown", "id": "ae9a2808", "metadata": {}, "source": [ "#### Doing the test with `scipy.stats`\n", "\n", "Calculating $\\chi^2$ test of independence with `scipy.stats` is fairly straightforward, although like most of what you get with `scipy`, there aren't a lot of frills. Just like with the goodness-of-fit test, you start by calculating a frequency table. Before, we used the `value_counts()` method to find the frequency of the different suits drawn in the `cards` data. Since we have two columns of data (choices by robots and choices by humans), we can use the `crosstab` function from `pandas` to do the work, and we can store the results in a variable called `observations`:" ] }, { "cell_type": "code", "execution_count": 23, "id": "0dbce09d", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
specieshumanrobot
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" ], "text/plain": [ "species human robot\n", "choice \n", "data 65 44\n", "flower 13 30\n", "puppy 15 13" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "observations = pd.crosstab(index=df[\"choice\"], columns=df[\"species\"],margins=False)\n", "observations" ] }, { "attachments": {}, "cell_type": "markdown", "id": "adf59800", "metadata": {}, "source": [ "Now that we have a frequency table, we can use the `chi2_contingency` function from `scipy.stats` to crunch the numbers:" ] }, { "cell_type": "code", "execution_count": 41, "id": "c8ac7eb0", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "chi2 = 10.721571792595633\n", "p = 0.004697213134214071\n", "degrees of freedom = 2\n", "expected = [[56.31666667 52.68333333]\n", " [22.21666667 20.78333333]\n", " [14.46666667 13.53333333]]\n" ] } ], "source": [ "from scipy.stats import chi2_contingency\n", "\n", "chi2, p, dof, ex = chi2_contingency(observations)\n", "\n", "print(\"chi2 = \", chi2)\n", "print(\"p = \", p)\n", "print(\"degrees of freedom = \", dof)\n", "print(\"expected = \", ex)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "894cbc23", "metadata": {}, "source": [ "`chi2_contingency` produces four different variables, `chi2`, `p`, `dof`, and `ex`. `chi2` is the $\\chi^2$ statistic, p is the $p$-value, `dof` is the degrees of freedom for the test, and `ex` tells you the \"expected\" values, that is, the frequency counts you would expect to see under the null hypothesis. The null hypothesis, by the way, is that the variables are independent of one another, that is, that the choice of data, flower, or puppy has nothing to do with whether the chooser is a robot or a human. The alternative, of course, is that they are _not_ independent; that is, that robots and humans will tend to make different choices. Or, put another way, that the frequency of flowers, puppies, and data chosen is _contingent_ on whether or not the choosers were humans or robots. That's why the function is called `chi2_contingency`, I suppose." ] }, { "attachments": {}, "cell_type": "markdown", "id": "05568966", "metadata": {}, "source": [ "#### Doing the test with `pingouin`\n", "\n", "Now that we've done the test with `scipy`, I want to introduce you to another way to do the same thing: the `pingouin` package. `pingouin` takes existing statistical functions in Python and wraps them up in a way that makes them (in my opinion) easier to deal with, and easiser to report afterwards. Unfortunately, there isn't a `pingouin` command for every statistical test that you might want to do. For example, as of the time of writing this, there is no `pingouin` version of the $\\chi^2$ goodness-of-fit test. However, there is one for the $\\chi^2$ test of independence (or association, or contingency, or whatever else you might like to call it), and this is what I will introduce you to now. In the coming chapters on other statistical tests, I will use the `pingouin` version wherever possible.\n", "\n", "To start with, you will need to install the `pingouin` package in whatever Python environment you are using. This can be done easily with either `pip` or `conda`. Instructions can be found on the [Pingouin webpage](https://pingouin-stats.org/index.html). If you are working in a Jupyter notebook, you should be able to simply type `pip install pingouin` in a cell, and be good to go.\n", "\n", "Unlike `scipy.stats.chi2_contingency`, `pingouin` doesn't require you to build a frequency table first; you can just plug your raw `pandas` dataframe in, and tell it which columns you would like it to work with. Our data is already in a dataframe called `df`, so we can simply write:" ] }, { "cell_type": "code", "execution_count": 27, "id": "b70d4c78", "metadata": {}, "outputs": [], "source": [ "import pingouin as pg\n", "\n", "expected, observed, stats = pg.chi2_independence(df, x='species', y='choice')" ] }, { "attachments": {}, "cell_type": "markdown", "id": "470f4690", "metadata": {}, "source": [ "As you can see, `pingouin` calculates the expected frequencies, the observed frequencies, and all the relevant statistics directly from the raw data in the dataframe. We can start by comparing the expected frequencies with the ones we calculated with `scipy`, and the observed frequencies we calculated \"manually\"\n", ":" ] }, { "cell_type": "code", "execution_count": 40, "id": "817dbfb2", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "scipy's calculation of expected frequencies:\n", " \n", "[[56.31666667 52.68333333]\n", " [22.21666667 20.78333333]\n", " [14.46666667 13.53333333]]\n", "-------\n", "pingouin's calculation of expected frequencies:\n", " \n", "choice data flower puppy\n", "species \n", "human 56.316667 22.216667 14.466667\n", "robot 52.683333 20.783333 13.533333\n" ] } ], "source": [ "print(\"scipy's calculation of expected frequencies:\")\n", "print(\" \")\n", "chi2, p, dof, ex = chi2_contingency(observations)\n", "print(ex)\n", "\n", "print(\"-------\")\n", "\n", "print(\"pingouin's calculation of expected frequencies:\")\n", "print(\" \")\n", "expected, observed, stats = pg.chi2_independence(df, x='species', y='choice')\n", "print(expected)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "70f1768d", "metadata": {}, "source": [ "Yup, the values are the same; `pingouin` just makes them a little easier to read. And if we check the observed values:" ] }, { "cell_type": "code", "execution_count": 35, "id": "72c86067", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "species human robot\n", "choice \n", "data 65 44\n", "flower 13 30\n", "puppy 15 13\n", " \n", "choice data flower puppy\n", "species \n", "human 65 13 15\n", "robot 44 30 13\n" ] } ], "source": [ "observed_manual = pd.crosstab(index=df[\"choice\"], columns=df[\"species\"],margins=False)\n", "expected, observed_pingouin, stats = pg.chi2_independence(df, x='species', y='choice')\n", "\n", "print(observed_manual)\n", "print(\" \")\n", "print(observed_pingouin)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "21b5637c", "metadata": {}, "source": [ "Again, the numbers are the same. So that's comforting. Now let's get to the good bit: the statistics!" ] }, { "cell_type": "code", "execution_count": 38, "id": "8943fcc0", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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testlambdachi2dofpvalcramerpower
0pearson1.00000010.7215722.00.0046970.2440580.842961
1cressie-read0.66666710.7577452.00.0046130.2444690.844244
2log-likelihood0.00000010.9221952.00.0042490.2463310.849966
3freeman-tukey-0.50000011.1315622.00.0038270.2486810.856988
4mod-log-likelihood-1.00000011.4217162.00.0033100.2519010.866249
5neyman-2.00000012.2804352.00.0021540.2611980.890668
\n", "
" ], "text/plain": [ " test lambda chi2 dof pval cramer power\n", "0 pearson 1.000000 10.721572 2.0 0.004697 0.244058 0.842961\n", "1 cressie-read 0.666667 10.757745 2.0 0.004613 0.244469 0.844244\n", "2 log-likelihood 0.000000 10.922195 2.0 0.004249 0.246331 0.849966\n", "3 freeman-tukey -0.500000 11.131562 2.0 0.003827 0.248681 0.856988\n", "4 mod-log-likelihood -1.000000 11.421716 2.0 0.003310 0.251901 0.866249\n", "5 neyman -2.000000 12.280435 2.0 0.002154 0.261198 0.890668" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "expected, observed, stats = pg.chi2_independence(df, x='species', y='choice')\n", "stats" ] }, { "attachments": {}, "cell_type": "markdown", "id": "41e3b205", "metadata": {}, "source": [ "Wow! Look at all that.... stuff! `pingouin` doesn't hold back with the information it provides you with, and it puts it all in a nice table, too. In fact, there is so much here that we're not going to get into all of it (although I will spend a bit of time explaining what [\"cramer\"](chisqeffectsize) is, below. For now, the important thing is the first row, which gives us the same values for the chi2 statistic, the degrees of freedom, and the $p$-value that we got from `scipy.stats.chi2_contingency()`. So you can take your pick: you'll get the same answer no matter which one you use. In fact, `pinguoin` is actually using `scipy` to do the dirty work of running the calculations: it's just wrapping them up with a nice little bow for you." ] }, { "attachments": {}, "cell_type": "markdown", "id": "f8974c6e", "metadata": {}, "source": [ "This output gives us enough information to write up the result:\n", "\n", "> Pearson's $\\chi^2$ revealed a significant association between species and choice ($\\chi^2(2) = 10.7, p < .01$): robots appeared to be more likely to say that they prefer flowers, but the humans were more likely to say they prefer data.\n", "\n", "Notice that, once again, I provided a little bit of interpretation to help the human reader understand what's going on with the data. Later on in my discussion section, I'd provide a bit more context. To illustrate the difference, here's what I'd probably say later on:\n", "\n", "> The fact that humans appeared to have a stronger preference for raw data files than robots is somewhat counterintuitive. However, in context it makes some sense: the civil authority on Chapek 9 has an unfortunate tendency to kill and dissect humans when they are identified. As such it seems most likely that the human participants did not respond honestly to the question, so as to avoid potentially undesirable consequences. This should be considered to be a substantial methodological weakness.\n", "\n", "This could be classified as a rather extreme example of a reactivity effect, I suppose. Obviously, in this case the problem is severe enough that the study is more or less worthless as a tool for understanding the difference preferences among humans and robots. However, I hope this illustrates the difference between getting a statistically significant result (our null hypothesis is rejected in favour of the alternative), and finding something of scientific value (the data tell us nothing of interest about our research hypothesis due to a big methodological flaw).\n", "\n", "### Postscript\n", "\n", "I later found out the data were made up, and I'd been watching cartoons instead of doing work." ] }, { "attachments": {}, "cell_type": "markdown", "id": "8a49d46a", "metadata": {}, "source": [ "(yates)=\n", "## The continuity correction\n", "\n", "Okay, time for a little bit of a digression. I've been lying to you a little bit so far. You probably expected that by now, and you were right. There's a tiny change that you need to make to your calculations whenever you only have 1 degree of freedom. It's called the \"continuity correction\", or sometimes the **_Yates correction_**. Remember what I pointed out earlier: the $\\chi^2$ test is based on an approximation, specifically on the assumption that binomial distribution starts to look like a normal distribution for large $N$. One problem with this is that it often doesn't quite work, especially when you've only got 1 degree of freedom (e.g., when you're doing a test of independence on a $2 \\times 2$ contingency table). The main reason for this is that the true sampling distribution for the $X^2$ statistic is actually discrete (because you're dealing with categorical data!) but the $\\chi^2$ distribution is continuous. This can introduce systematic problems. Specifically, when $N$ is small and when $df=1$, the goodness of fit statistic tends to be \"too big\", meaning that you actually have a bigger $\\alpha$ value than you think (or, equivalently, the $p$ values are a bit too small). Yates {cite:ps}`Yates1934` suggested a simple fix, in which you redefine the goodness of fit statistic as:\n", "\n", "$$\n", "X^2 = \\sum_{i} \\frac{(|E_i - O_i| - 0.5)^2}{E_i}\n", "$$\n", "\n", "Basically, he just subtracts off 0.5 everywhere. As far as I can tell from reading Yates' paper, the correction is basically a hack. It's not derived from any principled theory: rather, it's based on an examination of the behaviour of the test, and observing that the corrected version seems to work better. I feel obliged to explain this because you will sometimes see this correction, so it's kind of useful to know what it's about.\n", "\n", "Both `scipy.stats.chi2_contingency` and `pingouin` let you use Yates' correction, if you want to. So, for our data, we could have written either\n", "\n", "`chi2, p, dof, ex = chi2_contingency(observations, correction = True)`\n", "\n", "if we were using `scipy` or \n", "\n", "`expected, observed, stats = pg.chi2_independence(df, x='species', y='choice', correction = True)`\n", "\n", "if were using `pingouin`. Then again, in our specific case, it wouldn't make any difference, because in our test we had two degrees of freedom, and the correction is only run when there is only one degree of freedom." ] }, { "attachments": {}, "cell_type": "markdown", "id": "d2ab6d12", "metadata": {}, "source": [ "(chisqeffectsize)=\n", "## Effect size\n", "\n", "As we discussed [earlier](effectsize), it's becoming commonplace to ask researchers to report some measure of effect size. So, let's suppose that you've run your chi-square test, which turns out to be significant. So you now know that there is some association between your variables (independence test) or some deviation from the specified probabilities (goodness of fit test). Now you want to report a measure of effect size. That is, given that there is an association/deviation, how strong is it?\n", "\n", "There are several different measures that you can choose to report, and several different tools that you can use to calculate them. I won't discuss all of them; instead I will just mention the one that `pingouin` provides you with automatically: Cramer's $V$.\n", "\n", "By default, the two measures that people tend to report most frequently are the $\\phi$ statistic and the somewhat superior version, known as Cram\\'er's $V$. Mathematically, they're very simple. To calculate the $\\phi$ statistic, you just divide your $X^2$ value by the sample size, and take the square root:\n", "\n", "$$ \n", "\\phi = \\sqrt{\\frac{X^2}{N}}\n", "$$\n", "\n", "The idea here is that the $\\phi$ statistic is supposed to range between 0 (no at all association) and 1 (perfect association), but it doesn't always do this when your contingency table is bigger than $2 \\times 2$, which is a total pain. For bigger tables it's actually possible to obtain $\\phi>1$, which is pretty unsatisfactory. So, to correct for this, people usually prefer to report the $V$ statistic proposed by Cramer {cite:ps}`Cramer1946`. It's a pretty simple adjustment to $\\phi$. If you've got a contingency table with $r$ rows and $c$ columns, then define $k = \\min(r,c)$ to be the smaller of the two values. If so, then **_Cramer's $V$_** statistic is\n", "\n", "$$\n", "V = \\sqrt{\\frac{X^2}{N(k-1)}}\n", "$$\n", "\n", "And you're done. This seems to be a fairly popular measure, presumably because it's easy to calculate, and it gives answers that aren't completely silly: you know that $V$ really does range from 0 (no at all association) to 1 (perfect association). " ] }, { "attachments": {}, "cell_type": "markdown", "id": "4ea6549c", "metadata": {}, "source": [ "(chisqassumptions)=\n", "## Assumptions of the test(s)\n", "\n", "All statistical tests make assumptions, and it's usually a good idea to check that those assumptions are met. For the chi-square tests discussed so far in this chapter, the assumptions are:\n", "\n", "\n", "- *Expected frequencies are sufficiently large*. Remember how in the previous section we saw that the $\\chi^2$ sampling distribution emerges because the binomial distribution is pretty similar to a normal distribution? Well, like we [discussed earlier](probability), this is only true when the number of observations is sufficiently large. What that means in practice is that all of the expected frequencies need to be reasonably big. How big is reasonably big? Opinions differ, but the default assumption seems to be that you generally would like to see all your expected frequencies larger than about 5, though for larger tables you would probably be okay if at least 80\\% of the the expected frequencies are above 5 and none of them are below 1. However, from what I've been able to discover e.g.,{cite}`Cochran1954`, these seem to have been proposed as rough guidelines, not hard and fast rules; and they seem to be somewhat conservative {cite}`Larntz1978`. \n", "- *Data are independent of one another*. One somewhat hidden assumption of the chi-square test is that you have to genuinely believe that the observations are independent. Here's what I mean. Suppose I'm interested in proportion of babies born at a particular hospital that are boys. I walk around the maternity wards, and observe 20 girls and only 10 boys. Seems like a pretty convincing difference, right? But later on, it turns out that I'd actually walked into the same ward 10 times, and in fact I'd only seen 2 girls and 1 boy. Not as convincing, is it? My original 30 *observations* were massively non-independent... and were only in fact equivalent to 3 independent observations. Obviously this is an extreme (and extremely silly) example, but it illustrates the basic issue. Non-independence \"stuffs things up\". Sometimes it causes you to falsely reject the null, as the silly hospital example illustrates, but it can go the other way too. To give a slightly less stupid example, let's consider what would happen if I'd done the cards experiment slightly differently: instead of asking 200 people to try to imagine sampling one card at random, suppose I asked 50 people to select 4 cards. One possibility would be that *everyone* selects one heart, one club, one diamond and one spade (in keeping with the \"representativeness heuristic\"{cite}`Kahneman1973`). This is highly non-random behaviour from people, but in this case, I would get an observed frequency of 50 for all four suits. For this example, the fact that the observations are non-independent (because the four cards that you pick will be related to each other) actually leads to the opposite effect... falsely retaining the null.\n", "\n", "\n", "\n", "If you happen to find yourself in a situation where independence is violated, it may be possible to use the McNemar test (which we'll discuss) or the Cochran test (which we won't). Similarly, if your expected cell counts are too small, check out the Fisher exact test. It is to these topics that we now turn. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "e821a55e", "metadata": {}, "source": [ "(fisherexacttest)=\n", "## The Fisher exact test\n", "\n", "What should you do if your cell counts are too small, but you'd still like to test the null hypothesis that the two variables are independent? One answer would be \"collect more data\", but that's far too glib: there are a lot of situations in which it would be either infeasible or unethical do that. If so, statisticians have a kind of moral obligation to provide scientists with better tests. In this instance, Fisher (1922) kindly provided the right answer to the question. To illustrate the basic idea, let's suppose that we're analysing data from a field experiment, looking at the emotional status of people who have been accused of witchcraft; some of whom are currently being burned at the stake.[^note10] Unfortunately for the scientist (but rather fortunately for the general populace), it's actually quite hard to find people in the process of being set on fire, so the cell counts are awfully small in some cases. The `salem.csv` file illustrates the point:" ] }, { "cell_type": "code", "execution_count": 43, "id": "16910cf5", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
happyon.fire
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" ], "text/plain": [ " happy on.fire\n", "0 True False\n", "1 True False\n", "2 False False\n", "3 False True\n", "4 True False" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "df = pd.read_csv('https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/salem.csv')\n", "df.head()" ] }, { "cell_type": "code", "execution_count": 46, "id": "c6106c7f", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
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" ], "text/plain": [ "on.fire False True\n", "happy \n", "False 3 3\n", "True 10 0" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pd.crosstab(index=df[\"happy\"], columns=df[\"on.fire\"],margins=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "e413b1d9", "metadata": {}, "source": [ "Looking at this data, you'd be hard pressed not to suspect that people not on fire are more likely to be happy than people on fire. However, the chi-square test makes this very hard to test because of the small sample size. If I try to do so, Python gives me a warning message:" ] }, { "cell_type": "code", "execution_count": 48, "id": "df684133", "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/opt/anaconda3/envs/pythonbook2/lib/python3.10/site-packages/pingouin/contingency.py:151: UserWarning: Low count on observed frequencies.\n", " warnings.warn('Low count on {} frequencies.'.format(name))\n", "/opt/anaconda3/envs/pythonbook2/lib/python3.10/site-packages/pingouin/contingency.py:151: UserWarning: Low count on expected frequencies.\n", " warnings.warn('Low count on {} frequencies.'.format(name))\n" ] }, { "data": { "text/html": [ "
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testlambdachi2dofpvalcramerpower
0pearson1.0000003.3094021.00.0688850.4547940.444097
1cressie-read0.6666673.2658041.00.0707380.4517880.439356
2log-likelihood0.0000003.3218581.00.0683650.4556490.445448
3freeman-tukey-0.5000003.5070691.00.0611070.4681790.465301
4mod-log-likelihood-1.0000003.8500511.00.0497440.4905390.500918
5neyman-2.0000005.2766921.00.0216130.5742760.632005
\n", "
" ], "text/plain": [ " test lambda chi2 dof pval cramer power\n", "0 pearson 1.000000 3.309402 1.0 0.068885 0.454794 0.444097\n", "1 cressie-read 0.666667 3.265804 1.0 0.070738 0.451788 0.439356\n", "2 log-likelihood 0.000000 3.321858 1.0 0.068365 0.455649 0.445448\n", "3 freeman-tukey -0.500000 3.507069 1.0 0.061107 0.468179 0.465301\n", "4 mod-log-likelihood -1.000000 3.850051 1.0 0.049744 0.490539 0.500918\n", "5 neyman -2.000000 5.276692 1.0 0.021613 0.574276 0.632005" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pingouin as pd\n", "expected, observed_pingouin, stats = pg.chi2_independence(df, x='happy', y='on.fire')\n", "stats" ] }, { "attachments": {}, "cell_type": "markdown", "id": "e14880ca", "metadata": {}, "source": [ "Speaking as someone who doesn't want to be set on fire, I'd *really* like to be able to get a better answer than this. This is where **_Fisher's exact test_** {cite}`Fisher1922` comes in very handy. \n", "\n", "The Fisher exact test works somewhat differently to the chi-square test (or in fact any of the other hypothesis tests that I talk about in this book) insofar as it doesn't have a test statistic; it calculates the $p$-value \"directly\". I'll explain the basics of how the test works for a $2 \\times 2$ contingency table, though the test works fine for larger tables. As before, let's have some notation: \n", "\n", "[^note10]: This example is based on a joke article published in the *Journal of Irreproducible Results*." ] }, { "attachments": {}, "cell_type": "markdown", "id": "21db3e96", "metadata": {}, "source": [ "| |Happy |Sad |Total |\n", "|:---------------|:--------|:--------|:-------|\n", "|Set on fire |$O_{11}$ |$O_{12}$ |$R_{1}$ |\n", "|Not set on fire |$O_{21}$ |$O_{22}$ |$R_{2}$ |\n", "|Total |$C_{1}$ |$C_{2}$ |$N$ |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "3a17fc8d", "metadata": {}, "source": [ "In order to construct the test Fisher treats both the row and column totals ($R_1$, $R_2$, $C_1$ and $C_2$) are known, fixed quantities; and then calculates the probability that we would have obtained the observed frequencies that we did ($O_{11}$, $O_{12}$, $O_{21}$ and $O_{22}$) given those totals. In the notation that we developed back in the section on [probability](probability) this is written:\n", "\n", "$$\n", "P(O_{11}, O_{12}, O_{21}, O_{22} \\ | \\ R_1, R_2, C_1, C_2) \n", "$$\n", "\n", "and as you might imagine, it's a slightly tricky exercise to figure out what this probability is, but it turns out that this probability is described by a distribution known as the *hypergeometric distribution*. Now that we know this, what we have to do to calculate our $p$-value is calculate the probability of observing this particular table *or a table that is \"more extreme\"* [^note11]. Back in the 1920s, computing this sum was daunting even in the simplest of situations, but these days it's pretty easy as long as the tables aren't too big and the sample size isn't too large. The conceptually tricky issue is to figure out what it means to say that one contingency table is more \"extreme\" than another. The easiest solution is to say that the table with the lowest probability is the most extreme. This then gives us the $p$-value. \n", "\n", "Fisher's exact test can be computed in Python using the `fisher_exact` function from `scipy.stats` function, like this:\n", "\n", "[^note11]: Not surprisingly, the Fisher exact test is motivated by Fisher's interpretation of a $p$-value, not Neyman's!" ] }, { "cell_type": "code", "execution_count": 52, "id": "ef80efd5", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "p = 0.03571428571428571\n" ] } ], "source": [ "import pandas as pd\n", "from scipy.stats import fisher_exact\n", "\n", "df = pd.read_csv('https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/salem.csv')\n", "freq_table = pd.crosstab(index=df[\"happy\"], columns=df[\"on.fire\"],margins=False)\n", "\n", "oddsratio, pvalue = fisher_exact(freq_table) \n", "\n", "print(\"p = \", pvalue)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "7c36c097", "metadata": {}, "source": [ "The main thing we're interested in here is the $p$-value, which in this case is small enough ($p=.036$) to justify rejecting the null hypothesis that people on fire are just as happy as people not on fire. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "94657ce8", "metadata": {}, "source": [ "(mcnemar)=\n", "## The McNemar test\n", "\n", "Suppose you've been hired to work for the *Australian Generic Political Party* (AGPP), and part of your job is to find out how effective the AGPP political advertisements are. So, what you do, is you put together a sample of $N=100$ people, and ask them to watch the AGPP ads. Before they see anything, you ask them if they intend to vote for the AGPP; and then after showing the ads, you ask them again, to see if anyone has changed their minds. Obviously, if you're any good at your job, you'd also do a whole lot of other things too, but let's consider just this one simple experiment. One way to describe your data is via the following contingency table:\n", "\n", "| | Before | After | Total |\n", "|:-----|:------:|:-----:|:-----:|\n", "|Yes | 30 | 10 | 40 |\n", "|No | 70 | 90 | 160 |\n", "|Total | 100 | 100 | 200 |\n", "\n", "At first pass, you might think that this situation lends itself to the [Pearson $\\chi^2$ test of independence](chisqindependence). However, a little bit of thought reveals that we've got a problem: we have 100 participants, but 200 observations. This is because each person has provided us with an answer in *both* the before column and the after column. What this means is that the 200 observations aren't independent of each other: if voter A says \"yes\" the first time and voter B says \"no\", then you'd expect that voter A is more likely to say \"yes\" the second time than voter B! The consequence of this is that the usual $\\chi^2$ test won't give trustworthy answers due to the violation of the independence assumption. Now, if this were a really uncommon situation, I wouldn't be bothering to waste your time talking about it. But it's not uncommon at all: this is a *standard* repeated measures design, and none of the tests we've considered so far can handle it. Eek. \n", "\n", "The solution to the problem was published by McNemar{cite}`McNemar1947`. The trick is to start by tabulating your data in a slightly different way:\n", "\n", "| | Before: Yes | Before: No | Total |\n", "|:----------|:-----------:|:----------:|:-----:|\n", "|After: Yes | 5 | 5 | 10 |\n", "|After: No | 25 | 65 | 90 |\n", "|Total | 30 | 70 | 100 |\n", "\n", "This is exactly the same data, but it's been rewritten so that each of our 100 participants appears in only one cell. Because we've written our data this way, the independence assumption is now satisfied, and this is a contingency table that we *can* use to construct an $X^2$ goodness of fit statistic. However, as we'll see, we need to do it in a slightly nonstandard way. To see what's going on, it helps to label the entries in our table a little differently:\n", "\n", "| | Before: Yes | Before: No | Total |\n", "|:----------|:-----------:|:----------:|:-----:|\n", "|After: Yes | $a$ | $b$ | $a+b$ |\n", "|After: No | $c$ | $d$ | $c+d$ |\n", "|Total | $a+c$ | $b+d$ | $n$ |\n", "\n", "Next, let's think about what our null hypothesis is: it's that the \"before\" test and the \"after\" test have the same proportion of people saying \"Yes, I will vote for AGPP\". Because of the way that we have rewritten the data, it means that we're now testing the hypothesis that the *row totals* and *column totals* come from the same distribution. Thus, the null hypothesis in McNemar's test is that we have \"marginal homogeneity\". That is, the row totals and column totals have the same distribution: $P_a + P_b = P_a + P_c$, and similarly that $P_c + P_d = P_b + P_d$. Notice that this means that the null hypothesis actually simplifies to $P_b = P_c$. In other words, as far as the McNemar test is concerned, it's only the off-diagonal entries in this table (i.e., $b$ and $c$) that matter! After noticing this, the **_McNemar test of marginal homogeneity_** is no different to a usual $\\chi^2$ test. After applying the Yates correction, our test statistic becomes:\n", "\n", "$$\n", "X^2 = \\frac{(|b-c| - 0.5)^2}{b+c}\n", "$$\n", "\n", "or, to revert to the notation that we used earlier in this chapter:\n", "\n", "$$\n", "X^2 = \\frac{(|O_{12}-O_{21}| - 0.5)^2}{O_{12} + O_{21}}\n", "$$\n", "\n", "and this statistic has an (approximately) $\\chi^2$ distribution with $df=1$. However, remember that -- just like the other $\\chi^2$ tests -- it's only an approximation, so you need to have reasonably large expected cell counts for it to work." ] }, { "attachments": {}, "cell_type": "markdown", "id": "cc19786f", "metadata": {}, "source": [ "### Doing the McNemar test in Python\n", "\n", "Now that you know what the McNemar test is all about, lets actually run one. The `agpp.csv` file contains the raw data that I discussed previously, so let's have a look at it:" ] }, { "cell_type": "code", "execution_count": 53, "id": "72d3d2c3", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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idresponse_beforeresponse_after
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" ], "text/plain": [ " id response_before response_after\n", "0 subj.1 no yes\n", "1 subj.2 yes no\n", "2 subj.3 yes no\n", "3 subj.4 yes no\n", "4 subj.5 no no" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df = pd.read_csv('https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/agpp.csv')\n", "df.head()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "2f6b377e", "metadata": {}, "source": [ "The `df` data frame contains three variables, an `id` variable that labels each participant in the data set (we'll see why that's useful in a moment), a `response_before` variable that records the person's answer when they were asked the question the first time, and a `response_after` variable that shows the answer that they gave when asked the same question a second time. And, together with `pingouin`, this is all we need to run the McNemar test:" ] }, { "cell_type": "code", "execution_count": 58, "id": "7fcbd7d6", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
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" ], "text/plain": [ "response_after 0 1\n", "response_before \n", "0 65 5\n", "1 25 5" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pingouin as pg\n", "observed, stats = pg.chi2_mcnemar(df, 'response_before', 'response_after')\n", "observed" ] }, { "cell_type": "code", "execution_count": 59, "id": "08d8edf7", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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chi2dofp-approxp-exact
mcnemar12.03333310.0005230.000325
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" ], "text/plain": [ " chi2 dof p-approx p-exact\n", "mcnemar 12.033333 1 0.000523 0.000325" ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stats" ] }, { "attachments": {}, "cell_type": "markdown", "id": "cfe3bc25", "metadata": {}, "source": [ "And we're done. We've just run a McNemar's test to determine if people were just as likely to vote AGPP after the ads as they were before hand. The test was significant ($\\chi^2(1) = 12.03, p<.001$), suggesting that they were not. And in fact, it looks like the ads had a negative effect: people were less likely to vote AGPP after seeing the ads. Which makes a lot of sense when you consider the quality of a typical political advertisement." ] }, { "attachments": {}, "cell_type": "markdown", "id": "egyptian-incident", "metadata": {}, "source": [ "\n", "\n", "## Summary\n", "\n", "The key ideas discussed in this chapter are:\n", "\n", "\n", "- The [chi-square goodness of fit test](goftest) is used when you have a table of observed frequencies of different categories; and the null hypothesis gives you a set of \"known\" probabilities to compare them to. You can use the `stats.chisquare()` function from the `scipy` package to run this test. \n", "- The [chi-square test of independence](chisqindependence) is used when you have a contingency table (cross-tabulation) of two categorical variables. The null hypothesis is that there is no relationship/association between the variables. You can either use the `chi2_contingency()` function from the `stats.scipy` package, or you can use `chi2_independence()` function from `pingouin`. \n", "- [Effect size for a contingency table](chisqeffectsize) can be measured in several ways. In particular we noted the Cramer's $V$ statistic, which `pingouin.chi2_independence()`calculates for you.\n", "- Both versions of the Pearson test rely on [two assumptions](chisqassumptions): that the expected frequencies are sufficiently large, and that the observations are independent. The [Fisher exact test](fisherexacttest) can be used when the expected frequencies are small. The [McNemar test](mcnemar) can be used for some kinds of violations of independence. \n", "\n", "\n", "\n", "If you're interested in learning more about categorical data analysis, a good first choice would be {cite}`Agresti1996` which, as the title suggests, provides an *Introduction to Categorical Data Analysis*. If the introductory book isn't enough for you (or can't solve the problem you're working on) you could consider {cite}`Agresti2002`, *Categorical Data Analysis*. The latter is a more advanced text, so it's probably not wise to jump straight from this book to that one. " ] }, { "cell_type": "code", "execution_count": null, "id": "d138b2e7", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "celltoolbar": "Tags", "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.11" } }, "nbformat": 4, "nbformat_minor": 5 } { "cells": [ { "attachments": {}, "cell_type": "markdown", "id": "thousand-temperature", "metadata": {}, "source": [ "(ttest)=\n", "# Comparing Two Means" ] }, { "attachments": {}, "cell_type": "markdown", "id": "distant-trunk", "metadata": {}, "source": [ "\n", "In the previous chapter we covered the situation when your outcome variable is nominal scale and your predictor variable[^note1] is also nominal scale. Lots of real world situations have that character, and so you'll find that chi-square tests in particular are quite widely used. However, you're much more likely to find yourself in a situation where your outcome variable is interval scale or higher, and what you're interested in is whether the average value of the outcome variable is higher in one group or another. For instance, a psychologist might want to know if anxiety levels are higher among parents than non-parents, or if working memory capacity is reduced by listening to music (relative to not listening to music). In a medical context, we might want to know if a new drug increases or decreases blood pressure. An agricultural scientist might want to know whether adding phosphorus to Australian native plants will kill them.[^note2] In all these situations, our outcome variable is a fairly continuous, interval or ratio scale variable; and our predictor is a binary \"grouping\" variable. In other words, we want to compare the means of the two groups. \n", "\n", "The standard answer to the problem of comparing means is to use a $t$-test, of which there are several varieties depending on exactly what question you want to solve. As a consequence, the majority of this chapter focuses on different types of $t$-test: [one sample $t$-tests](onesamplettest), [student's](studentttest) and [welch's](welchttest), and [paired samples $t$-tests](pairedsamplesttest). After that, we'll talk a bit about [Cohen's $d$](cohensd), which is the standard measure of effect size for a $t$-test. The later sections of the chapter focus on the assumptions of the $t$-tests, and possible remedies if they are violated. However, before discussing any of these useful things, we'll start with a discussion of the $z$-test. \n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "subjective-damages", "metadata": {}, "source": [ "(onesamplettest)=\n", "## The one-sample $z$-test\n", "\n", "In this section I'll describe one of the most useless tests in all of statistics: the **_$z$-test_**. Seriously -- this test is almost never used in real life. Its only real purpose is that, when teaching statistics, it's a very convenient stepping stone along the way towards the $t$-test, which is probably the most (over)used tool in all statistics.\n", "\n", "### The inference problem that the test addresses\n", "\n", "To introduce the idea behind the $z$-test, let's use a simple example. A friend of mine, Dr Zeppo, grades his introductory statistics class on a curve. Let's suppose that the average grade in his class is 67.5, and the standard deviation is 9.5. Of his many hundreds of students, it turns out that 20 of them also take psychology classes. Out of curiosity, I find myself wondering: do the psychology students tend to get the same grades as everyone else (i.e., mean 67.5) or do they tend to score higher or lower? He emails me the `zeppo.csv` file, which I use to pull up the `grades` of those students, " ] }, { "cell_type": "code", "execution_count": 2, "id": "widespread-crisis", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
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" ], "text/plain": [ " grades\n", "0 50\n", "1 60\n", "2 60\n", "3 64\n", "4 66" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/zeppo.csv\")\n", "df.head()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "indonesian-entrance", "metadata": {}, "source": [ "and calculate the mean:" ] }, { "cell_type": "code", "execution_count": 3, "id": "0a014792", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "72.3" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df['grades'].mean()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "43557fb9", "metadata": {}, "source": [ "Hm. It *might* be that the psychology students are scoring a bit higher than normal: that sample mean of $\\bar{X} = 72.3$ is a fair bit higher than the hypothesised population mean of $\\mu = 67.5$, but on the other hand, a sample size of $N = 20$ isn't all that big. Maybe it's pure chance. \n", "\n", "To answer the question, it helps to be able to write down what it is that I think I know. Firstly, I know that the sample mean is $\\bar{X} = 72.3$. If I'm willing to assume that the psychology students have the same standard deviation as the rest of the class, then I can say that the population standard deviation is $\\sigma = 9.5$. I'll also assume that since Dr Zeppo is grading to a curve, the psychology student grades are normally distributed. \n", "\n", "Next, it helps to be clear about what I want to learn from the data. In this case, my research hypothesis relates to the *population* mean $\\mu$ for the psychology student grades, which is unknown. Specifically, I want to know if $\\mu = 67.5$ or not. Given that this is what I know, can we devise a hypothesis test to solve our problem? The data, along with the hypothesised distribution from which they are thought to arise, are shown in {numref}`fig-zeppo`. Not entirely obvious what the right answer is, is it? For this, we are going to need some statistics." ] }, { "cell_type": "code", "execution_count": 4, "id": "3c76df7a", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "zeppo-fig" } }, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "import seaborn as sns\n", "import scipy.stats as stats\n", "\n", "mu = 67.5\n", "sigma = 9.5\n", "x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)\n", "y = 100* stats.norm.pdf(x, mu, sigma)\n", "\n", "fig, ax = plt.subplots()\n", "ax1 = sns.histplot(df['grades'])\n", "\n", "ax2 = sns.lineplot(x=x,y=y, color='black')\n", "\n", "plt.ylim(bottom=-1)\n", "\n", "ax1.set_frame_on(False)\n", "ax1.axes.get_yaxis().set_visible(False)\n", "\n", "plt.close()\n", "from myst_nb import glue\n", "glue(\"zeppo-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "37263680", "metadata": {}, "source": [ "```{glue:figure} zeppo-fig\n", ":figwidth: 600px\n", ":name: fig-zeppo\n", "\n", "\n", "The theoretical distribution (solid line) from which the psychology student grades (blue bars) are supposed to have been generated.\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "typical-finder", "metadata": {}, "source": [ "### Constructing the hypothesis test\n", "\n", "The first step in constructing a hypothesis test is to be clear about what the null and alternative hypotheses are. This isn't too hard to do. Our null hypothesis, $H_0$, is that the true population mean $\\mu$ for psychology student grades is 67.5\\%; and our alternative hypothesis is that the population mean *isn't* 67.5\\%. If we write this in mathematical notation, these hypotheses become,\n", "\n", "$$\n", "\\begin{array}{ll}\n", "H_0: & \\mu = 67.5 \\\\\n", "H_1: & \\mu \\neq 67.5\n", "\\end{array}\n", "$$\n", "\n", "though to be honest this notation doesn't add much to our understanding of the problem, it's just a compact way of writing down what we're trying to learn from the data. The null hypotheses $H_0$ and the alternative hypothesis $H_1$ for our test are both illustrated in {numref}`fig-ztesthyp`. In addition to providing us with these hypotheses, the scenario outlined above provides us with a fair amount of background knowledge that might be useful. Specifically, there are two special pieces of information that we can add:\n", "\n", "1. The psychology grades are normally distributed.\n", "1. The true standard deviation of these scores $\\sigma$ is known to be 9.5.\n", "\n", "For the moment, we'll act as if these are absolutely trustworthy facts. In real life, this kind of absolutely trustworthy background knowledge doesn't exist, and so if we want to rely on these facts we'll just have make the *assumption* that these things are true. However, since these assumptions may or may not be warranted, we might need to check them. For now though, we'll keep things simple." ] }, { "cell_type": "code", "execution_count": 5, "id": "fifty-grenada", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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y4osv8tdff5nOEhEREXEqI0aMoH///gB89tln9OzZ03ARZMiQgdWrV1OuXDkuXLhA9erVOXz4sOksEUlDGjyJuJEBAwbwxRdfADBhwgTat29vNuhvcuXKRXh4OPny5eOPP/6gRo0aXLx40XSWiIiIiFMYM2YMb7/9NgCDBw/m3XffNVx0R6ZMmVizZg2lSpXir7/+onr16hw/ftx0loikEQ2eRNzEl19+yZAhQwAYOXIkXbp0MVz0b3nz5iU8PJxcuXJx4MABatasSWRkpOksEREREYc2Y8YM3njjDQDef/99PvjgA8NF/xYcHMyPP/5IsWLFOHXqFNWrV+fChQums0QkDWjwJOIG1qxZQ9++fQH4/PPP6dGjh+GieytYsCDh4eFkz56d3bt307FjR/TwTREREZG72759O507dwagd+/efPzxx8bO7ryf7Nmz259ifOzYMZo1a0ZCQoLpLBFJZRo8ibi4Y8eO0aJFC6xWK506deKdd95JsY/t6elJSEgIISEhKXpoZdGiRVm8eDHe3t7Mnz+f4cOHp9jHFhEREXEVFy9e5NVXXyU2Npb69evzxRdfOOzQKUnOnDlZtmwZgYGBRERE8N5775lOShWptU4WcUYWm7YSiLis6OhoXnjhBXbt2kX58uXZsGEDfn5+prMe2Lhx4+jevTseHh6sWbOGF1980XSSiIiIiENISEigdu3ahIeHU6RIEbZt20aGDBlMZz2wRYsWERISAsDcuXNp1qyZ4SIRSS3a8STiomw2G6+99hq7du0ia9asLFq0yKmGTgCvvfYa7du3x2q10qxZM06cOGE6SURERMQhvP/++4SHhxMQEMCSJUucaugE0LhxY/tup9DQUPbt22e4SERSi3Y8ibioMWPG8MYbb+Dp6cnatWupVq2a6aRkiYmJoVKlSuzYsYOnnnqKjRs3Ot0ATURERCQlLViwgKZNm9p/nbRzyNkkJiZSp04dfvzxRwoXLsy2bdvImDGj6SwRSWHa8STigjZv3sxbb70F3D5MPLWGTlFRUVgsFiwWC1FRUanyOdKlS8eiRYvIkiULO3bssD+xRURERMQdHTx4kA4dOgDw7rvvOu3QCW6fgzR37lzy5cvH4cOHadOmjcs8VCYt1skizkKDJxEXExUVRdu2bUlISKBZs2b06tXLdNIjy5cvH/PmzcPDw4PJkyezdOlS00kiIiIiaS4+Pp42bdoQFRXFiy++yNChQ00nPbIsWbKwePFi/Pz8WL58OZMmTTKdJCIpTIMnERfTr18/jhw5Qp48eZgwYYLDP9nkQb344ov2J/J17dqVy5cvGy4SERERSVuff/45O3bsIFOmTEyfPh0vLy/TSSniySefZNiwYQD07t1b53qKuBgNnkRcSEREBKNGjQJg8uTJTnfI5P0MHDiQEiVKcP78eXr06GE6R0RERCTN7Nmzh0GDBgEwatQocubMabgoZb355ptUrFiRyMhIQkNDsVqtppNEJIVo8CTiIiIjI+33+3ft2pWaNWsaLkp5fn5+TJ06FU9PT+bMmcOiRYtMJ4mIiIikuvj4eNq1a0d8fDwNGzakZcuWppNSnKenJ2FhYaRLl47w8HAmTJhgOklEUogGTyIu4t133+X48ePky5ePL774wnROqilfvrz90bvdunXj4sWLhotEREREUtewYcPYtWsXWbJkYfz48S5zlML/Kly4MJ999hkA77zzDkePHjVcJCIpQYMnERfw448/Mm7cOACmTJlC+vTpDRelrg8//JDSpUtz8eJFXn/9ddM5IiIiIqlm586dfPzxxwCMGTOG7NmzGy5KXa+//jpVqlQhKipKt9yJuAgNnkSc3I0bN+jYsSNw+xt19erV0+xze3p6UrduXerWrYunp2eafV5fX1+mTp2Kl5cXCxYsYN68eWn2uUVERETSSlxcHO3atSMhIYGQkBCaNm1qOinVeXh4MGXKFAICAli/fj2jR482nZQsptbJIo7IYrPZbKYjRCT5evXqxddff03BggXZvXs3gYGBppPSzMCBAxk0aBDZsmXjjz/+cLnD1EVERMS9ffrpp/Tr14+sWbOyf/9+smbNajopzYwbN47u3bsTEBDA77//Tq5cuUwniUgyafAk4sT279/PE088QWJiIqtXr6ZWrVqmk9JUXFwcZcqU4ffff6dXr16MGDHCdJKIiIhIijhz5gyPP/44UVFRTJ8+nTZt2phOSlNWq5WKFSvyyy+/0LJlS2bNmmU6SUSSSYMnESdls9moUaMG4eHhNGzYkCVLlphOMmL16tXUqVMHLy8vdu/eTYkSJUwniYiIiDyyli1bMmfOHJ5//nk2btzosgeK/5fffvuNp59+GpvNxoYNG6hUqZLpJBFJBp3xJOKkFi1aRHh4OH5+fsZ2+kRFRREQEEBAQABRUVFGGmrXrk2DBg1ISEjgzTffRLN0ERERcXYbNmxgzpw5WCwWRo0a5ZZDJ4Ann3ySzp07A9CjRw8SExMNFz04R1gnizgKDZ5EnFB0dDRvv/02AO+++y4FChQw2hIdHW3s8wOMGDECX19ffvrpJxYvXmy0RURERORRJCQk0KNHDwC6dOnCk08+abjIrKFDh5IpUyZ2797NhAkTTOc8FEdYJ4s4Ag2eRJzQp59+ysmTJ8mbNy99+/Y1nWNcwYIFeffddwHo3bu3vsGLiIiI0/r222/Zs2cPmTJlYujQoaZzjAsODmbIkCEAfPDBB1y6dMlwkYg8LA2eRJzM0aNH+fzzz4HbO338/f0NFzmG9957j7x583Ly5En7/31EREREnMmlS5f44IMPAPj444/JkiWL4SLH0LVrV8qUKcPVq1ft//cREeehwZOIk+nduzexsbG8+OKLvPrqq6ZzHIa/vz/Dhw8H4LPPPuP48eNmg0REREQe0gcffMDVq1d54okn6Nq1q+kch+Hl5cWoUaOA2zvCfvvtN8NFIvIwNHgScSLh4eEsXboULy8vRo4c6bYHTd5L48aNqV69Ordu3bLfeiciIiLiDPbu3cu3334LwKhRo/D09DRc5FgqV65MixYtsNlsvPXWW3qgjIgT0eBJxEnYbDbee+89AF577TVKlChhuMjxWCwWvv76aywWCwsWLGDHjh2mk0REREQeSP/+/bHZbDRp0oRKlSqZznFIn3/+Ob6+vvz888/88MMPpnNE5AFp8CTiJJYsWcK2bdsICAhwmHvbPTw8qFKlClWqVMHDwzG+nJQuXZrWrVsD8P777xuuEREREbm/TZs28f333+Pp6cnHH39sOsdh5c6d2/7Ev379+mG1Wg0X3ZsjrpNFTLHYtEdRxOElJCRQunRpDh06xIcffsjgwYNNJzm0Y8eO8fjjjxMfH89PP/1E9erVTSeJiIiI3JXNZqNy5cps3LiRLl26MGHCBNNJDu3y5csULFiQGzduMHv2bFq0aGE6SUTuQ6NXEScwffp0Dh06RJYsWXj77bdN5zi8AgUK8NprrwG3r4Zpvi4iIiKOatWqVWzcuBE/Pz8++ugj0zkOL0uWLPazPD/44APi4uIMF4nI/WjwJOLgbt26xYABA4DbQ5QMGTIYLnIO/fv3JyAggK1bt/Ldd9+ZzhERERH5F6vVSr9+/QDo0aMHuXLlMlzkHHr27Em2bNk4evQokydPNp0jIvehwZOIgxs3bhynT58md+7cdO/e3XTOP0RFRZE1a1ayZs1KVFSU6Zx/yJ49O7169QJuD6ESEhIMF4mIiIj809y5c9mzZw8ZMmSwP0RG7i8wMJAPP/wQgMGDBxMdHW246N8ceZ0sktY0eBJxYDdu3GDo0KEADBw4kHTp0hku+rdLly5x6dIl0xl31adPHzJnzszBgweZMWOG6RwRERERu7i4OPvw5N133yVz5syGi5xLly5dyJ8/P+fOnWPkyJGmc+7KkdfJImlJgycRBzZ8+HAuX77M448/Trt27UznOJ0MGTLYn2w3YMAAbt26ZbhIRERE5LZJkyZx9OhRsmfPTs+ePU3nOB0fHx+GDBkCwGeffcbVq1cNF4nIvWjwJOKgLl26xPDhwwH4+OOP8fLyMlzknLp3706uXLk4deoU48ePN50jIiIiQkxMjH1o8uGHHxIQEGC4yDm1aNGCUqVKce3aNT7//HPTOSJyDxo8iTior7/+mqioKMqVK0fjxo1N5zitdOnS2Z8Q88UXX2jXk4iIiBg3adIkzp07R758+ejcubPpHKfl6enJxx9/DMDo0aO5cuWK4SIRuRsNnkQc0LVr1xg1ahRw+yqYxWIxXOTc2rVrR+7cuTl79ixhYWGmc0RERMSNxcbG8tlnnwHw3nvv4ePjY7jIub3yyiuUKVOGyMhIhz3rScTdafAk4oBGjhzJjRs3KFWqFA0aNDCd4/R8fX159913Afj000+Jj483XCQiIiLuatq0aZw5c4acOXPSvn170zlOz8PDg/79+wPwzTffcOPGDcNFIvK/NHgScTA3b97k66+/BqB///54eDjuP1MPDw+efvppnn76aYfuBOjUqRPZs2fn5MmTesKdiIiIGBEfH88nn3wC3H6SnZ+fn+Ei19C4cWOKFSvGtWvXGDNmjOkcwLnWySKpzWKz2WymI0Tkjs8++4z33nuPokWLcuDAATw9PU0nuYzhw4fTp08fChcuzMGDB3Vgu4iIiKSpadOm0b59e7Jly8axY8fw9/c3neQyZs6cSZs2bQgODub48eM6sF3EgWj0KuJAoqOj7U+ye//99zV0SmFdu3YlS5YsHD58mHnz5pnOERERETeSmJjIsGHDAHj77bc1dEphzZs3p1ChQly6dIkJEyaYzhGRv9HgScSBfPvtt1y8eJECBQrQsmVL0zkuJzAwkN69ewMwdOhQrFar4SIRERFxFwsWLOCPP/4gc+bMdOvWzXSOy/Hy8qJfv37A7ScZx8TEGC4SkSQaPIk4iFu3bvH5558D0K9fP7y9vQ0X3V90dDT58+cnf/78REdHm855IG+88QYZM2bk4MGDLF682HSOiIiIuAGr1crHH38MQK9evUifPr3hItfUpk0b8ubNy7lz55g8ebLRFmdcJ4ukFg2eRBxEWFgYf/31F7lz56Zt27amcx6IzWbjxIkTnDhxAmc5Li4oKIg333wTgI8//thpukVERMR5LV26lP379xMUFMQbb7xhOsdl+fj40LdvX+D2ualxcXHGWpxxnSySWjR4EnEA8fHxfPbZZwD07dsXX19fw0WurWfPngQGBrJ7925WrFhhOkdERERcmM1mY+jQoQC8+eabZMyY0WyQiwsNDSVHjhycPn2a6dOnm84RETR4EnEICxcu5MSJE2TNmpWOHTuaznF5fz9b4csvvzRcIyIiIq4sIiKCHTt2kC5dOnr27Gk6x+X5+fnx9ttvA7efaKwzPUXM0+BJxDCbzWYffvTo0YN06dIZLnIPPXv2xMvLi/Xr17Nt2zbTOSIiIuKiktZ5oaGhBAcHG65xD507dyYoKIhDhw6xcuVK0zkibk+DJxHDIiIi+O2330iXLp2ecJKGcuXKZX9y4PDhww3XiIiIiCvav38/K1euxGKx0KtXL9M5biMoKIiuXbsC2t0u4gg0eBIxLOmbYYcOHXQVLI0lbcNesGABx48fNxsjIiIiLmfEiBEAvPrqqxQqVMhwjXt58803tbtdxEFo8CRikLNfBbNYLJQoUYISJUpgsVhM5zy0MmXKUKtWLaxWK19//bXpHBEREXEhf/31FzNnzgSgT58+hmvcT+7cuWnRogVgZne7s6+TRVKSxaZnO4oY07FjR6ZMmcKrr77KokWLTOe4pbVr11KrVi0CAgI4deoUmTJlMp0kIiIiLqB///4MGzaMF154gY0bN5rOcUu7d++mbNmyeHh4cOTIEfLnz286ScQtaceTiCG6CuYYatSoQZkyZYiKimLChAmmc0RERMQFREZGMm7cOEDrPJOeeOIJatasqd3tIoZp8CRiyOjRo4mLi+P555+nQoUKpnPclsVisS8IR44cSWxsrOEiERERcXZhYWFcvXqVIkWK8Morr5jOcWtJ67xJkyZx9epVwzUi7kmDJxEDXOUqWHR0NCVLlqRkyZJER0ebzkm2Zs2akStXLv766y9mz55tOkdEREScWEJCgv1Q8d69e+Pp6Wm4yL3VrFmT0qVLp/nudldZJ4ukBA2eRAxIugpWuHBh6tevbzon2Ww2GwcOHODAgQM483FxPj4+9OzZE7j9lEFnfi8iIiJi1uLFizl+/DjBwcG0bdvWdI7bM7W73VXWySIpQYMnkTSWmJjIN998A0CvXr10FcxBdOnShfTp03PgwAHWrFljOkdERESc1FdffQVA9+7d8ff3N1wjAM2bN7fvbp83b57pHBG3o8GTSBpbtWoVR44cIVOmTLRr1850jvy/DBkyEBoaCsCoUaMM14iIiIgz2rp1K1u2bMHHx4fu3bubzpH/5+Pjw+uvvw7c3vWkHUgiaUuDJ5E0NnLkSAA6duxIQECA4Rr5u6QFycqVKzl8+LDhGhEREXE2SRevmjVrRvbs2Q3XyN917twZX19fduzYwZYtW0zniLgVDZ5E0tDBgwdZu3YtHh4e9iGHOI4iRYpQt25dbDYbY8aMMZ0jIiIiTuTcuXP227h69OhhuEb+V3BwMC1btgTuXAgWkbShwZNIGho9ejQA9evXJ3/+/GZj5K7efPNNAKZMmcLNmzcN14iIiIiz+Pbbb4mPj6dChQqUL1/edI7cRdJAcOHChZw9e9ZwjYj70OBJJI1cu3aNadOmAXeGG87OYrGQL18+8uXLh8ViMZ2TImrWrEnRokW5ceMG06dPN50jIiIiTiAuLo5x48YB2u3kyMqVK0elSpVISEhg/Pjxqfq5XHGdLJJcGjyJpJGwsDCioqIoVaoUVatWNZ2TIvz9/Tl+/DjHjx93mae2eHh42BeMo0ePxmq1Gi4SERERR7do0SLOnTtHjhw5aNy4sekc+Q9J67zx48cTGxubap/HFdfJIsmlwZNIGkhMTLTfZtejRw9d9XBw7dq1I3369Bw6dIgff/zRdI6IiIg4uKQzg1577TV8fHwM18h/adiwIblz5+bixYv2M7lEJHVp8CSSBlatWsXRo0fJlCkTrVq1Mp0j95E+fXo6dOgA6PBJERER+W/btm1jy5YteHt707VrV9M5ch/e3t50794duL3Os9lshotEXJ8GTyJpIGl40alTJwICAgzXpJyYmBjKly9P+fLliYmJMZ2Tot544w0AVq5cyeHDhw3XiIiIiKMaNWoUAM2bNyd79uyGa+RBdO7cGV9fX3bs2MGWLVtS5XO48jpZ5GFp8CSSyg4cOMDatWvx8PCwX11xFVarle3bt7N9+3aXOwupSJEivPTSS9hsNsaMGWM6R0RERBzQuXPnmDt3LqBDxZ1JcHAwLVu2BFJvd7srr5NFHpYGTyKpLGloUb9+ffLnz282Rh5K0tMHp0yZQmRkpOEaERERcTQTJ04kPj6e5557jvLly5vOkYeQNChcuHAhZ8+eNVwj4to0eBJJRTdv3mT69OnAnVu3xHnUqlWLIkWKcOPGDWbPnm06R0RERBxIQkICEyZMALTOc0blypXjhRdeICEhgUmTJpnOEXFpGjyJpKLZs2cTGRlJ0aJFqV69uukceUgeHh72Q0LHjRunwydFRETEbvny5Zw5c4bg4GBCQkJM50gydOvWDbi9cy0hIcFwjYjr0uBJJJXYbDbGjRsH3H60rsViMVwkydG+fXt8fX3ZtWsXW7duNZ0jIiIiDmL8+PEAhIaG4uvra7hGkiMkJITg4GBOnz7NihUrTOeIuCwNnkRSyZYtW9i9ezd+fn60a9fOdI4kU5YsWWjWrBmAfZAoIiIi7u3IkSOsXr0ai8Vi3x0tzsfX15fQ0FBA6zyR1KTBk0gqSfrm1bx5czJnzmy4JvUEBwcTHBxsOiNVJW3DnjdvHleuXDFcIyIiIqYlne1Uu3ZtChYsaLhGHkXS4HD16tUcOXIkRT+2O6yTRR6EBk8iqeDy5cvMnz8fuH2bnasKCAjg4sWLXLx4kYCAANM5qebZZ5+lbNmy3Lp1i6lTp5rOEREREYNu3brFlClTgDsXp8R5FSxYkNq1awN3BoopwV3WySIPQoMnkVQQFhZGbGws5cqV45lnnjGdI4/IYrHYF5bjx4/HarUaLhIRERFTFi5cyOXLl8mTJw/16tUznSMpIGmdN2XKFG7dumW4RsT1aPAkksKsVqv9akm3bt10qLiLaNmyJenTp+fPP/8kPDzcdI6IiIgYknScQpcuXfD09DRcIymhXr165M6dm8uXL7No0SLTOSIuR4MnkRT2008/cfjwYYKCgmjZsqXpnFQVExND1apVqVq1KjExMaZzUlVgYCBt27YFdPikiIiIu9qzZw+bN2/Gy8uLTp06mc6RFOLl5UWXLl2AlFvnudM6WeR+NHgSSWFJ36zatm3r8vdzW61W1q9fz/r1693i9rOk87qWLl3KmTNnDNeIiIhIWkta5zVq1IjHHnvMcI2kpE6dOuHl5cWmTZvYu3fvI388d1sni/wXDZ5EUtCZM2dYtmwZ4NqHirurUqVKUbFiRRITE5k0aZLpHBEREUlDN2/eZObMmYDWea4oR44cNGzYENDudpGUpsGTSAqaNGkSiYmJVKpUiZIlS5rOkVSQdPjkxIkTSUhIMFwjIiIiaWXWrFlERkby+OOPU61aNdM5kgqSBoozZswgMjLScI2I69DgSSSFJCYmMnnyZEBXwVxZ48aNyZIlC2fOnOGHH34wnSMiIiJpZOLEiQB07dpVD49xUdWrV6dIkSJERkYyb9480zkiLkODJ5EUsnr1ak6dOkXmzJl59dVXTedIKvH19aVdu3YAfPvtt4ZrREREJC3s2LGD3377DR8fH9q0aWM6R1KJxWKxHxqvdZ5IytHgSSSFJF0Fa9u2LX5+foZrJDV17twZgBUrVuiQcRERETeQtM5r3LgxwcHBhmskNbVv3x5vb2+2bt3Knj17TOeIuAQNnkRSwF9//cX3338P3BlKuAt/f3/8/f1NZ6SpYsWKUalSJaxWK2FhYaZzREREJBVFRkYye/ZswP3Wee4oW7ZsNGjQALgzcEwud1wni9yNBk8iKSAsLIzExEReeOEFSpQoYTonzQQEBBAVFUVUVBQBAQGmc9JU0sJz0qRJekSuiIiIC5s3bx43b96kcOHCVK1a1XSOpIEuXboAtw8Zj46OTtbHcOd1ssj/0uBJ5BFZrVYmTZoE6CqYOwkJCSFjxoycOHGCtWvXms4RERGRVJK066Vz5846VNxNvPjiixQoUIDr16+zcOFC0zkiTk+DJ5FHFB4ezrFjx8iQIQNNmjQxnSNpJF26dPbDRR91G7aIiIg4pj179vDrr7/i5eVlf7iIuD4PDw/7IeNa54k8Og2eRB5R0hMvWrdu7Xb3cN+6dYt69epRr149bt26ZTonzSXtcFu6dCnnz583XCMiIiIpLWno0KBBA7Jnz264RtJShw4d8PT0ZOPGjRw4cOCh/767r5NF/k6DJ5FHcPHiRb777jvAPW+zS0xMZOXKlaxcuZLExETTOWmudOnSPPfccyQkJDB16lTTOSIiIpKCYmJimDlzJnDnzB9xHzly5ODll18GsB+r8TDcfZ0s8ncaPIk8gmnTphEfH88zzzzDE088YTpHDNAh4yIiIq5p4cKFXLt2jfz581OjRg3TOWJA0sBx+vTp2rUk8gg0eBJJJpvN9o/DJsU9NWvWjPTp03P48GEiIiJM54iIiEgKSTpOoWPHjnh46Mcmd1S7dm3y5MnD5cuXWbJkiekcEaelr6AiybRhwwb++OMPAgMDad68uekcMSQgIIBWrVoBOnxSRETEVRw6dIiNGzfi6elJhw4dTOeIIZ6ennTs2BHQOk/kUWjwJJJMkydPBqBFixYEBgYarhGTkp56smTJEq5cuWK4RkRERB5V0jqvbt265MqVy3CNmBQaGorFYmHdunUcOXLEdI6IU9LgSSQZrl+/zsKFCwHsV0HEfT355JM88cQTxMbGMmvWLNM5IiIi8gji4+OZPn06oHWeQJ48eahVqxYAYWFhhmtEnJMGTyLJMGfOHGJiYihZsiTPPPOM6RwxzGKx2BemkydPxmazGS4SERGR5Fq+fDkXLlwge/bs1K1b13SOOICkdd7UqVP1hDqRZNDgSSQZkrZfd+zYEYvFYrjGnICAAGw2GzabjYCAANM5RrVq1QpfX192797Nb7/9ZjpHREREkilpndeuXTu8vb0N14gjqF+/PlmyZOHMmTOsXr36gf6O1skid2jwJPKQ9uzZw/bt2/H29qZNmzamc8RBZM6cmUaNGgF3FqwiIiLiXM6cOcOqVauA22f7iAD4+vra1/1a54k8PA2eRB5S0jebBg0aEBwcbLhGHEnSNuzZs2cTExNjuEZEREQe1rRp07BarVSsWJHHH3/cdI44kKR13rJly7hw4YLhGhHnosGTyEOIjY1l5syZgA6bBLh16xZNmjShSZMm3Lp1y3SOcdWrVyd//vxcv36dRYsWmc4RERGRh2C1WpkyZQqgdZ78W6lSpXjmmWdISEhgxowZ93291skid2jwJPIQvvvuO65cuULu3LmpWbOm6RzjEhMTWbhwIQsXLtRBi4CHhwcdOnQAtA1bRETE2WzYsIEjR46QPn16mjRpYjpHHNDDPExG62SROzR4EnkIScOE9u3b4+npabhGHFH79u2xWCxERERw5MgR0zkiIiLygJLWec2bN9dh0HJXzZo1I126dBw8eJAtW7aYzhFxGho8iTygEydO8OOPPwLYd7WI/K+8efNSq1YtAPt2fREREXFs165dY+HChYBus5N7y5Ahg303nHa3izw4DZ5EHlBYWBg2m43q1atTsGBB0zniwJIWrFOnTiUhIcFwjYiIiNzPnDlzuHXrFiVLluSZZ54xnSMOLGmdN2/ePCIjIw3XiDgHDZ5EHkBiYiJhYWGAroLJ/dWvX58sWbJw9uxZVq9ebTpHRERE7iNp90rHjh2xWCyGa8SRVapUiSJFihAZGcn8+fNN54g4BQ2eRB5AeHg4J0+eJGPGjDRq1Mh0jjg4X19f2rRpA+h2OxEREUe3Z88eduzYgbe3N61btzadIw7OYrEQGhoKaJ0n8qA0eBJ5AEm7nVq2bEm6dOkM14gzSFqQfP/991y8eNFwjYiIiNxL0jqvfv36ZM2a1XCNOIO2bdvi4eHBpk2b+OOPP0zniDg8DZ5E7uPq1assXrwY0KHi/8vf35/IyEgiIyPx9/c3neNQSpcuzVNPPUV8fDyzZs0ynSMiIiJ3ERcXx8yZMwGt8+TB5cyZkzp16gB3Bpf/S+tkkTs0eBK5j7lz5xIbG2sfJMgdFouFgIAAAgICdB7CXSTteko6mF5EREQcy/Lly7l06RI5cuSgdu3apnPEiSSt86ZPn37Xh8lonSxyhwZPIveRdO92hw4d9E1DHkqLFi3w9fVlz5497Ny503SOiIiI/I+k3Spt27bFy8vLcI04k1deecX+MJk1a9aYzhFxaBo8ifyHffv2sX37dry8vHTY5F3ExsbSvn172rdvT2xsrOkch5MpUyb7YfQ6fFJERMSx/PXXX6xcuRLQbXby8Hx8fOw/H9ztdjutk0Xu0OBJ5D8kfRN55ZVXdNjkXSQkJDBt2jSmTZt21y3GcmchO3v2bG7dumW4RkRERJLMmDEDq9XK888/z+OPP246R5xQ0jpv6dKlXLp06R9/pnWyyB0aPIncQ3x8PDNmzAB0FUyS78UXXyR37txcvXqVZcuWmc4RERERwGaz2S8wap0nyfXEE09Qrlw54uPjmT17tukcEYelwZPIPaxYsYKLFy/y2GOP8dJLL5nOESfl6elJ+/btAd1uJyIi4ii2bNnCoUOH8Pf3p2nTpqZzxIn9/WEyInJ3GjyJ3EPSkKBNmzY6bFIeSdLgac2aNZw+fdpsjIiIiNiHBCEhIQQFBRmuEWfWsmVLfHx82LVrlx4mI3IPGjyJ3MW5c+d02KSkmEKFClG5cmVsNhvTp083nSMiIuLWoqOjmTt3LqB1njy6zJkz06BBA0C7nkTuRYMnkbuYOXMmiYmJPPfccxQvXtx0jriAv2/DttlshmtERETc1+LFi7l58yYFCxakcuXKpnPEBSSt82bNmqUn2InchQZPIv/DZrPZb7PTVTBJKSEhIQQGBnL48GE2btxoOkdERMRtJa3z2rdvj4eHfhySR1ezZk1y5crFlStX9DAZkbvQV1qR/7F161YOHjyIn58fzZo1M53j0Pz9/blw4QIXLlzA39/fdI5DCwgIsB9eqm3YIiIiZhw7dox169ZhsVho166d6RxxEZ6enrRt2xa4s87TOlnkDg2eRP7H1KlTAWjcuDEZMmQwG+PgLBYLWbNmJWvWrFgsFtM5Di9pB938+fOJjIw0XCMiIuJ+ks5afPHFF8mbN6/hGnElSQ+TWb16NWfPntU6WeRvNHhyULGxsYSFhdGwYUPy5MmDv78/FovlP/+zZcsW09l35UzvJSYmhjlz5gC6zU5S3gsvvECRIkWIiopi0aJFpnNERMQQZ1ob3Y8zvRer1Wq/wKh1nqS0okWL8sILL2C1WpkxY4bpHBGHYrHplFuHs3//fl599VX++OOPB/47FouFGzduEBgYmIplD8/Z3svcuXNp0aIFefPm5dixY7rv/z5iY2Pp3bs3ACNGjMDX19dwkeMbNmwY/fv3p0qVKkRERJjOERGRNOZsa6P/4mzvZd26dVSvXp2goCDOnTtHunTp0rxBXNvkyZPp1KkTxYoVY+fOnbz99tuA1skiGjw5mOPHj/Pcc89x/vx5AOrXr0/btm0pUKAAFy5cICwsjPnz59tfX6dOHXx9fQkODmbSpEmmsu/KGd9L7dq1WbNmDR9++CGDBw820uBMoqKi7AvHyMhIAgICDBc5vlOnTpEvXz5sNhtHjhyhYMGCppNERCSNOOPa6F6c8b20a9eO6dOn06VLFyZMmGCkQVzbjRs3eOyxx4iJiSE8PJzq1asDWieLYBOHYbVabS+88IINsAG2CRMm3PV1LVq0sL9mwYIFD/zxk/7Oo/wnLCzMId5Lajh58qTNYrHYANvhw4eNtjiLyMhI+///IiMjTec4jVq1atkA20cffWQ6RURE0ojWeWbXeTdu3LD5+/vbANsvv/xitEVcW5s2bWyArUOHDloni/w/3UfkQObPn8+mTZsAePvtt+nSpctdX9erVy/7rx31Vh1nfC8zZszAZrNRuXJlChUqZLRFXFvS4ZPTpk3DarWajRERkTThjGuje3HG9zJ//nyio6N5/PHHefbZZ422iGtLWuctXLjQbIiIA/EyHSB3jB49GoAsWbIwcODAe76ubNmyeHh4YLVaOXPmzAN//L179z5qIrlz536g16X2e0lpNpvN/uhTHTYpqa1hw4ZkyJCBEydOEBERYd+GLSIirkvrPHPrPOAfh4rrCWOSmqpWrUr+/Pk5fvy46RQRh6HBk4M4d+4cGzduBKBly5b/eeCit7e3/Zv4wxx+XapUqUfufBAp/V62bdvGgAED+OWXX4iLi6NkyZK89dZbtGzZMsWaN2/ezOHDhwkICCAkJCTFPq7I3aRLl44WLVowfvx4wsLCNHgSEXFxWueZXef9+eefbNy4EQ8PD9q0aZNiH1fkbjw8PGjXrh2DBg0ynSLiMHSrnYPYsGGD/de1atX6z9devnyZhIQEAPLkyZOqXcmRku8lIiKCihUr8vPPPxMSEkK3bt24dOkSrVq1YtiwYSnWnLTbqUmTJg73xBhxTUnbsBctWsT169fNxoiISKrSOs/sOi9pt1Pt2rXJmTNnin1ckXtp166d6QQRh6IdTw5i37599l8/8cQT//nabdu22X9dvnz5ZH2O5MqdOzcZM2Z84M/zKO8lISGBTp06YbFY2LBhA+XKlQNgwIABVKhQgQEDBtCkSROKFCnykO/in6KiouxPXdFtdpJWnnnmGYoXL87BgwdZsGABnTp1Mp0kIiKpROs8c+u8xMREpk+fDmidJ2mnQIECVKpUiZ9//tl0iohD0ODJQZw8edL+6xw5cvzna1esWAGAxWJ5qFt0Spcunby4vwkLC7Pv1LiXlHov4eHhHDlyhA4dOtgXIwDp06fnww8/pHnz5oSFhT3yFbHFixdz8+ZNChYsSKVKlR7pY7mbdOnScezYMfuv5cFZLBbat29P3759CQsL0+BJRMSFaZ1nbp33008/cfr0aTJlysQrr7zySB9L5GF06NCBn3/+mXz58uHn52c6R8Qo3WrnIP7+ZKvY2Nh7vu769evMmjULuL1d+H7f8E1IqfeS9PSTu23jTvq99evXP2qu/Ta79u3b67DJh+Th4UH+/PnJnz//Q51DIbe1adMGT09PNm/ezO+//246R0REUonWeebXeS1bttQP/5KmmjZtSmBgICdOnGDz5s2mc0SM0k+KDiJbtmz2X2/fvv2er3vvvfe4evUqAH379n2oz2Gz2R75P/e7CpaS7+XPP/8EuOsW60yZMhEcHGx/TXIdP36cdevWYbFYaNu27SN9LJGHlSNHDurUqQPAtGnTDNeIiEhq0TrPzDrv2rVrLFmyBOCB3ptISgoICKBp06bAnXPGRNyVBk8OomLFivZfDxs2DJvN9q/XDB8+nPHjxwPQtm1bqlatmlZ5DyWl3kvSgcsZMmS46+cJCgp65EOZk37Yr169Ovny5Xukj+WO4uLieOedd3jnnXeIi4szneOUkhbC06dPJzEx0WyMiIikCq3zqv7rNWmxzps7dy6xsbGUKlWKp5566pE+lsjDiouLIzo6GoB58+YRFRVluEjEHIvtbt8tJM0lJCRQokQJ+5Wd2rVr06NHD3LkyMHRo0eZOHEia9asAW5/w1+7dq3DbhdOqfdSq1Yt1q5dy59//knhwoX/9eeFChXi9OnT/7nN+79YrVYKFy7MsWPHmDlzJq1atUrWx3FnUVFR9qcARkZGEhAQYLjI+cTGxpIzZ06uXLnCDz/8QO3atU0niYhICtM6L+3XeQDPPfccv/76K8OHD6d3797J/jgiyfH3dTLcvuCtOyzEXWnw5ED27t1LtWrVuHz58j1f06FDB8aMGePwBzmnxHtp0qQJCxcuZPv27Xe9SpU1a1YsFgsXLlxIVmNERATVqlUjKCiIw4cP4+/vn6yP486ioqLInj07oMHTo+jRowejR4+mWbNmzJ0713SOiIikAq3z/im113kHDx6kRIkSeHp68scff9jXK/LwtL5Lnv8dPFWtWpV169YZLBIxR0+1cyClS5dm//79fP7556xcudL+1JBcuXJRrVo1unTp4jTbhFPivSTd8//nn3/+67VXr17l0qVLPP/888luTDpsslmzZv84r0AkrXXo0IHRo0fz3XffcfXqVTJlymQ6SUREUpjWef+UVuu8evXqUahQoWR/HOGut1PKw4uIiODo0aMULFjQdIpImtOOJ3FYq1evpk6dOnTo0IEpU6b848/mzZtH8+bN6devX7Ies3vz5k0ee+wxoqOj2bx58yMtbOQ27XhKPpvNRtmyZdmzZw9jx46lW7duppNERERSVWqu8xISEsiTJw/nzp1jyZIlNGrUKKWy3ZJ+XEyev+94ql69OuHh4QwYMICBAweaDRMxQIMncVgJCQk8/vjjnDlzhi1btlC2bFng9tCoQoUK/P777+zfv5+iRYs+9MeeMmUKHTt25PHHH+fgwYP2g//k4ehWu5Tz1Vdf0bt3b8qXL8/WrVtN54iIiKSq1FznrVixgpdffpng4GDOnDlDfHx8Cte7F63vkufvg6fJkyfTsWNH8uXLx9GjR/Hw0DO+xL3oVjtxWF5eXkyaNInatWtTqVIlWrRoQVBQEIsXL+bYsWN8/PHHyVqMwJ3t1+3bt8disegbqhjXqlUr3n33XbZt28b+/fspWbKk6SQREZFUkxbrvNatW+Pj44OPj09Kpos8tPr16xMUFMSJEyeIiIigevXqppNE0pRGreLQqlWrxsaNG6lYsSLz589n7NixZMmShZkzZ9K/f/9kfczDhw+zceNGPDw8aNOmTQoXiyRPtmzZqFevHgBTp041GyMiIpIGUmOdd/nyZZYtWwbcvsAo4gjSpUtH8+bNAa3zxD3pVjtxOx988AFDhw6lTp06rFq1ynSOU7NarRw8eBCA4sWLa9vwI1q6dCkNGzYke/bsnDp1Cm9vb9NJIiIiTmXUqFG8+eablCtXjt9++810jrix/10nb926lQoVKpAuXTrOnTtHUFCQ4UKRtKOfEsWtJCYmMm3aNOD2k8Tk0Xh4eFCyZElKliypoVMKqFu3LlmzZuX8+fOsXr3adI6IiIjTSdpNonWemPa/6+Rnn32WYsWKERMTw4IFC0zniaQp/aQobiU8PJzTp0+TMWNG6tevbzpH5B+8vb1p3bo1cOd8ChEREXkwe/bs4bfffsPb25sWLVqYzhH5B4vFYr/9U+s8cTcaPIlbSfoi37JlS/z8/AzXOL+4uDgGDhzIwIEDiYuLM53jEpIWJN9//z2XLl0yGyMiIuJEktZ59evXJzg42HCNuLu7rZPbtGmDh4cHmzZt4o8//jBcKJJ2dMaTuI1r166RI0cObt26xdatWylfvrzpJKf398fERkZG6umAKeSpp57it99+4+uvv6Znz56mc0RERBxeXFwcuXLl4tKlS3z//fe8/PLLppPEzd1rnVy3bl1WrVpFv379GDZsmMlEkTSjHU/iNubOncutW7coVaoUTz/9tOkckXsKDQ0FYMqUKejagIiIyP2tWLGCS5cu8dhjj1GnTh3TOSL3lLTOmzZtGomJiYZrRNKGBk/iNqZMmQLcPmzSYrEYrhG5txYtWuDj48OePXvYuXOn6RwRERGHl7TOa9u2LV5eXoZrRO7tlVdeIXPmzJw9e5Y1a9aYzhFJExo8iVvYv38/27Ztw8vLy354s4ijypw5M40aNQJ0+KSIiMj9nDt3jlWrVgF6mp04Pl9fXz1MRtyOBk/iFpK+qL/88stky5bNcI3I/SUtnGfNmsWtW7cM14iIiDiuGTNmkJiYSIUKFShWrJjpHJH7SlrnLV26lMuXLxuuEUl9GjyJy4uPj2fGjBmAroKJ86hRowa5c+fm6tWrLFu2zHSOiIiIQ7LZbP84TkHEGZQtW5ayZcsSFxfH7NmzTeeIpDoNnsTlrVy5kgsXLpA9e3Zeeukl0zkiD8TT05N27doB2oYtIiJyL7/++iuHDh0iXbp0NGvWzHSOyANLOmRc6zxxBxo8ictL+mLepk0bvL29Dde4Fj8/P7Zu3crWrVvx8/MzneNy2rdvD8CaNWs4c+aM2RgREREHlLTOCwkJISgoyHCNyB33Wye3bNkSHx8fdu7cye7duw0UiqQdDZ7EpZ0/f54VK1YA2n6dGjw9PSlfvjzly5fH09PTdI7LKVy4MJUqVcJqtTJ9+nTTOSIiIg4lOjqauXPnAlrnieO53zo5S5Ys1K9fH9CuJ3F9GjyJS5s5cyYJCQk8++yzlChRwnSOyENL2oY9ZcoUbDab4RoRERHHsXjxYm7cuEGBAgWoUqWK6RyRh5a0zps5cyZxcXGGa0RSjwZP4rJsNpv96oGugqWOuLg4vvjiC7744gt9s0wlISEhBAQEcPjwYTZt2mQ6R0RExGEkrfPat2+Ph4d+rBHH8iDr5Fq1apEzZ04uX77M999/n8aFImnHYtMldHFR27Zt45lnnsHPz49z586RIUMG00kuJyoqisDAQAAiIyMJCAgwXOSaQkNDCQsLIzQ0lMmTJ5vOERERMe748eMUKFAAi8XCsWPHyJcvn+kkkX940HVyv379+PTTT6lXrx7Lly9Py0SRNKNLA+Kykh6t27hxYw2dxKklbcOeN28ekZGRhmtERETMmzp1KgAvvviihk7i1JLuzFi1ahVnz541XCOSOjR4EpcUHR3N7NmzAd1mJ87vhRdeoEiRIkRFRTF//nzTOSIiIkYlJibqOAVxGUWLFqVixYpYrVamTZtmOkckVWjwJC7p74dNVqtWzXSOyCOxWCz/OGRcRETEnYWHh3Py5EkyZsxIo0aNTOeIPDI9TEZcnQZP4pKSzsHp0KGDDpsUl9CuXTs8PT3ZtGkThw4dMp0jIiJiTNI6r1WrVqRLl85wjcija9KkCYGBgRw+fJiff/7ZdI5IitNP5OJyjhw5QkREBBaLhfbt25vOEUkROXLk4KWXXgLuPMVHRETE3Vy5coUlS5YAd3aJiDi7wMBAmjdvDqAHyYhL0uBJXE7SrUi1atUiT548hmtEUk7Hjh0BmDZtGvHx8YZrRERE0t6sWbOIi4ujbNmyPPnkk6ZzRFJM0iB1wYIFXL9+3XCNSMrS4ElcSmJiov0pJ0k/pEvq8fPzY926daxbtw4/Pz/TOS6vXr16ZMuWjfPnz7Ny5UrTOSIiImnKZrPZd4NonSeO7mHXyc899xzFixcnJiaGefPmpUGhSNrR4ElcyurVqzl79ixZsmShfv36pnNcnqenJ1WrVqVq1ap4enqaznF53t7etG3bFtAh4yIi4n527tzJ7t278fX1pWXLlqZzRP7Tw66TLRaLfaCq2+3E1WjwJC4l6Yt069at8fX1NVwjkvKStmGvWLGCv/76y3CNiIhI2kla5zVq1IjMmTMbrhFJeW3atMHLy4utW7eyb98+0zkiKUaDJ3EZFy9eZNmyZYC2X6eV+Ph4xowZw5gxY3TmUBopXrw4FSpUIDExkenTp5vOERERSRMxMTHMmjUL0KHi4hySs07Oli0br7zyCqDd7eJaLDabzWY6QiQljBgxgrfffpvy5cuzdetW0zluISoqisDAQAAiIyMJCAgwXOQeJk+eTKdOnShatCiHDh3CYrGYThIREUlVs2fPplWrVuTLl4+jR4/i4aHr5+LYkrtOXrFiBS+//DLBwcGcOXMGHx+f1MwUSRP6ii0u4e+HTeoqmLi6pk2bEhAQwB9//MGmTZtM54iIiKS6pHVehw4dNHQSl1a7dm1y5MjBpUuX7HdziDg7fdUWl/Drr79y4MAB0qVLR4sWLUzniKSq9OnT07RpU0CHT4qIiOs7duwY4eHhWCwW2rdvbzpHJFV5eXnZ/3eudZ64Cg2exCUkfVEOCQkhQ4YMhmtEUl/SOWbz58/nxo0bhmtERERST9JZNzVq1CBfvnyGa0RSX9IdHKtXr+bUqVOGa0QenQZP4vRu3rzJnDlzAOjUqZPhGpG08fzzz1O8eHGio6Pt//sXERFxNQkJCfbBk9Z54i4KFy5M1apVsdlsOmRcXIIGT+L05s6dS1RUFEWLFqVSpUqmc0TShMVisS/AJ06caLhGREQkdaxatYqzZ88SHBxMgwYNTOeIpJnOnTsDt+/sSExMNFwj8mg0eBKnN2nSJOD2VTA93UvcSZs2bfD29mbHjh3s3LnTdI6IiEiKS7q40rZtW3x9fQ3XiKSdV199lUyZMnHq1CnWrl1rOkfkkWjwJE5tz549bN26FS8vL9q1a2c6x+34+vqyfPlyli9frsWgAVmzZqVRo0bAnQGsiIiIqzhz5gwrVqwAdJudOJ9HXSf7+fnRpk0bQLvbxflZbDabzXSESHL16NGD0aNH07hxYxYuXGg6RyTN/fjjj9SsWZMMGTJw9uxZ/P39TSeJiIikiKFDh/LBBx9QsWJFfv75Z9M5Imlu7969lClTBi8vL06fPk327NlNJ4kki3Y8idOKiYlh5syZwJ17oEXcTfXq1SlQoADXr19nwYIFpnNERERShNVqtT+1WLudxF2VLl2aZ599loSEBKZNm2Y6RyTZNHgSp7Vo0SKuXbtGvnz5qFmzpukctxQfH8/UqVOZOnUq8fHxpnPckoeHBx07dgS0DVtERFzHTz/9xLFjx8iQIQNNmjQxnSPy0FJqnZx0gX3SpEnoZiVxVrrVTpxWlSpV2LBhA4MGDeKjjz4yneOWoqKiCAwMBCAyMpKAgADDRe7p7Nmz5M2bl8TERA4cOEDx4sVNJ4mIiDySZs2aMX/+fLp3786YMWNM54g8tJRaJ0dGRpIjRw4iIyNZt24dVatWTcFKkbShHU/ilP744w82bNiAh4cHoaGhpnNEjMqZMyf16tUDdMi4iIg4v4sXL7JkyRJAxymIBAYG0qJFC0DrPHFeGjyJU0r6ovvSSy+RO3duwzUi5iWdfzFt2jRiY2MN14iIiCTf9OnTiY+P56mnnqJs2bKmc0SMSxrALly4kCtXrhiuEXl4GjyJ04mLi2Pq1KmADpsUSfLSSy+RM2dOLl++zHfffWc6R0REJFlsNpv9zELtdhK57emnn6ZMmTLExsbaH64k4kw0eBKns2zZMi5evMhjjz1mv71IxN15eXnZbzvVIeMiIuKsNm7cyO+//46/v7/99iIRd2exWOyD2IkTJ+qQcXE6GjyJ05kwYQIAHTp0wNvb23CNiOPo2LEjFouFn376iT///NN0joiIyENLWuc1b96coKAgwzUijqN169akS5eOffv28csvv5jOEXkoGjyJUzl8+DA//vgjFouFLl26mM4RcSj58+enTp06AHz77beGa0RERB7OpUuXWLBgAQCvvfaa4RoRx5IxY0aaNWsGwPjx4w3XiDwcDZ7EqST9MF2nTh3y589vNkbw9fVl/vz5zJ8/H19fX9M5AnTt2hWAqVOn6pBxERFxKtOmTSMuLo5y5crx9NNPm84ReSSpsU5OGsjOnz9fh4yLU7HYdIOoOInY2Fhy587NpUuX+O6772jQoIHpJBGHk5CQQP78+Tlz5gyzZ8/W+RgiIuIUbDYbjz/+OH/++ScTJkzQznaRu7DZbJQrV47du3fz1Vdf8dZbb5lOEnkg2vEkTmPx4sVcunSJXLly6VBxkXvw8vKyP+1R27BFRMRZrFu3jj///JPAwEBdNBG5B4vFYt/1NH78eB0yLk5DgydxGkmHTXbq1AkvLy/DNQK3d9csWLCABQsWkJCQYDpH/l+nTp3w8PBgw4YNHDx40HSOiIjIfSWt81q3bk369OkN14g8utRaJ7ds2ZKAgAB+//13NmzYkGIfVyQ16VY7cQoHDx6kRIkSeHh4cOLECXLnzm06SYCoqCgCAwMBiIyMJCAgwHCRJGnQoAHLli2jZ8+efP3116ZzRERE7un8+fPkzp2bhIQEdu7cSdmyZU0niTyy1Fwnd+nShYkTJ9K8eXPmzJmTYh9XJLVox5M4haRDxV9++WUNnUQeQNIh49OmTSMmJsZwjYiIyL2FhYWRkJDAM888o6GTyANIWuctWrSIixcvGq4RuT8NnsThxcTEMG3aNODOF1kR+W+1a9cmX758XLt2zf5oahEREUdjtVrtFxiTzq4Rkf/21FNP8fTTTxMfH8/UqVNN54jclwZP4vAWLFjA1atXyZcvH7Vr1zadI+IUPD096dy5M6BDxkVExHGtXbuWY8eOkSFDBpo1a2Y6R8RpJA1qJ0yYgNVqNVwj8t80eBKHl3TYZOfOnfH09DRcI+I8QkND8fLy4pdffmHv3r2mc0RERP4laZ3Xpk0b/P39DdeIOI/mzZsTFBTEkSNHCA8PN50j8p80eBKHtnfvXjZv3oyXlxehoaGmc0ScSo4cOWjQoAEA48aNM1wjIiLyT2fOnGHZsmWAjlMQeVgBAQG0adMG0DpPHJ8GT+LQxowZA0DDhg3JkSOH4RoR59OtWzcAZsyYwY0bNwzXiIiI3PHtt9+SmJhIxYoVKVWqlOkcEaeTdLvd0qVLOX36tOEakXvT4Ekc1vXr15k5cyYAr7/+uuEauRsfHx/CwsIICwvDx8fHdI7cRfXq1Xn88ceJjIxkxowZpnNEREQAiIuLsx8qrnWeuKK0WCeXKlWKypUrk5iYaP/3JOKILDabzWY6QuRuRo4cSc+ePSlZsiR79+7FYrGYThJxSkn/looXL87+/fv1b0lERIybN28ezZs3J3v27Jw8eVIXsESSaf78+TRr1kz/lsShaceTOCSbzcbYsWMB6N69u35QFnkE7dq1IyAggIMHD7J+/XrTOSIiIvZ1XpcuXfSDssgjaNSoETly5OD8+fMsWbLEdI7IXWnwJA4pPDyc33//ncDAQPuheeJ4EhISWLFiBStWrCAhIcF0jtxDhgwZaN26NXDn3DQRERFT9u7dy4YNG/D09NSh4uKy0mqd7O3tTZcuXQCt88Rx6VY7cUivvvoqS5YsoXv37voC6sCioqIIDAwEIDIykoCAAMNFci979+6lTJkyeHp6cuLECXLlymU6SURE3FS3bt0YP348jRs3ZuHChaZzRFJFWq6Tz5w5Q758+UhMTGTPnj2ULl061T6XSHJox5M4nFOnTrF06VLg9m12IvLoSpcuTaVKlXT4pIiIGHX9+nX7wy50qLhIysiVKxeNGjUCtOtJHJMGT+JwJkyYgNVqpWrVqpQsWdJ0jojLSFrgf/vtt8TFxRmuERERdzR9+nSioqIoUaIEVatWNZ0j4jKS1nkzZ87k+vXrhmtE/kmDJ3EocXFxTJw4EdBVMJGU1qhRIx577DHOnTvHd999ZzpHRETcjB4eI5J6qlSpQsmSJYmKimL69Ommc0T+QYMncSiLFi3iwoUL5MyZkwYNGpjOEXEpPj4+dO7cGdA2bBERSXvr1q3j0KFDeniMSCqwWCz2Y0rGjh2LjnIWR6LBkziUpB+Gu3Tpgre3t+EaEdfTtWtXPD092bBhA3v37jWdIyIibiRpndemTRuCgoIM14i4ntatWxMYGMihQ4cIDw83nSNip8GTOIydO3eyadMmvLy87I8EFZGUlStXLho2bAjA6NGjzcaIiIjbOHnypB4eI5LKgoKCaNu2LQCjRo0yXCNyhwZP4jC++eYbAJo0aUKOHDkM18iD8PHxYfTo0YwePRofHx/TOfKA3nzzTQBmzJjBlStXDNeIiIg7GDNmDImJiVSvXp1SpUqZzhFJdabWyT169ABg2bJlHD16NM0+r8h/sdh086c4gPPnz5M3b17i4uLYsmULzz77rOkkEZdls9l48skn2bVrF59++il9+/Y1nSQiIi4sKiqKPHnycPXqVZYuXUr9+vVNJ4m4tDp16rB69Wp69erFiBEjTOeIaMeTOIYJEyYQFxfHs88+q6GTSCqzWCz07NkTuH0FOiEhwXCRiIi4spkzZ3L16lUKFixIvXr1TOeIuLykdd7kyZO5efOm4RoRDZ7EAcTFxTFu3DjgzhdJcQ6JiYlEREQQERFBYmKi6Rx5CM2bNydr1qycOnWKJUuWmM4REREXZbPZGDlyJHD7FiBPT0/DRSJpw+Q6uXbt2hQtWpQbN24wbdq0NP3cInejwZMYN3/+fM6dO0fOnDkJCQkxnSMP4datW1SrVo1q1apx69Yt0znyEPz8/HjttdeAO+eriYiIpLQff/yRAwcOEBgYSIcOHUzniKQZk+tkDw8P+5meI0eOxGq1punnF/lfGjyJUTabzf5Db/fu3fH29jZcJOI+unXrhre3N5s2bWLHjh2mc0RExAUlrfM6dOhAhgwZDNeIuI927dqRIUMG/vzzT3744QfTOeLmNHgSo3755Re2b9+Or68vXbp0MZ0j4lZy5MhB06ZNAe16EhGRlPfnn3+yYsUKLBaL/UlbIpI2AgMD6dixI6B1npinwZMYlfRFsFWrVmTNmtVwjYj7STpXbe7cuZw7d85wjYiIuJJRo0YBULduXYoUKWK4RsT9vPHGG3h4eLBmzRoOHjxoOkfcmAZPYsypU6dYtGgRoEPFRUwpX748FSpUID4+ngkTJpjOERERF3Hjxg3CwsIArfNETClQoAD169cHsB/yL2KCBk9izNixY0lMTKRq1aqUKVPGdI6I20r6gWDcuHHExsYarhEREVcQFhZGZGQkJUqUoEaNGqZzRNxW0jpv+vTpXL161XCNuCsNnsSIqKgo++4KXQUTMevVV18lV65cnD9/njlz5pjOERERJ5eYmGjfXfHmm29isVgMF4m4rypVqlCmTBmio6O1u12M0eBJjAgLC+Pq1asUKlSIV155xXSOJJO3tzeff/45n3/+uZ5I6MS8vb3tj9wdPnw4NpvNcJGIiDizJUuWcPToUbJkyUKbNm1M54gY4SjrZIvFQu/evYHbt9vFxcUZaxH3ZbHpJwxJY4mJiRQpUoRjx44xZswYunfvbjpJxO1du3aNPHnyEBkZyQ8//EDt2rVNJ4mIiBOy2Ww899xzbN26lY8++ohBgwaZThJxe3FxcRQoUICzZ88ydepU2rVrZzpJ3Ix2PEmaW7JkCceOHSNLliy0b9/edI6IABkzZqRz584AfPnll4ZrRETEWW3atImtW7fi6+vL66+/bjpHRAAfHx/77vYvv/xSu9slzWnwJGnKZrPZf6jt3r07/v7+hovkUSQmJrJt2za2bdtGYmKi6Rx5RD179sTT05Mff/yRXbt2mc4REREnlLTOa9euHdmyZTNcI2KOo62Tu3btSmBgIPv27WPNmjWmc8TN6FY7SVObNm2iYsWK+Pr6cuLECbJnz246SR5BVFQUgYGBAERGRhIQEGC4SB5Vy5YtmTNnDq1bt2bGjBmmc0RExIn8/vvvFC9eHJvNxqFDh3j88cdNJ4kY44jr5F69evH1119Ts2ZNDZ8kTWnHk6SppKtgbdu21dBJxAG9/fbbAMydO5dTp04ZrhEREWfy1VdfYbPZqF+/voZOIg4oaXf72rVr2b17t+kccSMaPEma+eOPP1i6dCmA/ckKIuJYnnrqKapVq0ZCQoL9UdgiIiL3c+HCBaZNmwZAnz59DNeIyN3kz5+fJk2aALefZCySVjR4kjSTdBXslVdeoVixYqZzROQekn5gmDBhAtevXzdcIyIizmDs2LHcunWLZ555hooVK5rOEZF7SNrdPmfOHE6fPm24RtyFBk+SJi5evMjUqVOBO1/sRMQx1alTh+LFi3Pz5k0mTZpkOkdERBxcdHQ0Y8aMAW6v8ywWi+EiEbmXp59+mipVqmh3u6QpDZ4kTSRdBXv66aepXLmy6RwR+Q8eHh72AfHXX39NfHy84SIREXFk06dP59KlS+TPn59XX33VdI6I3Mffd7ffuHHDcI24Aw2eJNVFRkbap+l9+vTRVTARJ9CqVSuyZ8/O6dOnmT17tukcERFxUAkJCXzxxRfA7SdmeXl5GS4SkfupW7cuxYsX58aNG4wfP950jrgBDZ4k1U2cOJErV65QuHBhQkJCTOdICvL29mbAgAEMGDAAb29v0zmSgvz8/OwPAfj000+xWq2Gi0RExBHNnz+fo0ePEhwcTMeOHU3niDgMR14ne3h40LdvXwBGjBhBTEyM4SJxdRabzWYzHSGuKzY2loIFC3L27FkmTpxIp06dTCeJyAO6ceMG+fLl49q1ayxatEi3T4iIyD9YrVaeeOIJ9u3bx8cff0z//v1NJ4nIA4qPj6dw4cKcPHmSMWPG0L17d9NJ4sK040lS1YwZMzh79iw5c+akTZs2pnNE5CEEBQXxxhtvADBs2DB0nUJERP5uxYoV7Nu3j/Tp0/P666+bzhGRh+Dt7c0777wDwBdffKEzPSVVafAkqSYhIYFPP/0UuH22k6+vr+EiSWlWq5X9+/ezf/9+3Yrlonr27Im/vz87duxg7dq1pnNERMRB2Gw2hg0bBkD37t3JmDGj2SARB+MM6+SOHTuSLVs2jh8/zty5c03niAvT4ElSzcKFCzly5AhZsmShc+fOpnMkFcTExFCqVClKlSqle8NdVHBwMF26dAHgk08+MVwjIiKOYv369WzZsgU/Pz969eplOkfE4TjDOjldunT2f7+ffPKJww7IxPlp8CSpwmaz2X9IffPNNwkMDDRcJCLJ9fbbb+Pt7U1ERASbN282nSMiIg4gabdTaGgo2bNnN1wjIsnVrVs3goKCOHjwIEuXLjWdIy5KgydJFStXrmTPnj0EBgbaz4gREeeUO3du2rZtC2jXk4iIwPbt21m7di2enp72M2JExDllyJDB/vPaJ598ojM9JVVo8CQpzmazMXToUOD2BD1z5syGi0TkUfXt2xcPDw+WL1/Onj17TOeIiIhBSRchWrVqRf78+c3GiMgj69mzJ+nSpWPbtm389NNPpnPEBWnwJCluw4YN/PLLL/j6+uqefxEXUaRIEZo0aQJo15OIiDs7ePAgixcvxmKx8N5775nOEZEUkC1bNvuZvEkbCERSkgZPkuIGDx4MQIcOHciRI4fhGhFJKUk/YMyfP5/ff//dcI2IiJjw8ccfA9CwYUOKFy9uuEZEUkqfPn3w8vIiIiKCjRs3ms4RF6PBk6SoDRs2EB4ejre3t66CibiYsmXLUr9+faxWK0OGDDGdIyIiaezQoUPMmTMHgA8//NBwjYikpDx58tChQwcABg4caDZGXI4GT5Kikr5IhYaGki9fPrMxkuq8vb3p06cPffr0wdvb23SOpIGkf+Nz5szh0KFDZmNERCRNDR48GJvNRsOGDSlXrpzpHBGH5ozr5P79++Pl5cVPP/3Ezz//bDpHXIjFpmPrJYWsX7+eqlWr4u3tzeHDh8mbN6/pJBFJBQ0bNmTp0qW0aNGC2bNnm84REZE0cODAAUqVKoXNZmPnzp2ULVvWdJKIpIKuXbvy7bffUr16dR00LilGO54kxSTthOjYsaOGTiIubMCAAQDMnTuXgwcPGq4REZG0MGTIEGw2G40aNdLQScSFvf/++3h7exMeHs6GDRtM54iL0OBJUkRERAQRERF4e3vTr18/0zmSRqxWK8ePH+f48eNYrVbTOZJGypUrR8OGDbHZbPaHCYiIiOvav38/8+bNA+5cfBCR/+as6+R8+fIRGhoK6KwnSTm61U5SRNWqVVm/fj3dunVj7NixpnMkjURFRREYGAhAZGQkAQEBhoskrezatYty5cphsVjYt28fJUqUMJ0kIiKppHnz5sybN49XX32VRYsWmc4RcQrOvE4+efIkhQsXJj4+noiICKpUqWI6SZycdjzJI4uIiGD9+vX4+Phot5OImyhbtiyNGjXSricRERe3f/9+5s+fD2i3k4i7yJs3Lx07dgS060lShgZP8khsNpt9EdKpUyfy5MljuEhE0krSQmT+/Pns37/fbIyIiKSKQYMGYbPZaNy4MWXKlDGdIyJp5P3338fHx8d+pIrIo9DgSR7JunXr2LBhg3Y7ibihMmXK0LhxY2w2G4MGDTKdIyIiKWzv3r0sWLAA0G4nEXeTJ08eOnXqBNz+968TeuRRaPAkyWaz2ezDps6dO5M7d27DRSKS1gYMGIDFYmHBggXs2LHDdI6IiKSg/v37AxASEkLp0qUN14hIWuvXrx++vr5s2LCB1atXm84RJ6bBkyTbd999x9atW/H39+eDDz4wnSMiBpQuXZqWLVsCt7dki4iIa9i0aRPff/89np6efPzxx6ZzRMSA3Llz8/rrrwO3h1DO9HQ+cSwaPEmyJCQk2H/I7N27N4899pjhIhExZfDgwXh7e7NmzRrCw8NN54iIyCOy2Wy89957AISGhvL4448bLhIRU95//32CgoLYtWsX8+bNM50jTkqDJ0mW6dOnc+jQIbJkyUKfPn1M54ghXl5edO/ene7du+Pl5WU6RwwpWLAgXbt2BeC9997TGQAiIk5u5cqVbNy4ET8/P53tJJJMrrJOzpIlC++88w4AH3zwAXFxcYaLxBlZbPoJQR5STEwMRYsW5fTp0wwfPpzevXubThIRw86fP0+hQoWIiopi4cKFNG7c2HSSiIgkQ2JiIuXKlWPv3r28++67fPbZZ6aTRMSwyMhIChcuzPnz5xkzZgzdu3c3nSRORjue5KGNHTuW06dPkydPHn3REREAsmfPbh9C9+/fn4SEBMNFIiKSHHPmzGHv3r1kzJiRvn37ms4REQcQGBjIhx9+CNw+YiEqKspwkTgbDZ7koVy/fp1hw4YBMGjQIPz8/AwXiUk2m42LFy9y8eJF3V4l9OnTh+DgYH7//XemTp1qOkdERB5SbGys/YfLvn37kjlzZsNFIs7L1dbJnTt3pmDBgpw/f56vv/7adI44GQ2e5KF88cUXXLlyhRIlStC2bVvTOWJYdHQ02bJlI1u2bERHR5vOEcOCgoLsj94eOHAgMTExhotERORhfPvttxw/fpwcOXLw5ptvms4RcWqutk728fGxP+Hy888/5/Lly4aLxJlo8CQP7K+//uKrr74CYOjQoXh6ehouEhFH89prr5E3b17OnDnDqFGjTOeIiMgDunnzJkOGDAFgwIAB+Pv7Gy4SEUfTrFkzypYty40bN+x3wYg8CA2e5IF98MEHREdH89xzz9GgQQPTOSLigPz8/Bg8eDBwe0B98eJFw0UiIvIgPvnkEy5evEiRIkUIDQ01nSMiDsjDw4NPPvkEgFGjRnHkyBHDReIsNHiSB7Jz507CwsIAGDFiBBaLxXCRiDiqNm3aUK5cOW7cuKHHcIuIOIHjx48zYsQI4PaxCt7e3oaLRMRR1a5dm1q1ahEfH8+7775rOkechAZPcl82m41evXphs9lo0aIFFSpUMJ0kIg7Mw8PDflvuhAkT2Ldvn+EiERH5L3379iU2Npbq1atTv3590zki4sAsFgsjRozAw8ODxYsXExERYTpJnIAGT3Jf3333HevXr8fPz49PP/3UdI6IOIEqVarQuHFjrFYrvXv3domnuYiIuKJNmzYxf/58+0UD7WoXkfspWbIkr732GgC9e/cmMTHRcJE4Og2e5D/FxsbSp08f4Paj0vPmzWu4SEScxeeff46Pjw9r165l5cqVpnNEROR/WK1W3nrrLQA6duxImTJlzAaJiNMYNGgQGTJkYOfOnUybNs10jjg4DZ7kP40aNYqjR4+SI0cO+vbtazpHHIyXlxft2rWjXbt2eHl5mc4RB1OwYEH7DzRvv/028fHxZoNEROQfZs2axfbt20mfPr39iXYikjJcfZ0cHBzMRx99BED//v25efOm4SJxZBab7n+Qe7hw4QJFihThxo0bhIWF0b59e9NJIuJkbty4QeHChbl48SLffPMNb775pukkEREBoqKiKFq0KGfPnuXTTz/VBUYReWhxcXGULFmSw4cP8/777zN06FDTSeKgtONJ7mnAgAHcuHGDJ598krZt25rOEREnFBQUxMcffwzAwIEDuXLliuEiERGB20+vO3v2LAUKFKBnz56mc0TECfn4+PDll18CMHz4cE6cOGG4SByVdjzJXe3atYunnnoKq9XKhg0bqFSpkukkcUA2m43o6GgA/P39dSCp3FViYiLlypVj7969dO/enTFjxphOEhFxa8ePH6dEiRLExMSwYMECQkJCTCeJuBx3WSfbbDZefPFF1q1bR+PGjVm4cKHpJHFAGjzJv1itVl544QW2bNlC06ZNmTdvnukkcVBRUVEEBgYCEBkZSUBAgOEicVTr1q2jevXqWCwWtm3bxlNPPWU6SUTEbTVo0IBly5ZRpUoV1q1b57I/EIuY5E7r5D179vDkk0+SmJjIqlWrqFOnjukkcTC61U7+ZcqUKWzZsoXAwEBGjBhhOkdEXEC1atVo2bIlNpuNbt266bG7IiKGLFu2jGXLluHl5cXYsWM1dBKRR1amTBn7OZ5vvPEGt27dMlwkjkaDJ/mHS5cu2Q+XHDx4MLly5TJcJCKu4ssvvyQoKIht27YxceJE0zkiIm4nOjra/sPh22+/TYkSJQwXiYirGDhwIDlz5uTIkSN89tlnpnPEwWjwJP/w3nvvceXKFcqUKUOPHj1M54iIC8mRI4f9oPF+/fpx4cIFw0UiIu5l6NChnDhxgjx58vDhhx+azhERFxIUFMRXX30FwCeffMLhw4cNF4kj0eBJ7DZv3szkyZMBGDt2LF5eXoaLRMTVdOvWjXLlynHt2jXeffdd0zkiIm7j0KFDfPHFFwCMHDnSpc+bEREzmjRpQs2aNYmNjeWNN95Ax0lLEg2eBICEhAS6desGQGhoKC+88ILhIhFxRV5eXowbNw6LxcK0adPYsGGD6SQREZdns9l4/fXXiY+Pp169ejRo0MB0koi4IIvFwujRo/Hx8WH16tUsXrzYdJI4CA2eBIDRo0ezZ88eMmfOrHtyRSRVPfvss3Tu3BmA7t27Ex8fb7hIRMS1zZ07l/DwcPz8/Bg5cqQOFBeRVFO0aFH7mcE9e/bk5s2bhovEEWjwJJw4ccJ+n/+nn35KcHCw4SJxFp6enoSEhBASEoKnp6fpHHEiw4YNIzg4mP3799tv/RARkZR3+fJlevXqBUD//v0pWLCg4SIR9+DO6+R+/fpRoEABzpw5wwcffGA6RxyAxaYbL92azWajTp06rFmzhhdeeIENGzbg4aF5pIikvhkzZtC2bVt8fHzYtWsXxYsXN50kIuJy2rZty4wZMyhevDg7d+7E19fXdJKIuIHVq1dTp04dLBYLP//8s45ycXOaMLi5qVOnsmbNGvz8/JgyZYqGTiKSZlq3bk3dunWJi4sjNDSUxMRE00kiIi5lxYoVzJgxAw8PD8LCwjR0EpE0U7t2bTp06IDNZiM0NJSYmBjTSWKQpgxu7OzZs/at14MHD6Zo0aKGi0TEnVgsFiZMmEBQUBBbtmxh5MiRppNERFzG9evX6dq1KwC9evXi2WefNVwkIu5m+PDh5MiRgz/++INBgwaZzhGDdKudm7LZbDRo0IDvv/+e8uXLs3nzZry8vExniZOJiooiMDAQgMjISD2aWZJl4sSJdOnShXTp0rFnzx4KFy5sOklExOl16dKFiRMnUrhwYXbv3o2/v7/pJBG3onXybcuWLaNBgwZ4eHiwZcsWypcvbzpJDNCOJzc1Z84cvv/+e7y9vZkyZYqGTiJiTKdOnXjxxReJiYmhY8eOWK1W00kiIk7txx9/ZOLEiQBMnjxZQycRMaZ+/fq0aNECq9VKaGgosbGxppPEAA2e3ND58+fp0aMHAB999BGlSpUyXCQi7sxisTBx4kT8/f3ZsGED48ePN50kIuK0IiMj6dSpEwCvv/46lStXNlwkIu5u5MiRZM2alX379jFs2DDTOWKABk9uxmaz8frrr3PlyhXKli1L3759TSeJiFCgQAE+/fRTAN59912OHTtmuEhExDn17duXEydOkC9fPj755BPTOSIiBAcHM2bMGACGDRvGzp07DRdJWtPgyc1MmzaNRYsW4eXlxZQpU/D29jadJCIC3L4yX6lSJaKiomjdujUJCQmmk0REnMrKlSsZO3YscPv8vPTp0xsuEhG5LSQkhMaNG5OQkECrVq2Ijo42nSRpSIMnN3L48GH7LXZDhgyhXLlyhotERO7w8PBg+vTpBAUFsXnzZm3FFhF5COfPn6dDhw4A9OzZk5o1axouEhG5w2KxMH78eHLkyMHBgwd55513TCdJGtLgyU3Ex8fTunVrIiMjqVKliv6hi4hDyp8/v/1q/eDBg/nll18MF4mIOD6bzUZoaCgXLlygVKlS9luXRUQcSXBwMNOmTQNg7NixLF++3HCRpBUNntzEkCFD+PXXX8mYMSMzZszA09PTdJK4AE9PT+rWrUvdunX1vylJMa1ataJly5YkJibSunVrbt68aTpJRMShjR07lpUrV+Lr68vs2bPx8/MznSTi9rROvruaNWvSu3dvAEJDQzl//rzhIkkLFpvNZjMdIalr06ZNVK5cGavVyrx582jatKnpJBGR/3Tt2jXKli3LiRMnaN++PWFhYaaTREQc0oEDB3jqqae4desW33zzDW+++abpJBGR/xQbG8szzzzDnj17eOmll1ixYgUWi8V0lqQi7XhycdevX6d169ZYrVbatWunoZOIOIWMGTMyc+ZMPDw8mDp1KvPnzzedJCLicGJjY2nZsiW3bt2iTp069rM8RUQc2d93Z65atcr+xDtxXRo8uTCbzcZrr73G8ePHKViwICNHjjSdJCLywCpWrEj//v0B6Nq1K8eOHTNcJCLiWPr27cvu3bvJmjUrYWFh2jEgIk6jZMmSfPnllwD06dOHXbt2mQ2SVKXBkwsbPXo0c+fOxcvLi5kzZxIUFGQ6SVxMVFQUAQEBBAQEEBUVZTpHXNCHH35IhQoVuHbtGiEhIdy6dct0koiIQ5g/fz7ffPMNAFOmTOGxxx4zXCQif6d18v11796dl19+mdjYWBo3bsy1a9dMJ0kq0eDJRf3yyy/2Q9u+/PJLKlSoYLhIXFV0dDTR0dGmM8RFeXt7M2/ePIKDg/ntt990domICHDw4EE6duwIQL9+/Xj55ZcNF4nI3Wid/N8sFgvTpk0jf/78HD16lLZt22K1Wk1nSSrQ4MkFXbhwgSZNmpCQkEDTpk31g5qIOLU8efIwZ84cLBYLEydO1EHjIuLWIiMjady4MZGRkVSvXp3BgwebThIRSbbMmTOzcOFCfH19+f777/n8889NJ0kq0ODJxSQmJtKiRQvOnDlDsWLFmDRpku73FxGnV6NGDYYMGQLc3patcwBExB3ZbDY6d+7MwYMHyZkzJ3PmzMHLy8t0lojII3nqqacYPXo0AP379yc8PNxwkaQ0DZ5czEcffUR4eDgBAQEsWrSI9OnTm04SEUkR/fr1o169ety6dUvnAIiIW/r7+Z0LFiwgW7ZsppNERFJEx44d6dChA1arlebNm3PmzBnTSZKCNHhyIcuWLWPYsGEATJo0iRIlShguEhFJOR4eHsyYMcN+DkCbNm10DoCIuI1NmzbZz+8cPnw4zz//vOEiEZGUY7FYGDNmDE888QQXL16kSZMmxMbGms6SFKLBk4vYtWsXLVu2BKBHjx40b97ccJGISMrLlCkTixYtwtfXl+XLl/Pee++ZThIRSXXHjh2jUaNGJCQk0KxZM3r06GE6SUQkxaVLl45FixaRIUMGfvnlFzp37ozNZjOdJSlAgycX8Ndff/HKK68QFRVFjRo1GD58uOkkcRMeHh5UqVKFKlWq4OGhLyeSNp588kmmTp0KwBdffMHkyZPNBomIpKLr16/z8ssvc/HiRZ566ikmT56s8ztFnIDWyclTqFAhFixYgKenJzNmzOCTTz4xnSQpwGLTCNGpRUdHU7VqVbZt20axYsX45ZdfyJgxo+ksEZFUN2jQIAYOHIiXlxdr1qyhWrVqppNERFJUQkIC9erVY82aNeTKlYutW7eSM2dO01kiIqlu/PjxdOvWDYAFCxYQEhJiuEgehUavTsxqtdKuXTu2bdtGlixZWL58uYZOIuI2PvroI1q0aEFCQgKNGzfmjz/+MJ0kIpKi3nrrLdasWYO/vz/Lli3T0ElE3MZrr71Gz549AWjbti3btm0zXCSPQoMnJ/bRRx+xcOFCvL29Wbx4MYUKFTKdJCKSZiwWC1OmTOG5557j6tWrvPzyy1y5csV0lohIihg9ejRjxozBYrEwa9YsnnzySdNJIiJpavjw4dStW5eYmBjq16/PqVOnTCdJMmnw5KQmT57M0KFDAZg4cSKVK1c2XCTuKCoqiqxZs5I1a1aioqJM54gb8vPz47vvviNv3rz8+eefNGzYkJiYGNNZIiKPZOnSpfYr/Z9++ikNGzY0GyQiD03r5Efn6enJ3LlzKV26NOfOnaNevXpcvXrVdJYkgwZPTmjx4sV06dIFgPfff5927doZLhJ3dunSJS5dumQ6Q9xY9uzZWb58OUFBQfz88880bdqU+Ph401kiIskSERFBs2bNsFqthIaG8s4775hOEpFk0jr50aVPn57vv/+exx57jL179/LKK68QHR1tOksekgZPTubHH3+kRYsWWK1WOnXqxMcff2w6SUTEuNKlS/P999/j5+fH8uXLCQ0NxWq1ms4SEXkoO3bsoH79+sTGxtKgQQMmTJigJ9iJiNvLly8fq1evJmPGjGzatImQkBDi4uJMZ8lD0ODJifz66680bNiQuLg4QkJCGD9+vBYjIiL/r3LlyvbH786cOZNevXqhB7eKiLM4dOgQderU4ebNm1SrVo25c+fi5eVlOktExCGUKVOGFStWkC5dOlatWkW7du1ITEw0nSUPSIMnJ7F//37q1q1LVFQUNWvWZObMmXh6eprOEhFxKC+//DLTpk0DYOTIkQwZMsRwkYjI/Z08eZJatWpx6dIlnnrqKb777jv8/PxMZ4mIOJTnn3+exYsX4+3tzdy5c+nRo4cuMjoJDZ6cwOHDh6lVqxZXrlzh2WefZfHixfj6+prOEhFxSK1atWLkyJEADBgwgK+//tpskIjIf/jrr7+oVasWp06dolixYqxatYqgoCDTWSIiDqlOnTrMmDEDi8XCuHHj6Nevn4ZPTkCDJwf3+++/U7lyZc6ePUuJEiVYsWIFgYGBprNERBxajx49GDRoEAC9evXiyy+/NFwkIvJvp0+fpkqVKvz+++/kzZuXNWvWkDVrVtNZIiIOrVmzZowdOxaAzz77jHfffVfDJwenG8cd2IEDB6hevTrnz5+nZMmS/PTTT2TJksV0loidh4cHTz/9tP3XIo7kww8/JCEhgSFDhvDOO+8QFxfH+++/bzpLRAS4fXtdtWrVOHr0KPny5WPdunXkyZPHdJaIpBCtk1PXa6+9RkJCAj169ODLL78kPj6er776SmcgOyiLTaNBh7Rnzx5q1KjBxYsXKVOmDD/++KOugImIJMOQIUP46KOPABg4cCAfffSRFiUiYtSxY8eoXr06x48fp2DBgoSHh5MvXz7TWSIiTmfChAm89tprAHTv3p1Ro0Zp0OeANHhyQDt37qRmzZpcvnyZJ598kjVr1mink4jII/j000/p168fAP3792fIkCEaPomIEYcPH6Z69eqcOnWKIkWKEB4eTu7cuU1niYg4rSlTptCpUydsNhudO3dm/PjxGj45GA2eHMyGDRto0KAB165d45lnnmH16tVkzJjRdJaIiNMbMWIEb7/9NgBvvPEGX3/9tZ4OKiJpateuXdStW5e//vqLYsWK8dNPP5EzZ07TWSIiTm/GjBm0b98eq9VKixYtCAsL0wO5HIjGgA5k/vz51KxZk2vXrvHCCy+wZs0aDZ3EoUVHR5M/f37y589PdHS06RyR/9S7d29GjRoFwOjRo2nSpAkxMTGGq0TEXaxZs4ZKlSrx119/UapUKSIiIjR0EnFhWienrTZt2jBr1iy8vLyYM2cOL730EteuXTOdJf9PO54cgM1m46uvvrJfiW/UqBGzZs0iXbp0hstE/ltUVJT9KYuRkZEEBAQYLhK5v/nz59OmTRvi4uKoUKECy5YtIzg42HSWiLiwadOm0alTJxISEqhWrRqLFy/WxUURF6d1shlr166lcePG3Lx5k1KlSrFy5Uo9uMEBaMeTYYmJifTq1cs+dOrRowcLFizQ0ElEJJU0bdqUtWvXkjFjRn755RdeeOEFjh49ajpLRFyQzWbj448/pn379iQkJNCiRQtWrVqloZOISCqpWbMmGzZsIEeOHOzbt48KFSqwd+9e01luT4Mng27evEmTJk345ptvAPjyyy/55ptvdOaIiEgqq1y5Mps2bSJv3rz88ccfVKhQgc2bN5vOEhEXEhsbS+fOnfnwww8B6Nu3LzNnztSZIyIiqaxs2bJs2bKFEiVKcObMGSpWrMiqVatMZ7k1DZ4M+eOPP3juuedYsmQJPj4+zJkzh7fffltPWRIRSSMlSpTgl19+oWzZsly4cIGqVasybtw4dAe6iDyq06dPU6VKFSZPnozFYmH06NF8+umnesqSiEgayZs3Lxs3bqRy5crcuHGDevXqMXToUKxWq+k0t6TvfgYsX76c8uXLc+DAAXLmzElERATNmzc3nSUi4nZy5szJzz//TEhICPHx8XTv3p1OnTpx69Yt02ki4qQ2bNjAU089xa+//kqmTJlYuXIlr7/+uuksERG3kylTJtasWUPXrl2x2Wx88MEHNG7cmBs3bphOczsaPKUhq9XKwIEDeeWVV7hx4wYVK1Zkx44dVKhQwXSaiIjbCgwMZP78+Xz22Wd4eHgwZcoUKleuzKlTp0yniYgTsdlsjBw5khdffJELFy5QpkwZtm/fTp06dUyniYi4LV9fX8aPH8/EiRPx8fHhu+++49lnn+XQoUOm09yKBk9p5Pz589SrV49BgwYB8MYbb/DTTz/x2GOPGS4TST6LxUKJEiUoUaKEbhMVp2axWHj33Xf54YcfyJw5M9u2bePJJ59k+fLlptNExAlcu3aNli1b0rNnT/sh4ps3b6ZgwYKm00TEEK2THUunTp3YsGEDuXLl4tChQzzzzDPMmjXLdJbbsNh0mEWqW758OaGhoVy8eBE/Pz/Gjx9Pu3btTGeJiMhdHD9+nFdffZWdO3cC0L17d7744gv8/f0Nl4mII9qwYQNt2rTh5MmTeHp68sUXX/DWW2/pB00REQd0/vx5mjZtyoYNGwBo2bIlY8aM0dNGU5l2PKWi6OhoXn/9dV555RUuXrxo33KtoZOIiOPKnz8/mzdvplevXgCMHTuWp59+ml27dpkNExGHEh8fT//+/alatSonT56kUKFCbNq0iV69emnoJCLioLJnz85PP/3E4MGD8fT0ZPbs2ZQtW5aff/7ZdJpL046nVLJjxw7atGnDwYMHAejVqxfDhg3Dz8/PcJmIiDyoNWvW0K5dO86dO4e3tzdDhw6lV69eeHl5mU4TEYMOHTpE27Zt2bZtGwAdOnTgm2++IX369IbLRETkQW3ZsoVWrVpx9OhRPDw8eO+99/joo4/w9fU1neZytOMphUVGRtK7d2+eeeYZDh48SI4cOVizZg0jRozQ0ElcTnR0NCVLlqRkyZJER0ebzhFJcbVq1WLv3r00aNCA+Ph43n33XZ577jn7bXgi4l5iY2MZNGgQTzzxBNu2bSNjxozMnz+fKVOmaOgkIv+gdbLje+6559i1axft27fHarUybNgwypYta78NT1KOdjyloJUrV9K9e3dOnDgBQPPmzRk1ahTBwcGGy0RSR1RUFIGBgcDtoWtAQIDhIpHUYbPZmDJlCn369OHatWt4enrSq1cvBg4cqP/di7iJn3/+mS5dutifhPTSSy8xYcIE8uTJY7hMRByR1snOZdGiRbz++uucP38egM6dO/PZZ5+RKVMmw2WuQTueUsDZs2dp0aIF9erV48SJE+TPn59Vq1YxZ84cDZ1ERFyAxWKhY8eOHDx4kGbNmpGYmMiXX35JqVKlWLFiBbqGI+K6Ll++TNeuXalcuTKHDh0iW7ZszJ07lxUrVmjoJCLiIho3bszBgwfp0qULABMnTqR48eLMmTNH67wUoMHTI4iKimLQoEEUKVKEuXPn4uHhQZ8+fdi3bx916tQxnSciIinsscceY+7cuSxfvpy8efNy/PhxXn75ZerUqcPevXtN54lICoqNjWXEiBEULlyYb7/9Frh9BfzQoUM0a9ZMB4iLiLiYTJkyMWHCBDZs2ECxYsU4f/48LVu25IUXXmDLli2m85yaBk/JYLVamTp1KkWLFmXgwIFER0fz/PPPs23bNr744gttoxQRcXH16tVj//79vPPOO/j4+LBmzRrKli1Lly5dOHfunOk8EXkENpuNRYsWUbJkSd5++22uXbvGE088wfr16/n2229124WIiIurVKkSu3btYsiQIfj7+/PLL79QoUIFWrRowfHjx03nOSUNnh6CzWZj+fLlPP3003To0IGzZ89SoEAB5s+fz8aNG3nyySdNJ4qISBoJDAzk888/5+DBgzRp0gSr1crEiRMpUqQIAwcO5Nq1a6YTReQhRUREULlyZUJCQjhy5AiPPfYYkydPZseOHVSuXNl0noiIpBFfX18++OAD/vzzT0JDQ7FYLMydO5dixYrRp08fLly4YDrRqehw8Qdgs9n4/vvvGTx4MDt27AAgKCiIDz74gB49euhpdeK2dGiiyB2bNm2iV69e9serZ8iQgZ49e/LWW29ph4SIA7PZbKxbt45BgwbZn2SULl063nnnHd555x379zkRkYehdbJr2bVrF71792bdunXA7e8T3bp145133uGxxx4zXOf4NHj6D4mJiSxbtowhQ4bYH53t7+/P66+/zjvvvEPWrFkNF4qYFR0dTYkSJQA4cOAA/v7+hotEzLJarSxcuJDBgwezf/9+ANKnT8+bb75Jz5499X1DxIHYbDbWrl3LkCFD2LhxIwA+Pj506tSJfv36kTt3bsOFIuLMtE52PTabjVWrVjFo0CC2bt0KgJ+fH127dqVPnz76vvEfNHi6ixs3bhAWFsbIkSM5evQocPuWijfeeIPevXvrBwcREflPVquVxYsXM3jwYPuh476+vrRu3ZqePXtSunRpw4Ui7ismJoaZM2fyzTff2AfEvr6+dO7cmb59++oHBxER+U82m43Vq1czaNAg+6HjXl5ehISE8NZbb/Hss88aLnQ8Gjz9zZEjRxg9ejSTJ0/m5s2bwO2T7bt160avXr0IDg42XCgiIs7EarXy3Xff8cknn7B9+3b771evXp2ePXtSr149PD09DRaKuI/Tp08zbtw4JkyYwOXLlwEICAigU6dOvPvuu+TMmdNwoYiIOJOknbNDhw6136oN8Nxzz9GzZ09effVVfHx8DBY6DrcfPEVHR7No0SKmTJlCRESE/feLFStGz549adOmje7HFRGRR2Kz2di8eTPffPMNixYtwmq1ApArVy7atWtHhw4dKFy4sOFKEdcTGxvL999/z5QpU1i9erX9317+/Pnp0aMHoaGhZMyY0WykiIg4vZ07d/LNN98wZ84c4uLiAAgODqZNmzaEhoZSqlQpw4VmueXgyWq18ssvvzB9+nTmzJlj391ksVioXbs2b731FjVr1sTDQw/9E/kvMTEx9qf8bNiwgXTp0hkuEnF8J0+eZMyYMUyaNIkrV67Yf79y5cp06NCBRo0akSFDBoOFIs7NZrPx22+/MWPGDGbOnGnf3QRQpUoVevbsSf369bXbUERSldbJ7un8+fOMHz+e8ePHc+7cOfvvly9fng4dOtCkSRO3vJPKbQZPScOmBQsWsHDhQs6cOWP/swIFChAaGkrbtm3JmzevwUoR56KndYgk3712Yvj4+FCrVi2aNm1K/fr1NYQSeQBJw6YFCxawYMEC+xmdADlz5qR9+/a0b9+eIkWKGKwUEXeidbJ7S0hI4IcffmDKlCl8//33JCQkAODp6Un16tVp0qQJjRo1cpshlEsPnqKjo1m3bh0rV65k6dKl/xg2BQUF0bBhQzp06EDlypW1u0kkGfQNVSRlnD59mmnTpjF79mwOHDhg/30fHx9q1qxJvXr1qFu3Lvny5TNYKeJYYmNj+fnnn+3rvL8Pm9KlS8fLL79Mhw4dqFWrlnY3iUia0zpZkly4cIFZs2YxY8YMdu7caf99T09Pqlatyssvv0zdunUpUqQIFovFYGnqcanBk81m49ChQ/z444+sXLmSdevWERsba//zoKAg6tevT9OmTalVqxa+vr4Ga0Wcn76hiqS8/fv323dt/H0IBVCiRAnq1q1L7dq1ef755/VoZnErNpuNI0eO8NNPP7Fq1Sp+/PFHoqKi7H+eNGxq0qQJdevW1fckETFK62S5m8OHD7Nw4ULmz5//jyEUQKFChahbty516tShYsWKBAUFGapMeU49eLJarRw6dIj169cTERFBREQEFy5c+Mdr8ubNa79SXLNmTQ2bRFKQvqGKpK79+/fz/fffs3LlSjZv3kxiYqL9z7y9vXn22WepWrUqVatW5bnnntO/QXEpSYOm9evXs27dOiIiIv6xe/3/2rt/n6bWMA7gz7UaEXAwMYHEQeNiS6ltYrQocdDJwKQu7s4uDv4PDv4lxjjpYvDHZotDwUEkKEpYjCYoUiGicAdyj+mF3Iv3ejgUP5+Ehb7Ds7Tnm+9533MiInp7e2NoaCgJ6r4DwHYhJ/Nvpqamkpz35MmTWF5eTj7L5XJx4sSJJOcNDg62dRHVVsXT+/fvo16vR61Wi1qtFvV6PT5+/NiypqOjIwYHB+PChQsxNDQUhUJhx25Xg6y5oMLWmZubiwcPHsT9+/djZGQkZmdnWz7P5XLR398f1Wo1qtVqDAwMxLFjxxwxom3Mzc3F6Oho1Gq1ePr0adTr9fjw4UPLmr8K179yXqVSkfOAbUlO5md8/vw5Hj5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" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "ztesthyp-fig" } }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import seaborn as sns\n", "from scipy import stats\n", "from matplotlib import pyplot as plt\n", "\n", "mu = 0\n", "sigma = 1\n", "x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)\n", "y = 100* stats.norm.pdf(x, mu, sigma)\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 5))\n", "\n", "sns.lineplot(x=x,y=y, color='black', ax=axes[0])\n", "sns.lineplot(x=x,y=y, color='black', ax=axes[1])\n", "\n", "axes[0].set_frame_on(False)\n", "axes[1].set_frame_on(False)\n", "axes[0].get_yaxis().set_visible(False)\n", "axes[1].get_yaxis().set_visible(False)\n", "axes[0].get_xaxis().set_visible(False)\n", "axes[1].get_xaxis().set_visible(False)\n", "\n", "axes[0].axhline(y=0, color='black')\n", "axes[0].axvline(x=mu, color='black', linestyle='--')\n", "\n", "axes[1].axhline(y=0, color='black')\n", "axes[1].axvline(x=mu + sigma, color='black', linestyle='--')\n", "\n", "axes[0].hlines(y=23.6, xmin = mu-sigma, xmax = mu, color='black')\n", "axes[1].hlines(y=23.6, xmin = mu-sigma, xmax = mu, color='black')\n", "\n", "\n", "axes[0].text(mu,42, r'$\\mu = \\mu_0$', size=20, ha=\"center\")\n", "axes[1].text(mu + sigma, 42, r'$\\mu \\neq \\mu_0$', size=20, ha=\"center\")\n", "\n", "axes[0].text(mu-sigma - 0.2, 23.6, r'$\\sigma = \\sigma_0$', size=20, ha=\"right\")\n", "axes[1].text(mu-sigma - 0.2, 23.6, r'$\\sigma = \\sigma_0$', size=20, ha=\"right\")\n", "\n", "\n", "# show the figure in the book with caption\n", "plt.close()\n", "from myst_nb import glue\n", "glue(\"ztesthyp-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "horizontal-scientist", "metadata": {}, "source": [ "```{glue:figure} ztesthyp-fig\n", ":figwidth: 600px\n", ":name: fig-ztesthyp\n", "\n", "\n", "Graphical illustration of the null and alternative hypotheses assumed by the one sample $z$-test (the two sided version, that is). The null and alternative hypotheses both assume that the population distribution is normal, and additionally assumes that the population standard deviation is known (fixed at some value $\\sigma_0$). The null hypothesis (left) is that the population mean $\\mu$ is equal to some specified value $\\mu_0$. The alternative hypothesis is that the population mean differs from this value, $\\mu \\neq \\mu_0$.\n", "\n", "```\n", "\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "loving-christopher", "metadata": {}, "source": [ "The next step is to figure out what we would be a good choice for a diagnostic test statistic; something that would help us discriminate between $H_0$ and $H_1$. Given that the hypotheses all refer to the population mean $\\mu$, you'd feel pretty confident that the sample mean $\\bar{X}$ would be a pretty useful place to start. What we could do, is look at the difference between the sample mean $\\bar{X}$ and the value that the null hypothesis predicts for the population mean. In our example, that would mean we calculate $\\bar{X} - 67.5$. More generally, if we let $\\mu_0$ refer to the value that the null hypothesis claims is our population mean, then we'd want to calculate\n", "\n", "$$\n", "\\bar{X} - \\mu_0\n", "$$\n", "\n", "If this quantity equals or is very close to 0, things are looking good for the null hypothesis. If this quantity is a long way away from 0, then it's looking less likely that the null hypothesis is worth retaining. But how far away from zero should it be for us to reject $H_0$? \n", "\n", "To figure that out, we need to be a bit more sneaky, and we'll need to rely on those two pieces of background knowledge that I wrote down previously, namely that the raw data are normally distributed, and we know the value of the population standard deviation $\\sigma$. If the null hypothesis is actually true, and the true mean is $\\mu_0$, then these facts together mean that we know the complete population distribution of the data: a normal distribution with mean $\\mu_0$ and standard deviation $\\sigma$. Adopting the notation from Section \\@ref(normal), a statistician might write this as:\n", "\n", "$$\n", "X \\sim \\mbox{Normal}(\\mu_0,\\sigma^2)\n", "$$\n", "\n", "\n", "\n", "Okay, if that's true, then what can we say about the distribution of $\\bar{X}$? Well, as we discussed earlier, the sampling distribution of the mean $\\bar{X}$ is also normal, and has mean $\\mu$. But the standard deviation of this sampling distribution $\\mbox{SE}({\\bar{X}})$, which is called the *standard error of the mean*, is\n", "\n", "$$\n", "\\mbox{SE}({\\bar{X}}) = \\frac{\\sigma}{\\sqrt{N}}\n", "$$\n", "\n", "In other words, if the null hypothesis is true then the sampling distribution of the mean can be written as follows:\n", "\n", "$$\n", "\\bar{X} \\sim \\mbox{Normal}(\\mu_0,\\mbox{SE}({\\bar{X}}))\n", "$$\n", "\n", "Now comes the trick. What we can do is convert the sample mean $\\bar{X}$ into a [standard score](zcores). This is conventionally written as $z$, but for now I'm going to refer to it as $z_{\\bar{X}}$. (The reason for using this expanded notation is to help you remember that we're calculating standardised version of a sample mean, *not* a standardised version of a single observation, which is what a $z$-score usually refers to). When we do so, the $z$-score for our sample mean is \n", "\n", "$$\n", "z_{\\bar{X}} = \\frac{\\bar{X} - \\mu_0}{\\mbox{SE}({\\bar{X}})}\n", "$$\n", "\n", "or, equivalently\n", "\n", "$$\n", "z_{\\bar{X}} = \\frac{\\bar{X} - \\mu_0}{\\sigma / \\sqrt{N}}\n", "$$\n", "\n", "This $z$-score is our test statistic. The nice thing about using this as our test statistic is that like all $z$-scores, it has a standard normal distribution:\n", "\n", "$$\n", "z_{\\bar{X}} \\sim \\mbox{Normal}(0,1)\n", "$$\n", "\n", "(again, see the section on [z-scores](zcores)) if you've forgotten why this is true). In other words, regardless of what scale the original data are on, the $z$-statistic iteself always has the same interpretation: it's equal to the number of standard errors that separate the observed sample mean $\\bar{X}$ from the population mean $\\mu_0$ predicted by the null hypothesis. Better yet, regardless of what the population parameters for the raw scores actually are, the 5\\% critical regions for $z$-test are always the same, as illustrated in Figures \\@ref(fig:ztest1) and \\@ref(fig:ztest2). " ] }, { "cell_type": "code", "execution_count": 6, "id": "standard-shoulder", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "ztest-fig" } }, "output_type": "display_data" } ], "source": [ "mu = 0\n", "sigma = 1\n", "\n", "x = np.arange(-3,3,0.001)\n", "y = stats.norm.pdf(x, mu, sigma)\n", "\n", "\n", "fig, (ax0, ax1) = plt.subplots(1, 2, sharey = True, figsize=(15, 5))\n", "\n", "\n", "# Two-sided test\n", "crit = 1.96\n", "p_lower = x[xcrit]\n", "\n", "ax0.plot(x, y)\n", "\n", "ax0.fill_between(p_lower, 0, stats.norm.pdf(p_lower, mu, sigma),color=\"none\",hatch=\"///\",edgecolor=\"b\")\n", "ax0.fill_between(p_upper, 0, stats.norm.pdf(p_upper, mu, sigma), color=\"none\",hatch=\"///\",edgecolor=\"b\")\n", "ax0.set_title(\"Two sided test\", size = 20)\n", "ax0.text(-1.96,-.03, '-1.96', size=18, ha=\"right\")\n", "ax0.text(1.96,-.03, '1.96', size=18, ha=\"left\")\n", "\n", "# One-sided test\n", "crit = 1.64\n", "p_upper = x[x>crit]\n", "\n", "ax1.plot(x, y)\n", "ax1.set_title(\"One sided test\", size = 20)\n", "ax1.text(1.64,-.03, '1.64', size=18, ha=\"left\")\n", "ax1.fill_between(p_upper, 0, stats.norm.pdf(p_upper, mu, sigma), color=\"none\",hatch=\"///\",edgecolor=\"b\")\n", "\n", "ax0.set_frame_on(False)\n", "ax1.set_frame_on(False)\n", "\n", "ax0.get_yaxis().set_visible(False)\n", "ax1.get_yaxis().set_visible(False)\n", "ax0.get_xaxis().set_visible(False)\n", "ax1.get_xaxis().set_visible(False)\n", "\n", "# show the figure in the book with caption\n", "plt.close()\n", "from myst_nb import glue\n", "glue(\"ztest-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "vital-coach", "metadata": {}, "source": [ "```{glue:figure} ztest-fig\n", ":figwidth: 600px\n", ":name: fig-ztest\n", "\n", "\n", "Rejection regions for the two-sided z-test (left) and the one-sided z-test (right).\n", "\n", "```\n", "\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "governmental-organic", "metadata": {}, "source": [ "And what this meant, way back in the days where people did all their statistics by hand, is that someone could publish a table like this:" ] }, { "attachments": {}, "cell_type": "markdown", "id": "numerical-childhood", "metadata": {}, "source": [ "| || critical z value |\n", "| :-------------: | :------------: | :------------: |\n", "| desired a level | two-sided test | one-sided test |\n", "| .1 | 1.644854 | 1.281552 |\n", "| .05 | 1.959964 | 1.644854 |\n", "| .01 | 2.575829 | 2.326348 |\n", "| .001 | 3.290527 | 3.090232 |\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "metropolitan-principal", "metadata": {}, "source": [ "which in turn meant that researchers could calculate their $z$-statistic by hand, and then look up the critical value in a text book. That was an incredibly handy thing to be able to do back then, but it's kind of unnecessary these days, since it's trivially easy to do it with software like Python." ] }, { "attachments": {}, "cell_type": "markdown", "id": "valuable-authority", "metadata": {}, "source": [ "### A worked example using Python\n", "\n", "Now, as I mentioned earlier, the $z$-test is almost never used in practice. However, the test is so incredibly simple that it's really easy to do one manually. Let's go back to the data from Dr Zeppo's class. Having loaded the `grades` data, the first thing I need to do is calculate the sample mean:" ] }, { "cell_type": "code", "execution_count": 7, "id": "recovered-tuition", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "72.3" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "sample_mean = df['grades'].mean()\n", "sample_mean" ] }, { "attachments": {}, "cell_type": "markdown", "id": "annual-calendar", "metadata": {}, "source": [ "Then, I create variables corresponding to known population standard deviation ($\\sigma = 9.5$), and the value of the population mean that the null hypothesis specifies ($\\mu_0 = 67.5$):" ] }, { "cell_type": "code", "execution_count": 8, "id": "decimal-painting", "metadata": {}, "outputs": [], "source": [ "sd_true = 9.5\n", "mu_null = 67.5" ] }, { "attachments": {}, "cell_type": "markdown", "id": "sexual-debate", "metadata": {}, "source": [ "Let's also create a variable for the sample size. We could count up the number of observations ourselves, and type `N = 20` at the command prompt, but counting is tedious and repetitive. Let's get Python to do the tedious repetitive bit by using the `len()` function, which tells us how many elements there are in a vector:" ] }, { "cell_type": "code", "execution_count": 9, "id": "remarkable-convert", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "20" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N = len(df['grades'])\n", "N" ] }, { "attachments": {}, "cell_type": "markdown", "id": "literary-sandwich", "metadata": {}, "source": [ "Next, let's calculate the (true) standard error of the mean:" ] }, { "cell_type": "code", "execution_count": 10, "id": "universal-poland", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2.1242645786248002" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import math\n", "sem_true = sd_true / math.sqrt(N)\n", "sem_true" ] }, { "attachments": {}, "cell_type": "markdown", "id": "included-terrain", "metadata": {}, "source": [ "And finally, we calculate our $z$-score:" ] }, { "cell_type": "code", "execution_count": 11, "id": "changed-drilling", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "2.259605535157681" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "z_score = (sample_mean - mu_null) / sem_true\n", "z_score" ] }, { "attachments": {}, "cell_type": "markdown", "id": "electric-ticket", "metadata": {}, "source": [ "At this point, we would traditionally look up the value 2.26 in our table of critical values. Our original hypothesis was two-sided (we didn't really have any theory about whether psych students would be better or worse at statistics than other students) so our hypothesis test is two-sided (or two-tailed) also. Looking at the little table that I showed earlier, we can see that 2.26 is bigger than the critical value of 1.96 that would be required to be significant at $\\alpha = .05$, but smaller than the value of 2.58 that would be required to be significant at a level of $\\alpha = .01$. Therefore, we can conclude that we have a significant effect, which we might write up by saying something like this:\n", "\n", "> With a mean grade of 73.2 in the sample of psychology students, and assuming a true population standard deviation of 9.5, we can conclude that the psychology students have significantly different statistics scores to the class average ($z = 2.26$, $N=20$, $p<.05$). \n", "\n", "However, what if want an exact $p$-value? Well, back in the day, the tables of critical values were huge, and so you could look up your actual $z$-value, and find the smallest value of $\\alpha$ for which your data would be significant (which, as discussed earlier, is the very definition of a $p$-value). However, looking things up in books is tedious, and typing things into computers is awesome. So let's do it using Python instead. Now, notice that the $\\alpha$ level of a $z$-test (or any other test, for that matter) defines the total area \"under the curve\" for the critical region, right? That is, if we set $\\alpha = .05$ for a two-sided test, then the critical region is set up such that the area under the curve for the critical region is $.05$. And, for the $z$-test, the critical value of 1.96 is chosen that way because the area in the lower tail (i.e., below $-1.96$) is exactly $.025$ and the area under the upper tail (i.e., above $1.96$) is exactly $.025$. So, since our observed $z$-statistic is $2.26$, why not calculate the area under the curve below $-2.26$ or above $2.26$? In Python we can calculate this using the `NormalDist().cdf()` method. For the lower tail:" ] }, { "cell_type": "code", "execution_count": 12, "id": "responsible-count", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.011922871882469877" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from statistics import NormalDist\n", "lower_area = NormalDist().cdf(-z_score)\n", "lower_area" ] }, { "attachments": {}, "cell_type": "markdown", "id": "amino-linux", "metadata": {}, "source": [ "`NormalDist().cdf()` calculates the \"cumulative density function\" for a normal distribution. Translated to something slightly less opaque, this means that `NormalDist().cdf()` gives us the probability that a random variable X will be less than or equal to a given value. In our case, the given value for the lower tail of the distribution was our z-score, $2.259$. So `NormalDist().cdf(-z_score)` gives us the probability that a random value draw from a normal distribution would be less than or equal to $-2.259$.\n", "\n", "Of course, becauwe we didn't have any particular theory about whether psychology students would do better worse than other students, our test should be two-tailed, that is, we are interested not only in the probability that a random value would be less than or equal to $-2.259$, but also whether it might fall in the upper tail, that is be greater than or equal to $-2.259$.\n", "\n", "Since the normal distribution is symmetrical, the upper area under the curve is identical to the lower area, and we can simply add them together to find our exact $p$-value:" ] }, { "cell_type": "code", "execution_count": 13, "id": "spatial-interpretation", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.023845743764939753" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "lower_area = NormalDist().cdf(-z_score)\n", "upper_area = lower_area\n", "p_value = lower_area + upper_area\n", "p_value" ] }, { "attachments": {}, "cell_type": "markdown", "id": "social-tuesday", "metadata": {}, "source": [ "(zassumptions)=\n", "### Assumptions of the $z$-test\n", "\n", "As I've said before, all statistical tests make assumptions. Some tests make reasonable assumptions, while other tests do not. The test I've just described -- the one sample $z$-test -- makes three basic assumptions. These are:\n", "\n", "- *Normality*. As usually described, the $z$-test assumes that the true population distribution is normal.[^note3] is often pretty reasonable, and not only that, it's an assumption that [we can check](shapiro-wilk) if we feel worried about it. \n", "- *Independence*. The second assumption of the test is that the observations in your data set are not correlated with each other, or related to each other in some funny way. This isn't as easy to check statistically: it relies a bit on good experimetal design. An obvious (and stupid) example of something that violates this assumption is a data set where you \"copy\" the same observation over and over again in your data file: so you end up with a massive \"sample size\", consisting of only one genuine observation. More realistically, you have to ask yourself if it's really plausible to imagine that each observation is a completely random sample from the population that you're interested in. In practice, this assumption is never met; but we try our best to design studies that minimise the problems of correlated data. \n", "- *Known standard deviation*. The third assumption of the $z$-test is that the true standard deviation of the population is known to the researcher. This is just stupid. In no real world data analysis problem do you know the standard deviation $\\sigma$ of some population, but are completely ignorant about the mean $\\mu$. In other words, this assumption is *always* wrong. \n", "\n", "In view of the stupidity of assuming that $\\sigma$ is known, let's see if we can live without it. This takes us out of the dreary domain of the $z$-test, and into the magical kingdom of the $t$-test, with unicorns and fairies and leprechauns, and um..." ] }, { "attachments": {}, "cell_type": "markdown", "id": "loving-thunder", "metadata": {}, "source": [ "(onesamplettest)=\n", "\n", "## The one-sample $t$-test\n", "\n", "After some thought, I decided that it might not be safe to assume that the psychology student grades necessarily have the same standard deviation as the other students in Dr Zeppo's class. After all, if I'm entertaining the hypothesis that they don't have the same mean, then why should I believe that they absolutely have the same standard deviation? In view of this, I should really stop assuming that I know the true value of $\\sigma$. This violates the assumptions of my $z$-test, so in one sense I'm back to square one. However, it's not like I'm completely bereft of options. After all, I've still got my raw data, and those raw data give me an *estimate* of the population standard deviation: " ] }, { "cell_type": "code", "execution_count": 14, "id": "daily-artwork", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "9.520614752375916" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df['grades'].std()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "fifth-allergy", "metadata": {}, "source": [ "In other words, while I can't say that I know that $\\sigma = 9.5$, I *can* say that $\\hat\\sigma = 9.52$. \n", "\n", "Okay, cool. The obvious thing that you might think to do is run a $z$-test, but using the estimated standard deviation of 9.52 instead of relying on my assumption that the true standard deviation is 9.5. So, we could just type this new number into Python and out would come the answer. And you probably wouldn't be surprised to hear that this would still give us a significant result. This approach is close, but it's not *quite* correct. Because we are now relying on an *estimate* of the population standard deviation, we need to make some adjustment for the fact that we have some uncertainty about what the true population standard deviation actually is. Maybe our data are just a fluke ... maybe the true population standard deviation is 11, for instance. But if that were actually true, and we ran the $z$-test assuming $\\sigma=11$, then the result would end up being *non-significant*. That's a problem, and it's one we're going to have to address." ] }, { "cell_type": "code", "execution_count": 15, "id": "trained-costume", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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J/Pnzc/LkSapVq8Yff/xhOktERETEqYwaNYqBAwcC8Nlnn9GrVy/DRZA2bVrWrVtHmTJluHLlClWrVuX48eOms0QkBWnwJOJGPv74Y7744gsAJk2aRPv27c0G/Un27NnZtGkTuXPn5rfffqN69epcvXrVdJaIiIiIUxg3bhzvvPMOAEOGDKFfv36Gix5Inz4969evp3jx4vzxxx9UrVqV06dPm84SkRSiwZOImxg5ciRDhw4FYPTo0XTp0sVw0f/KlSsXmzZtInv27Bw5coQaNWoQERFhOktERETEoc2aNYs333wTgPfff58PPvjAcNH/CgoKYsOGDRQuXJhz585RtWpVrly5YjpLRFKABk8ibmD9+vX0798fgM8//5yePXsaLvpn+fLlY9OmTWTOnJn9+/fTsWNH9PBNERERkb+3e/duOnfuDECfPn345JNPjJ3d+TCZM2e2P8X41KlTNGvWjPj4eNNZIpLMNHgScXGnTp2iRYsWWK1WOnXqxLvvvptk7+3p6UlwcDDBwcFJemhloUKFWLp0Kd7e3ixcuJAvv/wyyd5bRERExFVcvXqV119/nZiYGOrXr88XX3zhsEOnRNmyZWPFihUEBAQQFhbGe++9ZzopWSTXOlnEGVls2kog4rKioqJ4+eWX2bdvH+XKlSM8PBw/Pz/TWY9swoQJdO/eHQ8PD9avX0+1atVMJ4mIiIg4hPj4eGrVqsWmTZsoWLAgu3btIm3atKazHtmSJUsIDg4GYP78+TRr1sxwkYgkF+14EnFRNpuNN954g3379pEpUyaWLFniVEMngDfeeIP27dtjtVpp1qwZZ86cMZ0kIiIi4hDef/99Nm3ahL+/P8uWLXOqoRNA48aN7budQkJCOHTokOEiEUku2vEk4qLGjRvHm2++iaenJz/88AOvvPKK6aQnEh0dTcWKFdmzZw9ly5Zl69atTjdAExEREUlKixYtomnTpvZfJ+4ccjYJCQnUrl2bDRs2UKBAAXbt2kW6dOlMZ4lIEtOOJxEXtH37dt5++23g/mHiyTV0ioyMxGKxYLFYiIyMTJbPkSpVKpYsWULGjBnZs2eP/YktIiIiIu7o6NGjdOjQAYB+/fo57dAJ7p+DNH/+fHLnzs3x48dp06aNyzxUJiXWySLOQoMnERcTGRlJ27ZtiY+Pp1mzZvTu3dt00lPLnTs3CxYswMPDg6lTp7J8+XLTSSIiIiIpLi4ujjZt2hAZGUm1atUYNmyY6aSnljFjRpYuXYqfnx+rVq1iypQpppNEJIlp8CTiYgYMGMCJEyfImTMnkyZNcvgnmzyqatWq2Z/I17VrV65fv264SERERCRlff755+zZs4f06dMzc+ZMvLy8TCclieeee47hw4cD0KdPH53rKeJiNHgScSFhYWGMGTMGgKlTpzrdIZMPM2jQIIoWLcrly5fp2bOn6RwRERGRFHPgwAEGDx4MwJgxY8iWLZvhoqT11ltvUaFCBSIiIggJCcFqtZpOEpEkosGTiIuIiIiw3+/ftWtXatSoYbgo6fn5+TF9+nQ8PT2ZN28eS5YsMZ0kIiIikuzi4uJo164dcXFxNGzYkJYtW5pOSnKenp6EhoaSKlUqNm3axKRJk0wniUgS0eBJxEX069eP06dPkzt3br744gvTOcmmXLly9kfvduvWjatXrxouEhEREUlew4cPZ9++fWTMmJGJEye6zFEK/61AgQJ89tlnALz77rucPHnScJGIJAUNnkRcwIYNG5gwYQIA06ZNI02aNIaLkteHH35IiRIluHr1Kj169DCdIyIiIpJs9u7dyyeffALAuHHjyJw5s+Gi5NWjRw8qV65MZGSkbrkTcREaPIk4uTt37tCxY0fg/jfqqlWrptjn9vT0pE6dOtSpUwdPT88U+7y+vr5Mnz4dLy8vFi1axIIFC1Lsc4uIiIiklNjYWNq1a0d8fDzBwcE0bdrUdFKy8/DwYNq0afj7+7NlyxbGjh1rOumJmFonizgii81ms5mOEJEn17t3b77++mvy5cvH/v37CQgIMJ2UYgYNGsTgwYN55pln+O2331zuMHURERFxbyNGjGDAgAFkypSJw4cPkylTJtNJKWbChAl0794df39/fv31V7Jnz246SUSekAZPIk7s8OHDlCpVioSEBNatW0fNmjVNJ6Wo2NhYSpYsya+//krv3r0ZNWqU6SQRERGRJHHhwgWeffZZIiMjmTlzJm3atDGdlKKsVisVKlTgp59+omXLlsyZM8d0kog8IQ2eRJyUzWajevXqbNq0iYYNG7Js2TLTSUasW7eO2rVr4+Xlxf79+ylatKjpJBEREZGn1rJlS+bNm8d//vMftm7d6rIHiv+bX375heeffx6bzUZ4eDgVK1Y0nSQiT0BnPIk4qSVLlrBp0yb8/PyM7fSJjIzE398ff39/IiMjjTTUqlWLBg0aEB8fz1tvvYVm6SIiIuLswsPDmTdvHhaLhTFjxrjl0Angueeeo3PnzgD07NmThIQEw0WPzhHWySKOQoMnEScUFRXFO++8A0C/fv3Imzev0ZaoqChjnx9g1KhR+Pr6snHjRpYuXWq0RURERORpxMfH07NnTwC6dOnCc889Z7jIrGHDhpE+fXr279/PpEmTTOc8FkdYJ4s4Ag2eRJzQiBEjOHv2LLly5aJ///6mc4zLly8f/fr1A6BPnz76Bi8iIiJO69tvv+XAgQOkT5+eYcOGmc4xLigoiKFDhwLwwQcfcO3aNcNFIvK4NHgScTInT57k888/B+7v9EmdOrXhIsfw3nvvkStXLs6ePWv/30dERETEmVy7do0PPvgAgE8++YSMGTMaLnIMXbt2pWTJkty8edP+v4+IOA8NnkScTJ8+fYiJiaFatWq8/vrrpnMcRurUqfnyyy8B+Oyzzzh9+rTZIBEREZHH9MEHH3Dz5k1KlSpF165dTec4DC8vL8aMGQPc3xH2yy+/GC4SkcehwZOIE9m0aRPLly/Hy8uL0aNHu+1Bk/+kcePGVK1alXv37tlvvRMRERFxBgcPHuTbb78FYMyYMXh6ehouciyVKlWiRYsW2Gw23n77bT1QRsSJaPAk4iRsNhvvvfceAG+88QZFixY1XOR4LBYLX3/9NRaLhUWLFrFnzx7TSSIiIiKPZODAgdhsNpo0aULFihVN5zikzz//HF9fX3788Ue+//570zki8og0eBJxEsuWLWPXrl34+/s7zL3tHh4eVK5cmcqVK+Ph4Rh/nZQoUYLWrVsD8P777xuuEREREXm4bdu2sXLlSjw9Pfnkk09M5zisHDly2J/4N2DAAKxWq+Gif+aI62QRUyw27VEUcXjx8fGUKFGCY8eO8eGHHzJkyBDTSQ7t1KlTPPvss8TFxbFx40aqVq1qOklERETkb9lsNipVqsTWrVvp0qULkyZNMp3k0K5fv06+fPm4c+cOc+fOpUWLFqaTROQhNHoVcQIzZ87k2LFjZMyYkXfeecd0jsPLmzcvb7zxBnD/apjm6yIiIuKo1q5dy9atW/Hz8+Ojjz4ynePwMmbMaD/L84MPPiA2NtZwkYg8jAZPIg7u3r17fPzxx8D9IUratGkNFzmHgQMH4u/vz86dO/nuu+9M54iIiIj8D6vVyoABAwDo2bMn2bNnN1zkHHr16sUzzzzDyZMnmTp1qukcEXkIDZ5EHNyECRM4f/48OXLkoHv37qZz/iIyMpJMmTKRKVMmIiMjTef8RebMmenduzdwfwgVHx9vuEhERETkr+bPn8+BAwdImzat/SEy8nABAQF8+OGHAAwZMoSoqCjDRf/LkdfJIilNgycRB3bnzh2GDRsGwKBBg0iVKpXhov917do1rl27Zjrjb/Xt25cMGTJw9OhRZs2aZTpHRERExC42NtY+POnXrx8ZMmQwXORcunTpQp48ebh06RKjR482nfO3HHmdLJKSNHgScWBffvkl169f59lnn6Vdu3amc5xO2rRp7U+2+/jjj7l3757hIhEREZH7pkyZwsmTJ8mcOTO9evUyneN0fHx8GDp0KACfffYZN2/eNFwkIv9EgycRB3Xt2jW+/PJLAD755BO8vLwMFzmn7t27kz17ds6dO8fEiRNN54iIiIgQHR1tH5p8+OGH+Pv7Gy5yTi1atKB48eLcunWLzz//3HSOiPwDDZ5EHNTXX39NZGQkZcqUoXHjxqZznFaqVKnsT4j54osvtOtJREREjJsyZQqXLl0id+7cdO7c2XSO0/L09OSTTz4BYOzYsdy4ccNwkYj8HQ2eRBzQrVu3GDNmDHD/KpjFYjFc5NzatWtHjhw5uHjxIqGhoaZzRERExI3FxMTw2WefAfDee+/h4+NjuMi5vfbaa5QsWZKIiAiHPetJxN1p8CTigEaPHs2dO3coXrw4DRo0MJ3j9Hx9fenXrx8AI0aMIC4uznCRiIiIuKsZM2Zw4cIFsmXLRvv27U3nOD0PDw8GDhwIwDfffMOdO3cMF4nIf9PgScTB3L17l6+//hqAgQMH4uHhuH9MPTw8eP7553n++ecduhOgU6dOZM6cmbNnz+oJdyIiImJEXFwcn376KXD/SXZ+fn6Gi1xD48aNKVy4MLdu3WLcuHGmcwDnWieLJDeLzWazmY4QkQc+++wz3nvvPQoVKsSRI0fw9PQ0neQyvvzyS/r27UuBAgU4evSoDmwXERGRFDVjxgzat2/PM888w6lTp0idOrXpJJcxe/Zs2rRpQ1BQEKdPn9aB7SIORKNXEQcSFRVlf5Ld+++/r6FTEuvatSsZM2bk+PHjLFiwwHSOiIiIuJGEhASGDx8OwDvvvKOhUxJr3rw5+fPn59q1a0yaNMl0joj8iQZPIg7k22+/5erVq+TNm5eWLVuaznE5AQEB9OnTB4Bhw4ZhtVoNF4mIiIi7WLRoEb/99hsZMmSgW7dupnNcjpeXFwMGDADuP8k4OjracJGIJNLgScRB3Lt3j88//xyAAQMG4O3tbbjo4aKiosiTJw958uQhKirKdM4jefPNN0mXLh1Hjx5l6dKlpnNERETEDVitVj755BMAevfuTZo0aQwXuaY2bdqQK1cuLl26xNSpU422OOM6WSS5aPAk4iBCQ0P5448/yJEjB23btjWd80hsNhtnzpzhzJkzOMtxcYGBgbz11lsAfPLJJ07TLSIiIs5r+fLlHD58mMDAQN58803TOS7Lx8eH/v37A/fPTY2NjTXW4ozrZJHkosGTiAOIi4vjs88+A6B///74+voaLnJtvXr1IiAggP3797N69WrTOSIiIuLCbDYbw4YNA+Ctt94iXbp0ZoNcXEhICFmzZuX8+fPMnDnTdI6IoMGTiENYvHgxZ86cIVOmTHTs2NF0jsv789kKI0eONFwjIiIiriwsLIw9e/aQKlUqevXqZTrH5fn5+fHOO+8A959orDM9RczT4EnEMJvNZh9+9OzZk1SpUhkucg+9evXCy8uLLVu2sGvXLtM5IiIi4qIS13khISEEBQUZrnEPnTt3JjAwkGPHjrFmzRrTOSJuT4MnEcPCwsL45ZdfSJUqlZ5wkoKyZ89uf3Lgl19+abhGREREXNHhw4dZs2YNFouF3r17m85xG4GBgXTt2hXQ7nYRR6DBk4hhid8MO3TooKtgKSxxG/aiRYs4ffq02RgRERFxOaNGjQLg9ddfJ3/+/IZr3Mtbb72l3e0iDkKDJxGDnP0qmMVioWjRohQtWhSLxWI657GVLFmSmjVrYrVa+frrr03niIiIiAv5448/mD17NgB9+/Y1XON+cuTIQYsWLQAzu9udfZ0skpQsNj3bUcSYjh07Mm3aNF5//XWWLFliOsct/fDDD9SsWRN/f3/OnTtH+vTpTSeJiIiICxg4cCDDhw/n5ZdfZuvWraZz3NL+/fspXbo0Hh4enDhxgjx58phOEnFL2vEkYoiugjmG6tWrU7JkSSIjI5k0aZLpHBEREXEBERERTJgwAdA6z6RSpUpRo0YN7W4XMUyDJxFDxo4dS2xsLP/5z3946aWXTOe4LYvFYl8Qjh49mpiYGMNFIiIi4uxCQ0O5efMmBQsW5LXXXjOd49YS13lTpkzh5s2bhmtE3JMGTyIGuMpVsKioKIoVK0axYsWIiooynfPEmjVrRvbs2fnjjz+YO3eu6RwRERFxYvHx8fZDxfv06YOnp6fhIvdWo0YNSpQokeK7211lnSySFDR4EjEg8SpYgQIFqF+/vumcJ2az2Thy5AhHjhzBmY+L8/HxoVevXsD9pww689ciIiIiZi1dupTTp08TFBRE27ZtTee4PVO7211lnSySFDR4EklhCQkJfPPNNwD07t1bV8EcRJcuXUiTJg1Hjhxh/fr1pnNERETESX311VcAdO/endSpUxuuEYDmzZvbd7cvWLDAdI6I29HgSSSFrV27lhMnTpA+fXratWtnOkf+X9q0aQkJCQFgzJgxhmtERETEGe3cuZMdO3bg4+ND9+7dTefI//Px8aFHjx7A/V1P2oEkkrI0eBJJYaNHjwagY8eO+Pv7G66RP0tckKxZs4bjx48brhERERFnk3jxqlmzZmTOnNlwjfxZ586d8fX1Zc+ePezYscN0johb0eBJJAUdPXqUH374AQ8PD/uQQxxHwYIFqVOnDjabjXHjxpnOERERESdy6dIl+21cPXv2NFwj/y0oKIiWLVsCDy4Ei0jK0OBJJAWNHTsWgPr165MnTx6zMfK33nrrLQCmTZvG3bt3DdeIiIiIs/j222+Ji4vjpZdeoly5cqZz5G8kDgQXL17MxYsXDdeIuA8NnkRSyK1bt5gxYwbwYLjh7CwWC7lz5yZ37txYLBbTOUmiRo0aFCpUiDt37jBz5kzTOSIiIuIEYmNjmTBhAqDdTo6sTJkyVKxYkfj4eCZOnJisn8sV18kiT0qDJ5EUEhoaSmRkJMWLF6dKlSqmc5JE6tSpOX36NKdPn3aZp7Z4eHjYF4xjx47FarUaLhIRERFHt2TJEi5dukTWrFlp3Lix6Rz5F4nrvIkTJxITE5Nsn8cV18kiT0qDJ5EUkJCQYL/NrmfPnrrq4eDatWtHmjRpOHbsGBs2bDCdIyIiIg4u8cygN954Ax8fH8M18m8aNmxIjhw5uHr1qv1MLhFJXho8iaSAtWvXcvLkSdKnT0+rVq1M58hDpEmThg4dOgA6fFJERET+3a5du9ixYwfe3t507drVdI48hLe3N927dwfur/NsNpvhIhHXp8GTSApIHF506tQJf39/wzVJJzo6mnLlylGuXDmio6NN5ySpN998E4A1a9Zw/PhxwzUiIiLiqMaMGQNA8+bNyZw5s+EaeRSdO3fG19eXPXv2sGPHjmT5HK68ThZ5XBo8iSSzI0eO8MMPP+Dh4WG/uuIqrFYru3fvZvfu3S53FlLBggV59dVXsdlsjBs3znSOiIiIOKBLly4xf/58QIeKO5OgoCBatmwJJN/udldeJ4s8Lg2eRJJZ4tCifv365MmTx2yMPJbEpw9OmzaNiIgIwzUiIiLiaCZPnkxcXBwvvvgi5cqVM50jjyFxULh48WIuXrxouEbEtWnwJJKM7t69y8yZM4EHt26J86hZsyYFCxbkzp07zJ0713SOiIiIOJD4+HgmTZoEaJ3njMqUKcPLL79MfHw8U6ZMMZ0j4tI0eBJJRnPnziUiIoJChQpRtWpV0znymDw8POyHhE6YMEGHT4qIiIjdqlWruHDhAkFBQQQHB5vOkSfQrVs34P7Otfj4eMM1Iq5LgyeRZGKz2ZgwYQJw/9G6FovFcJE8ifbt2+Pr68u+ffvYuXOn6RwRERFxEBMnTgQgJCQEX19fwzXyJIKDgwkKCuL8+fOsXr3adI6Iy9LgSSSZ7Nixg/379+Pn50e7du1M58gTypgxI82aNQOwDxJFRETEvZ04cYJ169ZhsVjsu6PF+fj6+hISEgJonSeSnDR4Ekkmid+8mjdvToYMGQzXJJ+goCCCgoJMZySrxG3YCxYs4MaNG4ZrRERExLTEs51q1apFvnz5DNfI00gcHK5bt44TJ04k6Xu7wzpZ5FFo8CSSDK5fv87ChQuB+7fZuSp/f3+uXr3K1atX8ff3N52TbF544QVKly7NvXv3mD59uukcERERMejevXtMmzYNeHBxSpxXvnz5qFWrFvBgoJgU3GWdLPIoNHgSSQahoaHExMRQpkwZypcvbzpHnpLFYrEvLCdOnIjVajVcJCIiIqYsXryY69evkzNnTurWrWs6R5JA4jpv2rRp3Lt3z3CNiOvR4EkkiVmtVvvVkm7duulQcRfRsmVL0qRJw++//86mTZtM54iIiIghiccpdOnSBU9PT8M1khTq1q1Ljhw5uH79OkuWLDGdI+JyNHgSSWIbN27k+PHjBAYG0rJlS9M5ySo6OpoqVapQpUoVoqOjTeckq4CAANq2bQvo8EkRERF3deDAAbZv346XlxedOnUynSNJxMvLiy5dugBJt85zp3WyyMNo8CSSxBK/WbVt29bl7+e2Wq1s2bKFLVu2uMXtZ4nndS1fvpwLFy4YrhEREZGUlrjOa9SoEVmyZDFcI0mpU6dOeHl5sW3bNg4ePPjU7+du62SRf6PBk0gSunDhAitWrABc+1Bxd1W8eHEqVKhAQkICU6ZMMZ0jIiIiKeju3bvMnj0b0DrPFWXNmpWGDRsC2t0uktQ0eBJJQlOmTCEhIYGKFStSrFgx0zmSDBIPn5w8eTLx8fGGa0RERCSlzJkzh4iICJ599lleeeUV0zmSDBIHirNmzSIiIsJwjYjr0OBJJIkkJCQwdepUQFfBXFnjxo3JmDEjFy5c4PvvvzedIyIiIilk8uTJAHTt2lUPj3FRVatWpWDBgkRERLBgwQLTOSIuQ4MnkSSybt06zp07R4YMGXj99ddN50gy8fX1pV27dgB8++23hmtEREQkJezZs4dffvkFHx8f2rRpYzpHkonFYrEfGq91nkjS0eBJJIkkXgVr27Ytfn5+hmskOXXu3BmA1atX65BxERERN5C4zmvcuDFBQUGGayQ5tW/fHm9vb3bu3MmBAwdM54i4BA2eRJLAH3/8wcqVK4EHQwl3kTp1alKnTm06I0UVLlyYihUrYrVaCQ0NNZ0jIiIiySgiIoK5c+cC7rfOc0fPPPMMDRo0AB4MHJ+UO66TRf6OBk8iSSA0NJSEhARefvllihYtajonxfj7+xMZGUlkZCT+/v6mc1JU4sJzypQpekSuiIiIC1uwYAF3796lQIECVKlSxXSOpIAuXboA9w8Zj4qKeqL3cOd1ssh/0+BJ5ClZrVamTJkC6CqYOwkODiZdunScOXOGH374wXSOiIiIJJPEXS+dO3fWoeJuolq1auTNm5fbt2+zePFi0zkiTk+DJ5GntGnTJk6dOkXatGlp0qSJ6RxJIalSpbIfLvq027BFRETEMR04cICff/4ZLy8v+8NFxPV5eHjYDxnXOk/k6WnwJPKUEp940bp1a7e7h/vevXvUrVuXunXrcu/ePdM5KS5xh9vy5cu5fPmy4RoRERFJaolDhwYNGpA5c2bDNZKSOnTogKenJ1u3buXIkSOP/Xp3XyeL/JkGTyJP4erVq3z33XeAe95ml5CQwJo1a1izZg0JCQmmc1JciRIlePHFF4mPj2f69Ommc0RERCQJRUdHM3v2bODBmT/iPrJmzUq9evUA7MdqPA53XyeL/JkGTyJPYcaMGcTFxVG+fHlKlSplOkcM0CHjIiIirmnx4sXcunWLPHnyUL16ddM5YkDiwHHmzJnatSTyFDR4EnlCNpvtL4dNintq1qwZadKk4fjx44SFhZnOERERkSSSeJxCx44d8fDQj03uqFatWuTMmZPr16+zbNky0zkiTkt/g4o8ofDwcH777TcCAgJo3ry56RwxxN/fn1atWgE6fFJERMRVHDt2jK1bt+Lp6UmHDh1M54ghnp6edOzYEdA6T+RpaPAk8oSmTp0KQIsWLQgICDBcIyYlPvVk2bJl3Lhxw3CNiIiIPK3EdV6dOnXInj274RoxKSQkBIvFwubNmzlx4oTpHBGnpMGTyBO4ffs2ixcvBrBfBRH39dxzz1GqVCliYmKYM2eO6RwRERF5CnFxccycORPQOk8gZ86c1KxZE4DQ0FDDNSLOSYMnkScwb948oqOjKVasGOXLlzedI4ZZLBb7wnTq1KnYbDbDRSIiIvKkVq1axZUrV8icOTN16tQxnSMOIHGdN336dD2hTuQJaPAk8gQSt1937NgRi8ViuMYcf39/bDYbNpsNf39/0zlGtWrVCl9fX/bv388vv/xiOkdERESeUOI6r127dnh7exuuEUdQv359MmbMyIULF1i3bt0jvUbrZJEHNHgSeUwHDhxg9+7deHt706ZNG9M54iAyZMhAo0aNgAcLVhEREXEuFy5cYO3atcD9s31EAHx9fe3rfq3zRB6fBk8ijynxm02DBg0ICgoyXCOOJHEb9ty5c4mOjjZcIyIiIo9rxowZWK1WKlSowLPPPms6RxxI4jpvxYoVXLlyxXCNiHPR4EnkMcTExDB79mxAh00C3Lt3jyZNmtCkSRPu3btnOse4qlWrkidPHm7fvs2SJUtM54iIiMhjsFqtTJs2DdA6T/5X8eLFKV++PPHx8cyaNeuhH691ssgDGjyJPIbvvvuOGzdukCNHDmrUqGE6x7iEhAQWL17M4sWLddAi4OHhQYcOHQBtwxYREXE24eHhnDhxgjRp0tCkSRPTOeKAHudhMlonizygwZPIY0gcJrRv3x5PT0/DNeKI2rdvj8ViISwsjBMnTpjOERERkUeUuM5r3ry5DoOWv9WsWTNSpUrF0aNH2bFjh+kcEaehwZPIIzpz5gwbNmwAsO9qEflvuXLlombNmgD27foiIiLi2G7dusXixYsB3WYn/yxt2rT23XDa3S7y6DR4EnlEoaGh2Gw2qlatSr58+UzniANLXLBOnz6d+Ph4wzUiIiLyMPPmzePevXsUK1aM8uXLm84RB5a4zluwYAERERGGa0ScgwZPIo8gISGB0NBQQFfB5OHq169PxowZuXjxIuvWrTOdIyIiIg+RuHulY8eOWCwWwzXiyCpWrEjBggWJiIhg4cKFpnNEnIIGTyKPYNOmTZw9e5Z06dLRqFEj0zni4Hx9fWnTpg2g2+1EREQc3YEDB9izZw/e3t60bt3adI44OIvFQkhICKB1nsij0uBJ5BEk7nZq2bIlqVKlMlwjziBxQbJy5UquXr1quEZERET+SeI6r379+mTKlMlwjTiDtm3b4uHhwbZt2/jtt99M54g4PA2eRB7i5s2bLF26FNCh4v8tderUREREEBERQerUqU3nOJQSJUpQtmxZ4uLimDNnjukcERER+RuxsbHMnj0b0DpPHl22bNmoXbs28GBw+d+0ThZ5QIMnkYeYP38+MTEx9kGCPGCxWPD398ff31/nIfyNxF1PiQfTi4iIiGNZtWoV165dI2vWrNSqVct0jjiRxHXezJkz//ZhMlonizygwZPIQyTeu92hQwd905DH0qJFC3x9fTlw4AB79+41nSMiIiL/JXG3Stu2bfHy8jJcI87ktddesz9MZv369aZzRByaBk8i/+LQoUPs3r0bLy8vHTb5N2JiYmjfvj3t27cnJibGdI7DSZ8+vf0weh0+KSIi4lj++OMP1qxZA+g2O3l8Pj4+9p8P/u52O62TRR7Q4EnkXyR+E3nttdd02OTfiI+PZ8aMGcyYMeNvtxjLg4Xs3LlzuXfvnuEaERERSTRr1iysViv/+c9/ePbZZ03niBNKXOctX76ca9eu/eWfaZ0s8oAGTyL/IC4ujlmzZgG6CiZPrlq1auTIkYObN2+yYsUK0zkiIiIC2Gw2+wVGrfPkSZUqVYoyZcoQFxfH3LlzTeeIOCwNnkT+werVq7l69SpZsmTh1VdfNZ0jTsrT05P27dsDut1ORETEUezYsYNjx46ROnVqmjZtajpHnNifHyYjIn9PgyeRf5A4JGjTpo0Om5Snkjh4Wr9+PefPnzcbIyIiIvYhQXBwMIGBgYZrxJm1bNkSHx8f9u3bp4fJiPwDDZ5E/salS5d02KQkmfz581OpUiVsNhszZ840nSMiIuLWoqKimD9/PqB1njy9DBky0KBBA0C7nkT+iQZPkqzCwsLo2rUrxYsXJzAwEB8fH3LlykWbNm0eekXgaV77tGbPnk1CQgIvvvgiRYoUSdbPJe7hz9uwbTab4RoREZGn56zrvKVLl3L37l3y5ctHpUqVkvVziXtIXOfNmTNHT7AT+RsWm34CkmRw7949KleuzM6dO//xY7y8vBg3bhxdunRJstcmBZvNRrFixTh69CiTJk1Kls/hKiIjIwkICAAgIiICf39/w0WOKzIykixZshAREUF4eDgVK1Y0nSQiIvJEnHmdB1C1alU2b97MkCFD+PDDD5Plc4h7SUhIIHfu3Fy4cIGFCxfSpEkTrZNF/kSDJ0kWt27dIn369Hh7exMcHEz9+vXJnz8/169fZ/ny5UyePJmEhAQ8PDzYs2cPpUuXTpLXJoWff/6ZF198ET8/Py5dukTatGmT9P1dic1msz86NigoCIvFYrjIsXXs2JFp06bRoUMHHTQuIiJOy5nXeadOnSJfvnxYLBZOnz5Nrly5kvT9xX29//77fPrpp7z66qusWbNG62SRP7OJJINbt27ZgoODbWfPnv3bfz5y5EgbYANsPXv2TLLXJoU33njDBthatWqV5O8t7u3HH3+0ATZ/f3/b3bt3TeeIiIg8EWde5w0aNMgG2KpXr57k7y3u7ddff7UBNg8PD9uFCxdM54g4FO14EiOuX79OUFAQALVr12bt2rUp8tqHiY6OJmvWrNy+fZsNGzZQrVq1JHtvEZvNxrPPPsvvv//O9OnTadeunekkERGRJOeo6zyr1Ur+/Pk5ffo0c+bMoWXLlkn23iIAFSpUYNu2bYwYMYL+/fubzhFxGDpc3IHFxMQQGhpKw4YNyZkzJ6lTp8Zisfzrf3bs2GE6+5FERETYf50hQ4YUe+3DLF++nNu3b5MrVy5eeeWVJH1vVxQTE0OPHj3o0aOHDlJ8BBaLhfbt2wN66omIiLvTOi/pX/swW7Zs4fTp0wQGBtKoUaMkfW8RePCUxOnTp3Pv3j2tk0X+nwZPDurw4cOULFmSkJAQli9fzvnz54mOjv7X11gsFooXL55ChU9n+PDh9l83btw4xV77MInDgHbt2uHhoT8eDxMfH8/48eMZP3488fHxpnOcQps2bbBYLGzZsoWTJ0+azhEREQO0zkue1z7M9OnTAWjevDmpUqVK0vcWAWjSpAmpUqXi2LFj/PTTT1oni/w//WTtgE6fPk21atX47bffAKhfvz6LFy9mz549rF27lqZNm/7l42vXrk2DBg0ICQmxPznBkX366ad8++23ANSrV4/XX389RV77MOfOneOHH34A0C1Qkmxy5sxJjRo1AJgxY4bhGhERSWla5yXPax/m7t27LF68GHiwK0UkqQUGBhIcHAzArFmzDNeIOBDDZ0zJf7FarbaXX37ZfqjipEmT/vbjWrRoYf+YRYsWPfL7J77maf4TGhr6xF/fiBEj7O/z4osv2iIiIlLktY9i2LBhNsBWqVKlJH1fVxYREWH//ySp//9wZXPnzrUBtty5c9sSEhJM54iISArROi95XvsopkyZYgNszz77rM1qtSbpe4v82caNG22ALU2aNFoni/w/7XhyMAsXLmTbtm0AvPPOO3Tp0uVvP6537972X4eFhaVE2lObNGkS7733HgDly5fn+++/x9/fP9lf+yhsNpv9NjtdBZPk1rBhQ9KmTcuZM2ec5s+viIg8Pa3zkv61jyrxNrsOHTrosfaSrKpUqUKePHm4e/eu6RQRh6Gn2jmYihUrsnXrVjJmzMjp06f/cUt1XFwcfn5+WK1WGjZsyLJlyx7p/Q8dOvTUjTly5CBdunSP9ZpTp05RpEgRYmJiKFGiBD/++CNp06ZN9tc+qm3btlGhQgX8/f25dOmSU2xldwSRkZH2/60iIiKSfJHoyrp168bEiRNp3bq1tmKLiLgJrfOS9rWP6vfff6dQoUJ4eHhw7tw5smXLlqTvL/LfBg0axODBg+3/XetkcXdepgPkgUuXLrF161YAWrZs+a/DD29vbzw8PLBarY91CLapQykXLVpkf5rDmDFjHmtB8TSvfVSJu52aNGmioZOkiPbt2zNx4kSWLFnC2LFjk+XfaxERcRxa5yX9ax9V4m6nWrVqaegkKaJdu3Z/GTyJuDvdaudAwsPD7b+uWbPmv37s9evX7U9HyJkzZ7J2JYVTp07Zf12+fPkUe+2jiIyMZOHChYBus5OUU758eYoUKUJ0dDSLFi0ynSMiIslM67ykf+2jSEhIYObMmYDWeZJy8ubNS8WKFU1niDgM7XhyIH/eHl2qVKl//dhdu3bZf12uXLkn+hxP6km2YP/5EcGP+/jap3nto1i6dCl3794lX758+gbxmFKlSmVfMOqxxI/HYrHQvn17+vfvT2hoKJ06dTKdJCIiyUjrvKR/7aPYuHEj58+fJ3369Lz22mtJ/v4i/6RDhw78+OOP5M6dGz8/P9M5IkZp8ORAzp49a/911qxZ//VjV69eDdz/4bVq1aqP/DlKlCjxZHF/EhoaSvv27Z/6fRxF4m127du312GTj8nDw4M8efKYznBabdq04f3332f79u38+uuvPPvss6aTREQkmWidZ0biOq9ly5b64V9SVNOmTXnrrbc4c+YM27dv1wVucWu61c6BWK1W+68T73X/O7dv32bOnDnA/XvVH7Z4cQSTJ08mOjr6L1e1UuK1D3P69Gk2b96MxWKhbdu2Sf7+Iv8ma9as1K5dG4AZM2YYrhERkeSkdV7Sv/Zhbt26ZT+Y3ZWGaeIc/P39adq0KfDgnDERd6XBkwN55pln7L/evXv3P37ce++9x82bNwHo37//Y30Om8321P95km/c3t7e+Pn5PdGVpqd57cMk/rBftWpVcufOneTv7+piY2N59913effdd4mNjTWd45QS/zzNnDmThIQEszEiIpJstM5L+tc+zPz584mJiaF48eKULVs2yd9f5N/ExsYSFRUFwIIFC4iMjDRcJGKOBk8OpEKFCvZfDx8+HJvN9j8f8+WXXzJx4kQA2rZtS5UqVVIq76lUqVIFi8XyRLeyPc1r/43VarUPnnTY5JOJi4tj5MiRjBw5kri4ONM5Tum1114jQ4YMXLhwgQ0bNpjOERGRZKJ1XtK/9mESd5l06NBBxylIiouLi2P+/PnA/YcZLVmyxHCRiDk648mB1KtXj4IFC/L777+zfv16Xn31VXr27EnWrFk5efIkkydPZv369cD9xcukSZMMFzu38PBwTp06RWBgIDVr1tRViCeg/82enq+vLy1btmTs2LGEhoZSq1Yt00kiIpIMtM5LWUePHuXnn3/G09OThg0bas3yFPz9/U0nuITQ0FAd7SFuS4MnB+Ll5cWSJUt45ZVXuH79OuvWrWPdunX/83EdOnRg3LhxOiDxKSUeNtmsWbO/bH8XSWkdOnRg7NixfPfdd9y8eZP06dObThIRkSSmdV7KSlzn1a1bl/z58xuucW5/tztPHl9YWBgnT54kX758plNEUpxutXMwJUqU4PDhw/Tp04fChQuTOnVqUqdOTcGCBenSpQu7d+9m2rRpenT9U7p79y6LFy8GdJudmFemTBlKlixJTEyMfUu2iIi4Hq3zUkZ8fDyzZs0CtM4Tx5D4dMqZM2caLhExw2LTCFvc0LRp0+jYsSPPPvssR48etR/8J48nMjKSzJkzAxAREaGt2E/hq6++ok+fPpQrV46dO3eazhEREXFaq1evpl69egQFBXHhwgWdQ/mUtL57MpGRkQQEBAAwdepUOnbsSO7cuTl58iQeHtr/Ie5Ft9qJW0rcft2+fXssFou+oYpxrVq1ol+/fuzatYvDhw9TrFgx00kiIiJOKXGd17p1a3x8fPDx8TFcJO6ufv36BAYGcubMGcLCwuw7oETchUat4naOHz/O1q1b8fDwoE2bNqZzRID7j9muW7cu8OApPCIiIvJ4rl+/zooVK4D7FxhFHEGqVKlo3rw5oHWeuCcNnsTtJP5lX7NmTbJnz242xsmlSpWKQ4cOcejQIZ1HkQQSz6GYNWuWbgsQERF5AnPnziUuLo4yZcpQqlQp0znixv57nZy4zlu8eDF37twxXCeSsjR4EreSkJDAjBkzAB02mRQ8PDwoVqwYxYoV073qSaBOnTpkypSJy5cv/+2TjkREROTfJV5g1DpPTPvvdfILL7xA4cKFiY6OZtGiRabzRFKUflIUt7Jp0ybOnz9PunTpqF+/vukckb/w9vamdevWwIPzKUREROTRHDhwgF9++QVvb29atGhhOkfkLywWi/32T63zxN1o8CRuJfEv+ZYtW+Ln52e4xvnFxsYyaNAgBg0aRGxsrOkcl5C4IFm5ciXXrl0zGyMiIuJEEtd59evXJygoyHCNuLu/Wye3adMGDw8Ptm3bxm+//Wa4UCTlWGw2m810hEhKuHXrFlmzZuXevXvs3LmTcuXKmU5yen9+TGxERISeDphEypYtyy+//MLXX39Nr169TOeIiIg4vNjYWLJnz861a9dYuXIl9erVM50kbu6f1sl16tRh7dq1DBgwgOHDh5tMFEkx2vEkbmP+/Pncu3eP4sWL8/zzz5vOEflHISEhAEybNg1dGxAREXm41atXc+3aNbJkyULt2rVN54j8o8R13owZM0hISDBcI5IyNHgStzFt2jTg/mGTFovFcI3IP2vRogU+Pj4cOHCAvXv3ms4RERFxeInrvLZt2+Ll5WW4RuSfvfbaa2TIkIGLFy+yfv160zkiKUKDJ3ELhw8fZteuXXh5edkPbxZxVBkyZKBRo0aADp8UERF5mEuXLrF27VpAT7MTx+fr66uHyYjb0eBJ3ELiX+r16tXjmWeeMVwj8nCJC+c5c+Zw7949wzUiIiKOa9asWSQkJPDSSy9RuHBh0zkiD5W4zlu+fDnXr183XCOS/DR4EpcXFxfHrFmzAF0FE+dRvXp1cuTIwc2bN1mxYoXpHBEREYdks9n+cpyCiDMoXbo0pUuXJjY2lrlz55rOEUl2GjyJy1uzZg1Xrlwhc+bMvPrqq6ZzRB6Jp6cn7dq1A7QNW0RE5J/8/PPPHDt2jFSpUtGsWTPTOSKPLPGQca3zxB1o8CQuL/Ev8zZt2uDt7W24xrX4+fmxc+dOdu7ciZ+fn+kcl9O+fXsA1q9fz4ULF8zGiIiIOKDEdV5wcDCBgYGGa0QeeNg6uWXLlvj4+LB37172799voFAk5WjwJC7t8uXLrF69GtD26+Tg6elJuXLlKFeuHJ6enqZzXE6BAgWoWLEiVquVmTNnms4RERFxKFFRUcyfPx/QOk8cz8PWyRkzZqR+/fqAdj2J69PgSVza7NmziY+P54UXXqBo0aKmc0QeW+I27GnTpmGz2QzXiIiIOI6lS5dy584d8ubNS+XKlU3niDy2xHXe7NmziY2NNVwjknw0eBKXZbPZ7FcPdBUsecTGxvLFF1/wxRdf6JtlMgkODsbf35/jx4+zbds20zkiIiIOI3Gd1759ezw89GONOJZHWSfXrFmTbNmycf36dVauXJnChSIpx2LTJXRxUbt27aJ8+fL4+flx6dIl0qZNazrJ5URGRhIQEABAREQE/v7+hotcU0hICKGhoYSEhDB16lTTOSIiIsadPn2avHnzYrFYOHXqFLlz5zadJPIXj7pOHjBgACNGjKBu3bqsWrUqJRNFUowuDYjLSny0buPGjTV0EqeWuA17wYIFREREGK4RERExb/r06QBUq1ZNQydxaol3Zqxdu5aLFy8arhFJHho8iUuKiopi7ty5gG6zE+f38ssvU7BgQSIjI1m4cKHpHBEREaMSEhJ0nIK4jEKFClGhQgWsViszZswwnSOSLDR4Epf058MmX3nlFdM5Ik/FYrH85ZBxERERd7Zp0ybOnj1LunTpaNSokekckaemh8mIq9PgSVxS4jk4HTp00GGT4hLatWuHp6cn27Zt49ixY6ZzREREjElc57Vq1YpUqVIZrhF5ek2aNCEgIIDjx4/z448/ms4RSXL6iVxczokTJwgLC8NisdC+fXvTOSJJImvWrLz66qvAg6f4iIiIuJsbN26wbNky4MEuERFnFxAQQPPmzQH0IBlxSRo8ictJvBWpZs2a5MyZ03CNSNLp2LEjADNmzCAuLs5wjYiISMqbM2cOsbGxlC5dmueee850jkiSSRykLlq0iNu3bxuuEUlaGjyJS0lISLA/5STxh3RJPn5+fmzevJnNmzfj5+dnOsfl1a1bl2eeeYbLly+zZs0a0zkiIiIpymaz2XeDaJ0nju5x18kvvvgiRYoUITo6mgULFqRAoUjK0eBJXMq6deu4ePEiGTNmpH79+qZzXJ6npydVqlShSpUqeHp6ms5xed7e3rRt2xbQIeMiIuJ+9u7dy/79+/H19aVly5amc0T+1eOuky0Wi32gqtvtxNVo8CQuJfEv6datW+Pr62u4RiTpJW7DXr16NX/88YfhGhERkZSTuM5r1KgRGTJkMFwjkvTatGmDl5cXO3fu5NChQ6ZzRJKMBk/iMq5evcqKFSsAbb9OKXFxcYwbN45x48bpzKEUUqRIEV566SUSEhKYOXOm6RwREZEUER0dzZw5cwAdKi7O4UnWyc888wyvvfYaoN3t4losNpvNZjpCJCmMGjWKd955h3LlyrFz507TOW4hMjKSgIAAACIiIvD39zdc5B6mTp1Kp06dKFSoEMeOHcNisZhOEhERSVZz586lVatW5M6dm5MnT+Lhoevn4tiedJ28evVq6tWrR1BQEBcuXMDHxyc5M0VShP7GFpfw58MmdRVMXF3Tpk3x9/fnt99+Y9u2baZzREREkl3iOq9Dhw4aOolLq1WrFlmzZuXatWv2uzlEnJ3+1haX8PPPP3PkyBFSpUpFixYtTOeIJKs0adLQtGlTQIdPioiI6zt16hSbNm3CYrHQvn170zkiycrLy8v+77nWeeIqNHgSl5D4l3JwcDBp06Y1XCOS/BLPMVu4cCF37twxXCMiIpJ8Es+6qV69Orlz5zZcI5L8Eu/gWLduHefOnTNcI/L0NHgSp3f37l3mzZsHQKdOnQzXiKSM//znPxQpUoSoqCj7v/8iIiKuJj4+3j540jpP3EWBAgWoUqUKNptNh4yLS9DgSZze/PnziYyMpFChQlSsWNF0jkiKsFgs9gX45MmTDdeIiIgkj7Vr13Lx4kWCgoJo0KCB6RyRFNO5c2fg/p0dCQkJhmtEno4GT+L0pkyZAty/Cqane4k7adOmDd7e3uzZs4e9e/eazhEREUlyiRdX2rZti6+vr+EakZTz+uuvkz59es6dO8cPP/xgOkfkqWjwJE7twIED7Ny5Ey8vL9q1a2c6x+34+vqyatUqVq1apcWgAZkyZaJRo0bAgwGsiIiIq7hw4QKrV68GdJudOJ+nXSf7+fnRpk0bQLvbxflZbDabzXSEyJPq2bMnY8eOpXHjxixevNh0jkiK27BhAzVq1CBt2rRcvHiR1KlTm04SERFJEsOGDeODDz6gQoUK/Pjjj6ZzRFLcwYMHKVmyJF5eXpw/f57MmTObThJ5ItrxJE4rOjqa2bNnAw/ugRZxN1WrViVv3rzcvn2bRYsWmc4RERFJElar1f7UYu12EndVokQJXnjhBeLj45kxY4bpHJEnpsGTOK0lS5Zw69YtcufOTY0aNUznuKW4uDimT5/O9OnTiYuLM53jljw8POjYsSOgbdgiIuI6Nm7cyKlTp0ibNi1NmjQxnSPy2JJqnZx4gX3KlCnoZiVxVrrVTpxW5cqVCQ8PZ/DgwXz00Uemc9xSZGQkAQEBAERERODv72+4yD1dvHiRXLlykZCQwJEjRyhSpIjpJBERkafSrFkzFi5cSPfu3Rk3bpzpHJHHllTr5IiICLJmzUpERASbN2+mSpUqSVgpkjK040mc0m+//UZ4eDgeHh6EhISYzhExKlu2bNStWxfQIeMiIuL8rl69yrJlywAdpyASEBBAixYtAK3zxHlp8CROKfEv3VdffZUcOXIYrhExL/H8ixkzZhATE2O4RkRE5MnNnDmTuLg4ypYtS+nSpU3niBiXOIBdvHgxN27cMFwj8vg0eBKnExsby/Tp0wEdNimS6NVXXyVbtmxcv36d7777znSOiIjIE7HZbPYzC7XbSeS+559/npIlSxITE2N/uJKIM9HgSZzOihUruHr1KlmyZLHfXiTi7ry8vOy3neqQcRERcVZbt27l119/JXXq1Pbbi0TcncVisQ9iJ0+erEPGxelo8CROZ9KkSQB06NABb29vwzUijqNjx45YLBY2btzI77//bjpHRETksSWu85o3b05gYKDhGhHH0bp1a1KlSsWhQ4f46aefTOeIPBYNnsSpHD9+nA0bNmCxWOjSpYvpHBGHkidPHmrXrg3At99+a7hGRETk8Vy7do1FixYB8MYbbxiuEXEs6dKlo1mzZgBMnDjRcI3I49HgSZxK4g/TtWvXJk+ePGZjBF9fXxYuXMjChQvx9fU1nSNA165dAZg+fboOGRcREacyY8YMYmNjKVOmDM8//7zpHJGnkhzr5MSB7MKFC3XIuDgVi003iIqTiImJIUeOHFy7do3vvvuOBg0amE4ScTjx8fHkyZOHCxcuMHfuXJ2PISIiTsFms/Hss8/y+++/M2nSJO1sF/kbNpuNMmXKsH//fr766ivefvtt00kij0Q7nsRpLF26lGvXrpE9e3YdKi7yD7y8vOxPe9Q2bBERcRabN2/m999/JyAgQBdNRP6BxWKx73qaOHGiDhkXp6HBkziNxMMmO3XqhJeXl+Eagfu7axYtWsSiRYuIj483nSP/r1OnTnh4eBAeHs7Ro0dN54iIiDxU4jqvdevWpEmTxnCNyNNLrnVyy5Yt8ff359dffyU8PDzJ3lckOelWO3EKR48epWjRonh4eHDmzBly5MhhOkmAyMhIAgICAIiIiMDf399wkSRq0KABK1asoFevXnz99demc0RERP7R5cuXyZEjB/Hx8ezdu5fSpUubThJ5asm5Tu7SpQuTJ0+mefPmzJs3L8neVyS5aMeTOIXEQ8Xr1aunoZPII0g8ZHzGjBlER0cbrhEREflnoaGhxMfHU758eQ2dRB5B4jpvyZIlXL161XCNyMNp8CQOLzo6mhkzZgAP/pIVkX9Xq1YtcufOza1bt+yPphYREXE0VqvVfoEx8ewaEfl3ZcuW5fnnnycuLo7p06ebzhF5KA2exOEtWrSImzdvkjt3bmrVqmU6R8QpeHp60rlzZ0CHjIuIiOP64YcfOHXqFGnTpqVZs2amc0ScRuKgdtKkSVitVsM1Iv9OgydxeImHTXbu3BlPT0/DNSLOIyQkBC8vL3766ScOHjxoOkdEROR/JK7z2rRpQ+rUqQ3XiDiP5s2bExgYyIkTJ9i0aZPpHJF/pcGTOLSDBw+yfft2vLy8CAkJMZ0j4lSyZs1KgwYNAJgwYYLhGhERkb+6cOECK1asAHScgsjj8vf3p02bNoDWeeL4NHgShzZu3DgAGjZsSNasWQ3XiDifbt26ATBr1izu3LljuEZEROSBb7/9loSEBCpUqEDx4sVN54g4ncTb7ZYvX8758+cN14j8Mw2exGHdvn2b2bNnA9CjRw/DNfJ3fHx8CA0NJTQ0FB8fH9M58jeqVq3Ks88+S0REBLNmzTKdIyIiAkBsbKz9UHGt88QVpcQ6uXjx4lSqVImEhAT7nycRR2Sx2Ww20xEif2f06NH06tWLYsWKcfDgQSwWi+kkEaeU+GepSJEiHD58WH+WRETEuAULFtC8eXMyZ87M2bNndQFL5AktXLiQZs2a6c+SODTteBKHZLPZGD9+PADdu3fXD8oiT6Fdu3b4+/tz9OhRtmzZYjpHRETEvs7r0qWLflAWeQqNGjUia9asXL58mWXLlpnOEflbGjyJQ9q0aRO//vorAQEB9kPzxPHEx8ezevVqVq9eTXx8vOkc+Qdp06aldevWwINz00REREw5ePAg4eHheHp66lBxcVkptU729vamS5cugNZ54rh0q504pNdff51ly5bRvXt3/QXqwCIjIwkICAAgIiICf39/w0XyTw4ePEjJkiXx9PTkzJkzZM+e3XSSiIi4qW7dujFx4kQaN27M4sWLTeeIJIuUXCdfuHCB3Llzk5CQwIEDByhRokSyfS6RJ6EdT+Jwzp07x/Lly4H7t9mJyNMrUaIEFStW1OGTIiJi1O3bt+0Pu9Ch4iJJI3v27DRq1AjQridxTBo8icOZNGkSVquVKlWqUKxYMdM5Ii4jcYH/7bffEhsba7hGRETc0cyZM4mMjKRo0aJUqVLFdI6Iy0hc582ePZvbt28brhH5Kw2exKHExsYyefJkQFfBRJJao0aNyJIlC5cuXeK7774znSMiIm5GD48RST6VK1emWLFiREZGMnPmTNM5In+hwZM4lCVLlnDlyhWyZctGgwYNTOeIuBQfHx86d+4MaBu2iIikvM2bN3Ps2DE9PEYkGVgsFvsxJePHj0dHOYsj0eBJHEriD8NdunTB29vbcI2I6+natSuenp6Eh4dz8OBB0zkiIuJGEtd5bdq0ITAw0HCNiOtp3bo1AQEBHDt2jE2bNpnOEbHT4Ekcxt69e9m2bRteXl72R4KKSNLKnj07DRs2BGDs2LFmY0RExG2cPXtWD48RSWaBgYG0bdsWgDFjxhiuEXlAgydxGN988w0ATZo0IWvWrIZr5FH4+PgwduxYxo4di4+Pj+kceURvvfUWALNmzeLGjRuGa0RExB2MGzeOhIQEqlatSvHixU3niCQ7U+vknj17ArBixQpOnjyZYp9X5N9YbLr5UxzA5cuXyZUrF7GxsezYsYMXXnjBdJKIy7LZbDz33HPs27ePESNG0L9/f9NJIiLiwiIjI8mZMyc3b95k+fLl1K9f33SSiEurXbs269ato3fv3owaNcp0joh2PIljmDRpErGxsbzwwgsaOokkM4vFQq9evYD7V6Dj4+MNF4mIiCubPXs2N2/eJF++fNStW9d0jojLS1znTZ06lbt37xquEdHgSRxAbGwsEyZMAB78JSnOISEhgbCwMMLCwkhISDCdI4+hefPmZMqUiXPnzrFs2TLTOSIi4qJsNhujR48G7t8C5OnpabhIJGWYXCfXqlWLQoUKcefOHWbMmJGin1vk72jwJMYtXLiQS5cukS1bNoKDg03nyGO4d+8er7zyCq+88gr37t0znSOPwc/PjzfeeAN4cL6aiIhIUtuwYQNHjhwhICCADh06mM4RSTEm18keHh72Mz1Hjx6N1WpN0c8v8t80eBKjbDab/Yfe7t274+3tbbhIxH1069YNb29vtm3bxp49e0zniIiIC0pc53Xo0IG0adMarhFxH+3atSNt2rT8/vvvfP/996ZzxM1p8CRG/fTTT+zevRtfX1+6dOliOkfErWTNmpWmTZsC2vUkIiJJ7/fff2f16tVYLBb7k7ZEJGUEBATQsWNHQOs8MU+DJzEq8S/BVq1akSlTJsM1Iu4n8Vy1+fPnc+nSJcM1IiLiSsaMGQNAnTp1KFiwoOEaEffz5ptv4uHhwfr16zl69KjpHHFjGjyJMefOnWPJkiWADhUXMaVcuXK89NJLxMXFMWnSJNM5IiLiIu7cuUNoaCigdZ6IKXnz5qV+/foA9kP+RUzQ4EmMGT9+PAkJCVSpUoWSJUuazhFxW4k/EEyYMIGYmBjDNSIi4gpCQ0OJiIigaNGiVK9e3XSOiNtKXOfNnDmTmzdvGq4Rd6XBkxgRGRlp312hq2AiZr3++utkz56dy5cvM2/ePNM5IiLi5BISEuy7K9566y0sFovhIhH3VblyZUqWLElUVJR2t4sxGjyJEaGhody8eZP8+fPz2muvmc6RJ+Tt7c3nn3/O559/ricSOjFvb2/7I3e//PJLbDab4SIREXFmy5Yt4+TJk2TMmJE2bdqYzhExwlHWyRaLhT59+gD3b7eLjY011iLuy2LTTxiSwhISEihYsCCnTp1i3LhxdO/e3XSSiNu7desWOXPmJCIigu+//55atWqZThIRESdks9l48cUX2blzJx999BGDBw82nSTi9mJjY8mbNy8XL15k+vTptGvXznSSuBnteJIUt2zZMk6dOkXGjBlp37696RwRAdKlS0fnzp0BGDlypOEaERFxVtu2bWPnzp34+vrSo0cP0zkiAvj4+Nh3t48cOVK72yXFafAkKcpms9l/qO3evTupU6c2XCRPIyEhgV27drFr1y4SEhJM58hT6tWrF56enmzYsIF9+/aZzhERESeUuM5r164dzzzzjOEaEXMcbZ3ctWtXAgICOHToEOvXrzedI25Gt9pJitq2bRsVKlTA19eXM2fOkDlzZtNJ8hQiIyMJCAgAICIiAn9/f8NF8rRatmzJvHnzaN26NbNmzTKdIyIiTuTXX3+lSJEi2Gw2jh07xrPPPms6ScQYR1wn9+7dm6+//poaNWpo+CQpSjueJEUlXgVr27athk4iDuidd94BYP78+Zw7d85wjYiIOJOvvvoKm81G/fr1NXQScUCJu9t/+OEH9u/fbzpH3IgGT5JifvvtN5YvXw5gf7KCiDiWsmXL8sorrxAfH29/FLaIiMjDXLlyhRkzZgDQt29fwzUi8nfy5MlDkyZNgPtPMhZJKRo8SYpJvAr22muvUbhwYdM5IvIPEn9gmDRpErdv3zZcIyIizmD8+PHcu3eP8uXLU6FCBdM5IvIPEne3z5s3j/PnzxuuEXehwZOkiKtXrzJ9+nTgwV92IuKYateuTZEiRbh79y5TpkwxnSMiIg4uKiqKcePGAffXeRaLxXCRiPyT559/nsqVK2t3u6QoDZ4kRSReBXv++eepVKmS6RwR+RceHh72AfHXX39NXFyc4SIREXFkM2fO5Nq1a+TJk4fXX3/ddI6IPMSfd7ffuXPHcI24Aw2eJNlFRETYp+l9+/bVVTARJ9CqVSsyZ87M+fPnmTt3rukcERFxUPHx8XzxxRfA/SdmeXl5GS4SkYepU6cORYoU4c6dO0ycONF0jrgBDZ4k2U2ePJkbN25QoEABgoODTedIEvL29ubjjz/m448/xtvb23SOJCE/Pz/7QwBGjBiB1Wo1XCQiIo5o4cKFnDx5kqCgIDp27Gg6R8RhOPI62cPDg/79+wMwatQooqOjDReJq7PYbDab6QhxXTExMeTLl4+LFy8yefJkOnXqZDpJRB7RnTt3yJ07N7du3WLJkiW6fUJERP7CarVSqlQpDh06xCeffMLAgQNNJ4nII4qLi6NAgQKcPXuWcePG0b17d9NJ4sK040mS1axZs7h48SLZsmWjTZs2pnNE5DEEBgby5ptvAjB8+HB0nUJERP5s9erVHDp0iDRp0tCjRw/TOSLyGLy9vXn33XcB+OKLL3SmpyQrDZ4k2cTHxzNixAjg/tlOvr6+hoskqVmtVg4fPszhw4d1K5aL6tWrF6lTp2bPnj388MMPpnNERMRB2Gw2hg8fDkD37t1Jly6d2SARB+MM6+SOHTvyzDPPcPr0aebPn286R1yYBk+SbBYvXsyJEyfImDEjnTt3Np0jySA6OprixYtTvHhx3RvuooKCgujSpQsAn376qeEaERFxFFu2bGHHjh34+fnRu3dv0zkiDscZ1smpUqWy//n99NNPHXZAJs5PgydJFjabzf5D6ltvvUVAQIDhIhF5Uu+88w7e3t6EhYWxfft20zkiIuIAEnc7hYSEkDlzZsM1IvKkunXrRmBgIEePHmX58uWmc8RFafAkyWLNmjUcOHCAgIAA+xkxIuKccuTIQdu2bQHtehIREdi9ezc//PADnp6e9jNiRMQ5pU2b1v7z2qeffqozPSVZaPAkSc5mszFs2DDg/gQ9Q4YMhotE5Gn1798fDw8PVq1axYEDB0zniIiIQYkXIVq1akWePHnMxojIU+vVqxepUqVi165dbNy40XSOuCANniTJhYeH89NPP+Hr66t7/kVcRMGCBWnSpAmgXU8iIu7s6NGjLF26FIvFwnvvvWc6R0SSwDPPPGM/kzdxA4FIUtLgSZLckCFDAOjQoQNZs2Y1XCMiSSXxB4yFCxfy66+/Gq4RERETPvnkEwAaNmxIkSJFDNeISFLp27cvXl5ehIWFsXXrVtM54mI0eJIkFR4ezqZNm/D29tZVMBEXU7p0aerXr4/VamXo0KGmc0REJIUdO3aMefPmAfDhhx8arhGRpJQzZ046dOgAwKBBg8zGiMvR4EmSVOJfUiEhIeTOndtsjCQ7b29v+vbtS9++ffH29jadIykg8c/4vHnzOHbsmNkYERFJUUOGDMFms9GwYUPKlCljOkfEoTnjOnngwIF4eXmxceNGfvzxR9M54kIsNh1bL0lky5YtVKlSBW9vb44fP06uXLlMJ4lIMmjYsCHLly+nRYsWzJ0713SOiIikgCNHjlC8eHFsNht79+6ldOnSppNEJBl07dqVb7/9lqpVq+qgcUky2vEkSSZxJ0THjh01dBJxYR9//DEA8+fP5+jRo4ZrREQkJQwdOhSbzUajRo00dBJxYe+//z7e3t5s2rSJ8PBw0zniIjR4kiQRFhZGWFgY3t7eDBgwwHSOpBCr1crp06c5ffo0VqvVdI6kkDJlytCwYUNsNpv9YQIiIuK6Dh8+zIIFC4AHFx9E5N856zo5d+7chISEADrrSZKObrWTJFGlShW2bNlCt27dGD9+vOkcSSGRkZEEBAQAEBERgb+/v+EiSSn79u2jTJkyWCwWDh06RNGiRU0niYhIMmnevDkLFizg9ddfZ8mSJaZzRJyCM6+Tz549S4ECBYiLiyMsLIzKlSubThInpx1P8tTCwsLYsmULPj4+2u0k4iZKly5No0aNtOtJRMTFHT58mIULFwLa7STiLnLlykXHjh0B7XqSpKHBkzwVm81mX4R06tSJnDlzGi4SkZSSuBBZuHAhhw8fNhsjIiLJYvDgwdhsNho3bkzJkiVN54hICnn//ffx8fGxH6ki8jQ0eJKnsnnzZsLDw7XbScQNlSxZksaNG2Oz2Rg8eLDpHBERSWIHDx5k0aJFgHY7ibibnDlz0qlTJ+D+n3+d0CNPQ4MneWI2m80+bOrcuTM5cuQwXCQiKe3jjz/GYrGwaNEi9uzZYzpHRESS0MCBAwEIDg6mRIkShmtEJKUNGDAAX19fwsPDWbdunekccWIaPMkT++6779i5cyepU6fmgw8+MJ0jIgaUKFGCli1bAve3ZIuIiGvYtm0bK1euxNPTk08++cR0jogYkCNHDnr06AHcH0I509P5xLFo8CRPJD4+3v5DZp8+fciSJYvhIhExZciQIXh7e7N+/Xo2bdpkOkdERJ6SzWbjvffeAyAkJIRnn33WcJGImPL+++8TGBjIvn37WLBggekccVIaPMkTmTlzJseOHSNjxoz07dvXdI4Y4uXlRffu3enevTteXl6mc8SQfPny0bVrVwDee+89nQEgIuLk1qxZw9atW/Hz89PZTiJPyFXWyRkzZuTdd98F4IMPPiA2NtZwkTgji00/Ichjio6OplChQpw/f54vv/ySPn36mE4SEcMuX75M/vz5iYyMZPHixTRu3Nh0koiIPIGEhATKlCnDwYMH6devH5999pnpJBExLCIiggIFCnD58mXGjRtH9+7dTSeJk9GOJ3ls48eP5/z58+TMmVN/6YgIAJkzZ7YPoQcOHEh8fLzhIhEReRLz5s3j4MGDpEuXjv79+5vOEREHEBAQwIcffgjcP2IhMjLScJE4Gw2e5LHcvn2b4cOHAzB48GD8/PwMF4lJNpuNq1evcvXqVd1eJfTt25egoCB+/fVXpk+fbjpHREQeU0xMjP2Hy/79+5MhQwbDRSLOy9XWyZ07dyZfvnxcvnyZr7/+2nSOOBkNnuSxfPHFF9y4cYOiRYvStm1b0zliWFRUFM888wzPPPMMUVFRpnPEsMDAQPujtwcNGkR0dLThIhEReRzffvstp0+fJmvWrLz11lumc0Scmqutk318fOxPuPz888+5fv264SJxJho8ySP7448/+OqrrwAYNmwYnp6ehotExNG88cYb5MqViwsXLjBmzBjTOSIi8oju3r3L0KFDAfj4449JnTq14SIRcTTNmjWjdOnS3Llzx34XjMij0OBJHtkHH3xAVFQUL774Ig0aNDCdIyIOyM/PjyFDhgD3B9RXr141XCQiIo/i008/5erVqxQsWJCQkBDTOSLigDw8PPj0008BGDNmDCdOnDBcJM5Cgyd5JHv37iU0NBSAUaNGYbFYDBeJiKNq06YNZcqU4c6dO3oMt4iIEzh9+jSjRo0C7h+r4O3tbbhIRBxVrVq1qFmzJnFxcfTr1890jjgJDZ7koWw2G71798Zms9GiRQteeukl00ki4sA8PDzst+VOmjSJQ4cOGS4SEZF/079/f2JiYqhatSr169c3nSMiDsxisTBq1Cg8PDxYunQpYWFhppPECWjwJA/13XffsWXLFvz8/BgxYoTpHBFxApUrV6Zx48ZYrVb69OnjEk9zERFxRdu2bWPhwoX2iwba1S4iD1OsWDHeeOMNAPr06UNCQoLhInF0GjzJv4qJiaFv377A/Uel58qVy3CRiDiLzz//HB8fH3744QfWrFljOkdERP6L1Wrl7bffBqBjx46ULFnSbJCIOI3BgweTNm1a9u7dy4wZM0zniIPT4En+1ZgxYzh58iRZs2alf//+pnPEwXh5edGuXTvatWuHl5eX6RxxMPny5bP/QPPOO+8QFxdnNkhERP5izpw57N69mzRp0tifaCciScPV18lBQUF89NFHAAwcOJC7d+8aLhJHZrHp/gf5B1euXKFgwYLcuXOH0NBQ2rdvbzpJRJzMnTt3KFCgAFevXuWbb77hrbfeMp0kIiJAZGQkhQoV4uLFi4wYMUIXGEXkscXGxlKsWDGOHz/O+++/z7Bhw0wniYPSjif5Rx9//DF37tzhueeeo23btqZzRMQJBQYG8sknnwAwaNAgbty4YbhIRETg/tPrLl68SN68eenVq5fpHBFxQj4+PowcORKAL7/8kjNnzhguEkelHU/yt/bt20fZsmWxWq2Eh4dTsWJF00nigGw2G1FRUQCkTp1aB5LK30pISKBMmTIcPHiQ7t27M27cONNJIiJu7fTp0xQtWpTo6GgWLVpEcHCw6SQRl+Mu62SbzUa1atXYvHkzjRs3ZvHixaaTxAFp8CT/w2q18vLLL7Njxw6aNm3KggULTCeJg4qMjCQgIACAiIgI/P39DReJo9q8eTNVq1bFYrGwa9cuypYtazpJRMRtNWjQgBUrVlC5cmU2b97ssj8Qi5jkTuvkAwcO8Nxzz5GQkMDatWupXbu26SRxMLrVTv7HtGnT2LFjBwEBAYwaNcp0joi4gFdeeYWWLVtis9no1q2bHrsrImLIihUrWLFiBV5eXowfP15DJxF5aiVLlrSf4/nmm29y7949w0XiaDR4kr+4du2a/XDJIUOGkD17dsNFIuIqRo4cSWBgILt27WLy5Mmmc0RE3E5UVJT9h8N33nmHokWLGi4SEVcxaNAgsmXLxokTJ/jss89M54iD0eBJ/uK9997jxo0blCxZkp49e5rOEREXkjVrVvtB4wMGDODKlSuGi0RE3MuwYcM4c+YMOXPm5MMPPzSdIyIuJDAwkK+++gqATz/9lOPHjxsuEkeiwZPYbd++nalTpwIwfvx4vLy8DBeJiKvp1q0bZcqU4datW/Tr1890joiI2zh27BhffPEFAKNHj3bp82ZExIwmTZpQo0YNYmJiePPNN9Fx0pJIgycBID4+nm7dugEQEhLCyy+/bLhIRFyRl5cXEyZMwGKxMGPGDMLDw00niYi4PJvNRo8ePYiLi6Nu3bo0aNDAdJKIuCCLxcLYsWPx8fFh3bp1LF261HSSOAgNngSAsWPHcuDAATJkyKB7ckUkWb3wwgt07twZgO7duxMXF2e4SETEtc2fP59Nmzbh5+fH6NGjdaC4iCSbQoUK2c8M7tWrF3fv3jVcJI5AgyfhzJkz9vv8R4wYQVBQkOEicRaenp4EBwcTHByMp6en6RxxIsOHDycoKIjDhw/bb/0QEZGkd/36dXr37g3AwIEDyZcvn+EiEffgzuvkAQMGkDdvXi5cuMAHH3xgOkccgMWmGy/dms1mo3bt2qxfv56XX36Z8PBwPDw0jxSR5Ddr1izatm2Lj48P+/bto0iRIqaTRERcTtu2bZk1axZFihRh7969+Pr6mk4SETewbt06ateujcVi4ccff9RRLm5OEwY3N336dNavX4+fnx/Tpk3T0ElEUkzr1q2pU6cOsbGxhISEkJCQYDpJRMSlrF69mlmzZuHh4UFoaKiGTiKSYmrVqkWHDh2w2WyEhIQQHR1tOkkM0pTBjV28eNG+9XrIkCEUKlTIcJGIuBOLxcKkSZMIDAxkx44djB492nSSiIjLuH37Nl27dgWgd+/evPDCC4aLRMTdfPnll2TNmpXffvuNwYMHm84Rg3SrnZuy2Ww0aNCAlStXUq5cObZv346Xl5fpLHEykZGRBAQEABAREaFHM8sTmTx5Ml26dCFVqlQcOHCAAgUKmE4SEXF6Xbp0YfLkyRQoUID9+/eTOnVq00kibkXr5PtWrFhBgwYN8PDwYMeOHZQrV850khigHU9uat68eaxcuRJvb2+mTZumoZOIGNOpUyeqVatGdHQ0HTt2xGq1mk4SEXFqGzZsYPLkyQBMnTpVQycRMaZ+/fq0aNECq9VKSEgIMTExppPEAA2e3NDly5fp2bMnAB999BHFixc3XCQi7sxisTB58mRSp05NeHg4EydONJ0kIuK0IiIi6NSpEwA9evSgUqVKhotExN2NHj2aTJkycejQIYYPH246RwzQ4MnN2Gw2evTowY0bNyhdujT9+/c3nSQiQt68eRkxYgQA/fr149SpU4aLREScU//+/Tlz5gy5c+fm008/NZ0jIkJQUBDjxo0DYPjw4ezdu9dwkaQ0DZ7czIwZM1iyZAleXl5MmzYNb29v00kiIsD9K/MVK1YkMjKS1q1bEx8fbzpJRMSprFmzhvHjxwP3z89LkyaN4SIRkfuCg4Np3Lgx8fHxtGrViqioKNNJkoI0eHIjx48ft99iN3ToUMqUKWO4SETkAQ8PD2bOnElgYCDbt2/XVmwRkcdw+fJlOnToAECvXr2oUaOG4SIRkQcsFgsTJ04ka9asHD16lHfffdd0kqQgDZ7cRFxcHK1btyYiIoLKlSvrD7qIOKQ8efLYr9YPGTKEn376yXCRiIjjs9lshISEcOXKFYoXL26/dVlExJEEBQUxY8YMAMaPH8+qVasMF0lK0eDJTQwdOpSff/6ZdOnSMWvWLDw9PU0niQvw9PSkTp061KlTR/9OSZJp1aoVLVu2JCEhgdatW3P37l3TSSIiDm38+PGsWbMGX19f5s6di5+fn+kkEbendfLfq1GjBn369AEgJCSEy5cvGy6SlGCx2Ww20xGSvLZt20alSpWwWq0sWLCApk2bmk4SEflXt27donTp0pw5c4b27dsTGhpqOklExCEdOXKEsmXLcu/ePb755hveeust00kiIv8qJiaG8uXLc+DAAV599VVWr16NxWIxnSXJSDueXNzt27dp3bo1VquVdu3aaegkIk4hXbp0zJ49Gw8PD6ZPn87ChQtNJ4mIOJyYmBhatmzJvXv3qF27tv0sTxERR/bn3Zlr1661P/FOXJcGTy7MZrPxxhtvcPr0afLly8fo0aNNJ4mIPLIKFSowcOBAALp27cqpU6cMF4mIOJb+/fuzf/9+MmXKRGhoqHYMiIjTKFasGCNHjgSgb9++7Nu3z2yQJCsNnlzY2LFjmT9/Pl5eXsyePZvAwEDTSeJiIiMj8ff3x9/fn8jISNM54oI+/PBDXnrpJW7dukVwcDD37t0znSQi4hAWLlzIN998A8C0adPIkiWL4SIR+TOtkx+ue/fu1KtXj5iYGBo3bsytW7dMJ0ky0eDJRf3000/2Q9tGjhzJSy+9ZLhIXFVUVBRRUVGmM8RFeXt7s2DBAoKCgvjll190domICHD06FE6duwIwIABA6hXr57hIhH5O1on/zuLxcKMGTPIkycPJ0+epG3btlitVtNZkgw0eHJBV65coUmTJsTHx9O0aVP9oCYiTi1nzpzMmzcPi8XC5MmTddC4iLi1iIgIGjduTEREBFWrVmXIkCGmk0REnliGDBlYvHgxvr6+rFy5ks8//9x0kiQDDZ5cTEJCAi1atODChQsULlyYKVOm6H5/EXF61atXZ+jQocD9bdk6B0BE3JHNZqNz584cPXqUbNmyMW/ePLy8vExniYg8lbJlyzJ27FgABg4cyKZNmwwXSVLT4MnFfPTRR2zatAl/f3+WLFlCmjRpTCeJiCSJAQMGULduXe7du6dzAETELf35/M5FixbxzDPPmE4SEUkSHTt2pEOHDlitVpo3b86FCxdMJ0kS0uDJhaxYsYLhw4cDMGXKFIoWLWq4SEQk6Xh4eDBr1iz7OQBt2rTROQAi4ja2bdtmP7/zyy+/5D//+Y/hIhGRpGOxWBg3bhylSpXi6tWrNGnShJiYGNNZkkQ0eHIR+/bto2XLlgD07NmT5s2bGy4SEUl66dOnZ8mSJfj6+rJq1Sree+8900kiIsnu1KlTNGrUiPj4eJo1a0bPnj1NJ4mIJLlUqVKxZMkS0qZNy08//UTnzp2x2WymsyQJaPDkAv744w9ee+01IiMjqV69Ol9++aXpJHETHh4eVK5cmcqVK+Phob9OJGU899xzTJ8+HYAvvviCqVOnmg0SEUlGt2/fpl69ely9epWyZcsydepUnd8p4gS0Tn4y+fPnZ9GiRXh6ejJr1iw+/fRT00mSBCw2jRCdWlRUFFWqVGHXrl0ULlyYn376iXTp0pnOEhFJdoMHD2bQoEF4eXmxfv16XnnlFdNJIiJJKj4+nrp167J+/XqyZ8/Ozp07yZYtm+ksEZFkN3HiRLp16wbAokWLCA4ONlwkT0OjVydmtVpp164du3btImPGjKxatUpDJxFxGx999BEtWrQgPj6exo0b89tvv5lOEhFJUm+//Tbr168nderUrFixQkMnEXEbb7zxBr169QKgbdu27Nq1y3CRPA0NnpzYRx99xOLFi/H29mbp0qXkz5/fdJKISIqxWCxMmzaNF198kZs3b1KvXj1u3LhhOktEJEmMHTuWcePGYbFYmDNnDs8995zpJBGRFPXll19Sp04doqOjqV+/PufOnTOdJE9IgycnNXXqVIYNGwbA5MmTqVSpkuEicUeRkZFkypSJTJkyERkZaTpH3JCfnx/fffcduXLl4vfff6dhw4ZER0ebzhIReSrLly+3X+kfMWIEDRs2NBskIo9N6+Sn5+npyfz58ylRogSXLl2ibt263Lx503SWPAENnpzQ0qVL6dKlCwDvv/8+7dq1M1wk7uzatWtcu3bNdIa4scyZM7Nq1SoCAwP58ccfadq0KXFxcaazRESeSFhYGM2aNcNqtRISEsK7775rOklEnpDWyU8vTZo0rFy5kixZsnDw4EFee+01oqKiTGfJY9Lgycls2LCBFi1aYLVa6dSpE5988onpJBER40qUKMHKlSvx8/Nj1apVhISEYLVaTWeJiDyWPXv2UL9+fWJiYmjQoAGTJk3SE+xExO3lzp2bdevWkS5dOrZt20ZwcDCxsbGms+QxaPDkRH7++WcaNmxIbGwswcHBTJw4UYsREZH/V6lSJfvjd2fPnk3v3r3Rg1tFxFkcO3aM2rVrc/fuXV555RXmz5+Pl5eX6SwREYdQsmRJVq9eTapUqVi7di3t2rUjISHBdJY8Ig2enMThw4epU6cOkZGR1KhRg9mzZ+Pp6Wk6S0TEodSrV48ZM2YAMHr0aIYOHWq4SETk4c6ePUvNmjW5du0aZcuW5bvvvsPPz890loiIQ/nPf/7D0qVL8fb2Zv78+fTs2VMXGZ2EBk9O4Pjx49SsWZMbN27wwgsvsHTpUnx9fU1niYg4pFatWjF69GgAPv74Y77++muzQSIi/+KPP/6gZs2anDt3jsKFC7N27VoCAwNNZ4mIOKTatWsza9YsLBYLEyZMYMCAARo+OQENnhzcr7/+SqVKlbh48SJFixZl9erVBAQEmM4SEXFoPXv2ZPDgwQD07t2bkSNHGi4SEflf58+fp3Llyvz666/kypWL9evXkylTJtNZIiIOrVmzZowfPx6Azz77jH79+mn45OB047gDO3LkCFWrVuXy5csUK1aMjRs3kjFjRtNZInYeHh48//zz9l+LOJIPP/yQ+Ph4hg4dyrvvvktsbCzvv/++6SwREeD+7XWvvPIKJ0+eJHfu3GzevJmcOXOazhKRJKJ1cvJ64403iI+Pp2fPnowcOZK4uDi++uornYHsoCw2jQYd0oEDB6hevTpXr16lZMmSbNiwQVfARESewNChQ/noo48AGDRoEB999JEWJSJi1KlTp6hatSqnT58mX758bNq0idy5c5vOEhFxOpMmTeKNN94AoHv37owZM0aDPgekwZMD2rt3LzVq1OD69es899xzrF+/XjudRESewogRIxgwYAAAAwcOZOjQoRo+iYgRx48fp2rVqpw7d46CBQuyadMmcuTIYTpLRMRpTZs2jU6dOmGz2ejcuTMTJ07U8MnBaPDkYMLDw2nQoAG3bt2ifPnyrFu3jnTp0pnOEhFxeqNGjeKdd94B4M033+Trr7/W00FFJEXt27ePOnXq8Mcff1C4cGE2btxItmzZTGeJiDi9WbNm0b59e6xWKy1atCA0NFQP5HIgGgM6kIULF1KjRg1u3brFyy+/zPr16zV0EocWFRVFnjx5yJMnD1FRUaZzRP5Vnz59GDNmDABjx46lSZMmREdHG64SEXexfv16KlasyB9//EHx4sUJCwvT0EnEhWmdnLLatGnDnDlz8PLyYt68ebz66qvcunXLdJb8P+14cgA2m42vvvrKfiW+UaNGzJkzh1SpUhkuE/l3kZGR9qcsRkRE4O/vb7hI5OEWLlxImzZtiI2N5aWXXmLFihUEBQWZzhIRFzZjxgw6depEfHw8r7zyCkuXLtXFRREXp3WyGT/88AONGzfm7t27FC9enDVr1ujBDQ5AO54MS0hIoHfv3vahU8+ePVm0aJGGTiIiyaRp06b88MMPpEuXjp9++omXX36ZkydPms4SERdks9n45JNPaN++PfHx8bRo0YK1a9dq6CQikkxq1KhBeHg4WbNm5dChQ7z00kscPHjQdJbb0+DJoLt379KkSRO++eYbAEaOHMk333yjM0dERJJZpUqV2LZtG7ly5eK3337jpZdeYvv27aazRMSFxMTE0LlzZz788EMA+vfvz+zZs3XmiIhIMitdujQ7duygaNGiXLhwgQoVKrB27VrTWW5NgydDfvvtN1588UWWLVuGj48P8+bN45133tFTlkREUkjRokX56aefKF26NFeuXKFKlSpMmDAB3YEuIk/r/PnzVK5cmalTp2KxWBg7diwjRozQU5ZERFJIrly52Lp1K5UqVeLOnTvUrVuXYcOGYbVaTae5JX33M2DVqlWUK1eOI0eOkC1bNsLCwmjevLnpLBERt5MtWzZ+/PFHgoODiYuLo3v37nTq1Il79+6ZThMRJxUeHk7ZsmX5+eefSZ8+PWvWrKFHjx6ms0RE3E769OlZv349Xbt2xWaz8cEHH9C4cWPu3LljOs3taPCUgqxWK4MGDeK1117jzp07VKhQgT179vDSSy+ZThMRcVsBAQEsXLiQzz77DA8PD6ZNm0alSpU4d+6c6TQRcSI2m43Ro0dTrVo1rly5QsmSJdm9eze1a9c2nSYi4rZ8fX2ZOHEikydPxsfHh++++44XXniBY8eOmU5zKxo8pZDLly9Tt25dBg8eDMCbb77Jxo0byZIli+EykSdnsVgoWrQoRYsW1W2i4tQsFgv9+vXj+++/J0OGDOzatYvnnnuOVatWmU4TESdw69YtWrZsSa9eveyHiG/fvp18+fKZThMRQ7ROdiydOnUiPDyc7Nmzc+zYMcqXL8+cOXNMZ7kNi02HWSS7VatWERISwtWrV/Hz82PixIm0a9fOdJaIiPyN06dP8/rrr7N3714AunfvzhdffEHq1KkNl4mIIwoPD6dNmzacPXsWT09PvvjiC95++239oCki4oAuX75M06ZNCQ8PB6Bly5aMGzdOTxtNZtrxlIyioqLo0aMHr732GlevXrVvudbQSUTEceXJk4ft27fTu3dvAMaPH8/zzz/Pvn37zIaJiEOJi4tj4MCBVKlShbNnz5I/f362bdtG7969NXQSEXFQmTNnZuPGjQwZMgRPT0/mzp1L6dKl+fHHH02nuTTteEome/bsoU2bNhw9ehSA3r17M3z4cPz8/AyXiYjIo1q/fj3t2rXj0qVLeHt7M2zYMHr37o2Xl5fpNBEx6NixY7Rt25Zdu3YB0KFDB7755hvSpEljuExERB7Vjh07aNWqFSdPnsTDw4P33nuPjz76CF9fX9NpLkc7npJYREQEffr0oXz58hw9epSsWbOyfv16Ro0apaGTuJyoqCiKFStGsWLFiIqKMp0jkuRq1qzJwYMHadCgAXFxcfTr148XX3zRfhueiLiXmJgYBg8eTKlSpdi1axfp0qVj4cKFTJs2TUMnEfkLrZMd34svvsi+ffto3749VquV4cOHU7p0aftteJJ0tOMpCa1Zs4bu3btz5swZAJo3b86YMWMICgoyXCaSPCIjIwkICADuD139/f0NF4kkD5vNxrRp0+jbty+3bt3C09OT3r17M2jQIP17L+ImfvzxR7p06WJ/EtKrr77KpEmTyJkzp+EyEXFEWic7lyVLltCjRw8uX74MQOfOnfnss89Inz694TLXoB1PSeDixYu0aNGCunXrcubMGfLkycPatWuZN2+ehk4iIi7AYrHQsWNHjh49SrNmzUhISGDkyJEUL16c1atXo2s4Iq7r+vXrdO3alUqVKnHs2DGeeeYZ5s+fz+rVqzV0EhFxEY0bN+bo0aN06dIFgMmTJ1OkSBHmzZundV4S0ODpKURGRjJ48GAKFizI/Pnz8fDwoG/fvhw6dIjatWubzhMRkSSWJUsW5s+fz6pVq8iVKxenT5+mXr161K5dm4MHD5rOE5EkFBMTw6hRoyhQoADffvstcP8K+LFjx2jWrJkOEBcRcTHp06dn0qRJhIeHU7hwYS5fvkzLli15+eWX2bFjh+k8p6bB0xOwWq1Mnz6dQoUKMWjQIKKiovjPf/7Drl27+OKLL7SNUkTExdWtW5fDhw/z7rvv4uPjw/r16yldujRdunTh0qVLpvNE5CnYbDaWLFlCsWLFeOedd7h16xalSpViy5YtfPvtt7rtQkTExVWsWJF9+/YxdOhQUqdOzU8//cRLL71EixYtOH36tOk8p6TB02Ow2WysWrWK559/ng4dOnDx4kXy5s3LwoUL2bp1K88995zpRBERSSEBAQF8/vnnHD16lCZNmmC1Wpk8eTIFCxZk0KBB3Lp1y3SiiDymsLAwKlWqRHBwMCdOnCBLlixMnTqVPXv2UKlSJdN5IiKSQnx9ffnggw/4/fffCQkJwWKxMH/+fAoXLkzfvn25cuWK6USnosPFH4HNZmPlypUMGTKEPXv2ABAYGMgHH3xAz5499bQ6cVs6NFHkgW3bttG7d2/749XTpk1Lr169ePvtt7VDQsSB2Ww2Nm/ezODBg+1PMkqVKhXvvvsu7777rv37nIjI49A62bXs27ePPn36sHnzZuD+94lu3brx7rvvkiVLFsN1jk+Dp3+RkJDAihUrGDp0qP3R2alTp6ZHjx68++67ZMqUyXChiFlRUVEULVoUgCNHjpA6dWrDRSJmWa1WFi9ezJAhQzh8+DAAadKk4a233qJXr176viHiQGw2Gz/88ANDhw5l69atAPj4+NCpUycGDBhAjhw5DBeKiDPTOtn12Gw21q5dy+DBg9m5cycAfn5+dO3alb59++r7xr/Q4Olv3Llzh9DQUEaPHs3JkyeB+7dUvPnmm/Tp00c/OIiIyL+yWq0sXbqUIUOG2A8d9/X1pXXr1vTq1YsSJUoYLhRxX9HR0cyePZtvvvnGPiD29fWlc+fO9O/fXz84iIjIv7LZbKxbt47BgwfbDx338vIiODiYt99+mxdeeMFwoePR4OlPTpw4wdixY5k6dSp3794F7p9s361bN3r37k1QUJDhQhERcSZWq5XvvvuOTz/9lN27d9t/v2rVqvTq1Yu6devi6elpsFDEfZw/f54JEyYwadIkrl+/DoC/vz+dOnWiX79+ZMuWzXChiIg4k8Sds8OGDbPfqg3w4osv0qtXL15//XV8fHwMFjoOtx88RUVFsWTJEqZNm0ZYWJj99wsXLkyvXr1o06aN7scVEZGnYrPZ2L59O9988w1LlizBarUCkD17dtq1a0eHDh0oUKCA4UoR1xMTE8PKlSuZNm0a69ats//Zy5MnDz179iQkJIR06dKZjRQREae3d+9evvnmG+bNm0dsbCwAQUFBtGnThpCQEIoXL2640Cy3HDxZrVZ++uknZs6cybx58+y7mywWC7Vq1eLtt9+mRo0aeHjooX8i/yY6Otr+lJ/w8HBSpUpluEjE8Z09e5Zx48YxZcoUbty4Yf/9SpUq0aFDBxo1akTatGkNFoo4N5vNxi+//MKsWbOYPXu2fXcTQOXKlenVqxf169fXbkMRSVZaJ7uny5cvM3HiRCZOnMilS5fsv1+uXDk6dOhAkyZN3PJOKrcZPCUOmxYtWsTixYu5cOGC/Z/lzZuXkJAQ2rZtS65cuQxWijgXPa1D5Mn9004MHx8fatasSdOmTalfv76GUCKPIHHYtGjRIhYtWmQ/oxMgW7ZstG/fnvbt21OwYEGDlSLiTrROdm/x8fF8//33TJs2jZUrVxIfHw+Ap6cnVatWpUmTJjRq1MhthlAuPXiKiopi8+bNrFmzhuXLl/9l2BQYGEjDhg3p0KEDlSpV0u4mkSegb6giSeP8+fPMmDGDuXPncuTIEfvv+/j4UKNGDerWrUudOnXInTu3wUoRxxITE8OPP/5oX+f9ediUKlUq6tWrR4cOHahZs6Z2N4lIitM6WRJduXKFOXPmMGvWLPbu3Wv/fU9PT6pUqUK9evWoU6cOBQsWxGKxGCxNPi41eLLZbBw7dowNGzawZs0aNm/eTExMjP2fBwYGUr9+fZo2bUrNmjXx9fU1WCvi/PQNVSTpHT582L5r489DKICiRYtSp04datWqxX/+8x89mlncis1m48SJE2zcuJG1a9eyYcMGIiMj7f88cdjUpEkT6tSpo+9JImKU1snyd44fP87ixYtZuHDhX4ZQAPnz56dOnTrUrl2bChUqEBgYaKgy6Tn14MlqtXLs2DG2bNlCWFgYYWFhXLly5S8fkytXLvuV4ho1amjYJJKE9A1VJHkdPnyYlStXsmbNGrZv305CQoL9n3l7e/PCCy9QpUoVqlSpwosvvqg/g+JSEgdNW7ZsYfPmzYSFhf1l9zpAlixZqFOnjn2hrj8DIuIotE6Whzl+/Lh9nbdlyxbi4uLs/8zT05OyZcva13kvv/yyUw+inGrwdPXqVXbu3MnPP//Mzz//zM6dO7l169ZfPsbPz4+XX36Z2rVrU6dOHYoUKeKy29X+r7075mmqDcMA/CgaleJgYoKJg8bFU0ptFy2aOOjKpC7uzi4O/gcHf4kxTroY1FHEoeBgJWiVsJiYEJEjRJR+A7GkQr6U7+NwKF5XwkLf4Vnac+c+73sO5M0FFXbO/Px8PH36NJ48eRJjY2MxNzfX8XlfX18MDw9HrVaLWq0WIyMjcfbsWUeM6Bnz8/MxMTER4+Pj8fLly3j16lV8+fKlY83vwvV3zqtWq3IesCvJyWzFt2/f4tmzZ/H48eMYGxvrOD4esfYitGKx2JHzSqVSHDhwIKeJt2ZXFk8rKyvx/v37mJqaisnJyajX6zE5ObnhLlfEWtF06dKldhN44cIFu5pgh7igQj5arVZ8+PC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" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "ttesthyp_onesample-fig" } }, "output_type": "display_data" } ], "source": [ "mu = 0\n", "sigma = 1\n", "x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)\n", "y = 100* stats.norm.pdf(x, mu, sigma)\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 5))\n", "\n", "sns.lineplot(x=x,y=y, color='black', ax=axes[0])\n", "sns.lineplot(x=x,y=y, color='black', ax=axes[1])\n", "\n", "axes[0].set_frame_on(False)\n", "axes[1].set_frame_on(False)\n", "axes[0].get_yaxis().set_visible(False)\n", "axes[1].get_yaxis().set_visible(False)\n", "axes[0].get_xaxis().set_visible(False)\n", "axes[1].get_xaxis().set_visible(False)\n", "\n", "\n", "axes[0].axhline(y=0, color='black')\n", "axes[0].axvline(x=mu, color='black', linestyle='--')\n", "\n", "axes[1].axhline(y=0, color='black')\n", "axes[1].axvline(x=mu + sigma, color='black', linestyle='--')\n", "\n", "axes[0].hlines(y=23.6, xmin = mu-sigma, xmax = mu, color='black')\n", "axes[1].hlines(y=23.6, xmin = mu-sigma, xmax = mu, color='black')\n", "\n", "\n", "axes[0].text(mu,42, r'$\\mu = \\mu_0$', size=20, ha=\"center\")\n", "axes[1].text(mu + sigma, 42, r'$\\mu \\neq \\mu_0$', size=20, ha=\"center\")\n", "\n", "axes[0].text(mu-sigma - 0.2, 23.6, r'$\\sigma = ??$', size=20, ha=\"right\")\n", "axes[1].text(mu-sigma - 0.2, 23.6, r'$\\sigma = ??$', size=20, ha=\"right\")\n", "\n", "# show the figure in the book with caption\n", "plt.close()\n", "from myst_nb import glue\n", "glue(\"ttesthyp_onesample-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "norwegian-programming", "metadata": {}, "source": [ "\n", "```{glue:figure} ttesthyp_onesample-fig\n", ":figwidth: 600px\n", ":name: fig-ttesthyp_onesample\n", "\n", "\n", "Graphical illustration of the null and alternative hypotheses assumed by the (two sided) one sample $t$-test. Note the similarity to the $z$-test. The null hypothesis is that the population mean $\\mu$ is equal to some specified value $\\mu_0$, and the alternative hypothesis is that it is not. Like the $z$-test, we assume that the data are normally distributed; but we do not assume that the population standard deviation $\\sigma$ is known in advance.\n", "\n", "```\n", "\n", "\n", "\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "cooperative-agent", "metadata": {}, "source": [ "(ttest_onesample)=\n", "### Introducing the $t$-test\n", "\n", "This ambiguity is annoying, and it was resolved in 1908 by a guy called William Sealy Gosset {cite}`Student1908`, who was working as a chemist for the Guinness brewery at the time. Because Guinness took a dim view of its employees publishing statistical analysis (apparently they felt it was a trade secret), he published the work under the pseudonym \"A Student\", and to this day, the full name of the $t$-test is actually **_Student's $t$-test_**. The key thing that Gosset figured out is how we should accommodate the fact that we aren't completely sure what the true standard deviation is.[^note4] The answer is that it subtly changes the sampling distribution. In the $t$-test, our test statistic (now called a $t$-statistic) is calculated in exactly the same way I mentioned above. If our null hypothesis is that the true mean is $\\mu$, but our sample has mean $\\bar{X}$ and our estimate of the population standard deviation is $\\hat{\\sigma}$, then our $t$ statistic is:\n", "\n", "$$\n", "t = \\frac{\\bar{X} - \\mu}{\\hat{\\sigma}/\\sqrt{N} }\n", "$$\n", "\n", "The only thing that has changed in the equation is that instead of using the known true value $\\sigma$, we use the estimate $\\hat{\\sigma}$. And if this estimate has been constructed from $N$ observations, then the sampling distribution turns into a $t$-distribution with $N-1$ **_degrees of freedom_** (df). The $t$ distribution is very similar to the normal distribution, but has \"heavier\" tails, as [discussed earlier](otherdists) and illustrated in {numref}`fig-ttestdist`. Notice, though, that as df gets larger, the $t$-distribution starts to look identical to the standard normal distribution. This is as it should be: if you have a sample size of $N = 70,000,000$ then your \"estimate\" of the standard deviation would be pretty much perfect, right? So, you should expect that for large $N$, the $t$-test would behave exactly the same way as a $z$-test. And that's exactly what happens! " ] }, { "cell_type": "code", "execution_count": 16, "id": "virgin-standing", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "ttestdist-fig" } }, "output_type": "display_data" } ], "source": [ "mu = 0\n", "variance = 1\n", "sigma = np.sqrt(variance)\n", "\n", "\n", "x = np.linspace(-4, 4, 100)\n", "y_norm = stats.norm.pdf(x, mu, sigma)\n", "\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 5))\n", "\n", "\n", "# t-distribution with 2 degrees of freedom\n", "y_t = stats.t.pdf(x, 2)\n", "sns.lineplot(x = x, y = y_norm, color = 'black', linestyle='--', ax = axes[0])\n", "sns.lineplot(x = x, y = y_t, color = 'black', ax = axes[0])\n", "\n", "# t-distribution with 10 degrees of freedom\n", "y_t = stats.t.pdf(x, 10)\n", "sns.lineplot(x = x, y = y_norm, color = 'black', linestyle='--', ax = axes[1])\n", "sns.lineplot(x = x, y = y_t, color = 'black', ax = axes[1])\n", "\n", "axes[0].text(0, 0.42, r'$df = 2$', size=20, ha=\"center\")\n", "axes[1].text(0, 0.42, r'$df = 10$', size=20, ha=\"center\")\n", "\n", "\n", "#sns.despine()\n", "axes[0].get_yaxis().set_visible(False)\n", "axes[1].get_yaxis().set_visible(False)\n", "axes[0].set_frame_on(False)\n", "axes[1].set_frame_on(False)\n", "\n", "plt.close()\n", "\n", "# show the figure in the book with caption\n", "plt.close()\n", "from myst_nb import glue\n", "glue(\"ttestdist-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "involved-disclaimer", "metadata": {}, "source": [ "\n", "```{glue:figure} ttestdist-fig\n", ":figwidth: 600px\n", ":name: fig-ttestdist\n", "\n", "The $t$ distribution with 2 degrees of freedom (left) and 10 degrees of freedom (right), with a standard normal distribution (i.e., mean 0 and std dev 1) plotted as dotted lines for comparison purposes. Notice that the $t$ distribution has heavier tails (higher kurtosis) than the normal distribution; this effect is quite exaggerated when the degrees of freedom are very small, but negligible for larger values. In other words, for large $df$ the $t$ distribution is essentially identical to a normal distribution.\n", "\n", "\n", "```\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "federal-teddy", "metadata": {}, "source": [ "### Doing the test in Python\n", "\n", "As you might expect, the mechanics of the $t$-test are almost identical to the mechanics of the $z$-test. So there's not much point in going through the tedious exercise of showing you how to do the calculations using low level commands: it's pretty much identical to the calculations that we did earlier, except that we use the estimated standard deviation (i.e., something like `se_est = statistics.stdev(grades)`), and then we test our hypothesis using the $t$ distribution rather than the normal distribution. And so instead of going through the calculations in tedious detail for a second time, I'll jump straight to showing you how $t$-tests are actually done in practice. \n", "\n", "The situation with $t$-tests is very similar to the one we encountered with [chi-squared tests](chisquare). `scipy.stats` includes a variety of methods for running different kinds of $t$-tests, but the `pingouin` package makes wraps these up and makes them easier (to my mind) to deal with. We'll start with the `scipy` version for the one-sample $t$-test, so you can get a flavor for how `scipy` deals with $t$-tests, and then use the `pingouin` alternatives for the rest.\n", "\n", "To run a one-sample $t$-test with `scipy`, use the `ttest_1samp` method. It's pretty straightforward to use: all you need to do is specify `a`, the variable containing the data, and `popmean`, the true population mean according to the null hypothesis. All you need to type is this:" ] }, { "cell_type": "code", "execution_count": 17, "id": "focused-connection", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(2.25471286700693, 0.03614521878144544)" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from scipy.stats import ttest_1samp\n", "t, p = ttest_1samp(a = df['grades'], popmean = 67.5)\n", "t, p" ] }, { "attachments": {}, "cell_type": "markdown", "id": "amazing-destiny", "metadata": {}, "source": [ "So that seems straightforward enough. Our calculation resulted in a $t$-statistic of 2.54, and a $p$-value of 0.036. Now what do we *do* with this output? Well, since we're pretending that we actually care about my toy example, we're overjoyed to discover that the result is statistically significant (i.e. $p$ value below .05), and we will probably want to report our result. We could report the result by saying something like this:\n", "\n", "> With a mean grade of 72.3, the psychology students scored slightly higher than the average grade of 67.5 ($t(19) = 2.25$, $p<.05$); the 95\\% confidence interval is [67.8, 76.8].\n", "\n", "where $t(19)$ is shorthand notation for a $t$-statistic that has 19 degrees of freedom.\n", "\n", "As you will have already noticed, `ttest_1samp()` does not give us all the information included in the report above. A $t$-statistic, yes, and a $p$-value as well. But what about 95% confidence intervals? Where can we find those, and what about the degrees of freedom? In a better, simpler, kinder world, `ttest_1samp()` would provide all this information for us, and it's honestly a bit shocking that it doesn't. But sadly that is not the world we live in, and we have a bit more work to do.\n", "\n", "First, off: degrees of freedom. In this case, that is not so difficult, because the degrees of freedom for a one-sample t-test is just $N-1$, so we can easily find that ourselves. We also already know how to find the sample mean of our data:" ] }, { "cell_type": "code", "execution_count": 18, "id": "confident-content", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Sample mean: 72.3\n", "Degrees of freedom: 19\n" ] } ], "source": [ "N = len(df['grades'])\n", "degfree = N-1\n", "sample_mean = df['grades'].mean()\n", "print('Sample mean:', sample_mean)\n", "print('Degrees of freedom:', degfree)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "amino-youth", "metadata": {}, "source": [ "Now at least we have the bare minimum of what is necessary to report our results. Still, it would be sweet if we could get those confidence intervals as well. `scipy` actually has all the tools we need, and why these are not just built into the `ttest_1samp()` method is beyond me. To find the confidence interval, we need to:\n", "\n", "1. To set a confidence level\n", "1. To know the sample mean\n", "1. To know the degrees of freedom for the test\n", "1. To know the standard error of the sample\n", "\n", "We discussed confidence intervals [earlier](ci), and now is not the time to go into detail on this, so for now I'm just going to give you some code that you can use to find 95% confidence intervals for a one-sample t-test, so that we can get on with reporting our data:" ] }, { "cell_type": "code", "execution_count": 19, "id": "related-marijuana", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(67.84421513791415, 76.75578486208585)" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from scipy import stats\n", "\n", "confidence_level = 0.95\n", "degrees_freedom = len(df['grades'])-1\n", "sample_mean = df['grades'].mean()\n", "sample_standard_error = df['grades'].sem()\n", "\n", "confidence_interval = stats.t.interval(confidence_level, degrees_freedom, sample_mean, sample_standard_error)\n", "confidence_interval" ] }, { "attachments": {}, "cell_type": "markdown", "id": "actual-county", "metadata": {}, "source": [ "Whew. Now at least we have everything we need for a full report of our results.\n", "\n", "That said, it's often the case that people don't report the confidence interval, or do so using a much more compressed form than I've done here. For instance, it's not uncommon to see the confidence interval included as part of the stat block, like this:\n", "\n", "> $t(19) = 2.25$, $p<.05$, CI$_{95} = [67.8, 76.8]$\n", "\n", "With that much jargon crammed into half a line, you know it must be really smart. [^note5]" ] }, { "attachments": {}, "cell_type": "markdown", "id": "c1014d22", "metadata": {}, "source": [ "Having gone through all that, let's take a look at the `pingouin` command to achieve the same thing." ] }, { "cell_type": "code", "execution_count": 20, "id": "126b3305", "metadata": { "tags": [ "remove-cell" ] }, "outputs": [ { "data": { "text/plain": [ "{'divide': 'warn', 'over': 'warn', 'under': 'ignore', 'invalid': 'warn'}" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# hopefully temporary code to silence pointless numpy warnings on macOS\n", "np.seterr(all=\"ignore\")\n" ] }, { "cell_type": "code", "execution_count": 21, "id": "85e0ebb3", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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Tdofalternativep-valCI95%cohen-dBF10power
T-test2.2519two-sided0.04[67.84, 76.76]0.51.7950.57
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" ], "text/plain": [ " T dof alternative p-val CI95% cohen-d BF10 power\n", "T-test 2.25 19 two-sided 0.04 [67.84, 76.76] 0.5 1.795 0.57" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "from pingouin import ttest\n", "ttest(df['grades'], 67.5).round(2)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "d36be179", "metadata": {}, "source": [ "Well. That was easy! All we need to do is feed the `ttest()` function from `pingouin` with the data and the population mean under the null hypothesis, and `pinguoin` does the rest. We get a nice table with the $t$-value, the degrees of freedom, the $p$-value, 95% confidence interval, effect size (Cohen's $d$), and a power estimate. BF10 refers to the \"bayes factor\", which we will meet (briefly) in the section on [Bayesian statistics](bayes)." ] }, { "attachments": {}, "cell_type": "markdown", "id": "protecting-better", "metadata": {}, "source": [ "(ttestoneassumptions)=\n", "### Assumptions of the one sample $t$-test\n", "\n", "Okay, so what assumptions does the one-sample $t$-test make? Well, since the $t$-test is basically a $z$-test with the assumption of known standard deviation removed, you shouldn't be surprised to see that it makes the same assumptions as the $z$-test, minus the one about the known standard deviation. That is\n", "\n", "- *Normality*. We're still assuming that the the population distribution is normal[^note_normality].\n", "- *Independence*. Once again, we have to assume that the observations in our sample are generated independently of one another. See the [earlier discussion](zassumptions) about the $z$-test for specifics.\n", "\n", "Overall, these two assumptions aren't terribly unreasonable, and as a consequence the one-sample $t$-test is pretty widely used in practice as a way of comparing a sample mean against a hypothesised population mean.\n", "\n", "[^note_normality]: A technical comment... in the same way that we can weaken the assumptions of the $z$-test so that we're only talking about the sampling distribution, we *can* weaken the $t$ test assumptions so that we don't have to assume normality of the population. However, for the $t$-test, it's trickier to do this. As before, we can replace the assumption of population normality with an assumption that the sampling distribution of $\\bar{X}$ is normal. However, remember that we're also relying on a sample estimate of the standard deviation; and so we also require the sampling distribution of $\\hat{\\sigma}$ to be chi-square. That makes things nastier, and this version is rarely used in practice: fortunately, if the population is normal, then both of these two assumptions are met., and as noted earlier, there are [standard tools](shapiro) that you can use to check to see if this assumption is met, and [other tests](wilcox) you can do in it's place if this assumption is violated." ] }, { "attachments": {}, "cell_type": "markdown", "id": "stuffed-retail", "metadata": {}, "source": [ "(studentttest)=\n", "## The independent samples $t$-test (Student test)\n", "\n", "Although the one sample $t$-test has its uses, it's not the most typical example of a $t$-test[^note6]. A much more common situation arises when you've got two different groups of observations. In psychology, this tends to correspond to two different groups of participants, where each group corresponds to a different condition in your study. For each person in the study, you measure some outcome variable of interest, and the research question that you're asking is whether or not the two groups have the same population mean. This is the situation that the independent samples $t$-test is designed for. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "corrected-victim", "metadata": {}, "source": [ "### The data\n", "\n", "Suppose we have 33 students taking Dr Harpo's statistics lectures, and Dr Harpo doesn't grade to a curve. Actually, Dr Harpo's grading is a bit of a mystery, so we don't really know anything about what the average grade is for the class as a whole. There are two tutors for the class, Anastasia and Bernadette. There are $N_1 = 15$ students in Anastasia's tutorials, and $N_2 = 18$ in Bernadette's tutorials. The research question I'm interested in is whether Anastasia or Bernadette is a better tutor, or if it doesn't make much of a difference. Dr Harpo emails me the course grades, in the `harpo.csv` file. As usual, I'll load the file and have a look at what variables it contains:" ] }, { "cell_type": "code", "execution_count": 22, "id": "specific-direction", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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gradetutor
065Anastasia
172Bernadette
266Bernadette
374Anastasia
473Anastasia
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" ], "text/plain": [ " grade tutor\n", "0 65 Anastasia\n", "1 72 Bernadette\n", "2 66 Bernadette\n", "3 74 Anastasia\n", "4 73 Anastasia" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/harpo.csv\")\n", "df.head()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "annual-correction", "metadata": {}, "source": [ "As we can see, there's a single data frame with two variables, `grade` and `tutor`. The `grade` variable is a numeric vector, containing the grades for all $N = 33$ students taking Dr Harpo's class; the `tutor` variable is a factor that indicates who each student's tutor was. The first five observations in this data set are shown above, and below is a nice little table with some summary statistics:" ] }, { "cell_type": "code", "execution_count": 23, "id": "2306c570", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
studentsmeanstd devN
0Anastasia's students74.538.99894215
1Bernadette's students69.065.77491818
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" ], "text/plain": [ " students mean std dev N\n", "0 Anastasia's students 74.53 8.998942 15\n", "1 Bernadette's students 69.06 5.774918 18" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "harpo_summary = pd.DataFrame(\n", " {'students': ['Anastasia\\'s students','Bernadette\\'s students'],\n", " 'mean': [df.loc[df['tutor'] == 'Anastasia']['grade'].mean().round(2), \n", " df.loc[df['tutor'] == 'Bernadette']['grade'].mean().round(2)],\n", " 'std dev': [df.loc[df['tutor'] == 'Anastasia']['grade'].std(), \n", " df.loc[df['tutor'] == 'Bernadette']['grade'].std()],\n", " 'N': [len(df.loc[df['tutor'] == 'Anastasia']),\n", " len(df.loc[df['tutor'] == 'Bernadette'])]\n", " })\n", "harpo_summary" ] }, { "attachments": {}, "cell_type": "markdown", "id": "pediatric-island", "metadata": {}, "source": [ "To give you a more detailed sense of what's going on here, I've plotted histograms showing the distribution of grades for both tutors {numref}`fig-harpohist`. Inspection of these histograms suggests that the students in Anastasia's class may be getting slightly better grades on average, though they also seem a little more variable." ] }, { "cell_type": "code", "execution_count": 24, "id": "narrow-expansion", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "harpohist_fig" } }, "output_type": "display_data" } ], "source": [ "fig, axes = plt.subplots(1, 2, figsize=(15, 5))\n", "Anastasia = pd.DataFrame(df.loc[df['tutor'] == 'Anastasia']['grade'])\n", "Bernadette = pd.DataFrame(df.loc[df['tutor'] == 'Bernadette']['grade'])\n", "\n", "sns.histplot(Anastasia['grade'], ax = axes[0], binwidth = 5)\n", "sns.histplot(Bernadette['grade'], ax = axes[1], binwidth = 5)\n", "\n", "axes[0].set_xlim(50,100)\n", "axes[1].set_xlim(50,100)\n", "\n", "axes[0].set_ylim(0,7)\n", "axes[1].set_ylim(0,7)\n", "\n", "axes[0].set_title('Anastasia')\n", "axes[1].set_title('Bernadette')\n", "\n", "sns.despine()\n", "\n", "# show the figure in the book with caption\n", "plt.close()\n", "from myst_nb import glue\n", "glue(\"harpohist_fig\", fig, display=False)\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "subject-withdrawal", "metadata": {}, "source": [ " ```{glue:figure} harpohist_fig\n", ":figwidth: 600px\n", ":name: fig-harpohist\n", "\n", "Histogram showing the overall distribution of grades for students in Anastasia and Bernadette's classes\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "fabulous-community", "metadata": {}, "source": [ "{numref}`fig-ttestci` is a simpler plot showing the means and corresponding confidence intervals for both groups of students." ] }, { "cell_type": "code", "execution_count": 25, "id": "expensive-session", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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ppzWj2+uBu4MLta87cEYPvrtJ76w+qCu5bGAJAEVBmAFKQXglP82JvU9zY5srvKKvU1tugU1/2/CDOk1L0oo9p9jAEgBuEWEGKEUd766s1aPba3yXevLxdN7A8sfMaxr+6W71n71Dh366ZFCFAGA+hBmglHl7uGt4hwitGxulhxuHFmrfdvRndZuxWZO/+l5Z19jAEgB+D2EGMEiVQB/9rf+9+vTZlqpX2XkDywKbXXO/PqaO72zUwl3pbGAJAL+BMAMYrHXtSloxoq1eeyRS5X+1geW57Fy9+K896jVzq/acvGhMgQDg4ggzgAvwcHfT021qasO4aPVtXq1Qe2r6RcX8/WtNWLRH5y/nGlAhALguwgzgQir5e+ut3vdo8bDWalzN6tRmt0uf7UxX9NsbNH/bceUX2AyqEgBcC2EGcEFNq9+lJcPaaGqvRqrg57yBZda1fE1cuk8P/3WLdh47b1CFAOA6ihxmPv74Y7Vp00ZVqlRRWlqaJCk+Pl5Lly4ttuKAO5mbm0WPt6iuDWOjFds6XG4W5/YDpy+p7z+2aeTn3+inrGvGFAkALqBIYWbmzJkaM2aMHnroIV28eFEFBb9smhcYGKj4+PjirA+441l9PTWpRwOtGNFOLWpWKNS+NPVHdXxno2YlHVFuPlNPAO48RQozf/3rXzV79my98sorcnf/74O/mjdvru+++67YigPwX/VDA/TFkPs14/Emqhzg7dR2ObdAU1ceUNf4TUo6dNagCgHAGEUKM8eOHVPTpk0Lnff29tbly2yaB5QUi8WimCZVtX5stIZG1Zanu/Pc09FzlzVo7k49O3+X0s9fMahKAChdRQozNWvWVGpqaqHzK1euVGRk5O3WBOB3+Hl76KVud2v1qPaKqhtUqH3t9z/pgelJmr72kK7mFhhQIQCUHo/fv6Sw8ePHa/jw4bp27Zrsdrt27typzz77TFOmTNGHH35Y3DUCuIFaQf6a9/R9Stx/RpOX71P6+auOttx8m95bd1hfppzUqw/XV5cGIbJYLL/RGwCYU5HCzNNPP638/Hy9+OKLunLlivr376+qVatqxowZevzxx4u7RgC/wWKxqHNkZbWrU0kfbDqq9zf+oGt5/10InHHxqoYu2K12dSrptUcaKCLY38BqAaD4Wex2+21t+nLu3DnZbDYFBwcXV03FKisrS1arVZmZmQoICDC6HJSgNlPXK+PiLyMTVQN99PVLHQ2uyBgnL1zRGyv2a+Xe04XaPNwserpNuEY8UEfly3kaUB0AFL/bfmhepUqVXDbIAHeianf5auaAZlowuGWhUZh8m12zNx9Tx2lJWrT7pG7zdxkAcAk3Pc3UtGnTm55v3717d5ELAlA82tappJUj2+mjrccVn3hY2Tn5jrazl3I0ZuG3+nTHCU3q0UANq1p/oycAcG03PTLTs2dPxcTEKCYmRl26dNGRI0fk7e2t6OhoRUdHq1y5cjpy5Ii6dOlSkvUCuAWe7m56pl0trR8XpV73Vi3Uvivtgnr8bYv+uOQ7XWADSwAmVaQ1M88884xCQ0P1pz/9yen8a6+9pvT0dM2dO7fYCrxdrJm5c7Bm5velpJ3XxKX7tO/HrEJtgb6eGvdgPT3Rorrcf713AgC4sCKFGavVql27dqlOnTpO5w8fPqzmzZsrMzOz2Aq8XYSZOwdh5uYU2Oz6PPmE3l59UBev5BVqb1AlQJNjGqhZjcJbJwCAKyrSAmAfHx9t2bKl0PktW7aoXLlyt10UgJLj7mbRky1raMPYaA24v3qhDSz3/Zilx2Zu05iFqTpziQ0sAbi+Ij1nZtSoUXr++eeVkpKi+++/X5K0fft2zZ07VxMnTizWAgGUjLv8vPTnno30+H3V9dqyfUpJu+DUvmh3htbs+0mjOtXRoNbh8nS/7ZsfAaBEFPk5MwsXLtSMGTO0f/9+SVL9+vU1cuRI9e3bt1gLvF1MM905mGYqOrvdrsXfZGjKygM6eymnUHtEsL8mPdJAbetUMqA6APhtt/3QPFdHmLlzEGZu36Vrefrr+h80d8sx5dsKfzR0axiiV7rXV7W7fA2oDgCuj3FjAA7ly3nq5Yfqa9Wodmp3nVGYlXtPq9P0JL237rCu5bGBJQDXUKQwU1BQoHfeeUctWrRQSEiIKlSo4HQAMLeI4PKaH9dCswY0U9VAH6e2a3k2TV97SJ3fTdLa73/iKcIADFekMPP6669r+vTp6tu3rzIzMzVmzBj16tVLbm5umjRpUjGXCMAIFotFXRuGKHFMlEY8UEdeHs4fF+nnr+rZ+bsUm5Cso2ezDaoSAIoYZj755BPNnj1b48aNk4eHh5544gl9+OGHmjhxorZv317cNQIwkI+Xu8Z0rqt1Y6L0YGTlQu1Jh86qS/wmTV15QJf/Z8sEACgtRQozp0+fVqNGjSRJ/v7+jofkPfzww1qxYkXxVQfAZYRV8NUHA5vro7gWqlXJz6ktr8CuWUlH1HHaRi1NzWDqCUCpKlKYqVatmk6dOiVJioiI0Jo1ayRJycnJ8vb2Lr7qALicqLpBWjWqvSZ0u1t+Xu5ObT9l5Wjk56nq98F27T9VeMsEACgJRQozjz76qNatWydJGjlypF599VXVqVNHAwcOVFxcXLEWCMD1eHm46bmo2lo/Llo9m1Qp1L7z2Hl1f2+zXlu6V5nX2TIBAIpTsTxnZseOHfr6668VERGhHj16FEddxYbnzNw5eM6McXYeO6+JS/fqwOlLhdoq+HnpxS711Ld5mNzYwBJACbjlkZm8vDw9/fTTOnr0qONcy5YtNWbMGJcLMgBKR4uaFbT8D201OaaBAso575Jy/nKuXlr0nXq+/7W+OXHhBj0AQNHdcpjx9PTU4sWLS6IWACbm4e6mga3CtWFctJ5oESbLrwZh9pzM1KPvb9X4f36rc9mFt0wAgKIq8pqZJUuWFHMpAMqCiv7emtKrsZYOb6MmYYGF2v+ZclId3tn4y5YJBbbSLxBAmVOkXbMjIiL0pz/9SVu3blWzZs3k5+d8m+aIESOKpTgA5tW4WqAWPd9aX+4+qb+sOqBz2bmOtkvX8jV5+ff6Ijldk3o0UKvaFQ2sFIDZFWkBcM2aNW/cocXitJ7GaCwAvnOwANh1ZV3LU/zaw/po23EVXGcDy+6NQ/XKQ/VV5VdbJwDAzSjSyMyxY8eKuw4AZVhAOU9NfCRSj7cI02tL92nb0Z+d2lfsOaX1+8/ohY4ReqZdTXl7uN+gJwAorEhhZsyYMdc9b7FYVK5cOUVERCgmJuZ3N50MDw9XWlpaofPDhg1TfHy8/vjHP+rf//63jh49KqvVqk6dOmnq1KmqUqXwcy0AuL66lcvr02db6t/fndafV3yvU5nXHG1X8wr09uqD+ueudE18JFId7y68dQIAXE+Rppk6dOig3bt3q6CgQPXq1ZPdbtfhw4fl7u6uu+++WwcPHpTFYtGWLVsUGRl5w37Onj2rgoICx9d79+5V586dtWHDBjVt2lS9e/fWs88+q3vuuUcXLlzQqFGjlJ+fr127dt10rUwz3TmYZjKXK7n5en/DEX2w6ahyr7MQ+IG7g/Xqw5EK/9XWCQDwa0UKM/Hx8dq8ebMSEhIcASErK0uDBw9W27Zt9eyzz6p///66evWqVq9efdP9jho1SsuXL9fhw4dl+fV9nfplu4QWLVooLS1N1atXv6k+CTN3DsKMOR0/d1l/Wv691h04U6jNy91Nz7avqeEdIuTrVaSBZAB3gCLdmv3222/rT3/6k1M4CAgI0KRJk/TWW2/J19dXEydOVEpKyk33mZubqwULFiguLu66QUaSMjMzZbFYFBgYeMN+cnJylJWV5XQAcF3hlfw0J/Y+zY1trvCKvk5tuQU2/X3DET0wLUkr9pxiA0sA11WkMJOZmakzZwr/FnX27FlHeAgMDFRubm6ha25kyZIlunjxomJjY6/bfu3aNb300kvq37//b46wTJkyRVar1XGEhYXddA0AjNPx7spaPbq9xnepJx9P5wXApzKvafinu9V/9g4d+qnwlgkA7mxFCjMxMTGKi4vT4sWLdfLkSWVkZGjx4sUaPHiwevbsKUnauXOn6tate9N9zpkzR926dbvu4t68vDw9/vjjstlsev/993+znwkTJigzM9NxpKen39L3BsA43h7uGt4hQuvGRunhxqGF2rcd/VndZmzW5K++V9Y1NrAE8IsirZnJzs7W6NGjNX/+fOXn50uSPDw8NGjQIL377rvy8/NTamqqJKlJkya/219aWppq1aqlRYsWKSYmxqktLy9Pffv21dGjR7V+/XpVrHhrD9dizcydgzUzZc/WI+f0+rLvdfA6ozGV/L30Yte71fveamxgCdzhbmvX7OzsbB09elR2u121a9eWv79/kfqZNGmS/vGPfyg9PV0eHv9d5PefIHP48GFt2LBBQUFBt9w3YebOQZgpm/ILbPp4e5qmrz2kS9fyC7U3CQvU5JgGalwtsPSLA+ASijTN9B/+/v5q3Lix7rnnniIHGZvNpoSEBA0aNMgpyOTn56t3797atWuXPvnkExUUFOj06dM6ffr0La3FAWBuHu5uerpNTW0YF62+zasVak9Nv6iYv3+tCYv26PxlPhuAO9FthZnikJiYqBMnTiguLs7p/MmTJ7Vs2TKdPHlSTZo0UWhoqOPYunWrQdUCMEolf2+91fseLR7WWo2rWZ3a7Hbps53pin57g+ZvO84GlsAd5rammcyAaaY7B9NMdw6bza6Fu9L11uqD1x2NuTukvCbHNFSLmr/9FHIAZYPhIzMAcKvc3Cx6vEV1bRgbrdjW4fr1+t8Dpy+p7z+2aeTn3+j0/2yZAKBsIswAMC2rr6cm9WigFSPaXXcUZmnqj+o4baNmJR1Rbj5TT0BZxfPBAZhe/dAAfTHkfi379ke9+e/9+ikrx9F2JbdAU1ce0MLkdL3Wo4Gi6t76XZEwXu+ZWx0bk4Zay+lfz7c2uCK4EkZmAJQJFotFMU2qav3YaA2Nqi1Pd+e5p6PnLmvQ3J16dv4upZ+/YlCVKKpTmdeUcfGqMi5eddptHZAIMwDKGD9vD73U7W6tHtX+uqMwa7//SQ9MT9L0tYd0NbfAgAoBFDfCDIAyqVaQv+Y9fZ9mD2yusAo+Tm25+Ta9t+6wOk1P0qq9bGAJmB1hBkCZZbFY1DmystaOjtKYznVVztP5Iy/j4lUNXbBbA+fu1A9nsg2qEsDtIswAKPPKebprxAN1lDgmSt0ahhRq33z4nLrGb9IbK77XJTawBEyHMAPgjlHtLl/NHNBMCwa3VESw8xYs+Ta7Zm8+po7TkrRo90mmngATIcwAuOO0rVNJK0e20x+715e/t/MTKs5eytGYhd+q96xt2puRaVCFAG4FYQbAHcnT3U3PtKul9eOi1OveqoXaU9Iu6JG/bdEri7/TBTawBFwaYQbAHS24fDlN79tEXz7fSg2qOO/fZrdLn+w4oQ7TNmrB9jQV2Jh6AlwRYQYAJDWrUUHLXmirNx5tqEBfT6e2i1fy9Mcle9Xjb1uUknbeoAoB3AhhBgD+P3c3i55sWUMbxkZrwP3VC21gue/HLD02c5vGfJGqM1k8hRZwFYQZAPiVu/y89OeejbTshbZqVuOuQu2LvslQx2lJ+nDzUeUVsIElYDTCDADcQMOqVv1raCtN73uPgsp7O7Vl5+Trzyv2q9uMzdpy+JxBFQKQCDMA8JssFot63VtN68dGaUj7WvL41dzTD2eyNWDODj2/IEUnL7CBJWAEwgwA3ITy5Tz18kP1tWpUO7WrU6lQ+8q9p9VpepJmJB7WtTw2sARKE2EGAG5BRHB5zY9roVkDmqlqoPMGltfybHo38ZA6v5uktd//xFOEgVJCmAGAW2SxWNS1YYgSx0RpxAN15OXh/FGafv6qnp2/S7EJyTp6lg0sgZJGmAGAIvLxcteYznW1bkyUHoysXKg96dBZdYnfpKkrD+hyTr4BFQJ3BsIMANymsAq++mBgc30U10K1Kvk5teUV2DUr6Yg6TtuopakZTD0BJYAwAwDFJKpukFaNaq+Xut0tXy93p7afsnI08vNU9ftgu/afyjKoQqBsIswAQDHy8nDT0KjaWj82WjFNqhRq33nsvLq/t1mvLd2rzCt5BlQIlD2EGQAoASHWcprxeFMtfK6V7g4p79Rms0sfbUtTh2kb9fnOE7KxgSVwWwgzAFCCWtSsoOV/aKvJMQ0UUM7Dqe385Vy9tOg79Xz/a31z4oJBFQLmR5gBgBLm4e6mga3CtWFctJ5oESbLrzaw3HMyU4++v1Xj//mtzl7KMaZIwMQIMwBQSir6e2tKr8ZaOryNmoQFFmr/Z8pJdXxno+ZuOaZ8NrAEbhphBgBKWeNqgVr0fGu93buxKvl7ObVdysnX5OXfq/t7W7TtyM8GVQiYC2EGAAzg5mZRn+ZhWj8uWnFtasr9VxtYHvzpkp6YvV3DP92tHy9eNahKwBwIMwBgoIBynpr4SKRWjmynVrUqFmpfseeUHpiWpL9v+EE5+WxgCVwPYQYAXEDdyuX16bMt9ff+9yrUWs6p7Wpegd5efVBd3t2k9Qd+MqhCwHURZgDARVgsFnVvHKp1Y6P0QocIebk7f0Qf//mK4ubt0uB5yTp+7rJBVQKuhzADAC7G18tD47rU05rR7fXA3cGF2tcdOKMH392kt1cf0JVcNrAECDMA4KLCK/lpTux9mhvbXOEVfZ3acgts+vuGI3pgWpKW7/mRDSxxRyPMAICL63h3Za0e3V7ju9STj6fzBpanMq/phU+/Uf/ZO3Top0sGVQgYizADACbg7eGu4R0itG5slB5uHFqofdvRn9VtxmZN/up7ZV1jA0vcWQgzAGAiVQJ99Lf+9+rTZ1uqXmXnDSwLbHbN/fqYOr6zUQt3pbOBJe4YhBkAMKHWtStpxYi2eu2RSJX/1QaW57Jz9eK/9qjXzK3ac/KiMQUCpYgwAwAm5eHupqfb1NSGcdHq27xaofbU9IuK+fvXeunLPfo5mw0sUXYRZgDA5Cr5e+ut3vdo8bDWalzN6tRmt0ufJ6erwzsbNX/bcTawRJlEmAGAMqJp9bu0ZFgbTe3VSBX8nDewzLqWr4lL9+nhv27RzmPnDaoQKBmEGQAoQ9zcLHq8RXVtGBut2Nbh+tX+lTpw+pL6/mObRn7+jU5nXjOmSKCYGRpmwsPDZbFYCh3Dhw+XJNntdk2aNElVqlSRj4+PoqOjtW/fPiNLBgBTsPp6alKPBloxop1a1KxQqH1p6o/qOG2jZiUdUW4+U08wN0PDTHJysk6dOuU41q5dK0nq06ePJOmtt97S9OnT9be//U3JyckKCQlR586ddekSD4YCgJtRPzRAXwy5XzMeb6LKAd5ObVdyCzR15QF1jd+kpENnDaoQuH2GhpmgoCCFhIQ4juXLl6t27dqKioqS3W5XfHy8XnnlFfXq1UsNGzbURx99pCtXrujTTz+9YZ85OTnKyspyOgDgTmaxWBTTpKrWj43W0Kja8nR3nns6eu6yBs3dqWfn71L6+SsGVQkUncusmcnNzdWCBQsUFxcni8WiY8eO6fTp03rwwQcd13h7eysqKkpbt269YT9TpkyR1Wp1HGFhYaVRPgC4PD9vD73U7W6tHtVeUXWDCrWv/f4nPTA9SdPXHtLV3AIDKgSKxmXCzJIlS3Tx4kXFxsZKkk6fPi1Jqly5stN1lStXdrRdz4QJE5SZmek40tPTS6xmuJZQazlVDfRR1UAfhVrLGV0O4LJqBflr3tP3afbA5gqr4OPUlptv03vrDqvT9CSt2nuKDSxhCh6/f0npmDNnjrp166YqVao4nbdYnIdD7XZ7oXP/y9vbW97e3jdsR9n1r+dbG10CYBoWi0WdIyurXZ1K+mDTUb2/8Qddy/vvQuCMi1c1dMFutY2opEk9IhURXP43egOM5RIjM2lpaUpMTNQzzzzjOBcSEiJJhUZhzpw5U2i0BgBQNOU83TXigTpKHBOlbg1DCrVv+eGcusZv1hsrvtclNrCEi3KJMJOQkKDg4GB1797dca5mzZoKCQlx3OEk/bKuJikpSa1b8xs4ABSnanf5auaAZlowuKUigv2d2vJtds3efEwdpyVp0e6TTD3B5RgeZmw2mxISEjRo0CB5ePx31stisWjUqFF68803tXjxYu3du1exsbHy9fVV//79DawYAMqutnUqaeXIdvpj9/ry93ZeiXD2Uo7GLPxWvWdt096MTIMqBAozfM1MYmKiTpw4obi4uEJtL774oq5evaphw4bpwoULatmypdasWaPy5Zm7BYCS4unupmfa1VKPJlU0deUBLdqd4dSeknZBj/xti/q3qK5xD9bTXb/aOgEobRZ7GR8vzMrKktVqVWZmpgICAowuBwBMJyXtvCYu3ad9PxZ+blegr6fGPVhPT7SoLvdf751QjNpMXa+Mi1clSVUDffT1Sx1L7L1gPoZPMwEAXFuzGhW07IW2euPRhgr09XRqu3glT39cslc9/rZFKWlsYAljEGYAAL/L3c2iJ1vW0Iax0Rpwf/VCG1ju+zFLj83cpjFfpOpMFhtYonQRZgAAN+0uPy/9uWcjLXuhrZrVuKtQ+6JvMtRxWpJmbzqqvAI2sETpIMwAAG5Zw6pW/WtoK03ve4+Cyjs/qDQ7J19v/Hu/us3YrC2HzxlUIe4khBkAQJFYLBb1urea1o+N0pD2teTxq7mnH85ka8CcHXp+QYpOXmADS5QcwgwA4LaUL+eplx+qr1Wj2qldnUqF2lfuPa1O05M0I/GwruWxgSWKH2EGAFAsIoLLa35cC80a0ExVA503sLyWZ9O7iYfU+d0krdl3mqcIo1gRZgAAxcZisahrwxAljonSiAfqyMvD+b+Z9PNXNeTjFMUmJOvo2WyDqkRZQ5gBABQ7Hy93jelcV+vGROnByMKbAycdOqsu8Zs0deUBXc7JN6BClCWEGQBAiQmr4KsPBjbXR3EtVKuSn1NbXoFds5KOqOO0jVqamsHUE4qMMAMAKHFRdYO0alR7vdTtbvl6uTu1/ZSVo5Gfp6rfB9u1/1ThLROA30OYAQCUCi8PNw2Nqq31Y6MV06RKofadx86r+3ub9drSvcq8kuc4f/FKrrL/ZyoqOydf57JzSqVmmAMbTQIADLHz2HlNXLpXB05fKtRWwc9L4x+sp4yLVzR78zHl5Ds/TdjL3U0DW9XQS93uloc7v5ff6QgzAADD5BfY9OnOE3pn9UFlXbv1hcA97qmi+H5N5FaCO3bD9RFnAQCG8XB308BW4dowLlpPtAiT5RYzybJvf9TKvadLpjiYBmEGAGC4iv7emtKrsZYOb6MmYYG39NqPth4vkZpgHoQZAIDLaFwtUF8Muf+WRmh2Hj+vrGt5v38hyizCDADApVzLs+lWV3NeKsJ6G5QdhBkAgEvx83aXp/utLZ4JKOdRQtXADAgzAACX4uHupgfuLrwFwo20q1NJ5ct5lmBFcHWEGQCAyxnUOvzmr21189eibCLMAABcTqvaFfVc+1q/e13/ltX1QP3gUqgIroxJRgCAS3qp290KKu+tv67/QZlXne9WKu/toSHta2l4hwhZbvXhNChzeAIwAMClXcsrUKsp63Th/+/XdJevp7b8X0f5efP7OH7BNBMAwKWV83SXr9d/g4uvlwdBBk4IMwAAwNQIMwAAwNQIMwAAwNQIMwAAwNQIMwAAwNQIMwAAwNQIMwAAwNQIMwAAwNQIMwAAwNQIMwAAwNQIMwAAwNQIMwAAwNQIMwAAwNQIMwAAwNQIMwAAwNQIMwAAwNQIMwAAwNQMDzMZGRkaMGCAKlasKF9fXzVp0kQpKSmO9uzsbL3wwguqVq2afHx8VL9+fc2cOdPAigEAgCvxMPLNL1y4oDZt2qhDhw5auXKlgoODdeTIEQUGBjquGT16tDZs2KAFCxYoPDxca9as0bBhw1SlShXFxMQYVzwAAHAJhoaZv/zlLwoLC1NCQoLjXHh4uNM127Zt06BBgxQdHS1JGjJkiP7xj39o165dhBkAAGDsNNOyZcvUvHlz9enTR8HBwWratKlmz57tdE3btm21bNkyZWRkyG63a8OGDTp06JC6dOly3T5zcnKUlZXldAAAgLLL0DBz9OhRzZw5U3Xq1NHq1as1dOhQjRgxQvPnz3dc89577ykyMlLVqlWTl5eXunbtqvfff19t27a9bp9TpkyR1Wp1HGFhYaX17QAAAAMYOs1ks9nUvHlzvfnmm5Kkpk2bat++fZo5c6YGDhwo6Zcws337di1btkw1atTQpk2bNGzYMIWGhqpTp06F+pwwYYLGjBnj+DorK4tAAwBAGWZomAkNDVVkZKTTufr16+vLL7+UJF29elUvv/yyFi9erO7du0uSGjdurNTUVL3zzjvXDTPe3t7y9vYu+eIBAIBLMHSaqU2bNjp48KDTuUOHDqlGjRqSpLy8POXl5cnNzblMd3d32Wy2UqsTAAC4LkNHZkaPHq3WrVvrzTffVN++fbVz50598MEH+uCDDyRJAQEBioqK0vjx4+Xj46MaNWooKSlJ8+fP1/Tp040sHQAAuAhDw8x9992nxYsXa8KECZo8ebJq1qyp+Ph4Pfnkk45rPv/8c02YMEFPPvmkzp8/rxo1auiNN97Q0KFDDawcAAC4CovdbrcbXURJysrKktVqVWZmpgICAowuBwBQBG2mrlfGxauSpKqBPvr6pY4GVwRXYvh2BgAAALeDMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEyNMAMAAEzNw+gCAAD4PaHWctf9MyBJFrvdbje6iJKUlZUlq9WqzMxMBQQEGF0OAAAoZkwzAQAAUyPMAAAAUyPMAAAAUyPMAAAAUzM8zGRkZGjAgAGqWLGifH191aRJE6WkpDhds3//fvXo0UNWq1Xly5fX/fffrxMnThhUMQAAcCWG3pp94cIFtWnTRh06dNDKlSsVHBysI0eOKDAw0HHNkSNH1LZtWw0ePFivv/66rFar9u/fr3LluDUPAAAYfGv2Sy+9pK+//lqbN2++4TWPP/64PD099fHHH99Unzk5OcrJyXF8nZWVpbCwMG7NBgCgjDJ0mmnZsmVq3ry5+vTpo+DgYDVt2lSzZ892tNtsNq1YsUJ169ZVly5dFBwcrJYtW2rJkiU37HPKlCmyWq2OIywsrBS+EwAAYBRDR2b+M1U0ZswY9enTRzt37tSoUaP0j3/8QwMHDtTp06cVGhoqX19f/fnPf1aHDh20atUqvfzyy9qwYYOioqIK9cnIDAAAdxZDw4yXl5eaN2+urVu3Os6NGDFCycnJ2rZtm3788UdVrVpVTzzxhD799FPHNT169JCfn58+++yz330PngAMAEDZZug0U2hoqCIjI53O1a9f33GnUqVKleTh4fGb1wAAgDuboWGmTZs2OnjwoNO5Q4cOqUaNGpJ+Gbm57777fvMaAABwZzP01uzRo0erdevWevPNN9W3b1/t3LlTH3zwgT744APHNePHj1e/fv3Uvn17x5qZr776Shs3bjSucAAA4DIM3zV7+fLlmjBhgg4fPqyaNWtqzJgxevbZZ52umTt3rqZMmaKTJ0+qXr16ev311xUTE3NT/WdmZiowMFDp6emsmQEAwGTKly8vi8Xym9cYHmZK2smTJ7k9GwAAk7qZG3jKfJix2Wz68ccfbyrZwfz+cys+I3FA2cPP953pZv7/NnTNTGlwc3NTtWrVjC4DpSwgIIAPO6CM4ucbv2b4RpMAAAC3gzADAABMjTCDMsXb21uvvfaavL29jS4FQDHj5xs3UuYXAAMAgLKNkRkAAGBqhBkAAGBqhBkAAGBqhBngJkyaNElNmjQxugwAvyE6OlqjRo0yugwYgDCDErF161a5u7ura9eupf7eJfGBNm7cOK1bt65Y+wRcWWxsrCwWi+OoWLGiunbtqj179hhdWrGZN2+eAgMDC50PDw9XfHx8qdeDoiPMoETMnTtXf/jDH7RlyxadOHHC6HJum7+/vypWrGh0GUCp6tq1q06dOqVTp05p3bp18vDw0MMPP1zk/nJzc4uxOuC/CDModpcvX9bChQv1/PPP6+GHH9a8efMcbRs3bpTFYtG6devUvHlz+fr6qnXr1jp48KDjmiNHjigmJkaVK1eWv7+/7rvvPiUmJjq9x/vvv686deqoXLlyqly5snr37i3pl98mk5KSNGPGDMdvlMePH1dBQYEGDx6smjVrysfHR/Xq1dOMGTOc+ty4caNatGghPz8/BQYGqk2bNkpLS5NUeJopOTlZnTt3VqVKlWS1WhUVFaXdu3cX898kYCxvb2+FhIQoJCRETZo00f/93/8pPT1dZ8+elSRlZGSoX79+uuuuu1SxYkXFxMTo+PHjjtfHxsaqZ8+emjJliqpUqaK6devq+PHjslgsWrRokTp06CBfX1/dc8892rZtm+N1P//8s5544glVq1ZNvr6+atSokT777DOn2i5fvqyBAwfK399foaGhmjZtWqH6c3Nz9eKLL6pq1ary8/NTy5YttXHjRkm//Lw//fTTyszMdHxWTJo0SdHR0UpLS9Po0aMd5/9j69atat++vXx8fBQWFqYRI0bo8uXLxfg3jqIizKDYffHFF6pXr57q1aunAQMGKCEhQb9+nNErr7yiadOmadeuXfLw8FBcXJyjLTs7Ww899JASExP1zTffqEuXLnrkkUccIzy7du3SiBEjNHnyZB08eFCrVq1S+/btJUkzZsxQq1at9Oyzzzp+owwLC5PNZlO1atW0cOFCff/995o4caJefvllLVy4UJKUn5+vnj17KioqSnv27NG2bds0ZMiQG25udunSJQ0aNEibN2/W9u3bVadOHT300EO6dOlSSfyVAobLzs7WJ598ooiICFWsWFFXrlxRhw4d5O/vr02bNmnLli3y9/dX165dnUZg1q1bp/3792vt2rVavny54/wrr7yicePGKTU1VXXr1tUTTzyh/Px8SdK1a9fUrFkzLV++XHv37tWQIUP01FNPaceOHY7Xjx8/Xhs2bNDixYu1Zs0abdy4USkpKU41P/300/r666/1+eefa8+ePerTp4+6du2qw4cPq3Xr1oqPj1dAQIDjs2LcuHFatGiRqlWrpsmTJzvOS9J3332nLl26qFevXtqzZ4+++OILbdmyRS+88EJJ/rXjZtmBYta6dWt7fHy83W632/Py8uyVKlWyr1271m632+0bNmywS7InJiY6rl+xYoVdkv3q1as37DMyMtL+17/+1W632+1ffvmlPSAgwJ6VlXXda6OiouwjR4783TqHDRtmf+yxx+x2u93+888/2yXZN27ceN1rX3vtNfs999xzw77y8/Pt5cuXt3/11Ve/+76AGQwaNMju7u5u9/Pzs/v5+dkl2UNDQ+0pKSl2u91unzNnjr1evXp2m83meE1OTo7dx8fHvnr1akcflStXtufk5DiuOXbsmF2S/cMPP3Sc27dvn12Sff/+/Tes56GHHrKPHTvWbrfb7ZcuXbJ7eXnZP//8c0f7zz//bPfx8XH87P/www92i8Viz8jIcOrngQcesE+YMMFut9vtCQkJdqvVWui9atSoYX/33Xedzj311FP2IUOGOJ3bvHmz3c3N7Tc/u1A6GJlBsTp48KB27typxx9/XJLk4eGhfv36ae7cuU7XNW7c2PHn0NBQSdKZM2ck/TJ8/OKLLyoyMlKBgYHy9/fXgQMHHCMznTt3Vo0aNVSrVi099dRT+uSTT3TlypXfrW3WrFlq3ry5goKC5O/vr9mzZzv6rFChgmJjYx2jQDNmzHD8RnY9Z86c0dChQ1W3bl1ZrVZZrVZlZ2eXifVBwH906NBBqampSk1N1Y4dO/Tggw+qW7duSktLU0pKin744QeVL19e/v7+8vf3V4UKFXTt2jUdOXLE0UejRo3k5eVVqO/f+gwoKCjQG2+8ocaNG6tixYry9/fXmjVrHD9fR44cUW5urlq1auXoo0KFCqpXr57j6927d8tut6tu3bqO+vz9/ZWUlORU381KSUnRvHnznPrq0qWLbDabjh07dsv9oXh5GF0AypY5c+YoPz9fVatWdZyz2+3y9PTUhQsXHOc8PT0df/7PVI7NZpP0y/Dx6tWr9c477ygiIkI+Pj7q3bu3Y+i6fPny2r17tzZu3Kg1a9Zo4sSJmjRpkpKTk697Z4IkLVy4UKNHj9a0adPUqlUrlS9fXm+//bbTsHVCQoJGjBihVatW6YsvvtAf//hHrV27Vvfff3+h/mJjY3X27FnFx8erRo0a8vb2VqtWrVjgiDLFz89PERERjq+bNWsmq9Wq2bNny2azqVmzZvrkk08KvS4oKMipj+v5rc+AadOm6d1331V8fLwaNWokPz8/jRo1yvHzZb+JXXhsNpvc3d2VkpIid3d3pzZ/f//fff31+nvuuec0YsSIQm3Vq1e/5f5QvAgzKDb5+fmaP3++pk2bpgcffNCp7bHHHtMnn3yihg0b/m4/mzdvVmxsrB599FFJv8zV/++iQumXEZ9OnTqpU6dOeu211xQYGKj169erV69e8vLyUkFBQaE+W7durWHDhjnOXe+3s6ZNm6pp06aaMGGCWrVqpU8//fS6YWbz5s16//339dBDD0mS0tPTde7cud/93gAzs1gscnNz09WrV3Xvvffqiy++UHBwsAICAor1fTZv3qyYmBgNGDBA0i9B4vDhw6pfv74kKSIiQp6entq+fbsjSFy4cEGHDh1SVFSUpF9+lgsKCnTmzBm1a9fuuu9zvc+KG52/9957tW/fPqdwB9fBNBOKzfLly3XhwgUNHjxYDRs2dDp69+6tOXPm3FQ/ERERWrRokVJTU/Xtt9+qf//+jt/Y/vM+7733nlJTU5WWlqb58+fLZrM5hpjDw8O1Y8cOHT9+XOfOnZPNZlNERIR27dql1atX69ChQ3r11VeVnJzs6PPYsWOaMGGCtm3bprS0NK1Zs0aHDh1yfHher8aPP/5Y+/fv144dO/Tkk0/Kx8fnNv72ANeTk5Oj06dP6/Tp09q/f7/+8Ic/KDs7W4888oiefPJJVapUSTExMdq8ebOOHTumpKQkjRw5UidPnryt942IiNDatWu1detW7d+/X88995xOnz7taPf399fgwYM1fvx4rVu3Tnv37lVsbKzc3P77X1rdunX15JNPauDAgVq0aJGOHTum5ORk/eUvf9G///1vSb98VmRnZ2vdunU6d+6cY7o6PDxcmzZtUkZGhuOXlP/7v//Ttm3bNHz4cKWmpurw4cNatmyZ/vCHP9zW94riQZhBsZkzZ446deokq9VaqO2xxx5TamrqTd2+/O677+quu+5S69at9cgjj6hLly669957He2BgYFatGiROnbsqPr162vWrFn67LPP1KBBA0m/PODO3d1dkZGRCgoK0okTJzR06FD16tVL/fr1U8uWLfXzzz87jdL4+vrqwIEDeuyxx1S3bl0NGTJEL7zwgp577rnr1jh37lxduHBBTZs21VNPPaURI0YoODj4Vv/KAJe2atUqhYaGKjQ0VC1btlRycrL++c9/Kjo6Wr6+vtq0aZOqV6+uXr16qX79+oqLi9PVq1dve6Tm1Vdf1b333qsuXbooOjpaISEh6tmzp9M1b7/9ttq3b68ePXqoU6dOatu2rZo1a+Z0TUJCggYOHKixY8eqXr166tGjh3bs2KGwsDBJUuvWrTV06FD169dPQUFBeuuttyRJkydP1vHjx1W7dm3HlFnjxo2VlJSkw4cPq127dmratKleffVVx3ofGMtiv5nJRwAAABfFyAwAADA1wgwAADA1wgwAADA1wgwAADA1wgwAADA1wgwAADA1wgwAADA1wgwAADA1wgwAADA1wgwAQ0VHR2vUqFG39JpJkyapSZMmJVIPAPMhzAC4Y+Xl5RldAoBiQJgBYJjY2FglJSVpxowZslgsslgsmjdvngIDA52uW7JkiSwWiyRp3rx5ev311/Xtt986vUaSTpw4oZiYGPn7+ysgIEB9+/bVTz/95OjnPyM6c+fOVa1ateTt7S22pwPMz8PoAgDcuWbMmKFDhw6pYcOGmjx5siRpxYoVv/mafv36ae/evVq1apUSExMlSVarVXa7XT179pSfn5+SkpKUn5+vYcOGqV+/ftq4caPj9T/88IMWLlyoL7/8Uu7u7iX2vQEoPYQZAIaxWq3y8vKSr6+vQkJCJOl3A4aPj4/8/f3l4eHheI0krV27Vnv27NGxY8cUFhYmSfr444/VoEEDJScn67777pMk5ebm6uOPP1ZQUFAJfVcAShvTTADKhP379yssLMwRZCQpMjJSgYGB2r9/v+NcjRo1CDJAGUOYAeBS3NzcCq1juZmFuna73bGu5rfO+/n53X6RAFwKYQaAoby8vFRQUOD4OigoSJcuXdLly5cd51JTU3/zNdIvozAnTpxQenq649z333+vzMxM1a9fv2SKB+ASCDMADBUeHq4dO3bo+PHjOnfunFq2bClfX1+9/PLL+uGHH/Tpp5867lb639ccO3ZMqampOnfunHJyctSpUyc1btxYTz75pHbv3q2dO3dq4MCBioqKUvPmzY355gCUCsIMAEONGzdO7u7uioyMVFBQkLKysrRgwQL9+9//VqNGjfTZZ59p0qRJTq957LHH1LVrV3Xo0EFBQUH67LPPZLFYtGTJEt11111q3769OnXqpFq1aumLL74w5hsDUGosdh6yAAAATIyRGQAAYGqEGQAAYGqEGQAAYGqEGQAAYGqEGQAAYGqEGQAAYGqEGQAAYGqEGQAAYGqEGQAAYGqEGQAAYGqEGQAAYGr/D9abM2eU/k67AAAAAElFTkSuQmCC", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "ttestci-fig" } }, "output_type": "display_data" } ], "source": [ "fig, axes = plt.subplots(1, 1)\n", "sns.pointplot(x = 'tutor', y = 'grade', data = df)\n", "sns.despine()\n", "\n", "# show the figure in the book with caption\n", "plt.close()\n", "from myst_nb import glue\n", "glue(\"ttestci-fig\", fig, display=False)\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "painted-running", "metadata": {}, "source": [ " ```{glue:figure} ttestci-fig\n", ":figwidth: 600px\n", ":name: fig-ttestci\n", "\n", "Plot showing the mean grade for the students in Anastasia's and Bernadette's tutorials. Error bars depict 95% confidence intervals around the mean. On the basis of visual inspection, it does look like there's a real difference between the groups, though it's hard to say for sure.\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "outer-victim", "metadata": {}, "source": [ "### Introducing the test\n", "\n", "The **_independent samples $t$-test_** comes in two different forms, Student's and Welch's. The original Student $t$-test -- which is the one I'll describe in this section -- is the simpler of the two, but relies on much more restrictive assumptions than the Welch $t$-test. Assuming for the moment that you want to run a two-sided test, the goal is to determine whether two \"independent samples\" of data are drawn from populations with the same mean (the null hypothesis) or different means (the alternative hypothesis). When we say \"independent\" samples, what we really mean here is that there's no special relationship between observations in the two samples. This probably doesn't make a lot of sense right now, but it will be clearer when we come to talk about the paired samples $t$-test later on. For now, let's just point out that if we have an experimental design where participants are randomly allocated to one of two groups, and we want to compare the two groups' mean performance on some outcome measure, then an independent samples $t$-test (rather than a paired samples $t$-test) is what we're after.\n", "\n", "Okay, so let's let $\\mu_1$ denote the true population mean for group 1 (e.g., Anastasia's students), and $\\mu_2$ will be the true population mean for group 2 (e.g., Bernadette's students),[^note7] and as usual we'll let $\\bar{X}_1$ and $\\bar{X}_2$ denote the observed sample means for both of these groups. Our null hypothesis states that the two population means are identical ($\\mu_1 = \\mu_2$) and the alternative to this is that they are not ($\\mu_1 \\neq \\mu_2$). Written in mathematical-ese, this is...\n", "\n", "$$\n", "\\begin{array}{ll}\n", "H_0: & \\mu_1 = \\mu_2 \\\\\n", "H_1: & \\mu_1 \\neq \\mu_2\n", "\\end{array}\n", "$$" ] }, { "cell_type": "code", "execution_count": 26, "id": "turkish-olive", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "ttesthyp_fig" } }, "output_type": "display_data" } ], "source": [ "mu1 = 0\n", "sigma1 = 1\n", "mu2 = 2\n", "\n", "x1 = np.linspace(mu1 - 4*sigma, mu1 + 4*sigma, 100)\n", "y1 = 100* stats.norm.pdf(x1, mu1, sigma)\n", "x2 = np.linspace(mu2 - 4*sigma, mu2 + 4*sigma, 100)\n", "y2 = 100* stats.norm.pdf(x2, mu2, sigma)\n", "\n", "\n", "fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 5))\n", "\n", "\n", "sns.lineplot(x=x1,y=y, color='black', ax = ax1)\n", "\n", "sns.lineplot(x=x1,y=y, color='black', ax = ax2)\n", "sns.lineplot(x=x2,y=y2, color='black', ax = ax2)\n", "\n", "ax1.text(0, 43, 'null hypothesis', size=20, ha=\"center\")\n", "ax2.text(0, 43, 'alternative hypothesis', size=20, ha=\"center\")\n", "\n", "ax1.set_frame_on(False)\n", "ax2.set_frame_on(False)\n", "ax1.get_yaxis().set_visible(False)\n", "ax2.get_yaxis().set_visible(False)\n", "ax1.get_xaxis().set_visible(False)\n", "ax2.get_xaxis().set_visible(False)\n", "ax1.axhline(y=0, color='black')\n", "ax2.axhline(y=0, color='black')\n", "\n", "# show the figure in the book with caption\n", "plt.close()\n", "from myst_nb import glue\n", "glue(\"ttesthyp_fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "complete-backup", "metadata": {}, "source": [ " ```{glue:figure} ttesthyp_fig\n", ":figwidth: 600px\n", ":name: fig-ttesthyp\n", "\n", "Graphical illustration of the null and alternative hypotheses assumed by the Student $t$-test. The null hypothesis assumes that both groups have the same mean $\\\\mu$, whereas the alternative assumes that they have different means $\\mu_1$ and $\\mu_2$. Notice that it is assumed that the population distributions are normal, and that, although the alternative hypothesis allows the group to have different means, it assumes they have the same standard deviation\n", "```\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "chubby-allen", "metadata": {}, "source": [ "To construct a hypothesis test that handles this scenario, we start by noting that if the null hypothesis is true, then the difference between the population means is *exactly* zero, \n", "$\\mu_1 - \\mu_2 = 0$\n", "As a consequence, a diagnostic test statistic will be based on the difference between the two sample means. Because if the null hypothesis is true, then we'd expect \n", "\n", "$$\n", "\\bar{X}_1 - \\bar{X}_2\n", "$$\n", "\n", "to be *pretty close* to zero. However, just like we saw with our one-sample tests (i.e., the one-sample $z$-test and the one-sample $t$-test) we have to be precise about exactly *how close* to zero this difference should be. And the solution to the problem is more or less the same one: we calculate a standard error estimate (SE), just like last time, and then divide the difference between means by this estimate. So our **_$t$-statistic_** will be of the form\n", "\n", "$$\n", "t = \\frac{\\bar{X}_1 - \\bar{X}_2}{\\mbox{SE}}\n", "$$\n", "\n", "We just need to figure out what this standard error estimate actually is. This is a bit trickier than was the case for either of the two tests we've looked at so far, so we need to go through it a lot more carefully to understand how it works." ] }, { "attachments": {}, "cell_type": "markdown", "id": "opponent-yugoslavia", "metadata": {}, "source": [ "### A \"pooled estimate\" of the standard deviation\n", "\n", "In the original \"Student $t$-test\", we make the assumption that the two groups have the same population standard deviation: that is, regardless of whether the population means are the same, we assume that the population standard deviations are identical, $\\sigma_1 = \\sigma_2$. Since we're assuming that the two standard deviations are the same, we drop the subscripts and refer to both of them as $\\sigma$. How should we estimate this? How should we construct a single estimate of a standard deviation when we have two samples? The answer is, basically, we average them. Well, sort of. Actually, what we do is take a *weighed* average of the *variance* estimates, which we use as our **_pooled estimate of the variance_**. The weight assigned to each sample is equal to the number of observations in that sample, minus 1. Mathematically, we can write this as\n", "\n", "$$\n", "\\begin{array}{rcl}\n", "w_1 &=& N_1 - 1\\\\\n", "w_2 &=& N_2 - 1\n", "\\end{array}\n", "$$\n", "\n", "Now that we've assigned weights to each sample, we calculate the pooled estimate of the variance by taking the weighted average of the two variance estimates, ${\\hat\\sigma_1}^2$ and ${\\hat\\sigma_2}^2$ \n", "\n", "$$\n", "\\hat\\sigma^2_p = \\frac{w_1 {\\hat\\sigma_1}^2 + w_2 {\\hat\\sigma_2}^2}{w_1 + w_2}\n", "$$\n", "\n", "Finally, we convert the pooled variance estimate to a pooled standard deviation estimate, by taking the square root. This gives us the following formula for $\\hat\\sigma_p$,\n", "\n", "$$\n", "\\hat\\sigma_p = \\sqrt{\\frac{w_1 {\\hat\\sigma_1}^2 + w_2 {\\hat\\sigma_2}^2}{w_1 + w_2}}\n", "$$\n", "\n", "And if you mentally substitute $w_1 = N_1 -1$ and $w_2 = N_2 -1$ into this equation you get a very ugly looking formula; a very ugly formula that actually seems to be the \"standard\" way of describing the pooled standard deviation estimate. It's not my favourite way of thinking about pooled standard deviations, however.[^note8]" ] }, { "attachments": {}, "cell_type": "markdown", "id": "excited-soccer", "metadata": {}, "source": [ "### The same pooled estimate, described differently\n", "\n", "I prefer to think about it like this. Our data set actually corresponds to a set of $N$ observations, which are sorted into two groups. So let's use the notation $X_{ik}$ to refer to the grade received by the $i$-th student in the $k$-th tutorial group: that is, $X_{11}$ is the grade received by the first student in Anastasia's class, $X_{21}$ is her second student, and so on. And we have two separate group means $\\bar{X}_1$ and $\\bar{X}_2$, which we could \"generically\" refer to using the notation $\\bar{X}_k$, i.e., the mean grade for the $k$-th tutorial group. So far, so good. Now, since every single student falls into one of the two tutorials, then we can describe their deviation from the group mean as the difference\n", "\n", "$$\n", "X_{ik} - \\bar{X}_k\n", "$$\n", "\n", "So why not just use these deviations (i.e., the extent to which each student's grade differs from the mean grade in their tutorial?) Remember, a variance is just the average of a bunch of squared deviations, so let's do that. Mathematically, we could write it like this:\n", "\n", "$$\n", "\\frac{\\sum_{ik} \\left( X_{ik} - \\bar{X}_k \\right)^2}{N}\n", "$$\n", "\n", "where the notation \"$\\sum_{ik}$\" is a lazy way of saying \"calculate a sum by looking at all students in all tutorials\", since each \"$ik$\" corresponds to one student.^[A more correct notation will be introduced in chapter on ANOVA(anova).] But, as we saw in the chapter on [estimation](estimation), calculating the variance by dividing by $N$ produces a biased estimate of the population variance. And previously, we needed to divide by $N-1$ to fix this. However, as I mentioned at the time, the reason why this bias exists is because the variance estimate relies on the sample mean; and to the extent that the sample mean isn't equal to the population mean, it can systematically bias our estimate of the variance. But this time we're relying on *two* sample means! Does this mean that we've got more bias? Yes, yes it does. And does this mean we now need to divide by $N-2$ instead of $N-1$, in order to calculate our pooled variance estimate? Why, yes...\n", "\n", "$$\n", "\\hat\\sigma^2_p = \\frac{\\sum_{ik} \\left( X_{ik} - \\bar{X}_k \\right)^2}{N -2}\n", "$$\n", "\n", "Oh, and if you take the square root of this then you get $\\hat{\\sigma}_p$, the pooled standard deviation estimate. In other words, the pooled standard deviation calculation is nothing special: it's not terribly different to the regular standard deviation calculation. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "rocky-funds", "metadata": {}, "source": [ "(indsamplesttest_formula)=\n", "### Completing the test\n", "\n", "Regardless of which way you want to think about it, we now have our pooled estimate of the standard deviation. From now on, I'll drop the silly $p$ subscript, and just refer to this estimate as $\\hat\\sigma$. Great. Let's now go back to thinking about the bloody hypothesis test, shall we? Our whole reason for calculating this pooled estimate was that we knew it would be helpful when calculating our *standard error* estimate. But, standard error of *what*? In the one-sample $t$-test, it was the standard error of the sample mean, $\\mbox{SE}({\\bar{X}})$, and since $\\mbox{SE}({\\bar{X}}) = \\sigma / \\sqrt{N}$ that's what the denominator of our $t$-statistic looked like. This time around, however, we have *two* sample means. And what we're interested in, specifically, is the the difference between the two $\\bar{X}_1 - \\bar{X}_2$. As a consequence, the standard error that we need to divide by is in fact the **_standard error of the difference_** between means. As long as the two variables really do have the same standard deviation, then our estimate for the standard error is\n", "\n", "$$\n", "\\mbox{SE}({\\bar{X}_1 - \\bar{X}_2}) = \\hat\\sigma \\sqrt{\\frac{1}{N_1} + \\frac{1}{N_2}}\n", "$$\n", "\n", "and our $t$-statistic is therefore \n", "\n", "$$\n", "t = \\frac{\\bar{X}_1 - \\bar{X}_2}{\\mbox{SE}({\\bar{X}_1 - \\bar{X}_2})}\n", "$$\n", "\n", "Just as we saw with our one-sample test, the sampling distribution of this $t$-statistic is a $t$-distribution (shocking, isn't it?) as long as the null hypothesis is true, and all of the assumptions of the test are met. The degrees of freedom, however, is slightly different. As usual, we can think of the degrees of freedom to be equal to the number of data points minus the number of constraints. In this case, we have $N$ observations ($N_1$ in sample 1, and $N_2$ in sample 2), and 2 constraints (the sample means). So the total degrees of freedom for this test are $N-2$. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "competent-eating", "metadata": {}, "source": [ "### Doing the test in Python\n", "\n", "Now, you can run an independent samples $t$-test with a method from `scipy`, and since the method for a one-sample $t$-test was `ttest_1samp`, it may come as no big surprise to you that the method for an independent-samples $t$-test is called `ttest_ind`. But from here on out, I think it will be easier for everyone (certainly easier for me!) if we use `pingouin` to ease the process of running these tests. How will it make it easier, you ask? Well, for one thing, remember how we just used the `ttest()` function from `pingouin` to run a one-sample $t$-test? Well we can use _the exact same function_ to run _all_ of the $t$-tests in this chapter. How's that for making things easier? Also, `pingouin` unifies the output format, so we get the same familiar table for each one of the tests.\n", "\n", "So let's give it a try. An important point here, and one that can cause a lot of frustration if you don't realize what is going on, is that `ttest()` wants the data from the two groups served up as two separate variables (this is true of the `scipy` version as well, by the way). Since Dr Harpo sent the data to us in _long format_, that is, with all the grades in one column, and a second column telling us who the tutor was for each student, we will need to do something to break the grade data into two. So step one will be creating a new dataframe, with one column per tutor (\"wide\" format):" ] }, { "cell_type": "code", "execution_count": 27, "id": "graphic-settlement", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
AnastasiaBernadette
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" ], "text/plain": [ " Anastasia Bernadette\n", "0 65.0 NaN\n", "1 NaN 72.0\n", "2 NaN 66.0\n", "3 74.0 NaN\n", "4 73.0 NaN" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "Harpo_wide = pd.DataFrame(\n", " {'Anastasia': df.loc[df['tutor'] == 'Anastasia']['grade'],\n", " 'Bernadette': df.loc[df['tutor'] == 'Bernadette']['grade']})\n", "\n", "Harpo_wide.head()\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "5d490fe6", "metadata": {}, "source": [ "Now, you will have noticed right away that this new dataframe has a bunch of things that aren't numbers in it. In fact, \"NaN\" stands for \"Not a Number\". But after a moment's reflection, this makes perfect sense: the students were divided up between Anastasia and Bernadette, and so of course now that we have a row for each student, if a student has Anastasia as a tutor, they can't _also_ have Bernadette as a tutor. Luckily, `ttest()` is smart enough to see what is going on, and deal with it appropriately. So, now that we have all of our ducks in order, let's do the test:" ] }, { "cell_type": "code", "execution_count": 28, "id": "short-wages", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Tdofalternativep-valCI95%cohen-dBF10power
T-test2.11543231two-sided0.042529[0.2, 10.76]0.7395611.7550.53577
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" ], "text/plain": [ " T dof alternative p-val CI95% cohen-d BF10 \\\n", "T-test 2.115432 31 two-sided 0.042529 [0.2, 10.76] 0.739561 1.755 \n", "\n", " power \n", "T-test 0.53577 " ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from pingouin import ttest\n", "\n", "ttest(Harpo_wide['Anastasia'], Harpo_wide['Bernadette'], correction = False)\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "operational-packaging", "metadata": {}, "source": [ "You probably noticed that in addition to telling `ttest()` which means I wanted to compare, I also added the argument `correction = False` to the command. This wasn't strictly necessary in this case, because by default this argument is set to `True`. By saying `correction = False`, what we're really doing is telling Python to use the *Student* independent samples $t$-test, and not the *Welch* independent samples $t$-test. More on this later, when we get to [Welch](welchttest). For now, let's just get the descriptive statistics for Anastasia and Bernadette's students so we can report our results:" ] }, { "cell_type": "code", "execution_count": 29, "id": "6f6a0f60", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
AnastasiaBernadette
count15.00000018.000000
mean74.53333369.055556
std8.9989425.774918
min55.00000056.000000
25%69.00000066.250000
50%76.00000069.000000
75%79.00000073.000000
max90.00000079.000000
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" ], "text/plain": [ " Anastasia Bernadette\n", "count 15.000000 18.000000\n", "mean 74.533333 69.055556\n", "std 8.998942 5.774918\n", "min 55.000000 56.000000\n", "25% 69.000000 66.250000\n", "50% 76.000000 69.000000\n", "75% 79.000000 73.000000\n", "max 90.000000 79.000000" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "Harpo_wide.describe()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "amino-overhead", "metadata": {}, "source": [ "The difference between the two groups is significant (just barely), so we might write up the result using text like this:\n", "\n", "> The mean grade in Anastasia's class was 74.5\\% (std dev = 9.0), whereas the mean in Bernadette's class was 69.1\\% (std dev = 5.8). A Student's independent samples $t$-test showed that this 5.4\\% difference was significant ($t(31) = 2.1$, $p<.05$), suggesting that a genuine difference in learning outcomes has occurred.\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "unable-providence", "metadata": {}, "source": [ " \n", "### Positive and negative $t$ values\n", "\n", "Before moving on to talk about the assumptions of the $t$-test, there's one additional point I want to make about the use of $t$-tests in practice. The first one relates to the sign of the $t$-statistic (that is, whether it is a positive number or a negative one). One very common worry that students have when they start running their first $t$-test is that they often end up with negative values for the $t$-statistic, and don't know how to interpret it. In fact, it's not at all uncommon for two people working independently to end up with Python outputs that are almost identical, except that one person has a negative $t$ values and the other one has a positive $t$ value. Assuming that you're running a two-sided test, then the $p$-values will be identical. On closer inspection, the students will notice that the confidence intervals also have the opposite signs. This is perfectly okay: whenever this happens, what you'll find is that the two versions of the Python output arise from slightly different ways of running the $t$-test. What's happening here is very simple. The $t$-statistic that Python is calculating here is always of the form \n", "\n", "$$\n", "t = \\frac{\\mbox{(mean 1)} -\\mbox{(mean 2)}}{ \\mbox{(SE)}}\n", "$$\n", "\n", "If \"mean 1\" is larger than \"mean 2\" the $t$ statistic will be positive, whereas if \"mean 2\" is larger then the $t$ statistic will be negative. Similarly, the confidence interval that R reports is the confidence interval for the difference \"(mean 1) minus (mean 2)\", which will be the reverse of what you'd get if you were calculating the confidence interval for the difference \"(mean 2) minus (mean 1)\".\n", "\n", "Okay, that's pretty straightforward when you think about it, but now consider our $t$-test comparing Anastasia's class to Bernadette's class. Which one should we call \"mean 1\" and which one should we call \"mean 2\". It's arbitrary. However, you really do need to designate one of them as \"mean 1\" and the other one as \"mean 2\". Not surprisingly, the way that Python handles this is also pretty arbitrary. In earlier versions of the book I used to try to explain it, but after a while I gave up, because it's not really all that important, and to be honest I can never remember myself. Whenever I get a significant $t$-test result, and I want to figure out which mean is the larger one, I don't try to figure it out by looking at the $t$-statistic. Why would I bother doing that? It's foolish. It's easier just look at the actual group means!\n", "\n", "\n", "Here's the important thing. Because it really doesn't matter what Python printed out, I usually try to *report* the $t$-statistic in such a way that the numbers match up with the text. Here's what I mean... suppose that what I want to write in my report is \"Anastasia's class had higher grades than Bernadette's class\". The phrasing here implies that Anastasia's group comes first, so it makes sense to report the $t$-statistic as if Anastasia's class corresponded to group 1. If so, I would write \n", "\n", ">Anastasia's class had *higher* grades than Bernadette's class ($t(31)= 2.1, p=.04$). \n", "\n", "(I wouldn't actually emphasise the word \"higher\" in real life, I'm just doing it to emphasise the point that \"higher\" corresponds to positive $t$ values). On the other hand, suppose the phrasing I wanted to use has Bernadette's class listed first. If so, it makes more sense to treat her class as group 1, and if so, the write up looks like this:\n", "\n", "> Bernadette's class had *lower* grades than Anastasia's class ($t(31)= -2.1, p=.04$). \n", "\n", "Because I'm talking about one group having \"lower\" scores this time around, it is more sensible to use the negative form of the $t$-statistic. It just makes it read more cleanly.\n", "\n", "One last thing: please note that you *can't* do this for other types of test statistics. It works for $t$-tests, but it wouldn't be meaningful for chi-square testsm $F$-tests or indeed for most of the tests I talk about in this book. So don't overgeneralise this advice! I'm really just talking about $t$-tests here and nothing else!" ] }, { "attachments": {}, "cell_type": "markdown", "id": "younger-monitor", "metadata": {}, "source": [ "(studentassumptions)= \n", "### Assumptions of the test\n", "\n", "As always, our hypothesis test relies on some assumptions. So what are they? For the Student t-test there are three assumptions, some of which we saw previously in the context of the one sample $t$-test (see Section \\@ref(ttestoneassumptions)):\n", "\n", "\n", "- *Normality*. Like the one-sample $t$-test, it is assumed that the data are normally distributed. Specifically, we assume that both groups are normally distributed. In Section \\@ref(shapiro) we'll discuss how to test for normality, and in Section \\@ref(wilcox) we'll discuss possible solutions.\n", "- *Independence*. Once again, it is assumed that the observations are independently sampled. In the context of the Student test this has two aspects to it. Firstly, we assume that the observations within each sample are independent of one another (exactly the same as for the one-sample test). However, we also assume that there are no cross-sample dependencies. If, for instance, it turns out that you included some participants in both experimental conditions of your study (e.g., by accidentally allowing the same person to sign up to different conditions), then there are some cross sample dependencies that you'd need to take into account.\n", "- *Homogeneity of variance* (also called \"homoscedasticity\"). The third assumption is that the population standard deviation is the same in both groups. You can test this assumption using the Levene test, which I'll talk about later on in the book (Section \\@ref(levene)). However, there's a very simple remedy for this assumption, which I'll talk about in the next section." ] }, { "attachments": {}, "cell_type": "markdown", "id": "demographic-roommate", "metadata": {}, "source": [ "(welchttest)=\n", "## The independent samples $t$-test (Welch test)\n", "\n", "The biggest problem with using the Student test in practice is the third assumption listed in the previous section: it assumes that both groups have the same standard deviation. This is rarely true in real life: if two samples don't have the same means, why should we expect them to have the same standard deviation? There's really no reason to expect this assumption to be true. We'll talk a little bit about how you can check this assumption later on because it does crop up in a few different places, not just the $t$-test. But right now I'll talk about a different form of the $t$-test {cite}`Welch1947` that does not rely on this assumption. A graphical illustration of what the **_Welch $t$ test_** assumes about the data is shown in Figure {numref}`fig-ttesthyp2`, to provide a contrast with the Student test version in {numref}`fig-ttesthyp`. I'll admit it's a bit odd to talk about the cure before talking about the diagnosis, but as it happens the Welch test is often the default $t$-test, so this is probably the best place to discuss it. \n", "\n", "The Welch test is very similar to the Student test. For example, the $t$-statistic that we use in the Welch test is calculated in much the same way as it is for the Student test. That is, we take the difference between the sample means, and then divide it by some estimate of the standard error of that difference:\n", "\n", "$$\n", "t = \\frac{\\bar{X}_1 - \\bar{X}_2}{\\mbox{SE}({\\bar{X}_1 - \\bar{X}_2})}\n", "$$\n", "\n", "The main difference is that the standard error calculations are different. If the two populations have different standard deviations, then it's a complete nonsense to try to calculate a pooled standard deviation estimate, because you're averaging apples and oranges.[^note9] But you can still estimate the standard error of the difference between sample means; it just ends up looking different:\n", "\n", "$$\n", "\\mbox{SE}({\\bar{X}_1 - \\bar{X}_2}) = \\sqrt{ \\frac{{\\hat{\\sigma}_1}^2}{N_1} + \\frac{{\\hat{\\sigma}_2}^2}{N_2} }\n", "$$\n", "\n", "The reason why it's calculated this way is beyond the scope of this book. What matters for our purposes is that the $t$-statistic that comes out of the Welch test is actually somewhat different to the one that comes from the Student test. \n", "\n", "The second difference between Welch and Student is that the degrees of freedom are calculated in a very different way. In the Welch test, the \"degrees of freedom \" doesn't have to be a whole number any more, and it doesn't correspond all that closely to the \"number of data points minus the number of constraints\" heuristic that I've been using up to this point. The degrees of freedom are, in fact...\n", "\n", "$$\n", "\\mbox{df} = \\frac{ ({\\hat{\\sigma}_1}^2 / N_1 + {\\hat{\\sigma}_2}^2 / N_2)^2 }{ ({\\hat{\\sigma}_1}^2 / N_1)^2 / (N_1 -1 ) + ({\\hat{\\sigma}_2}^2 / N_2)^2 / (N_2 -1 ) } \n", "$$\n", "\n", "... which is all pretty straightforward and obvious, right? Well, perhaps not. It doesn't really matter for our purposes. What matters is that you'll see that the \"df\" value that pops out of a Welch test tends to be a little bit smaller than the one used for the Student test, and it doesn't have to be a whole number. " ] }, { "cell_type": "code", "execution_count": 30, "id": "suspected-defendant", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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//nyUKlUq1e3c3d0xduxYAICnp2e6+5wxYwaKFSuWqePPmzcPhQsXfu//69evj9q1awMAjh8/nup9XV1d09334MGDtZ/527Zty1Sej2Fubo6ePXsCSP5cffbsWarbrV27FgBgY2ODzp07v3Xb7t27cfnyZQDAqlWrkC9fvlT30bp1a3z66acA3pzbfCxra2t4eXnB0tLyvdsmTJig/f/Unv9hw4YBACIiIrBly5ZU96+8T9RqNfr3759ullmzZqFQoULv/X+TJk20503nzp3D2bNn37p9wYIF2mOk9ViGDx+uPU/csmULHj16lG6WzBo/fnyqF+dxdHTE4MGDAQCXL1/Gq1ev3ro9O37mHj9+DAAoU6YMChYsmGZGe3t7mJll/quwsl/gzXlfaiwsLJAnT55M75fIVLDwRAapXLlyad7m6Oio/fvr169zIo5OpfdYgTePN63HqpwABQcHv3eCdO7cOQQFBQEAhgwZku5xSpYsmebJHgDkz58fJUqUAADtPgEgLi4ON2/eBADtiXJa3N3doVar39tHVnzse+XcuXPavzs7O793Fb2UfypVqqTdNuWJSceOHbVfoLp06YKmTZtizpw5OH/+PJKSkj76MZUsWRINGjQAAMyZMwcVK1bEd999h8OHD2fqKn1ERIYiLi4OoaGhuHbtGoKCghAUFAQhhPb2S5cu6ezYyoCBjY0N2rVrl+62yhfRhw8f4t69e6luY2lpie7du2fq2Hnz5k33mMrASWqDTu8SQuDx48e4ceOG9jkMCgqCi4sLAN0+hwDQr18/AMlX0N20adN7t8fGxmLr1q0AgE6dOsHOzu6t25XXoWzZsmkOxCmU1+Hs2bNZ+pxt0aIFChQokOptuXPn1hb2Unv+GzRogLJlywJIvQAWHx8PHx8fAEC7du3SPA4AODg4oFOnTmnenvLcLeWAXWJiIo4ePap9LEWLFk1zH0rxKjExEUeOHElzuw/xsQPE2fEzpwzOXbt2DWfOnPmw4OlIOeiX1cImkSli4YkMko2NTZq3pRy9yMpJh75I77ECbx5vWo+1U6dOyJ8/P4D3PyiVUSIrK6t0TxIApHtipFBGll68eKH9v/Dw8PduT4tarYaTk9N7+8iKj32vhIWFfdTxUhZ+nJycsGPHDhQuXBhCCPj7+2Py5MmoUaMGHB0d0a1bN+3I3odat24d6tatCyD55Ornn39Gs2bNkDdvXjRq1AiLFy/WztgiIjIkUVFRmD59Otzc3GBra4vixYujYsWKqFy5MipXrvzW7Ny0ZtBkB2UAIjo6GhYWFukOQLRv3157v5QDECm5urrC2to6U8d2dXVNdzZGRoNOQPJMofbt28Pe3h7Ozs4oW7as9jmsXLkydu/eDUC3zyEA1KpVS1uoUWY2pbRz505EREQASL1gobwO//33X7qvgUqlwrhx4wAkF3eych6R1UG/oUOHAgAOHz6MkJCQt27bsWMHnj9/DiDjQT93d3dYWFikeXvVqlW1M5lSDtjdvn1bez6S0aBfytv1ZdAvKz9zvXv3hlqtRlxcHOrVq4cOHTpg8eLFuHr16ltF6w9Vv3597QysiRMnolatWpg+fTpOnjyJ+Pj4j94vkalg4YnIyFlaWmLAgAEAgE2bNiEyMhJA8gjj+vXrAQCdO3d+60QgNcoyuvRk9IGeHfvIKUohytLSEleuXMn0n3dHJhs0aIDg4GCsWbMGffr0QZEiRQC8mYLfoUMHtG7d+oNnKhUuXBgnT57EwYMHMWbMGFSsWBEqlQoJCQk4duwYRo8ejUqVKuHGjRvZ84QQEeWAu3fvonLlyvjqq69w+fLlDAeQYmJidJYlOwYgUnJwcMj0PjI76KTRaN67TQiBYcOGoX379ti9e3eGs791+Rwq+vTpAwA4efIk7t69+9ZtSjEqX758aNmy5Xv3ze7XITOyOug3cOBAqNVqCCGwcuXKt25TBv2cnZ3Rpk2bdI+T0aCfhYWF9vwtZaEt5d8zGvRLuYzPGAb9ypUrh3Xr1sHBwQGJiYnYtWuX9pyoQIEC6N+/f5pLVNOjVquxc+dOlC9fHkDyrLqvvvoK9erVQ968edGmTRv4+PgYxaA3kS6kXUInMlIqlQpCiFRP1lKKiorKoUS6N2zYMMyaNQuRkZHw9fXFoEGDsG3bNu1spIxG3ADgyZMnGW6jnDCkLGKlPNFOaxRYkZiYqD3pyagQpmvKzKv4+Hg4OTm9NcX6Q1lbW6Nv377akdzbt29j9+7dmD9/Pm7cuAE/Pz98/fXXmDNnzgfvu1mzZmjWrBkA4Pnz5zh48CCWLl2Kw4cP49atW+jZsycCAwM/OjsRUU7q378/7ty5A5VKhcGDB6NXr14oX7488ufPDysrKwDJxRZzc3MAuh2sUL5AlixZEjt27Mj0/UqWLJnq/yuZdc3T0xMrVqwAkDwjZuLEiahduzYKFy4MGxsbbY4BAwZg9erVOTLg07dvX/z4448QQmDdunWYNm0agORZ0Xv37gWQ3E9RWW6fkvI61KtXD4sXL870MZWlhDIUKFAAHTt2xObNm+Ht7Y1vv/0WKpUKDx8+xP79+wEkP//pzWYCcm7QT19k189ct27d0Lx5c2zYsAF+fn44fvw4nj59imfPnmHNmjVYs2YNBg4cCE9Pzw/q81ShQgVcuXIFO3fuxM6dO3H06FHcunULMTEx2LdvH/bt24fZs2djz549mVopQGRKWHgik5M7d25ERES8tQTsXRqNRtuXyBiUK1cO9erVw4kTJ+Dl5YVBgwZpR9yKFSumbSyZnjt37uD58+fagsy7nj59qh3FTNnzyMrKCq6urrh58yb++eefdI8RGBiIhISE9/YB5PyJU8qlHPv370+3sfqHKlWqFMaPH4+BAweiYsWKuH//PjZu3PhRhaeUnJyc0LNnT/Ts2ROdOnXCjh07cPHiRdy8eTPDRrNERLJdv34dAQEBAIBp06bh119/TXW79D6/s5PyeffkyROUK1cuwyKBvli2bBkAoHTp0jh58iRy5cqV6nY59TwCyUsHa9WqhTNnzmDt2rXawpOvr692mVJaS/6dnJzw5MkTPH369L1zA302bNgwbN68GXfu3MHRo0fRuHFjrFq1SltcyY5Bv8TERO3rmHLALuXfMxr0S3m7vgz6ZcfPnL29PUaMGIERI0YASG5LsGPHDsybNw8PHz7EypUr4e7ujs8+++yD9mtubo7OnTtrm+A/evQIe/fuxcKFC3H+/HmcP38eI0eO1PYtI6JkXGpHJkcZFUnZPPpde/bsee9KG4ZOaR557Ngx+Pv749ChQwCAQYMGZWq0RwiBVatWpXm7t7e3dtTt3UKW8u9r167h9OnTae5j+fLl791HkbIvRlxcXIZ5s6pNmzbakdc5c+YgMTEx24+RJ08e1KxZE0D299hQZkHpYt9ERLpw9epV7d979eqV5nbpfX5nRmYHMpQBiOjoaJw4cSJLx8xJyvPYqVOnNItOQghcuHAhJ2NpC0tXr17VXqVOabJdokQJbd/Cdymvw40bN97rl6TPWrZsqb2CodJjU7kSbb169VCmTJkM93Hx4sV0zz8uXbqkLdylLMqVKlVKu9wto0G/lA249WXQTxc/cxUqVMCXX36J06dPw9bWFgCwcePGLO/X2dkZQ4YMwalTp1CtWjUAwK5du3JkCSuRIWHhiUxOo0aNACR/EKf2ofbo0SNMmDAhp2PpXPfu3WFvbw8gudeCRqPRLmXIrJ9//hn//fffe///77//akemnZ2d3+tzNHr0aG1xa8SIEakW9fbv369dGlCrVi1tQUbh5OSkbaB569atTGf+WIULF9Y+N5cuXcLIkSPTPfkLCwt7q3AGJF86Or1LE7969Up7wpfW0ozUXLx4Md1LiAshtFe3UalU2qsNEhHps5S/Y9Prz/Mhy61SowxkZDSIkfKz7I8//sjSMXOS8jym9xzu2LEDDx8+THc/mX2eMqtXr17aZX5r167F/fv3cezYMQDJRam0ihwdO3bU/t2QXgczMzPtrCZfX1/s3btXew6lNB/PyIsXL7Bz5840b1dmrwNvD9hZWFhoz3cPHDiQ5pUWgTeDfubm5mjcuPFbt+X0oF9O/MwVLVpUW/TLzoE5tVqtfc4TExPx8uXLbNs3kTFg4YlMzogRI2BhYQEhBDp06IC5c+fi3LlzOHnyJGbOnAl3d3dEREQY3dIkGxsb9O7dG8CbadVNmjTJdFHC1dUVGo0GderUwYwZM3D69GmcPn0aM2bMQN26dbXFpHnz5mkLRIrKlStjypQpAIArV66gWrVqWLp0Kc6ePYujR4/i888/R/v27ZGUlARLS0ssWbLkveNbWFhoi1Genp5Yt24d/v33XwQHByM4ODjbGmKmNGvWLO3on6enJ9zc3PDXX38hICAAFy9exJEjR7BgwQJ06dIFRYsWfe/L0Lp161C8eHG0a9cOf/31Fw4dOoTAwEAcO3YMCxcuRN26dfHgwQMAycW5zLp48SLc3d1Rq1Yt/Pzzz9i9ezfOnz+P06dPY926dWjVqpX2RLVTp05Z6k9FRJRTUn7uvtuQWbFo0SJs27YtS8dRfieGhYWl23i7Zs2a2mbXe/bswffff5/ufu/evYt169ZlKVt2UJ7HnTt3prqc7tatWxgzZkyG+1Gep+wa7ClQoIC2OLJu3Tr4+Pho+20qzcdT061bN21D50WLFmkHqdISFBSUbrEmJw0ZMgRmZmaIjo7WFqHs7OzQvXv3TO9j8uTJqS65O3r0KJYuXQoAqF69+nsDdmPHjgUAJCQkYMiQIaleec3T01Pbc6pbt27vnS+k/HdODPplx8/ctm3b0i363Lt3D9evXwfwYYN+x48fR3BwcJq3x8fH4+jRowCSX2PlitJElMwwFqsTZaOKFSvijz/+wOTJkxEeHo5Jkya9dbuDgwO2bduG7777zqj6PAHJ/QZSFkcyO+IGJDfpnDNnDnr06KHtzZCSmZkZ/vjjD3Tr1i3V+8+YMQNRUVFYuHAhbt++jZEjR763jb29PTZu3IiqVaumuo9p06ahQ4cOeP78+Xsnqd9//z1++OGHTD+ezLCzs8PRo0fRt29f7Nu3D9euXcPEiRPT3D5Pnjzv/V9CQgL27NmDPXv2pHm/sWPHYvz48R+c7+zZszh79myat9evXz/DE3QiIn3h7u6OSpUqISgoCIsWLcLLly/Rt29fODs74969e1izZg18fX21PQs/loeHB4Dkfo6jRo3C+PHj4eTkpJ1x88knn2i39fLyQo0aNfDo0SP89NNP8PPzw5AhQ1C5cmVYW1vj+fPnuHz5Mvbt24fDhw+jc+fO2kEeWQYMGID//e9/ePDgATw8PPDFF1+gYsWKiI2NxeHDhzF37lzExcWhWrVq6S638/DwwJ07d7Bjxw4sWbIE9erV086AyZMnz0c1T+7Xrx/8/Pxw7949TJ8+HUDy616hQoU072Nubo4NGzbAw8MDkZGRGDZsGDZt2oQ+ffqgbNmyUKvVCAsLQ2BgIHbt2oWTJ09iypQp6NChwwfny25FixZFy5YtsW/fPu2gX48ePWBnZ5ep+7u5ueHatWuoXr06pk2bhlq1aiEuLg579uzRtgGwsLDAggUL3rtvu3bt0L17d2zatAkHDx5E7dq1MWXKFJQvXx7h4eFYv369dsaUo6MjZs+e/d4+ihUrhiJFiuD+/fv4888/UbhwYZQtW1bbe6lgwYLInTv3xz49qcrqz9zcuXPRt29ftGvXDk2bNkX58uVhb2+P8PBwnDt3DvPmzdMug/uQQb9Dhw7h559/RoMGDdCuXTtUqVIF+fPnR0xMDG7cuIHFixdrf56GDRtmMD3hiHKMIDIQ33//vQAgMnrb+vv7a7fz9/dPc7t9+/aJVq1aCQcHB2FlZSVKliwpxo4dK0JCQoQQQjRq1EgAEI0aNXrvvnfu3NEew8vL673bvby8tLffuXPnAx7lG8WLFxcAxMCBA9PdbuDAgQKAKF68eKb2W6FCBQFA5M2bV8TExGS4/bvPw7Vr18TgwYNF8eLFhaWlpShQoIDo1q2bOHnyZKaOf+zYMdG3b19RrFgxYWVlJfLkySOqVq0qvvrqKxEWFpbh/Q8fPiw6deokXFxchFqt1j7P33//vXabzD7/Gb2OKR06dEgMHjxYuLq6Cjs7O2FhYSEcHR1FzZo1xdixY8WePXtEYmLiW/cJDw8XmzdvFqNGjRI1atQQhQsXFpaWliJXrlyiTJkyYtCgQSIgICDNY6b22IQQIi4uTvj7+4uvvvpKNGjQQJQsWVLY2NgIS0tLUaRIEdGxY0fh4+MjkpKS0n1MRET6JjAwUDg4OGh//737p3LlyuLhw4dp/n4UIuPPgKSkJFGnTp00j/Guu3fvipo1a6a5fco/gwcPfu/+6Z1PfOy26Z0TxcfHi5YtW6aZMVeuXGLjxo0Znj8EBgYKKyurVPeR8tzkQz5LX79+LWxsbN7a159//pnBs5Ls0qVLwtXVNVOvw48//pipfb4rvfdVSh/ymvr6+r6V7cSJExneJ+U54LJly4SFhUWqj9PS0lKsW7cuzf3ExMSILl26pPtcubi4iMDAwDT3sXDhwjTvm/L1zs7z9Kz8zCmvTXp/zM3NxW+//fbecdN7L6d8fOn96dq1a6bOr4lMDQtPRCYkIiJCe8I3evToTN3nQ06uiIiIsiokJESMGjVKFC9eXKjVauHo6Chq1aol/vzzT+0XuqwUnoRI/jz85ptvhJubm7CzsxMqlSrdL80ajUZs3bpV9OrVS1vsV6vVIn/+/MLDw0NMmTJFHD16VGg0mvfum9OFJyGESEhIEH///beoUaOGsLGxEbly5RKffPKJGDVqlPj333+FEJkbuLpw4YLo3bu3drAoq4UnIYTo3bu3dnszMzNx//79DO+T8nGtXLlSdO7cWRQtWlRYW1sLS0tL4ezsLBo3biy++eYbcf78+Uzv7126KDzFx8cLe3t7AUCULVs2UzneHXw8deqU6NGjh3BxcRGWlpaicOHCYsCAAeLq1auZ2t+OHTtE165dtfd3cHAQtWvXFtOnTxevX7/O8P6bN28WLVu2FAUKFHirCKarwpMQH/8z9+TJE7F27VoxaNAgUbVqVVGoUCFhYWEh7OzsRKVKlcSYMWPE5cuXUz1meu/lqKgosWfPHjFp0iRRp04dUaxYMWFtbS2sra1FiRIlRM+ePcXu3bvTfexEpkwlxP9fhoqIjN6yZcu0l5U9d+4cqlevnuF9GjdujKNHj6JRo0Y4cuSIjhMSERERGY/g4GBt363ff/8dX3zxRYb3KVGiBEJCQjBw4EDtlfCIiAwZm4sTmYikpCTMmTMHQHITyswUnYiIiIjo43l5eQFIvkjKgAEDJKchIpKDXc+IjNiLFy/w4sULPH/+HHPnzsW///4LAKk2ByciIiKi7BMREaG98lznzp1RqFAhyYmIiORg4YnIiP3999/48ccf3/q/du3apXnlOSIiIiL6eGFhYYiIiMCjR4/w448/4tmzZ1CpVJg6darsaERE0rDwRGQCLCwsULx4cfTu3ZuznYiIiIh05IsvvsDKlSvf+r/Ro0ejRo0akhIREcnHwhOREfvhhx/www8/ZGkfbChORERE9GEsLS1RunRpDB8+HOPHj5cdh4hIKl7VjoiIiIiIiIiIdIJXtSMiIiIiIiIiIp1g4YmIiIiIiIiIiHSChSciIiIiIiIiItIJFp6IiIiIiIiIiEgnWHgiIiIiIiIiIiKdYOGJiIiIiIiIiIh0goUnIiIiIiIiIiLSCRaeiIiIiIiIiIhIJ1h4IiIiIiIiIiIinWDhiYiIiIiIiIiIdIKFJyIiIiIiIiIi0gkWnoiIiIiIiIiISCdYeCIiIiIiIiIiIp1g4YmIiIiIiIiIiHSChSciIiIiIiIiItIJFp6IiIiIiIiIiEgnWHgiIiIiIiIiIiKdYOGJiIiIiIiIiIh0goUnIiIiIiIiIiLSCRaeiIiIiIiIiIhIJ1h4IiIiIiIiIiIinWDhiYiIiIiIiIiIdIKFJyIiIiIiIiIi0gkWnoiIiIiIiIiISCdYeCIiIiIiIiIiIp1g4YmIiIiIiIiIiHSChSciIiIiIiIiItIJFp6IiIiIiIiIiEgnWHgiIiIiIiIiIiKdYOGJiD6It7c3VCoVVCoVQkJC0tzu33//1W63YcOGHExIREREpoTnJkRE+o2FJyL6IJcuXQIAODg4oHjx4mlud/HiRe3f3dzcdB2LiIiITBTPTYiI9BsLT0T0QZSTtqpVq2ZqOxsbG5QpU0a3oYiIiMhk8dyEiEi/sfBERB/k8uXLAAB3d/d0twsMDAQAVK5cGWZm/FVDREREusFzEyIi/cbfuESUaaGhoXjx4gWAjE/ulGnvGY0+EhEREX0snpsQEek/Fp6IKNOUEzYg/ZO2hw8fIiwsLMPtiIiIiLKC5yZERPqPhSciyjSlN4K1tTXKlSuX5nbKVHaAJ3dERESkO9l1brJmzRqMHDkSNWrUgJWVFVQqFby9vbM5LRGRabKQHYCIDIcyqli5cmVYWKT960M5CTQzM0OVKlVyIhoRERGZoOw6N/nmm28QEhKCfPnywdnZGSEhITrJS0RkijjjiYgyLbNXjVFGFV1dXWFjY6PjVERERGSqsuvcZPny5bh79y6ePn2KUaNGZXdMIiKTxhlPRJQpkZGRuH37NoCMT+7Onj2bqe2IiIiIPlZ2nps0b948O6MREVEKnPFERJly6dIlCCEAAKVLl05zu6CgIISGhgIA3NzcciQbERERmR6emxARGQYWnogoU1JeNSYqKirN7f744w/t3znjiYiIiHSF5yZERIaBhSciypSUJ3ebN29OdZvFixdj9erV2n9XrFhR57mIiIjINPHchIjIMLDHExFlSsqrwfj4+CA+Ph6ffvop8ufPj3v37mHDhg3Yu3cv1Go1EhISAABbtmxBoUKF0KtXL4nJiYiIyBjx3ISIyDCohLIwmogoDRqNBrlz50Z0dDTGjRsHHx8fvHjx4r3t2rVrB1dXV8ydO1f7f0OHDsXy5ctzMC0REREZO12em8yYMQPTpk2Dl5cXBg0apIP0RESmhUvtiChDN27cQHR0NIDkq74EBASgU6dOyJs3L2xtbVGtWjX8/fff2LFjByZOnIiqVavC1tYWDRs2xNChQyWnJyIiImPDcxMiIsPBpXZElKGUPRSqVKmCkiVLYtu2baluW7x4cQQGBuZQMiIiIjJFPDchIjIcnPFERBlSeijkyZMHJUqUkJqFiIiIiOcmRESGgzOeiChDyqhipUqVoFKpJKchIiIiU5fd5ybLly9HQEAAAODKlSva/zty5AgAoHPnzujcuXOWj0NEZIpYeCKiDCmjilWqVJEbhIiIiAjZf24SEBCAlStXvvV/J06cwIkTJwAAJUqUYOGJiOgjsfBEROl6+vQpHj16BICFJyIiIpJPF+cm3t7e8Pb2zpZ9ERHR29jjiYjS9W7zTiIiIiKZeG5CRGRYVEIIITsEEREREREREREZH854IiIiIiIiIiIinWCPJyL6IPHx8Th9+jSOHj2KiIiIt24zMzODm5sbmjdvjgIFCkhKSERERJQsMTERe/bswYkTJ6DRaAAAFhYWaNasGZo2bQozM47DExHpGpfaEVGGgoODsWvXLhw8eBBHjhxBVFRUhvdRClAtW7ZEs2bNYG5ungNJiYiIiIA7d+5g6dKl8Pb2xuPHj1PdpkSJEhgyZAiGDx+OQoUK5XBCIiLTwcITEaUpJCQE33//PVatWoWUvyry5cuHZs2aoWjRom9tHxMTg4CAgLeafgJAhQoV8Ntvv6Fjx45QqVQ5kp2IiIhM0+rVqzFy5EjExMQAAPLnz4+uXbsid+7cAIDnz59jy5YtePXqFQDAwcEBa9euRZs2baRlJiIyZiw8EdF7nj59il9//RWLFi1CfHw8AKBZs2Zo3bo1mjdvjipVqqQ7NT0sLAyHDh3CwYMHsXXrVoSHhwMA6tSpgxkzZqBRo0Y58jiIiIjIdMTFxWHixIlYvHgxAKBBgwaYOHEi2rdvD0tLy7e2jY6OxubNmzF79mxcvHgRKpUK3377Lb777jvO0iYiymYsPBGRlkajwcyZM/HLL78gMjISQHLBafr06ahZs+ZH7fPly5eYOXMm5s6di+joaABAmzZtsGTJkvdmTBERERF9jGfPnqFdu3Y4c+YMVCoVvvvuO3z33XcZ9nCKi4vD5MmTsXDhQgBAq1atsGXLFtjY2OREbCIik8DCExEBSC4Q9e/fH7t27QIAVK9eHTNmzEDz5s2zZf+PHj3CL7/8gqVLlyIxMRH58+fHhg0b0KRJk2zZPxEREZmm2NhYNGvWDCdPnvzoZXMpl+d17twZvr6+nPlERJRNWHgiIgQFBaFLly4IDg6GtbU15s+fjyFDhuikH9PNmzfRs2dPBAYGwtzcHH/88QcmTZrE3k9ERET0wTQaDXr37o2NGzcib968OHHiBCpUqPBR+woICEDz5s21S/bmzJmTzWmJiEwTrx9KZOI2bNiA2rVrIzg4GMWLF8eJEycwdOhQnRWCXF1dceLECfTv3x9JSUmYMmUK+vTpk6kr5RERERGl9NVXX2Hjxo1Qq9XYunXrRxedAKB+/fpYuXIlAGDu3Ln4+++/sysmEZFJY+GJyEQJIfDtt9+iV69eiI6ORosWLXD+/HlUq1ZN58fOlSsXVq5cifnz58PCwgLr169H3bp18ejRI50fm4iIiIyDt7c3fv/9dwCAp6cnGjdunOV99uzZE9OnTwcATJw4Efv27cvyPomITB2X2hGZqB9++AE//vgjAGDq1Kn49ddfpfQyOHHiBD799FM8fvwYFStWxJEjR5AvX74cz0FERESGIzQ0FJUqVcLr16/x/fff44cffsi2fQshMGLECCxfvhwuLi64evUq8ubNm237JyIyNSw8EZmg33//HV9++SWA5Knkn332mdQ8t27dQsOGDfHw4UO4u7vj8OHDPMEjIiKiVAkh0KZNG/j5+cHDwwPHjh3L9sGzmJgYuLm54ebNmxg6dCiWL1+erfsnIjIlXGpHZGLmzZunLTpNnz5detEJAEqXLo1Dhw4hf/78CAwMRJs2bfD69WvZsYiIiEgPeXt7w8/PD1ZWVlixYoVOZmznypULK1asgEqlwooVK7B///5sPwYRkalg4YnIhCxfvhwTJkwAAHz77bfaApQ+KFeuHA4ePAgHBwecPn0aHTp0QHR0tOxYREREpEcePHiASZMmAQB++uknlCtXTmfHatCgAcaNGwcAGD58OAfFiIg+EpfaEZmIDRs2oHfv3hBCYMqUKZg5c6bOrlyXFefOnUOzZs0QERGBVq1aYdeuXbCwsJAdi4iIiPRAly5dsG3bNtSsWRMnT57U+TlCVFQUKleujDt37mDs2LGYP3++To9HRGSMWHgiMgGXLl1C3bp1ERMTg9GjR2PBggV6WXRSnDhxAi1btkR0dDQ+//xzzJw5U3YkIiIikuzYsWNo1KgRzM3NcfHiRVSqVClHjnvo0CE0b94c5ubmuHr1KsqWLZsjxyUiMhZcakdk5MLDw9G1a1fExMSgdevWmDdvnl4XnQCgXr16WLVqFQDgzz//hK+vr+REREREJJMQAlOnTgWQvOwtp4pOANCsWTO0b98eSUlJ+Oabb3LsuERExoIznoiMmEajQadOnbBr1y6UKFEC58+fh6Ojo+xYmfbFF19g5syZsLOzw5kzZ1C+fHnZkYiIiEiCbdu2oUuXLrCxsUFwcDCcnZ1z9PhXrlyBm5sbhBA4c+YMatasmaPHJyIyZJzxRGTEfvvtN+zatQtWVlbYvHmzQRWdgOT8jRs3RmRkJLp06YKIiAjZkYiIiCiHJSYm4quvvgIATJw4MceLTgBQuXJl9O/fHwDw5ZdfgmP3RESZxxlPREbKz88Pbdq0gRACnp6eGDx4sOxIHyUsLAzVqlXDgwcP0K1bN2zatEnvlwoSERFR9vH09MTQoUPh6OiI27dvw97eXkqOkJAQlClTBvHx8fDz80PLli2l5CAiMjSc8URkhEJCQtCnTx8IITBixAiDLToBQIECBeDr6wu1Wo3Nmzdj1qxZsiMRERFRDomNjcX3338PAPjqq6+kFZ0AoHjx4hgzZgyA5FlPGo1GWhYiIkPCGU9ERkaj0aBp06Y4evQoatasiePHj8PKykp2rCxbtGgRxowZAwsLC5w7dw5ubm6yIxEREZGOLV26FCNHjkSRIkVw8+ZNWFtbS83z7NkzlCpVCq9fv8bu3bvRtm1bqXmIiAwBZzwRGZlFixbh6NGjsLW1xfr1642i6AQAo0aNQpcuXZCYmIhBgwYhISFBdiQiIiLSIY1Go53pPGXKFOlFJwDIly8fRowYAQCYOXOm5DRERIaBM56IjMjt27dRpUoVREVFYf78+Rg7dqzsSNnqyZMnqFChAl68eIGffvoJ3377rexIREREpCPbt29H586dkTdvXoSGhiJ37tyyIwEA7t27h1KlSiExMRFnz55FjRo1ZEciItJrnPFEZCQ0Gg2GDh2KqKgoNGrUCKNHj5YdKdsVLFgQ8+fPBwD8/PPPuHz5suREREREpCvKjKJRo0bpTdEJAIoWLYpevXoBAP7880/JaYiI9B9nPBEZCaUHko2NDS5fvozSpUvLjqQTQgh07doV27ZtQ7Vq1XD69Gmo1WrZsYiIiCgbnTp1Ch4eHlCr1QgJCYGzs7PsSG+5dOkSqlatCjMzMwQHB6NkyZKyIxER6S3OeCIyAnfv3sX//vc/AMCMGTOMtugEACqVCosWLYKDgwMuXLiAP/74Q3YkIiIiymbKbKd+/frpXdEJANzc3NCiRQtoNBrMmTNHdhwiIr3GGU9EBk6j0aBFixY4fPgwGjRogCNHjsDMzPhrymvXrkW/fv2gVqtx/vx5VK5cWXYkIiIiygY3b95E2bJlIYTA1atXUaFCBdmRUnXgwAG0bNkSNjY2uHfvHhwdHWVHIiLSS8b/7ZTIyHl7e+Pw4cPIlSsXPD09TaLoBAB9+vRBx44dkZCQgGHDhkGj0ciORERERNlg/vz5EEKgbdu2elt0AoDmzZvDzc0N0dHRWLFihew4RER6yzS+oRIZqVevXmHatGkAgJ9++gmffPKJ5EQ5R1lyZ2dnhzNnzmDVqlWyIxEREVEWxcTEaD/Tx48fLzlN+lQqFcaNGwcAWLp0KbiQhIgodSw8ERmwn376CWFhYShbtiwmTJggO06Oc3FxwXfffQcA+PLLLxERESE5EREREWXFpk2b8PLlSxQvXhwtWrSQHSdDvXr1Qu7cuREcHAx/f3/ZcYiI9BILT0QG6vr16/j7778BAHPnzoWlpaXkRHJ89tlncHV1xZMnT/DLL7/IjkNERERZsHTpUgDA8OHDYW5uLjlNxuzs7NC3b18Ab7ITEdHb2FycyAAJIdCmTRv4+fmhQ4cO2LFjh+xIUu3evRvt27eHWq1GUFAQypQpIzsSERERfaCrV6+iUqVKMDc3R2hoKFxcXGRHypTAwEBUq1YNarUaDx48QP78+WVHIiLSK5zxRGSAdu3aBT8/P1haWmL27Nmy40jXrl07tGnTBgkJCZg0aZLsOERERPQRlBlDHTp0MJiiEwC4u7ujRo0aSEhIgLe3t+w4RER6h4UnIgMTFxenLa5MmjTJpBqKp2fOnDlQq9XYs2cPdu/eLTsOERERfYCUTcVHjhwpOc2HUzKzyTgR0ftYeCIyMHPmzMGtW7fg7OyMr7/+WnYcvVG2bFl89tlnAJILcvHx8ZITERERUWb5+voaVFPxd/Xq1Qt2dnYIDg7GkSNHZMchItIrLDwRGZBHjx5pG2j//vvvyJ07t+RE+uXbb79FwYIFcfPmTW3jdSIiItJ/y5cvBwAMGzbMIJqKvytlk/Fly5ZJTkNEpF/YXJzIgIwbNw4LFixA7dq1cfLkSZiZsXb8Lk9PTwwdOhQODg64c+cO7O3tZUciIiKidNy9exclS5aESqVCaGgoihQpIjvSRzl79ixq1aqFXLly4cmTJxwgJCL6f/zWSmQg7ty5o226+fvvv7PolIaBAweifPnyCA8Px59//ik7DhEREWXAx8cHANCkSRODLToBQI0aNVCmTBnExMRg69atsuMQEekNfnMlMhA//PADEhIS0LJlSzRq1Eh2HL1lbm6uXY44Z84chIWFSU5EREREaRFCYPXq1QCAfv36SU6TNSqVSvsY1qxZIzkNEZH+4FI7IgNw9epVVK5cGUIInD17FjVq1JAdSa8JIVCrVi2cO3cOEydOxJw5c2RHIiIiolRcuHAB1atXh7W1NR4/fmzwS+Rv376N0qVLw8zMDPfu3YOLi4vsSERE0nHGE5EB+PbbbyGEQNeuXVl0ygSVSoXffvsNALBw4UKEhoZKTkRERESpUWYGdezY0eCLTgBQqlQpeHh4QKPRYP369bLjEBHpBRaeiPTc2bNnsXXrVpiZmeHnn3+WHcdgNG/eHI0bN0Z8fDx++ukn2XGIiIjoHYmJiVi3bh0Aw19mlxKX2xERvY1L7Yj0XMuWLXHgwAEMHDgQ3t7esuMYlFOnTsHDwwPm5ua4evUqypYtKzsSERER/b/9+/ejVatWcHJywqNHj6BWq2VHyhbPnz+Hs7MzEhISEBQUhIoVK8qOREQkFWc8Eekxf39/HDhwAGq1Gj/88IPsOAanbt266NChA5KSkvDdd9/JjkNEREQpKDOCevXqZTRFJwBwcnJC27ZtAQBr166VnIaISD4Wnoj0lBACX331FQBg5MiRKFGihNxABuqXX36BSqXCxo0bERgYKDsOERERAYiOjsaWLVsAGNcyO4XymNauXQuNRiM5DRGRXCw8EempAwcO4PTp08iVKxe+/vpr2XEMVpUqVdC7d28AYI8sIiIiPbF3715ERUWhRIkSqF27tuw42a59+/aws7NDaGgozp49KzsOEZFULDwR6alffvkFQPJsp0KFCklOY9i++eYbqFQqbN26FUFBQbLjEBERmbxNmzYBALp37w6VSiU5TfaztrZGhw4dALx5rEREpoqFJyI9dPz4cRw/fhyWlpb4/PPPZccxeOXLl0fXrl0BANOnT5echoiIyLTFxMRg165dAIBPP/1UchrdUR6br68veD0nIjJlLDwR6aFff/0VADBo0CAULlxYchrjoCxXXL9+PYKDgyWnISIiMl1+fn6IiopCsWLFULNmTdlxdKZNmzawtbVFSEgIzp8/LzsOEZE0LDwR6Zlz587Bz88P5ubmmDp1quw4RsPd3R1t27aFRqPB77//LjsOERGRyVKWnn366adGucxOkStXLrRr1w4Al9sRkWlj4YlIz/z2228AgD59+qBUqVKS0xgXZdbTypUrce/ePclpiIiITE9sbCx27twJILm/k7FTHiOX2xGRKWPhiUiPBAUFYevWrVCpVJg2bZrsOEbHw8MDjRs3RkJCAmbOnCk7DhERkcnZv38/Xr9+jSJFiqBWrVqy4+hcmzZtkCtXLty+fRuBgYGy4xARScHCE5EeURpfd+3aFeXLl5ecxjgps56WLVuGJ0+eSE5DRERkWnx9fQEA3bp1g5mZ8X8VsbW1Rdu2bQG8eexERKbG+H/bExmI4OBgrF+/HsCb4ghlv2bNmqFWrVqIjY3FnDlzZMchIiIyGXFxcdi+fTsA01hmp1Ae66ZNm7jcjohMEgtPRHri999/h0ajQZs2beDu7i47jtFSqVTawt6CBQvw4sULyYmIiIhMw6FDhxAREQEXFxfUrVtXdpwc065dO1hbWyM4OBiXL1+WHYeIKMex8ESkBx49eoSVK1cC4GynnNC+fXtUrlwZkZGRWLx4sew4REREJmHbtm0AgM6dO5vEMjuFnZ0dWrZsCQDaGV9ERKbEdH7jE+mx+fPnIyEhAR4eHqhXr57sOEbPzMwM//vf/wAA8+bNQ1xcnORERERExk2j0WivZtepUyfJaXKe8phZeCIiU6QSXGhMJFVUVBSKFi2K8PBwbN68GV27dpUdySTEx8ejZMmSePjwIby8vDBo0CDZkYiIiIzW6dOnUbduXeTJkwdPnz6FpaWl7Eg5KiwsDIUKFYIQAqGhoShatKjsSEREOYYznogk8/b2Rnh4OEqXLm2SI4CyWFpaYsKECQCA2bNns9knERGRDu3YsQMA0KZNG5MrOgFAgQIF4OHhAQDamV9ERKaChSciiZKSkrRXVps4cSLMzc0lJzItI0aMgK2tLa5cuYIDBw7IjkNERGS0lCVmHTt2lJxEHuWxc7kdEZkaFp6IJNq5cydu3boFBwcHDB48WHYck+Pg4IChQ4cCSJ71RERERNkvODgY165dg4WFBdq2bSs7jjTKzHZ/f3+8evVKchoiopzDwhORRLNmzQIAjBw5Era2tpLTmKbPPvsMZmZm8PPzQ1BQkOw4RERERkeZ4dOoUSPkzZtXbhiJypYti7JlyyIhIQH79u2THYeIKMew8EQkyZkzZxAQEAC1Wo3x48fLjmOySpUqhS5dugDgrCciIiJdUPo7sZflm+dAeU6IiEwBC09EkiiznXr37g0XFxfJaUzblClTAABr167F48ePJachIiIyHs+ePUNAQAAA0+7vpFCegz179iAhIUFyGiKinMHCE5EEd+/eha+vLwBg8uTJktNQ3bp1UbduXcTHx2P+/Pmy4xARERmN3bt3Q6PRwM3NDcWLF5cdR7o6deogf/78ePnyJY4dOyY7DhFRjmDhiUiCv/76CxqNBs2bN4ebm5vsOIQ3s54WLVqEqKgoyWmIiIiMw86dOwFwtpPC3NwcHTp0AMDldkRkOlh4Isphr1+/hqenJwDOdtInnTt3RsmSJfHixQv4+PjIjkNERGTwEhISsH//fgDQFlsIaN++PQBg7969kpMQEeUMFp6Ictjq1asREREBV1dXtGrVSnYc+n/m5uYYO3YsAGDevHkQQkhOREREZNgCAgLw+vVr5M+fH9WrV5cdR280a9YMarUaN2/exM2bN2XHISLSORaeiHKQEELbQ2jcuHEwM+OPoD4ZMmQIcuXKhStXruD48eOy4xARERm0PXv2AADatGnDc54U8uTJgwYNGgB48xwRERkzfgIQ5SB/f3/8+++/sLW1xcCBA2XHoXc4ODigX79+AMAm40RERFmkFFXatWsnOYn+UZ4TFp6IyBSw8ESUg+bNmwcAGDhwIOzt7SWnodSMGzcOALBlyxbcv39fchoiIiLDdPfuXVy7dg3m5uZo0aKF7Dh6p23btgCAI0eOIDIyUnIaIiLdYuGJKIeEhIRor16i9BIi/VOlShU0bNgQSUlJWLJkiew4REREBkmZyePh4QEHBwfJafRP2bJlUbJkScTHx+Pw4cOy4xAR6RQLT0Q5ZNGiRdBoNGjWrBkqVKggOw6lY/z48QCApUuXIi4uTnIaIiIiw7N7924AXGaXFpVKpX1ulOeKiMhYsfBElANiYmKwfPlyAG+WcpH+6tSpEwoXLoywsDBs2rRJdhwiIiKDEhMTo53Foywpo/cpz82ePXty9Gq63t7eUKlUUKlUCAkJSXO7f//9V7vdhg0bciwfERkfFp6IcsD69evx/PlzFCtWDB06dJAdhzKgVqsxevRoAG/6chEREVHmHDlyBLGxsShSpAgqVaokO47eaty4MXLlyoX79+8jKCgox4576dIlAMkXVSlevHia2128eFH7dzc3N13HIiIjxsITkY4JIbTFizFjxsDc3FxyIsqM4cOHw9LSEmfOnMGZM2dkxyEiIjIYKZfZqVQqyWn0V65cudC0aVMAObvcTikoVa1aNVPb2djYoEyZMroNRURGjYUnIh07deoUAgMDYWVlhaFDh8qOQ5lUoEAB9OjRAwAwf/58yWmIiIgMgxBC21icy+wylnK5XU65fPkyAMDd3T3d7QIDAwEAlStXhpkZvzYS0cfjbxAiHVuwYAEAoE+fPsiXL5/kNPQhlCbjGzZswNOnTyWnISIi0n/BwcG4c+cO1Gq1djYPpa1NmzYAkgcqX716pfPjhYaG4sWLFwAyLjwpS/IymhlFRJQRFp6IdOjp06fw9fUFkLzMjgxLrVq1UKNGDcTHx8Pb21t2HCIiIr23b98+AED9+vVhZ2cnOY3+K1myJFxdXZGYmKhtyK5LSjEJSL+g9PDhQ4SFhWW4HRFRZrDwRKRD3t7eiI+PR/Xq1VGjRg3ZcegjjBw5EgCwdOlSaDQayWmIiIj0m5+fHwCgVatWkpMYDuW5Up47XVL6NllbW6NcuXJpbqcsswPeLzw9ePAAc+fORcuWLVGsWDFYWlqiUKFC6NatG/755x9dxCYiA8fCE5GOaDQaLF26FAAwatQoyWnoY/Xq1Qt58uRBcHBwjoxEEhERGaq4uDj4+/sDYOHpQ6QsPAkhdHosZcZT5cqVYWFhkeZ2SoHKzMwMVapUeeu2efPmYdKkSbh9+zZatGiBKVOmoH79+ti+fTs8PDywceNGneUnIsPEwhORjhw+fBjBwcHIkycPevXqJTsOfSQ7Ozv0798fALB48WLJaYiIiPTXiRMnEB0djYIFC75XrKC0NW7cGGq1Gnfv3sXNmzd1eqzMXtFOmfHk6uoKGxubt26rVasWjh07huDgYKxYsQLTp0+Hr68v/P39YW5ujtGjRyMuLk4X8YnIQLHwRKQjSpGif//+7HFg4JTldtu3b8ejR48kpyEiItJPylKxli1b8ipoH8DOzg7169cHoNvldpGRkbh9+zaAjAtPZ8+eTXO7rl27okGDBu/9f4MGDdCkSRO8ePECV65cyXJeIjIe/EQg0oFHjx5h27ZtAN4ULchwVa5cGR4eHkhMTISnp6fsOERERHqJ/Z0+Xk70ebp06ZJ2KV/p0qXT3C4oKAihoaEAADc3tw86hlqtBoB0l/ERkelh4YlIBzw9PZGUlAQPDw9UrlxZdhzKBkqfrqVLlyIpKUlyGiIiIv3y+PFjbf+gFi1aSE5jeJTCk7+/v86WqaW8ol1UVFSa2/3xxx/av3/IFe1CQ0Nx8OBBFCpUiOe/RPQWFp6IsllSUhKbihuhTz/9FI6OjggNDc2Rq84QEREZkv379wMAqlWrhgIFCkhOY3iqVKmCggULIjo6GidOnNDJMVIWnjZv3pzqNosXL8bq1au1/65YsWKm9p2QkID+/fsjLi4Of/zxB8zNzbMWloiMCgtPRNls3759CA0NhaOjIz799FPZcSib5MqVC4MGDQLAJuNERETvUgZlWrduLTmJYTIzM9P5cruUV6rz8fFB9+7dsWHDBhw+fBgrV65E27ZtMXr0aO1yOQDYsmUL1q9fn+5+NRoNhgwZgmPHjmH48OHai7IQESlUQtfX7CQyMR07dsTOnTsxadIkzJ49W3Ycykb//fcfypUrBzMzM9y9exdFixaVHYmIiEg6jUaDggUL4tmzZzh69CgaNmwoO5JB8vHxQd++feHm5qYtEmUXjUaD3LlzIzo6GuPGjYOPjw9evHjx3nbt2rWDq6sr5s6dq/2/oUOHYvny5anuVwiBYcOGwdPTE/369cPKlSvZWJ6I3sPfCkTZKDQ0FLt37wbApuLGqGzZsmjSpAk0Gk2aJ2BERESmJjAwEM+ePYOdnR3q1q0rO47BatGiBVQqFS5duoTHjx9n675v3LiB6OhoAEDz5s0REBCATp06IW/evLC1tUW1atXw999/Y8eOHZg4cSKqVq0KW1tbNGzYEEOHDk11nxqNBkOHDoWnpyd69+4Nb29vFp2IKFX8zUCUjVasWAGNRoMmTZqgbNmysuOQDih9u5YvX47ExETJaYiIiOQ7cOAAAKBp06ZvLdOiD5M/f364u7sDAA4ePJit+07Z36lKlSooX748tm3bhvDwcERGRuL8+fMYP348zMzMULx4cQQGBiIyMhJHjx5NtZio0WgwbNgweHl5oWfPnli9ejX7OhFRmlh4IsomSUlJ8PT0BAAMHz5cchrSlc6dOyNfvnx4+PAh9u3bJzsOERGRdEqRpHnz5pKTGD7lOczuwpOydC9PnjwoUaJElvalzHTy8vJC9+7dsWbNGhadiChdFrIDEBkLPz8/3L9/H46OjujSpYvsOKQjlpaWGDhwIGbNmoVly5ahffv2siMRERFJExMTg4CAAADJS8Uoa1q0aIE//vgDBw4cgBACKpUqW/arzHiqVKlSlvf5008/wdvbG3Z2dihTpgx++eWX97bp3LkzqlatmqXjEJHxYOGJKJsoPX/69+8Pa2tryWlIl4YOHYpZs2Zh9+7dePjwIVxcXGRHIiIikuL48eOIi4tDkSJF2GYgG9SvXx/W1tZ4+PAh/v33X1SoUCFb9qvMeKpSpUqW93X37l0AQGRkJH799ddUtylRogQLT0SkxaV2RNng8ePH2LlzJwBg2LBhktOQrpUvXx7169dHUlISVq5cKTsOERGRNMqSMKUxNmWNtbU1GjRoACD7lts9ffoUjx49ApA9hSdvb28IIdL9M2jQoCwfh4iMBwtPRNlg1apVSExMRJ06dVCpUiXZcSgHKAVGpaE8ERGRKVIai7O/U/ZRnkvluc2qdxuLExHlNJUQQsgOQWTIhBAoW7Ysbt68ieXLl6d5yVkyLlFRUXBxcUFERAQOHz6MJk2ayI5ERESUo8LCwlCwYEEAwJMnT1CgQAHJiYxDYGAgqlWrBjs7O7x48YJXCiQig8cZT0RZdOzYMdy8eRN2dnbo2bOn7DiUQ2xtbdGnTx8AwLJlyySnISIiynmHDx8GALi5ubHolI3c3NyQL18+REZG4p9//pEdh4goy1h4Isoipal47969YWdnJzkN5aThw4cDADZv3oznz59LTkNERJSzuMxON8zMzNCsWTMA2bfcjohIJhaeiLIgPDwcvr6+ANhU3BRVq1YN7u7uiI+Px9q1a2XHISIiyjFCCG1RpEWLFpLTGB/lOWXhiYiMAQtPRFng4+OD2NhYVK5cGTVr1pQdhyRQCo7Lli0DW+YREZGpuHnzJu7duwdLS0vtVdgo+yiFpzNnzuDVq1eS0xARZQ0LT0QfSQih7e0zbNgwXkLYRPXp0wfW1tYICgrCmTNnZMchIiLKEcpMnPr168PGxkZyGuNTrFgxlClTBklJSThy5IjsOEREWcLCE9FHOn/+PC5dugQrKyv069dPdhySJG/evOjevTuAN/2+iIiIjN3BgwcBsL+TLinPrfJcExEZKhaeiD6Sp6cnAKBr165wdHSUnIZkUpbbrV+/HlFRUZLTEBER6VbKWThKE2zKfspze+jQIclJiIiyhoUnoo8QExMDHx8fAMCQIUMkpyHZGjRogE8++QSRkZHaZvNERETG6sKFC3j58iXs7e1RvXp12XGMVuPGjaFSqfDvv//i4cOHsuMQEX00Fp6IPsLWrVvx6tUrFCtWDE2bNpUdhyRTqVQYPHgwgDcz4YiIiIyVMgOncePGMDc3l5zGeDk6OqJatWoAgMOHD0tOQ0T08Vh4IvoISnFh8ODBMDPjjxEBAwYMgJmZGY4dO4bg4GDZcYiIiHRGKTxxmZ3ucbkdERkDfmMm+kB3797VjjoNGjRIbhjSG0WKFEGrVq0AAN7e3nLDEBER6UhsbCwCAgIAsPCUE1IWnoQQktMQEX0cFp6IPtDKlSshhECzZs1QokQJ2XFIjyj9vry9vZGUlCQ5DRERUfY7deoUYmNj4ezsjPLly8uOY/Tq168PS0tL3Lt3jzOqichgsfBE9AE0Gg28vLwAsKk4va9Dhw5wcnLCgwcPcODAAdlxiIiIsp2y5Ktp06ZQqVSS0xg/Gxsb1K1bFwCX2xGR4WLhiegD+Pv7IyQkBPb29ujSpYvsOKRnrKys0LdvXwBsMk5ERMaJ/Z1ynqw+T0II3L59GytWrEC/fv1QsmRJuLi4oGTJkihXrhzc3NwwYMAA+Pr64vXr1zmajYgMi0pwsTBRpvXt2xc+Pj4YNWoUFi1aJDsO6aFLly6hatWqUKvVePjwIfLlyyc7EhERUbZ49eoVnJyckJSUhJCQEBQrVkx2JJNw8uRJ1KtXD46Ojnj69KnOL2yTkJAAT09P/PHHH7h9+3am7mNpaYkmTZpg+PDh6Nq1K2fDEdFbWHgiyqTw8HA4OzsjLi4OZ86cQc2aNWVHIj1VvXp1XLhwAX/99RcmTJggOw4REVG22LFjBzp16gRXV1fcuHFDdhyTkZCQACcnJ7x+/Rrnz59HtWrVdHIcjUaDDRs24LvvvtP2k7KwsEDt2rXRpEkTNG7cGE5OToiLi0NcXBxev34Nf39/bN++/a3+Uw0bNsRff/2FqlWr6iQnERkeLrUjyqT169cjLi4OlSpVQo0aNWTHIT2m9P/icjsiIjImXGYnh1qtRqNGjQDobrnd5cuXUb16dfTp0wfBwcEoUKAA/v77b7x8+RIBAQH4+eef0axZM1StWhW1a9dGw4YN0a5dO/z555+4ceMG/v33X3z99dfIlSsXjh07hurVq2PkyJF4+vSpTvISkWFh4Ykok1I2Fef0YUpP7969YWVlhUuXLiEwMFB2HCIiomxx+PBhAMmNxSln6bLP0/r161G3bl1cvHgRefLkwS+//IJbt25h/PjxsLW1zfD+KpUK5cqVwy+//ILr16+jV69e0Gg0WLp0KapWrYoLFy5ke2YiMiwsPBFlQlBQEM6ePQsLCwv069dPdhzSc46Ojtrm8ytWrJCchoiIKOuePHmCoKAgAECTJk0kpzE9SrEvICAA8fHx2bLPxMRE/O9//0Pv3r0RHR2NFi1aIDg4GF9//TXs7Ow+ap/FihXDunXrcPz4cZQvXx4PHz5EgwYNsG3btmzJTESGiYUnokzw9vYGALRv3x758+eXG4YMwuDBgwEA69atQ1xcnOQ0REREWXPkyBEAgJubGy+cIUGlSpWQL18+REVF4ezZs1ne38uXL9GmTRv8+eefAICpU6di79692XaeW79+fZw6dQqtWrVCdHQ0unbtij/++ANsL0xkmlh4IspAQkICVq9eDeBNMYEoI82aNUPhwoXx4sUL7Ny5U3YcIiKiLOEyO7nMzMy0M82U1+JjvX79Gm3atMHBgwdha2uLjRs3YsaMGTA3N8+OqFr29vbYtWsXxo4dCyEEpk6dijFjxrD4RGSCWHgiysC+ffsQFhaGAgUKoE2bNrLjkIEwNzfHgAEDALyZMUdERGSoWHiST3nus1J4iomJQadOnXD69Gk4ODggICAA3bt3z66I77GwsMD8+fMxb948mJmZYfHixZg2bZrOjkdE+omFJ6IMKEWDfv36Qa1Wyw1DBmXQoEEAkouXjx49khuGiIjoI4WGhiI4OBjm5uZo2LCh7DgmSyk8nTx5EjExMR98//j4eHTv3h3+/v7InTs3/Pz8ULVq1WxOmbpx48Zh2bJlAIDff/8dc+bMyZHjEpF+YOGJKB3Pnj3TLpMaOHCg5DRkaMqUKQMPDw8kJSVhzZo1suMQERF9FH9/fwBAjRo1kCdPHslpTJerqysKFy6M+Ph4nDx58oPum5SUhP79+2P37t3IlSsXdu3ahZo1a+ooaeqGDBmC6dOnAwAmT57McyMiE8LCE1E6fHx8kJCQgGrVqqFKlSqy45ABUmY9eXt7s6cBEREZJC6z0w8qleqjl9tNmzYNGzduhFqtxtatW6XNXJs6dSomTpwIILl36t69e6XkIKKcxcITUTqUZXZsKk4fq0ePHrC2tsa1a9dw7tw52XGIiIg+iBCChSc98jGFp61bt2LmzJkAgNWrV6NVq1Y6yZYZKpUKs2bNQt++fZGYmIiePXsiODhYWh4iyhksPBGl4dKlSwgMDIRarUbv3r1lxyEDZW9vj65duwIAvLy8JKchIiL6MMHBwbh//z4sLS3h4eEhO47JU65sd/bsWURERGS4/c2bN7WzrydPnoyePXvqMl6mmJmZwdPTEw0aNMDr16/Ro0cPxMbGyo5FRDrEwhNRGlauXAkA6NixI5ycnCSnIUOmzJhbt24dT6yIiMigKDNr6tatCxsbG8lpqHjx4ihdujSSkpJw/PjxdLeNjo5Gt27dEBERgfr162PGjBk5lDJjlpaW8PHxgZOTEwIDAzFlyhTZkYhIh1h4IkpFQkKCtuEhl9lRVjVp0gRFixbFy5cvsWPHDtlxiIiIMo3L7PRPZpbbCSEwatQoXLlyBQULFsSGDRv07urMRYoUwerVqwEACxcuhK+vr+RERKQrLDwRpWLPnj14+vQpChYsKHUdPBkHc3NzDBgwAACX2xERkeEQQmivaMfCk/5QXgvltUnNypUrsXr1apiZmWH9+vVwcXHJqXgfpE2bNpg6dSoAYOjQobh165bkRESkCyw8EaVCaSrev39/WFhYyA1DRmHgwIEAgP379+PBgweS0xAREWXs6tWrePr0KWxsbFCrVi3Zcej/KX2eAgMD8fz58/duf/DggfbKcT///DMaN26cg+k+3M8//wwPDw9ERESgV69eSExMlB2JiLIZC09E73j69Cl27doFANpmjERZ5erqivr160Oj0WinlRMREekzZSlX/fr1YWlpKTkNKQoWLIgKFSoAAI4ePfrWbcoSu1evXqFmzZr44osvZET8IGq1GuvXr0fevHlx7tw5zJ07V3YkIspmLDwRvcPHxweJiYmoUaMGKlasKDsOGRFl1tPKlSshhJCchoiIKH1cZqe/0lpu5+Pjg127dkGtVsPLy8tgZu4XLVoUs2bNAgB89913XHJHZGRYeCJ6h3I1O852ouzWvXt35MqVC9evX8fZs2dlxyEiIkpTUlISjhw5AuDN0i7SH8prkrLB+JMnTzBhwgQAycUbQxtAHTx4MJo2bYqYmBiMHDmSg3RERoSFJ6IULl++jMDAQFhaWqJXr16y45CRsbe3R9euXQG86SNGRESkjy5duoSXL18iT548qFatmuw49I5GjRpBpVLh2rVrePLkCQBg3LhxePHiBapWrapt2G1IVCoVlixZAmtraxw6dEg7GExEho+FJ6IUlA+4Dh06wMnJSXIaMkbKcrv169cjNjZWchoiIqLUKTNpGjZsaDDLtUyJk5MT3NzcACQvt9u6dSt8fX1hYWEBLy8vqNVqyQk/zieffIIff/wRADB58mRtUY2IDBsLT0T/LyEhAWvWrAHAZXakO02bNkWRIkUQHh6OnTt3yo5DRESUKvZ30n/Ka3PgwAFMmjQJAPDFF1+gatWqElNl3eTJk+Hu7o7w8HB89tlnsuMQUTZg4Yno//n5+SEsLAwFChRAq1atZMchI2Vubo7+/fsDAKeQExGRXkpISMCxY8cAsL+TPlNem23btiEkJARFihTB119/LTlV1llYWGD58uUwMzPDhg0btL3GiMhwsfBE9P+Unjv9+vUz2OnJZBiU5Xb79u3D48ePJachIiJ62/nz5xEZGQlHR0dUqVJFdhxKQ8OGDWFmZoYXL14AAGbOnAkbGxvJqbJHtWrVMHLkSADJM6CSkpIkJyKirGDhiQjA8+fPtcuelKIAka6ULVsWderUQVJSEtauXSs7DhER0VuU/k6NGzeGmRm/LuirPHnywMHBAQBQpkwZ9OzZU3Ki7PXjjz/C3t4egYGBWLVqlew4RJQF/CQhQnKj5/j4eLi7u3Nkj3KE0kfM29ublwsmIiK9wv5OhuHEiRN4/vw5gORBLZVKJTlR9sqfPz+++eYbAMBXX32FyMhIyYmI6GOx8ESEN712ONuJckrPnj1hZWWFoKAgBAYGyo5DREQEAIiLi0NAQAAA9nfSZxqN5q3G25cuXTLKgazx48ejdOnSePz4MWbMmCE7DhF9JBaeyORdu3YNZ8+ehYWFBfr06SM7DpmIvHnzonPnzgDe9BcjIiKS7Z9//kFsbCwKFiyI8uXLy45DaVi5ciXOnz+PPHnywMLCAqGhobh9+7bsWNnOysoKM2fOBADMmjULoaGhkhMR0cdg4YlMnjLbqV27dsifP7/kNGRKlBl2Pj4+iI+Pl5yGiIjoTX+nJk2aGN3SLWMRGxuL77//HgDw7bffok6dOgDeLJE0Np07d0ajRo0QGxuLL7/8UnYcIvoILDyRSUtMTMTq1asBvOm5Q5RTWrRoAWdnZzx//hy7d++WHYeIiIj9nQzAkiVLcO/ePRQpUgTjxo3TvlbGWnhSqVSYM2cOVCoV1q1bxxYFRAaIhScyaQcPHsSjR4/g5OSEtm3byo5DJsbCwgL9+vUD8GbmHRERkSzR0dE4deoUAPZ30leRkZH49ddfAQDfffcdrK2tta/V4cOHjbLPEwC4u7ujd+/eAJIfNxEZFhaeyKQpvXX69u0LS0tLuWHIJCnL7Xbv3o2nT59KTkNERKbs5MmTSEhIQNGiRVG6dGnZcSgVc+fOxdOnT/HJJ59oZ+vXqVMH1tbWePz4Mf777z+5AXXo+++/h7m5OXbt2oXTp0/LjkNEH4CFJzJZL1++xLZt2wDwanYkT8WKFVGzZk0kJibCx8dHdhwiIjJh7O+k3168eKFttP3TTz9BrVYDAKytreHh4QHgzWtojMqUKaM9Z//2228lpyGiD8HCE5msDRs2IC4uDpUrV4a7u7vsOGTClJMoXt2OiIhkYn8n/fbHH38gIiICVapUQc+ePd+6zdj7PCm+/fZbqNVqHDx4EEeOHJEdh4gyiYUnMlnKl/yBAwdyVI+k6tWrFywtLXHx4kVcunRJdhwiIjJBr1+/xtmzZwGwv5M+evToEf7++28AwK+//gozs7e/ximvmb+/PzQaTY7nyyklSpTA8OHDASQXoYy1pxWRsWHhiUzSf//9h9OnT8Pc3Bx9+/aVHYdMnJOTEzp06ACATcaJiEiO48ePIykpCaVLl0axYsVkx6F3TJ8+HTExMahbty7atWv33u01a9aEra0tnj9/jqCgIAkJc87XX38Na2trBAQEYP/+/bLjEFEmsPBEJkn5ct+6dWsUKlRIchoiaBuErlmzBgkJCXLDEBGRyVGWaHG2k/558uQJli1bBiC5t1NqM/XVajUaNGgAwPiX27m4uGDs2LEAgG+++YaznogMAAtPZHKSkpKwevVqAG++7BPJ1qpVKxQsWBBPnz7Fvn37ZMchIiITozSlZn8n/TN79mzExsaiTp06aNasWZrbKa+dMTcYV0ydOhW2trY4d+4c/Pz8ZMchogyw8EQm5/Dhw7h//z4cHBy0y5uIZFOr1dpln2wyTkREOSk8PByBgYEAgMaNG8sNQ295/vw5Fi5cCCB5dk96fUmV2WpHjx5FUlJSjuSTJX/+/Bg1ahQA4LfffpOchogywsITmRxlmV3v3r1hZWUlOQ3RG8rV7Xbu3Innz59LTkNERKbi6NGjEEKgfPnycHZ2lh2HUvj7778RGRmJqlWrom3btulu6+7uDnt7e7x69UpbSDRmkydPhqWlJY4fP47jx4/LjkNE6WDhiUzKq1evsGXLFgBvvuQT6YsqVarA3d0dCQkJWLdunew4RERkItjfST9FRERor2T39ddfZ3gVZnNzczRq1AiAaSy3c3FxweDBgwFw1hORvmPhiUzKpk2bEBMTg/Lly6NmzZqy4xC9R+k7xuV2RESUU5QiBQtP+mXhwoV4+fIlypcvj65du2bqPkqfJ2NvMK744osvYGZmhn379uHChQuy4xBRGlh4IpOiLLMbNGhQhqNGRDL06dMHarUa58+fN/rLIRMRkXxPnjzRft6w8KQ/oqKiMGvWLADAV199BTOzzH1tUwpPx48fR3x8vM7y6YtSpUqhd+/eADjriUifsfBEJiM4OBgBAQEwMzNDv379ZMchSlW+fPnQvn17AG8KpURERLpy5MgRAEDVqlXh5OQkNwxpLV++HM+ePUOpUqXQq1evTN+vYsWKyJ8/P6KionD27FkdJtQfX375JQBgy5Yt+PfffyWnIaLUsPBEJkNZutSqVSu4uLjIDUOUDmW53erVq5GQkCA3DBERGTVlmZ0yU4bkS0hIwOzZswEAU6dOhYWFRabva2Zmpp25Zgp9ngCgUqVK6Ny5M4QQ+P3332XHIaJUsPBEJiEpKemtZXZE+qxNmzbInz8/njx5Aj8/P9lxiIjIiLHwpH98fX0RGhqKAgUKYMCAAR98f+W1NJXCEwBMmzYNALBmzRqEhoZKTkNE72LhiUyCv78/7t+/j7x586Jjx46y4xClS61Wa5eDssk4ERHpSmhoKIKDg2Fubo4GDRrIjkMAhBCYOXMmAGD8+PGwtrb+4H0oM55OnjyJmJiYbM2nr2rVqoUmTZogKSlJeyVAItIfLDyRSVC+vPfu3fujPsCJcpoyM2/Hjh14/vy53DBERGSUlCuf1ahRA3ny5JGchoDk1yQwMBA2NjYYPXr0R+3D1dUVhQsXRnx8PE6ePJnNCfXXlClTAADLli1DRESE5DRElBILT2T0Xr16hc2bNwMABg8eLDkNUeZUqVIF1apVQ0JCAnx8fGTHISIiI8Rldvrnzz//BAAMGTLko5u9q1Qqk1xu16ZNG5QrVw4RERFYvny57DhElAILT2T0Nm7ciNjYWFSoUAE1atSQHYco05RZT1xuR0RE2U0IwcKTngkKCsLevXthZmaGiRMnZmlfplh4MjMz0856+uuvv3iBFiI9wsITGT3lS/ugQYOgUqnkhiH6AL1794ZarcaFCxdw+fJl2XGIiMiIBAcH4/79+7C0tISHh4fsOARg1qxZAICuXbuidOnSWdqX0ufp7NmzJrXsrF+/fihQoABCQ0Ph6+srOw4R/T8Wnsio3bhxAydPnoSZmZm2WTORociXLx86dOgAANqrMhIREWUHZSZM3bp1YWNjIzkNPXz4EGvXrgUAfP7551neX/HixVG6dGkkJSXh+PHjWd6fobC2tsbYsWMBJBfyhBCSExERwMITGTnly3rr1q3h7OwsOQ3Rh1OW261Zs4ZTxomIKNtwmZ1+mTdvHhISEtCgQQPUrl07W/ZpisvtAGD06NGwtrbG+fPncezYMdlxiAgsPJERS0pK0hae2FScDFXr1q1RsGBBhIWFYe/evbLjEBGREdBoNNor2rHwJF90dDSWLFkCAJg8eXK27ddUC0/58+fXDtwpyxeJSC4WnshoHTp0CA8ePICDg4N2uRKRoVGr1dplomwyTkRE2SEoKAhPnz6FjY0NatWqJTuOyVuzZg3Cw8NRsmTJbD1nVfo8Xbx4Ec+ePcu2/RqCSZMmQaVSYefOnbhx44bsOEQmj4UnMlpeXl4Akhs0W1lZSU5D9PGUUbudO3fi6dOncsMQEZHBO3ToEACgYcOGsLS0lJzGtAkh8PfffwMAxo8fD3Nz82zbd8GCBVGpUiUA0M5wMxVlypRBu3btAAALFiyQnIaIWHgioxQeHo6tW7cCAIYMGSI5DVHWVKpUCTVq1EBiYqK28SgREdHHUgpPzZo1k5yEDh06hKtXr8LOzk4n56zKa6y85qZkwoQJAJIHo03pyn5E+oiFJzJK69atQ1xcHCpXroxq1arJjkOUZUqfMk9PT16hhYiIPlpCQgKOHj0KgIUnffDXX38BSJ7dbG9vn+37N+XCU/PmzVG+fHm8fv2a7QqIJGPhiYySssxuyJAhUKlUktMQZZ2yZPTKlSu4cOGC7DhERGSgzp07h8jISDg5OcHNzU12HJMWHByM3bt3A0heZqcLjRo1grm5OYKDgxEaGqqTY+grlUqlfV7nzZsHjUYjORGR6WLhiYzO5cuXce7cOajVavTt21d2HKJs4eDggK5duwJInvVERET0MZSZL02aNIGZGb8KyDRv3jwIIdC2bVuUKVNGJ8fIkycPatasCcD0rm4HAP3794e9vT2Cg4Oxb98+2XGITBY/bcjoKLOdOnTogPz580tOQ5R9lOV2Pj4+iI2NlZyGiIgMEfs76YeIiAjtOavSi0hXTHm5nZ2dHYYOHQoA2ibuRJTzWHgioxIfH481a9YAYFNxMj5NmzZFsWLF8PLlS2zbtk12HCIiMjDR0dE4efIkgOTPFJLHy8sLr1+/Rrly5dCyZUudHkt5rQ8dOmSSfSLHjh0LlUoFPz8/XL9+XXYcIpPEwhMZlV27duHZs2dwdnZGq1atZMchylbm5uYYOHAgAC63IyKiD3fixAnEx8ejSJEicHV1lR3HZGk0GixYsABA8mwnXfcj9fDwgLW1NR49emSShZdSpUqhY8eOAID58+dLTkNkmlh4IqOiTFkeMGAALCwsJKchyn6DBg0CABw8eNDkmoQSEVHWpFxmx4uvyHPw4EHcvHkTefLkQf/+/XV+PGtra9SrVw+AaS63A94sZ/T29sarV68kpyEyPSw8kdF4+PAh9uzZA+BNLxwiY1OqVCk0btwYQgisXLlSdhwiIjIg7O+kH5TZTgMHDoSdnV2OHNOU+zwByc30K1asiKioKKxatUp2HCKTw8ITGY3Vq1dDo9HAw8MDZcuWlR2HSGeU/mVeXl68NDAREWVKeHg4zp8/D4CFJ5lCQkKwa9cuAMCYMWNy7LjKa37kyBEkJSXl2HH1hUql0j7fCxcuNMleV0QysfBERkEIoV1mx6biZOy6deuG3Llz486dOzh27JjsOEREZACOHDkCIQTKlSsHFxcX2XFM1uLFi6HRaNCsWTOUK1cux45bvXp12Nvb4+XLl9oCpKnp168f7OzscP36dRw+fFh2HCKTwsITGYVTp07hv//+g42NDXr06CE7DpFO2djYoFevXgDYZJyIiDLn4MGDAIAWLVpITmK6YmNjsXz5cgDJV1rLSebm5m9d3c4U5cmTBwMGDACQPOuJiHIOC09kFJQv359++ily584tOQ2R7ikz+3x9fdkkk4iIMqQUnpo3by45ienatGkTnj17hqJFi6JDhw45fnzltVfeC6ZIWW63fft23L9/X3IaItPBwhMZvNevX2P9+vUAgGHDhklOQ5QzateujQoVKiAmJgbr1q2THYeIiPRYaGgobty4AXNzczRq1Eh2HJOlNBUfOXKklKsvK4WngIAAREdH5/jx9UHFihXRqFEjJCUlYcmSJbLjEJkMFp7I4G3YsAFRUVEoU6YM6tevLzsOUY5QqVTaQqsybZ+IiCg1ygyXWrVqwd7eXnIa03Tu3Dn8888/UKvV0gZKXV1dUbRoUcTHxyMgIEBKBn2gzHpatmwZ4uPjJachMg0sPJHBW7FiBQBg6NChUKlUktMQ5Zz+/ftDrVbj/PnzuHjxouw4RESkp7jMTj6lp1D37t1RsGBBKRlUKhWX2wHo0qULChUqhCdPnmDLli2y4xCZBBaeyKBdvXoVp0+fhoWFhbZZIJGpyJcvHzp37gzgTQGWiIgoJY1Gw8KTZOHh4dq2EDndVPxdLDwBarUaI0aMAMAm40Q5hYUnMmjKl+327dujUKFCktMQ5byhQ4cCANasWYOYmBjJaYiISN9cuXIFT58+ha2tLerUqSM7jklauXIlYmJiUKVKFdStW1dqlmbNmgEAAgMD8fTpU6lZZBoxYgTMzc1x/PhxXLlyRXYcIqPHwhMZrLi4OKxatQoAm4qT6WrevDmKFSuGly9fYuvWrbLjEBGRnlFmtjRq1AiWlpaS05geIQQWL14MABg9erT0thAFCxZElSpVAACHDx+WmkWmwoULa2eNK68PEekOC09ksHbs2IHnz5+jcOHCaNWqlew4RFKYm5tjyJAhALjcjoiI3sdldnL5+/vjv//+g52dHfr27Ss7DgAut1OMGjUKALB69WpERkZKTkNk3Fh4IoOlXMlr0KBBUi5JS6QvBg8eDJVKhcOHD+PWrVuy4xARkZ6Ii4vDsWPHALDwJIsym6Zfv37InTu35DTJlPfCgQMHIISQnEaepk2bwtXVFa9fv4aPj4/sOERGjYUnMkghISE4cOAAAGhnexCZqmLFiqFly5YAAC8vL8lpiIhIX5w6dQrR0dEoWLAgKlWqJDuOyXn06JF2Gfzo0aMlp3mjYcOGUKvVCAkJMekBKzMzM4wcORJAcoHQlItwRLrGwhMZJC8vLwgh0LRpU5QqVUp2HCLplCbjXl5eSExMlJyGiIj0gbKUqlmzZtJ7C5miFStWIDExER4eHtq+SvrA1tYWHh4eAKAdyDVVgwYNgpWVFQIDA3H27FnZcYiMFgtPZHCSkpK0szrYVJwoWceOHZEvXz48fPgQfn5+suMQEZEe2L9/PwBoZ8VSzklKSsLSpUsB6NdsJ0WLFi0AsPDk5OSEHj16AAAWLVokOQ2R8WLhiQzOwYMHERoaCgcHB3Tp0kV2HCK9YGVlhQEDBgAAli1bJjkNERHJ9vz5c5w7dw7AmyID5Zzdu3fj3r17cHJywqeffio7znuUYuShQ4dMfqa0Uhhcv349wsPDJachMk4sPJHBUUaP+vXrB2tra8lpiPSHMgNw165dePjwoeQ0REQk06FDhyCEQKVKleDi4iI7jslRmooPHjxYL89Xq1WrBkdHR0RERODMmTOy40hVp04dVKlSBbGxsVi1apXsOERGiYUnMiiPHj3C9u3bAUDbDJCIkpUvXx7169dHUlISPD09ZcchIiKJuMxOnrt372Lfvn0A9Pd81dzcXHt1O+W9YqpUKpV21hObjBPpBgtPZFC8vLyQlJSEevXqoWLFirLjEOkd5QR3+fLlSEpKkpyGiIhkEEJo+/2x8JTzli1bBiEEmjdvjk8++UR2nDQp7w32hgT69u0LOzs7XL9+HUePHpUdh8josPBEBkOj0Wh714wYMUJyGiL91K1bNzg4OCAkJMTkG4YSEZmq69ev4/79+7CyskKDBg1kxzEpCQkJWLFiBQBg1KhRktOkTyk8nTlzxuR7G+XOnRt9+/YFACxZskRyGiLjw8ITGYyDBw/i7t27yJs3L7p37y47DpFeypUrl7bJuNIPjYiITIuydKphw4awsbGRnMa0bN++HU+ePEGhQoXQsWNH2XHSVbRoUZQvXx4ajQaHDx+WHUc6Zdb45s2bERYWJjkNkXFh4YkMhjL60L9/f+TKlUtyGiL9NXz4cADAjh072GSciMgEsb+TPMr56tChQ6FWqyWnyZjyHjH1Pk8A4O7ujpo1ayIhIQHe3t6y4xAZFRaeyCA8evQIO3bsAMBldkQZqVixIurVq4ekpCR4eXnJjkNERDkoLi4OR44cAcDCU04LDg7GwYMHoVKptINA+i5lnyc21X6zPHLp0qXQaDSS0xAZDxaeyCB4e3sjMTERHh4eqFSpkuw4RHpPmS6+bNkynjgREZmQkydPIjo6GgULFkTlypVlxzEpyhL31q1bo3jx4pLTZE6jRo2gVqsREhKC4OBg2XGk69mzJ/LkyYNbt27h0KFDsuMQGQ0Wnkjvsak40Yf79NNPkTdvXjYZJyIyMSmX2alUKslpTEdcXJx2lrG+NxVPydbWFvXr1wfA5XZA8vPRv39/AGwyTpSdWHgivXfw4EHcuXMH9vb2bCpOlEkpm4zzxImIyHT4+fkB4DK7nLZ161Y8e/YMhQsXRtu2bWXH+SApl9vRm1nj27dvx6NHjySnITIOLDyR3lOmLffv359XZiH6AMoMwR07dvDEiYjIBDx+/BiBgYEAgBYtWkhOY1oWL14MIPkCHxYWFpLTfJhWrVoBAA4fPoy4uDjJaeSrXLkyPDw8kJiYCE9PT9lxiIwCC0+k1x4/fozt27cD4DI7og+Vssk4T5yIiIyfslSqevXqKFiwoOQ0puP69es4evQozMzMMHToUNlxPljVqlVRqFAhREVF4cSJE7Lj6IWUvTKTkpIkpyEyfCw8kV5bvnw5EhMTUbduXTbIJPoISsF26dKlPHEiIjJye/fuBZDc3JpyjjI7v3379ihSpIjkNB9OpVJpZz0p7yFT1717dzg4OCAkJIRLEImyAQtPpLcSExO1H+RjxoyRnIbIMPXo0QOOjo4IDQ3Fnj17ZMchIiIdSUpK0s54YuEp58TExGDlypUA3sySMUTKe2bfvn2Sk+iHXLlyYeDAgQDYK5MoO7DwRHpr9+7duHfvHvLly4dPP/1Udhwig2RtbY0hQ4YAABYuXCg5DRER6cq5c+fw4sUL2Nvbo06dOrLjmAxfX1+8ePECxYsX184aMkQtWrSAmZkZgoKCcP/+fdlx9IIya3zXrl18ToiyiIUn0luLFi0CAAwZMgTW1taS0xAZLuWyzn5+frh165bkNEREpAvKEqkWLVoYXHNrQ6bMhhk+fDjMzc0lp/l4Tk5OqFWrFgDOelKUL18eDRs2hEajwYoVK2THITJoLDyRXgoODoafnx9UKpVBT1sm0gelS5dG69atIYTgdHEiIiOlFAvatGkjOYnpCAoKwokTJ2BhYaGdXWzIlPcOC09vKN9DlL6zRPRxWHgivaRckrZNmzYoVaqU5DREhm/06NEAAE9PT8TGxkpOQ0RE2en58+c4c+YMABj0ci9DowzmdOzYEc7OzpLTZJ3S5+nAgQNISEiQnEY/dOvWDfny5cP9+/fZK5MoC1h4Ir0TExMDLy8vAG++LBNR1rRr1w7FihXD8+fPsWnTJtlxiIgoG+3fvx9CCFSpUgWFCxeWHcckREdHY/Xq1QDeLGk3dDVq1EC+fPkQERGB06dPy46jF6ysrDBo0CAAbDJOlBUsPJHe2bhxo7ZJI6eLE2UPc3NzbZNMNhknIjIuytIoXs0u52zYsAGvXr1CqVKl0KxZM9lxsoWZmRlatmwJgMvtUlLOn/bu3YuQkBDJaYgMEwtPpHeUL8WjRo0y6CaNRPpm6NChUKvVOH36NAIDA2XHISKibKDRaODn5weAhaecpMx+GTFiBMzMjOcrlfIeYuHpDVdXVzRr1gxCCCxfvlx2HCKDZDy/JckonD9/HmfOnIFarTaKJo1E+qRQoULo2rUrgDdXjSQiIsMWGBiIJ0+ewM7ODvXq1ZMdxyQEBgbin3/+gVqtxuDBg2XHyVZKj7ALFy7g0aNHktPoj5RNxtn/iujDsfBEekX5Mty9e3cUKFBAchoi4zNmzBgAwNq1a/Hq1SvJaYiIKKt2794NAGjRogUsLS0lpzENymynbt26Gd35aoECBVCzZk0AyUvLKFmnTp1QsGBBPH78GDt27JAdh8jgsPBEeiM8PBw+Pj4A3nw5JqLs1aBBA1SsWBHR0dHw9vaWHYeIiLJo165dAID27dtLTmIaXr9+jbVr1wIwnqbi71LeS8p7iwBLS0vtagw2GSf6cCw8kd5YsWIFYmJi4ObmBg8PD9lxiIySSqXC2LFjAQDz58+HRqORnIiIiD7WkydPcPbsWQBA27ZtJacxDWvXrkVkZCTKlSuHhg0byo6jE+3atQMAHDhwAHFxcZLT6I8RI0ZApVLhwIEDCA4Olh2HyKCw8ER6ISkpCQsWLAAATJgwASqVSnIiIuPVv39/2NvbIzg4mM1DiYgMmLIUqnr16ihUqJDkNMZPCIHFixcDSJ7tZKznq+7u7ihUqBAiIyNx/Phx2XH0RokSJbRX3F66dKnkNESGhYUn0gu7du3C3bt34eTkhN69e8uOQ2TU7OzstNPF582bJzkNERF9LKW/kzJDhXTrn3/+waVLl2BtbY0BAwbIjqMzZmZm2hl0XG73NmV5paenJ2eDEX0AFp5IL/z9998AgOHDhyNXrlyS0xAZv7Fjx0KlUmHfvn3477//ZMchIqIPFB8fj/379wNg4SmnKLOdevXqBQcHB8lpdEt5TynFTUrWtm1bFC1aFM+fP4evr6/sOEQGg4Unki4oKAiHDx+GmZkZRo8eLTsOkUkoXbq0tnno/PnzJachIqIPFRAQgIiICBQoUAA1atSQHcfohYeHY8OGDQCAkSNHSk6jey1atIBarUZwcDBu3LghO47eMDc3x/DhwwG8KUQSUcZYeCLplKU+Xbp0QbFixSSnITIdEyZMAAB4e3sjIiJCchoiIvoQykyUtm3bwsyMp/S6tmrVKsTGxsLNzQ21a9eWHUfncufOjUaNGgHgcrt3DR06FObm5ggICEBQUJDsOEQGgZ9SJFV4eDhWr14N4M2XYCLKGc2aNUP58uURGRkJb29v2XGIiOgDsL9TzhFCYNGiRQCSZzsZa1Pxd3G5XepcXFzQqVMnAMCSJUskpyEyDCw8kVQrVqxATEwMqlSpggYNGsiOQ2RSVCoVxo8fDyB55qFGo5GciIiIMuPWrVv477//YGFhgRYtWsiOY/T8/f3x33//wc7ODv369ZMdJ8cohadjx45xZvQ7lPYgq1atQlRUlOQ0RPqPhSeSJikpSdtbZsKECSYzekSkT/r37w97e3sEBwdj3759suMQEVEmKEufGjRoAHt7e8lpjJ8y26l///7InTu35DQ5x9XVFa6urkhMTISfn5/sOHqladOm+OSTTxAREQEfHx/ZcYj0HgtPJM3OnTsREhICR0dH9OnTR3YcIpNkZ2eHoUOHAnhzdUkiItJv27dvBwB07NhRchLj9/DhQ2zduhUATPIiOMp7bMeOHZKT6BczMzOMGjUKALBw4UIIISQnItJvLDyRNMqX3OHDhyNXrlyS0xCZrrFjx0KlUsHPzw/Xr1+XHYeIiNIRHh6OY8eOAWDhKScsX74cSUlJqF+/PipXriw7To5T3mO7d+9GYmKi5DT6ZdCgQbC2tsbFixfxzz//yI5DpNdYeCIpLl68CH9/f5ibm2PMmDGy4xCZtFKlSqFDhw4AgLlz58oNQ0RE6dqzZw+SkpJQqVIllCpVSnYco5aYmIilS5cCgMmer3p4eMDJyQnh4eEICAiQHUevODk5oVevXgCSZz0RUdpYeCIpZs+eDQDo3r07ihUrJjkNEU2ePBkAsHLlSjx79kxyGiIiSouyzE65qhbpzs6dO/HgwQPkz58fXbt2lR1HCgsLC7Rv3x4Al9ulRilIbtiwgedPROlg4Yly3IMHD7Bu3ToAwJQpUySnISIAaNiwIapXr47Y2FhtE1UiItIvcXFx2gtBcJmd7imzWIYNGwYrKyvJaeRR3mvbt29nL6N31KxZEzVq1EB8fDw8PT1lxyHSWyw8UY6bN28eEhMT0aBBA9SoUUN2HCICoFKptLOe5s+fj9jYWMmJiIjoXUeOHMHr16/h7OzMcygdu3HjBg4ePAiVSoURI0bIjiNVy5YtYWVlhdu3b+PatWuy4+gdZdbT4sWLkZSUJDkNkX5i4YlyVGRkJJYsWQKAs52I9E337t1RpEgRhIWF8dLARER6SFlm16FDB5iZ8TRelxYvXgwAaNeuHUqUKCE3jGR2dnZo1qwZgDfvQXqjZ8+ecHBwwJ07d+Dn5yc7DpFe4icW5SgvLy+8fPkSrq6u2mbGRKQf1Go1JkyYACC5Dxun0xMR6Q8hhLbHDvs76VZUVBS8vLwAAKNHj5acRj8o7zn2eXqfjY0NBg8eDIBNxonSohL8ZkE5JCkpCWXKlMHt27excOFCfpAT6aGXL1+iaNGiiIyMxN69e9G6dWvZkYiICMD58+dRo0YN2Nra4tmzZ7C2tpYdyWgtW7YMI0aMQKlSpXDz5k3OLgPw8OFDFC5cWPt3Z2dnyYn0y82bN1GmTBmoVCrcunULJUuWlB2JSK/wtyjlmO3bt+P27dtwdHTEwIEDZccholTkzZsXw4YNA/Dm6pNERCSfssSpVatWLDrpkBAC8+fPBwCMHTuWRaf/5+Liglq1agFIvtofvc3V1RUtW7aEEIIXaSFKBX+TUo6ZNWsWgOQpyzY2NpLTEFFaJkyYADMzMxw4cACXL1+WHYeIiABs27YNAK9mp2sBAQG4fPkycuXKpV0+RcmU5XbKe5HeNm7cOADA8uXLER0dLTkNkX5h4YlyxOnTp3Hy5ElYWlpqfykTkX4qWbIkunXrBgCYM2eO5DRERHTz5k1cuXIFFhYW7JGpY8psp379+sHBwUFyGv3StWtXAMDBgwfx8uVLuWH0UNu2bVGiRAmEh4dj3bp1suMQ6RUWnihHKLOd+vbti0KFCklOQ0QZUa46uXbtWjx8+FByGiIi07ZlyxYAQJMmTeDo6Cg5jfF68OCB9rnmQOn7ypUrh/LlyyMhIQG7d++WHUfvmJubY+zYsQCAefPm8SItRCmw8EQ6d+PGDWzevBkAMHnyZMlpiCgzateujfr16yMhIYGznoiIJFOKIcpsVNKNpUuXIjExEQ0bNkSVKlVkx9FLyntQObentw0ZMgS5cuXCpUuXEBAQIDsOkd5g4Yl0bubMmRBCoEOHDqhUqZLsOESUSV9++SUAYPHixQgPD5echojINN27dw9nzpyBSqXS9tih7BcfH48lS5YA4Gyn9CjL7fbt24eoqCjJafSPo6Mj+vbtC+DNsk0iYuGJdOzBgwdYuXIlgDdfYonIMLRt2xaVK1dGZGQkFixYIDsOEZFJ2rp1KwCgfv36bFegQ76+vnjy5AlcXFzQuXNn2XH0VtWqVVGyZEnExMRg3759suPopfHjxwNInhX24MEDyWmI9AMLT6RTc+bMQUJCAho2bAgPDw/ZcYjoA6hUKm3B+K+//uIVWoiIJFCWNCkzTUg3lNkpo0aNglqtlpxGf6lUKu17UVkCSm+rUqUKGjZsiKSkJCxevFh2HCK9oBLsekY68uLFCxQvXhyRkZHYs2cP2rRpIzsSEX2gxMRElClTBnfu3MG8efO4/ICIKAc9efIEzs7OEEIgJCQExYoVkx3JKJ05cwa1a9eGWq1GaGgoZ5Zl4NSpU/Dw8ECePHkQFhYGKysr2ZH0zqZNm9CjRw8UKFAAoaGhfI7I5HHGE+nMggULEBkZCTc3N7Ru3Vp2HCL6CBYWFvjf//4HILlfW0JCguRERESmY/v27RBCoEaNGiw66dBff/0FAOjduzeLTplQu3ZtuLi4ICIiAocOHZIdRy917twZhQsXRlhYGDZs2CA7DpF0LDyRTkRFRWk/xL/88kuoVCrJiYjoYw0aNEg7Yrd+/XrZcYiITAavZqd7Dx8+xMaNGwEAn332meQ0hsHMzAxdunQBwKvbpUWtVmPMmDEAgLlz54KLjMjUsfBEOrFixQo8f/4cpUqVwqeffio7DhFlQa5cuTBp0iQAwIwZM6DRaCQnIiIyfi9evNDOJmF/J91ZuHAhEhMT0aBBA1SrVk12HIOhvCe3bdvG2dBpGDlyJKytrREYGIiAgADZcYikYuGJsl1CQgL+/PNPAMAXX3wBCwsLyYmIKKtGjx6NPHny4Nq1a9i1a5fsOERERm/r1q1ITEyEm5sbypQpIzuOUYqJicGSJUsAcLbTh2rYsCEKFCjwVoGU3ubk5IQBAwYASL7gEpEpY+GJsp2Pjw/u3buHggULYuDAgbLjEFE2sLe3104Znz59OqeMExHpmLK0uWfPnpKTGC8fHx88e/YMxYsXR6dOnWTHMSgWFhbaVQ3sYZQ2paC5bds23L59W3IaInlYeKJslZiYiN9++w0AMGnSJFhbW0tORETZ5bPPPoO1tTVOnz7N0U0iIh0KCwvD4cOHAbDwpCtCCG0/0nHjxnGG/kdQ3ptbt25FXFyc5DT6qUKFCmjZsiWEEJg/f77sOETSsPBE2Wr9+vW4ceMGHB0dtbMjiMg4FCpUCCNGjAAA/Pjjj5z1RESkI5s3b4ZGo0GNGjVQqlQp2XGMkr+/P65cuQJbW1sMHTpUdhyDVL9+fbi4uODVq1fw8/OTHUdvKX0yly9fjoiICMlpiORg4YmyTVJSEn755RcAwOeff47cuXNLTkRE2W3q1KmwsrJCQECAdjSeiIiyl7J0ibOddEeZ7TRw4EA4ODhITmOYzMzM0L17dwBcbpeeli1boly5cnj9+jW8vLxkxyGSQiU4ZE3ZxMfHB3379oWjoyPu3r3LwhORkZowYQLmzZuHBg0a4OjRo1CpVLIjEREZjYcPH6JIkSIQQiAkJATFihWTHcno/PfffyhfvjyEELh+/TrKli0rO5LBOnXqFDw8PGBnZ4ewsDDkypVLdiS9tHjxYowePRqlSpXCjRs3YG5uLjsSUY7ijCfKFklJSfj5558BAJMnT2bRiciITZ06FZaWljh+/DiOHDkiOw4RkVHZtGkThBDw8PBg0UlH5syZAyEEOnbsyKJTFtWpUwfFixdHZGQk9uzZIzuO3urfvz8cHBxw+/Zt7NixQ3YcohzHwhNli02bNuH69etwcHDA+PHjZcchIh0qXLgwhg8fDiC51xMREWUfLrPTrbCwMKxcuRIAMGXKFMlpDJ9KpUKPHj0AcLldemxtbTFq1CgAwJ9//ik5DVHOY+GJsiwpKQk//fQTgOTmeXny5JGciIh07csvv4SlpSWOHj2Ko0ePyo5DRGQUQkNDcerUKahUKu2l6il7LVy4ELGxsahZsyYaNGggO45RUIqku3btQmRkpOQ0+mv8+PGwtLTEyZMncfLkSdlxiHIUC0+UZb6+vvj333+RN29eTJgwQXYcIsoBRYoU0V4FiLOeiIiyh4+PDwCgUaNGcHFxkZzG+ERHR2PBggUAki+Ewx6F2aNatWr45JNPEBMTg23btsmOo7ecnZ3Rr18/AMDMmTMlpyHKWSw8UZZoNBptb6dJkybB3t5eciIiyinTpk2DWq2Gv78/jh8/LjsOEZFBE0Jg9erVAKD9ckrZa9WqVXj27BlKlCiBrl27yo5jNFQqlfY9u2bNGslp9Nvnn38OANi+fTtu3LghOQ1RzmHhibLE19cXV69ehb29PWc7EZmYokWLamc9ff/995LTEBEZtosXL+LatWuwsrJCt27dZMcxOklJSZg9ezaA5MFSCwsLyYmMS9++fQEABw4cwKNHjySn0V/ly5dH+/btIYTQvh+JTAELT/TREhMT8e233wJI/gDPmzev3EBElOOmTZsGS0tL+Pv74+DBg7LjEBEZLGWmSMeOHXlOpQM7d+7EzZs3kTdvXgwZMkR2HKPzySefoG7dutBoNFi3bp3sOHpNmfXk7e2NsLAwyWmIcgYLT/TRvL29cePGDeTLlw+TJ0+WHYeIJChWrBhGjx4NAPjqq68ghJCciIjI8CQmJmr7O3GZnW4oVxIbPXo07OzsJKcxTlxulzkNGzZEzZo1ERcXh/nz58uOQ5QjVILfEugjxMbGwtXVFffv38esWbNYeCIyYWFhYShVqhSioqKwZcsWdOnSRXYkIiKDsn//frRq1QpOTk54+PAhLC0tZUcyKsePH0fDhg1haWmJu3fvwtnZWXYko/T8+XMUKlQIiYmJCAoKQsWKFWVH0lubNm1Cjx494OjoiNDQUNja2sqORKRTnPFEH2XRokW4f/8+ihQpgjFjxsiOQ0QSFShQAJMmTQIAfPPNN0hKSpKciIjIsCgzRHr27Mmikw5Mnz4dADBo0CAWnXTIyckJbdu2BcBZTxnp2rUrSpUqhRcvXmDFihWy4xDpHGc80Qd7/fo1SpUqhWfPnmHZsmUYNmyY7EhEJNmrV6+0J1De3t4YOHCg7EhERAYhKioKBQsWRFRUFE6dOoU6derIjmRULl68CHd3d5iZmeHGjRsoXbq07EhGzdfXF927d0fRokVx9+5dmJlxnkNaFi1ahDFjxqBIkSK4desWi85k1PibgD7Y7Nmz8ezZM5QpUwaDBg2SHYeI9ID9/7V359E13uv7x987g4i5horQGltDWwk1jwk9RLXmeTjUrCg1VksoWmqeqYp5KoKWGIqamzhqDEJRs5KDIojI8Pz+yNf+nZ6etqj47L1zvdayakVW11VNdvZzPfdzfzJn5qOPPgKST7iLi4sznEhExDmsWbOGe/fuUahQIcqWLWs6jst5NO3UtGlTlU7PwTvvvEPmzJm5ePEiO3bsMB3Hob333nv4+Phw6dIlTYiJy1PxJE/k+vXrjBs3DoDhw4frKFoRsevevTu+vr6cP3+eWbNmmY4jIuIU5s6dCyQvZrbZbIbTuJaffvqJFStWANhvjkjKSps2LY0bNwaSDyKSP5Y2bVr69OkDwKhRo7SqQFyaiid5IiNHjiQmJoYSJUrQqFEj03FExIF4e3sTHBwMwIgRI7h7967hRCIiju3cuXNs3boVm82mKfIUMHr0aCzL4p133qF48eKm46Qa7dq1A5IXaN+5c8dwGsfWpUsXXnjhBU6dOsXKlStNxxFJMSqe5LFdvHiRadOmAfD555/rmW0R+Z127dpRqFAhoqOjmThxouk4IiIO7dG0U/Xq1cmbN6/hNK7l0qVLLFiwAICBAwcaTpO6lCtXjiJFihAbG8uyZctMx3FoGTJkoGfPnkDy9ZXWL4urUnMgj+2TTz4hLi6OKlWqULNmTdNxRMQBeXp6Mnz4cAC++OILrl27ZjiRiIhjSkxMtBdP7du3N5zG9YwfP574+HiqVq1KhQoVTMdJVWw2m/1rWie2/bUePXqQIUMGjhw5QlhYmOk4IilCxZM8lh9//JGFCxcCMG7cOO0gEJE/1KRJE8qUKcPdu3ftj96JiMhvbd26lYsXL/LCCy9Qr14903FcSnR0NF9++SWgaSdTWrdujYeHB//61784evSo6TgOLWvWrHTt2hWAzz77TFNP4pJUPMlfsiyL3r17A8k/REqVKmU4kYg4Mjc3N8aPHw/A7NmziYyMNJxIRMTxPJoEadmyJWnTpjWcxrWMHTuW+/fvU7p0aWrUqGE6TqqUM2dO3nnnHQDmzJljOI3j6927N15eXkRERLBt2zbTcUSeORVP8pdWr17Nrl278Pb25rPPPjMdR0ScQMWKFWncuDFJSUn06dNHd+9ERP7DjRs3WLNmDaDH7J616Oho+07SoUOHakrfoEdf2wsXLuThw4eG0zg2Hx8fOnToACR/3ep9k7gaFU/yp+Li4ujfvz8Affv25aWXXjKcSEScxahRo0iTJg2bN29m48aNpuOIiDiMxYsX8/DhQ0qWLIm/v7/pOC7lP6edatWqZTpOqhYUFESuXLm4fv06a9euNR3H4X300Ud4eXmxa9cuvv/+e9NxRJ4pFU/yp6ZNm8aZM2fw8fGxF1AiIo+jQIEC9pNa+vTpQ0JCguFEIiLmWZbF7Nmzgf9/7Lw8G5p2ciweHh60adMGwP41L38sT548dOrUCYDg4GBNPYlLsVn6ipY/cOPGDQoVKsStW7eYPXu2RsFF5IndunWLQoUKcePGDaZPn25fnikiklrt2bOHSpUq4e3tzeXLl3nhhRdMR3IZ/fv3Z8yYMZQuXZq9e/eqeHIAZ86coVChQthsNk6dOkXBggVNR3JoV65coWDBgjx48IBNmzZpR5m4DE08yR8aNmwYt27donjx4rRt29Z0HBFxQlmyZOHTTz8Fku/e3b5923AiERGzpk+fDkCLFi1UOj1DmnZyTAULFqRmzZpYlmU/aVD+mK+vL126dAE09SSuRRNP8j8dP34cPz8/EhIS2Lx5M2+99ZbpSCLipOLj4ylevDgnTpygd+/ejBs3znQkEREjoqOjyZMnD/Hx8ezfv5+SJUuajuQyNO3kuL799lvq1q1LtmzZuHTpkk5x/AtXr16lQIECxMbGsn79eu0qE5egiSf5Hcuy6N69OwkJCdSpU0elk4j8LZ6enkyYMAGASZMmcfToUcOJRETMCAkJIT4+nrJly6p0eoZ++eUXTTs5sNq1a/Pyyy9z48YNVqxYYTqOw/Px8eH9998HYMiQIZp6Epeg4kl+Z9myZWzbto20adMyadIk03FExAUEBQVRv359EhMT6datm95EiUiqk5iYaH/U6NFFpTwbI0aM4P79+5QrV07TIQ7I3d2dzp07A///UVP5c/379yddunTs27ePdevWmY4j8repeJLfuHPnDn369AHgk08+IV++fGYDiYjLmDBhAt7e3uzcuZMlS5aYjiMi8lxt2LCB8+fPkzVrVpo0aWI6jss4c+YMs2bNAmDUqFGadnJQ7du3x9PTk4iICA4ePGg6jsN78cUX6dGjBwAff/wxiYmJhhOJ/D0qnuQ3Pv30U3755RcKFSpE3759TccREReSN29eBg0aBEDfvn21aFxEUpVHkx7t2rXTjptnKDg4mISEBIKCgqhatarpOPIHcubMScOGDQGYMWOG4TTOYcCAAWTJkoWjR4+yePFi03FE/hYtFxe7yMhISpQoQWJiIhs2bCAoKMh0JBFxMXFxcRQvXpyffvqJnj17MnHiRNORRERS3OnTp3n11VexLIvTp0/rSPln5NChQ5QoUQKAAwcO2H8vjmnXrl1UqVKFdOnScfHiRbJmzWo6ksMbPXo0AwYM4OWXX+bkyZMqrcVpaeJJgOSF4t26dSMxMZEGDRqodBKRFOHl5cWUKVMAmDJlCocPHzacSEQk5U2aNAnLsnj77bdVOj1Dn3zyCQDNmjVT6eQEKlWqhJ+fH/fv3+err74yHccp9OjRg9y5c3PhwgVNiolT08STALBo0SJat25NunTpiIqK4uWXXzYdSURcWOPGjVm5ciUVK1Zk165d2skhIi7r119/JU+ePNy/f5+tW7dSrVo105Fcws6dO6latSru7u5ERUXxyiuvmI4kj2HBggW0adMGX19fzp49S5o0aUxHcnghISF06NCBrFmz8vPPP5M5c2bTkUSemCaehJs3b9oXig8ePFilk4ikuPHjx5MuXTr27NnDnDlzTMcREUkxs2bN4v79+xQvXpzAwEDTcVyCZVkMHDgQgA4dOqh0ciLNmjXDx8eHK1eusGLFCtNxnEKbNm0oUqQIN2/eZMyYMabjiDwVFU9C7969iY6OpmjRovTu3dt0HBFJBV566SWGDh0KQJ8+fbhy5YrZQCIiKeDhw4dMnjwZSH6/penOZyM0NJQffvgBb29vgoODTceRJ5AmTRq6d+8OJN+E0sM3f83Dw4ORI0cCyX9nv/zyi+FEIk9OxVMqt2nTJubPn4/NZiMkJETjriLy3Hz44Ye8+eab3L59m27duunNp4i4nBUrVnDlyhV8fHxo1qyZ6Tgu4cGDB/Tr1w+Afv364evraziRPKnOnTvj7e3NgQMH2Llzp+k4TqFu3bqUL1+e2NhYBg8ebDqOyBNT8ZSKxcTE0KlTJwA++OADypcvbziRiKQmHh4ehISE4OHhwZo1awgNDTUdSUTkmbEsi/HjxwPQvXt3vLy8DCdyDZMmTeLcuXP4+vrSv39/03HkKWTPnp02bdoA2L9H5M/ZbDbGjh0LwJw5czh48KDhRCJPRsvFU7EePXowdepU8uXLR2RkJBkyZDAdSURSoeDgYIYPH86LL77I8ePHyZYtm+lIIiJ/2/bt2wkMDMTb25sLFy6QPXt205Gc3rVr13jllVeIiYlh/vz5/POf/zQdSZ7SyZMnKVKkCDabjRMnTvDqq6+ajuQUWrRowdKlS6lcuTI7duzQ47viNDTxlErt3r2badOmAfDVV1+pdBIRYz755BOKFi1KdHS09syJiMt4tJOlTZs2Kp2ekeDgYGJiYihVqhStWrUyHUf+hsKFC/Puu+9iWRZffPGF6ThO44svvsDb25tdu3axcuVK03FEHpsmnlKhBw8e4O/vz8mTJ2nXrh0hISGmI4lIKhceHk7FihWxLIsNGzYQFBRkOpKIyFPbt28fZcqUwd3dnVOnTpE/f37TkZxeZGQk/v7+JCUlsWvXLipVqmQ6kvxN4eHhVKhQAQ8PD86cOaOTtR/T0KFD+fTTT8mbNy9RUVF4e3ubjiTylzTxlAoNGzaMkydP4uPjY39WWETEpPLly9OzZ08geenonTt3DCcSEXl6n332GQAtW7ZU6fQMWJZF7969SUpKonHjxiqdXET58uWpVq0aCQkJjB492nQcp9G/f3/y5MnD+fPnGTdunOk4Io9FE0+pzA8//EDlypVJSkoiNDSUBg0amI4kIgLAvXv3eOONNzh79izvvfcec+bMMR1JROSJRUZGUrx4cWw2G8ePH6dIkSKmIzm9lStX0rhxY7y8vIiKilKZ50K+//57qlevjpeXF+fOncPHx8d0JKewZMkSWrZsSbp06fjpp5/InTu36Ugif0oTT6lITEwMrVq1IikpidatW6t0EhGHkj59eubPn4/NZmPu3Lk65U5EnNKj3U4NGzZU6fQMxMTE0KtXLwA++ugjlU4uJjAwkHLlyhEXF6cT7p5A8+bNqVChAvfv36dPnz6m44j8JU08pSLt2rVj7ty55M2bl8OHD5M5c2bTkUREfufjjz9m5MiRZM2alcjISHx9fU1HEhF5LKdOnaJIkSIkJSVx8OBB/P39TUdyen379mXcuHEULFiQyMhI7bNxQWFhYbzzzjukT5+e8+fP63Tbx3TgwAFKly5NUlISmzZtokaNGqYjifwhTTylEqGhocydOxebzcbChQtVOomIwxo6dCglS5bk5s2btG3blqSkJNORREQey+eff05SUhK1a9dW6fQMREZGMnHiRACmTJmi0slFvf322/j7+3Pv3j37/2/5ayVLlqRHjx4AvP/++8TGxhpOJPLHVDylAleuXKFTp05A8ohy5cqVDScSEfljadKkYfHixXh7e7N582amTp1qOpKIyF86efIkCxYsAGDQoEGG0zi/pKQkunbtSmJiIg0bNqRWrVqmI0kKsdls9u+ZiRMn8u9//9twIucxbNgwfH19OXPmDKNGjTIdR+QPqXhycUlJSbRt25abN29SsmRJhg4dajqSiMhfKlKkiP3Uzf79+3Ps2DHDiURE/lxwcDBJSUm8++67lCtXznQcpzd//nz27NlD+vTpmTBhguk4ksIaNGjAm2++yd27d1WgPIFMmTIxadIkAEaNGsXJkycNJxL537TjycVNmjSJXr164e3tzYEDB7TkUkSchmVZ1K5dmw0bNuDn50dERARp06Y1HUtE5HcOHjxIyZIlsdlsHDp0iOLFi5uO5NT+/e9/U6xYMa5fv86YMWPo27ev6UjyHGzatImgoCC8vLw4ffo0efLkMR3JKViWxTvvvMP69esJDAxk69at2Gw207FEfkMTTy5s79699OvXD4CxY8eqdBIRp2Kz2ZgzZw7Zs2fn8OHD9lONREQczaPHhJo3b67S6Rno0aMH169f54033qBnz56m48hzUqNGDapUqUJcXBzDhw83Hcdp2Gw2pk6dStq0adm2bRsLFy40HUnkdzTx5KJu3LhBiRIluHjxIo0aNWL58uVqvkXEKX333XcEBQVhWRYLFy6kVatWpiOJiNjt3r2bypUr4+7uzokTJyhUqJDpSE5t9erVNGjQAHd3d/bu3cubb75pOpI8R3v27KFSpUr6fnoKo0aNYuDAgWTJkoVjx47pVGBxKJp4ckFJSUm0atWKixcv8sorrxASEqLSSUScVo0aNQgODgagc+fO2vckIg7Dsiw+/vhjANq3b6+L5L/pxo0bdO3aFUje76fSKfWpWLEitWvXJjEx0f6zXx5P3759KVWqFLdu3aJz585ovkQciSaeXNCIESMYPHgwadOmZe/evRr5FhGnl5iYSK1atdi8eTOFCxdm3759ZMyY0XQsEUnlHk3naCfNs9G6dWsWLVpE0aJFOXDggPb6pVKHDh2iZMmSWJZFREQEZcuWNR3JaRw7doySJUvy8OFD5s2bR5s2bUxHEgE08eRytm7dypAhQwCYMWOGSicRcQnu7u4sXryY3Llzc/LkSTp16qQ7eSJiVFxcnH3pdd++fVU6/U3r1q1j0aJFuLm5MXfuXJVOqZi/v7+9MOnVq5d+3j+B1157jU8//RSAnj17cvnyZcOJRJKpeHIhV65coUWLFiQlJdGuXTvatm1rOpKIyDOTI0cOli9fjoeHB8uWLWP69OmmI4lIKjZp0iR+/vlncuXKxUcffWQ6jlO7ceMGnTt3BqB3796acBE+//xz0qdPT0REBEuXLjUdx6n07duXMmXKcPv2bTp27KjiThyCiicXERsbS7169YiOjqZ48eJMnTrVdCQRkWeuQoUKjB49Gki+C7p9+3azgUQkVbp27RojRowAkhf6ZsiQwXAi52VZFh07duTKlSsULlyYYcOGmY4kDiBXrlz2/WkDBgzg3r17hhM5Dw8PD+bNm4eXlxcbNmwgJCTEdCQRFU+uwLIs2rVrx759+8iaNSurVq3C29vbdCwRkRTRq1cvmjVrRkJCAg0bNuTMmTOmI4lIKjNo0CBiYmIoXbq0Ttr8m7766itWr16Np6cnS5Ys0XtYsevduzd58+bl0qVLjB071nQcp1K0aFF7Od6zZ0+ioqIMJ5LUTsvFXcDw4cMJDg7Gw8ODzZs3ExAQYDqSiEiKio2NpWrVquzbt48iRYoQERFB5syZTccSkVTgwIEDlCpVCsuy2LNnDxUqVDAdyWlFRUXx5ptvEhsby9ixY+nTp4/pSOJgVqxYQZMmTfD29iYqKoq8efOajuQ0kpKSqFmzJlu2bKF48eLs3btXu9PEGE08ObkVK1bYjxqdMWOGSicRSRW8vb355ptvyJ07NydOnKBp06YkJCSYjiUiLi4hIcF+uEGLFi1UOv0NDx48oHnz5sTGxvKPf/yDDz/80HQkcUCNGjWiatWqxMbG0q1bN+0regJubm4sWLCAHDlycOTIEfthCCImqHhyYj/++KP9xIcPP/yQDh06GE4kIvL85MqVi7Vr15IuXTo2bdqkO+UikuKmTp3K/v37yZIlC+PHjzcdx6kNHDiQw4cPkz17dubPn4+bmy5L5PdsNhszZszA09OTsLAwQkNDTUdyKrly5WLBggUATJs2jTVr1pgNJKmWXuGd1OXLl6lbty6xsbG8/fbbjBkzxnQkEZHnrkSJEixcuBCAyZMnM3PmTMOJRMRVXbhwgUGDBgEwevRocubMaTiR81q9ejUTJ04EYO7cueTKlctsIHFoRYsWZeDAgQB88MEH3L5923Ai5xIUFGSfdmrXrh0XL140nEhSI+14ckI3b96kcuXKHD9+nGLFihEeHk6mTJlMxxIRMeazzz5j0KBBuLm5sWLFCho0aGA6koi4EMuyqFu3LmvXrqVSpUrs2LFDEzpP6cSJE5QpU4aYmBh69erFhAkTTEcSJ/DgwQP8/Pz46aef6Nq1K9OnTzcdyak8fPiQSpUqsW/fPsqVK8f27dvx8vIyHUtSERVPTubevXu89dZbREREkDt3bvbs2aMleyKS6lmWRadOnZg9ezZp0qRh48aNBAYGmo4lIi5i5cqVNG7cGE9PTw4dOkSxYsVMR3JKMTExlClThhMnTlClShW2bNmCp6en6VjiJLZv305gYCA2m43du3drx9oT+vnnnylVqhS//vorHTp0YNasWdhsNtOxJJXQrRonEh8fT6NGjYiIiOCFF15g06ZNKp1ERPj/OyDq16/Pw4cPqVu3LgcOHDAdS0RcwNWrV+natSsAAwYMUOn0lCzL4r333uPEiRP4+vqyfPlylU7yRAICAnjvvfewLIs2bdpw794905GcSoECBVi2bBlubm7Mnj2bL7/80nQkSUVUPDmJpKQk2rZty8aNG0mXLh1hYWG89tprpmOJiDgMDw8PlixZQkBAADExMQQFBXHq1CnTsUTEiVmWRceOHbl+/Tp+fn4MHjzYdCSnNWbMGEJDQ/H09CQ0NFQ7suSpjB8/njx58nD69Gmd0vYUatSowciRI4HkfVm7d+82nEhSCz1q5wQsy6JXr15MnjwZDw8Pvv32W2rVqmU6loiIQ7pz5w4BAQEcPHiQvHnzsmfPHnLnzm06log4odmzZ9OxY0fSpEnD/v37ef31101Hckrr1q2jbt26JCUlMXPmTDp37mw6kjixrVu38tZbbwGwfv16XRc9IcuyaN68OV9//TU5c+bkxx9/JE+ePKZjiYvTxJODsyyLAQMGMHnyZADmzZunF1cRkT+RKVMmNmzYQKFChTh//jzVqlXjypUrpmOJiJM5c+YMvXr1ApIPMFDp9HT2799P06ZNSUpKokOHDnTq1Ml0JHFy1atXp2fPnkDyKW3Xr183nMi52Gw2QkJCKF68ONeuXaNOnTrExMSYjiUuThNPDsyyLPr27cv48eMBmDp1Kt26dTOcSkTEOZw9e5aAgAAuXLjAK6+8wrZt2zT5JCKPJT4+nqpVqxIeHk7VqlXZunUr7u7upmM5nfPnz1OuXDmuXr1KjRo1WLdunfY6yTMRGxvLm2++SVRUFPXq1WPVqlValP2Ezp49S7ly5YiOjtb3p6Q4TTw5KMuy+PDDD+2l07Rp01Q6iYg8gfz587N9+3by5s3LqVOnCAgI4NKlS6ZjiYgTGDBgAOHh4WTOnJl58+apdHoKt27donbt2ly9epU33niDFStW6KJWnhlvb28WLVpEmjRpWLNmDRMmTDAdyenkz5+fsLAw0qVLx3fffUfHjh3RTIqkFBVPDsiyLHr27MmkSZMAmDlzJu+//77hVCIizudR+ZQvXz5Onz5N1apVuXDhgulYIuLAVq5cab+InT9/Pvny5TMbyAnFxcXRqFEjjh07hq+vL2FhYWTKlMl0LHExJUuWtH+v9u/fX4uyn0KpUqVYsWIF7u7uzJ8/n+DgYNORxEWpeHIwSUlJdO/enSlTpgAwa9YsLWAUEfkb8uXLx44dOyhQoAA///wzAQEBnDt3znQsEXFAP/30E+3atQOSL2Tr1q1rOJHziY+Pp1mzZmzdupUMGTIQFhbGSy+9ZDqWuKiuXbvSokULEhMTadKkCdeuXTMdyem8/fbbzJw5E4ARI0Ywffp0w4nEFWnHkwOJi4ujTZs2fP3119hsNmbPnm1/8yMiIn/PxYsXCQwM5MyZM/j4+LBhwwb8/f1NxxIRB3H37l0qVKhAZGQkVapUYevWrXh4eJiO5VQSExNp1aoVy5Ytw8vLi3Xr1tlPHxNJKXfv3qVMmTJERUURGBjIpk2b9FjnUxgyZAjDhg0Dkk/0bN++veFE4ko08eQgbt++TVBQEF9//TWenp4sWrRIpZOIyDP00ksvsXPnTt544w2uXr1qv7AUEUlMTKRFixZERkaSM2dOli1bptLpCT06tW7ZsmV4enoSGhqq0kmeiwwZMhAaGkr69OnZtm0bPXr00K6ipzB06FD7aYEdO3Zk/vz5hhOJK1Hx5AAuX75M5cqV2b59OxkzZmT9+vW0aNHCdCwREZfj6+vLzp07CQgIICYmhlq1arFkyRLTsUTEsL59+7J27VrSpk3LmjVryJUrl+lITsWyLHr06MG8efNwc3Nj6dKl1K5d23QsSUWKFi3KkiVLsNlsfPnll/YDmuTx2Ww2JkyYQLdu3bAsi/fee0/vkeSZUfFkWFRUlH2s28fHh507d+rukIhICsqSJQsbN26kSZMmxMfH07JlS8aOHWs6logYMn36dCZOnAgkLxMvV66c2UBOJjExkY4dOzJ9+nRsNhsLFiygYcOGpmNJKlSnTh174dSvXz/WrFljNpATstlsTJ48mU6dOmFZFq1bt2bp0qWmY4kLUPFk0IYNGyhfvjwXLlzg1Vdf5YcfftC+ERGR58DLy4ulS5fSq1cvIPkNaqdOnYiLizMbTESeq3Xr1vHBBx8A8Nlnn9GkSRPDiZxLXFwczZo1IyQkBDc3N+bMmUPLli1Nx5JUrGfPnnTt2hXLsmjZsiURERGmIzkdNzc3ZsyYQbt27UhKSqJly5ZMnTrVdCxxclouboBlWYwcOZJBgwZhWRYVK1ZkzZo1ZM+e3XQ0EZFUxbIsxo8fT79+/bAsi/Lly7Ny5Up8fX1NRxORFLZ161Zq165NXFwcbdu2Zc6cOdhsNtOxnMa9e/do0KAB3333HWnSpGHp0qU0aNDAdCwREhISePfdd9m4cSNZsmRh27Zturn/FBITE+nZsyfTpk0DIDg4mKFDh+p1Up6Kiqfn7O7du7Rt25bQ0FAAunTpwqRJk0iTJo3hZCIiqdfGjRtp3rw5t27dIleuXISGhlK+fHnTsUQkhezZs4caNWpw//596tWrx/Lly3UK1hO4fv06derUITw8nPTp07NmzRqtihCHcu/ePWrWrMmePXvIkSMHO3bsoGjRoqZjOR3Lshg+fDhDhgwBkq9dp06diru7u+Fk4mxUPD1Hp0+fpn79+hw9ehRPT0+mTZtGx44dTccSERGSX6Pr1avHsWPH7K/RHTp00J09ERezf/9+qlWrxp07d6hZsybffPMNXl5epmM5jaNHj/Luu+9y7tw5XnjhBdavX6+9WOKQbt++TfXq1dm/f7/9cJGCBQuajuWUZs6cyfvvv49lWdSpU4dFixaRMWNG07HEiWjH03OyaNEiSpYsydGjR/Hx8WH79u0qnUREHEihQoUIDw+nQYMGxMfH06lTJ1q2bMnt27dNRxORZ2T37t320qlKlSqsWrVKpdMTWLduHeXLl+fcuXMUKFCA3bt3q3QSh5U5c2Y2bdrE66+/zpUrV6hcuTLHjh0zHcspdenSheXLl+Pl5cW3335L+fLlOXPmjOlY4kRUPKWw27dv06pVK1q3bk1MTAyVKlVi//79VKhQwXQ0ERH5LxkzZmTlypV8/vnnuLu7s3TpUvz8/Pjhhx9MRxORv2njxo3UqFGDO3fuULlyZdauXUu6dOlMx3IKlmUxduxY6tSpw927dwkICOBf//oXxYoVMx1N5E9ly5aNzZs38/rrr/PLL79QpUoV9u3bZzqWU2rUqBE7duwgV65cHDt2jNKlS7N582bTscRJqHhKQeHh4ZQoUYLFixfj7u7OsGHD2LZtm5bWiog4MJvNxsCBA9m9ezf58+fn/PnzVK5cmU8//ZSEhATT8UTkKSxfvpw6deoQGxvL22+/zcaNG8mUKZPpWE7h119/pWHDhvZDGDp16sSmTZvIli2b6Wgij8XHx4cdO3ZQtmxZbt68SbVq1fj+++9Nx3JKZcuW5ccff6Rs2bL8+uuvBAUFMXr0aJKSkkxHEwen4ikFxMXFMXjwYCpXrszZs2fJly8fO3fuZPDgwXh4eJiOJyIij6FcuXIcOnSIVq1akZSUxNChQ6lSpQonTpwwHU1EHpNlWXz++ec0bdqU+Ph4mjVrxurVqzXp9JjCw8Px9/dn9erVeHp6MmXKFGbOnKlDccTpZM2alS1btlC9enXu3r1LUFAQISEhpmM5JV9fX7Zv306bNm1ISkpiwIAB1KxZk19++cV0NHFgKp6esd27d+Pv78+IESNITEykefPmHDp0SI/WiYg4oUyZMrFw4UIWLlxIxowZCQ8Px8/Pj+HDh/Pw4UPT8UTkT8TGxtKyZUs++eQTAHr06MGiRYtUmjyGxMRERo0aReXKlblw4QIFCxYkPDyc7t2768AFcVoZMmRg3bp1NGnShPj4eDp06MCHH36oaeankDZtWubOncusWbPw9vZmy5YtFC9enHXr1pmOJg5Kp9o9I7dv32bgwIHMmDEDSB7pnDJlCg0bNtQPaBERF3D+/Hm6du3Khg0bAHjttdeYPXu2FuuKOKALFy7QqFEj9u3bh4eHB1OnTqVz586mYzmF48eP06FDB8LDwwFo3rw5M2fO1KOJ4jIsy2LEiBEEBwcDULNmTRYtWkT27NkNJ3NOUVFRNG/enMOHDwPQuXNnvvjiCzJnzmw4mTgSTTz9TZZlsWzZMl577TV76dShQweOHz9Oo0aNVDqJiLiIvHnzEhYWxpIlS8iRIwfHjh2jQoUKdOvWjRs3bpiOJyL/Z9WqVfj5+bFv3z6yZs3K5s2bVTo9hvj4eEaMGEGJEiUIDw8nY8aMhISEsHjxYpVO4lJsNhuDBw9mxYoVeHt7s2nTJvz8/Ni+fbvpaE6paNGi7N27l169egHw5ZdfUqxYMVavXm02mDgUTTz9DXv37uXDDz+03xF65ZVXmDVrFgEBAWaDiYhIirpx4wZ9+vRh/vz5AGTJkoXg4GC6deumx3hEDImNjaVv375Mnz4dgDJlyrB06VIKFChgOJnj27lzJz169ODIkSMA1K5dm5kzZ5InTx7DyURS1uHDh2natCknT560F1Lay/v0tm3bRufOnTl16hQA9evXZ/LkyXotEU08PY2LFy/SqlUrypUrR3h4OOnTp2f48OEcPnxYpZOISCqQLVs25s2bx/fff4+fnx+3bt2id+/evP7663z77bfono7I87Vz5078/PzspVP//v3ZtWuXSqe/cPbsWRo3bkzVqlU5cuQI2bJlY/Hixaxdu1YXipIq+Pn5sX//ft577z0sy2LYsGH2w0XkyQUGBnL48GE+/vhjPDw8WL16NYULF2bIkCHcvXvXdDwxSBNPT+DatWuMHj2a6dOn8+DBA2w2G23atOGzzz7D19fXdDwRETEgMTGRuXPn8sknnxAdHQ1A5cqV+fTTTwkMDDScTsS13b59mwEDBvDll18CkCtXLubOnUvNmjUNJ3Nsv/76K6NHj2bChAnExcXh5uZGx44dGT58ODly5DAdT8SIJUuW0K1bN27duoW7uzv9+/cnODiYtGnTmo7mlCIjI3n//ffZvXs3ADlz5mT48OG0a9cOd3d3w+nkeVPx9BiuXbvGmDFjmD59OrGxsUDyRcWECRN48803DacTERFHcOfOHUaOHGm/kAOoWrUqQ4cO1TSsyDP2qPAdNGgQ165dA6Bjx46MHj2aLFmymA3nwG7evMmECROYPHkyd+7cAaB69epMmDCBN954w3A6EfN++eUXevToQWhoKAD58+fniy++0O7ep2RZFqtXr2bAgAGcPn0agFdffZWPP/6YFi1a4OnpaTihPC8qnv7EhQsXmDRpEjNmzLAXTmXLlmXo0KHUrFlTLz4iIvI7ly5dYtSoUXz11Vc8fPgQgICAAD766CNq1Kihnx0if4NlWWzZsoU+ffoQGRkJaMfm47h27RpTpkxh8uTJxMTEAPDGG28wYsQI3n33Xb0uifyX1atX0717d65cuQJAxYoVGTduHGXLljWczDk9fPiQGTNmMGzYMG7evAkkl3offfQRbdq0wcvLy3BCSWkqnv6HiIgIJkyYQGhoKImJiUDygsqhQ4cSFBSkH84iIvKXLl26xMiRI5k9e7a9gCpWrBi9evWiVatWeHt7G04o4jweFU7Dhg2zP7aRJUsWhgwZwvvvv6+l/n9g//79TJo0ia+//tr+OlS8eHGGDBlCvXr1cHPTuleRP3Lv3j3GjBnD6NGj7UMItWrVIjg4mHLlyhlO55xiYmKYMWMG48aNs68nyJkzJx07dqRz587aLefCVDz9n7i4OFavXs3kyZPtp9RB8oK0fv36qXASEZGncvHiRcaNG0dISIh9sWb27Nnp0qULHTt25OWXXzacUMRxJSQksGbNGsaNG0dERAQAXl5edOnShcGDB5MtWzbDCR1PTEwMK1asICQkhB9++MH+8fLly9OvXz/q1q2rwknkCVy+fJlBgwaxcOFC+1DCW2+9RZ8+fahRo4a+n57C/fv3+eqrrxgzZgyXL18GwN3dnXr16tG5c2eqVaumPVAuJtUXT0eOHCEkJIRFixbZx/7SpElD8+bN6dWrF/7+/mYDioiIS7h9+zYhISFMnjyZ8+fPA2Cz2fjHP/5B+/btqVu3rkbNRf5PdHQ0ISEhTJ8+nUuXLgGQNm1aunTpQr9+/XSoy39JSEhg+/btzJ8/n1WrVnH//n0APD09adq0KR988AGlS5c2nFLEuZ0+fZqRI0eyYMECEhISgORHfbt3784///lP7Zd7CvHx8axZs4Zp06axY8cO+8d9fX1p0aIFrVu3pnjx4gYTyrOSKounq1evEhoayrx58/jxxx/tH8+dOzft27ena9eu+Pj4GEwoIiKuKiEhgW+++YZp06axbds2+8ezZctGy5Ytad68OWXLltWUraQ6Dx48YN26dcyfP58NGzbYJwty5MhBp06d6N69u96f/Ye4uDi2bNnCqlWr+Oabb7hx44b9zwoXLkybNm1o27YtuXLlMphSxPWcPXuWyZMnM2fOHPuSfi8vL+rWrUubNm2oUaMGHh4ehlM6n8jISGbMmMGyZcv49ddf7R8vVqwY9erVo27dupQqVUoTZk4q1RRP0dHRrFq1iuXLl7Njxw6SkpKA5DtBderUoX379tSoUUMjfSIi8tycOXOGuXPnMm/ePPuoOcDLL79MkyZNaNKkCaVKlVIJJS7r7t27bNy4kVWrVhEWFma/iIPk/ZrdunWjSZMmOs6c5D1Xp0+f5rvvvmPz5s1s3brV/vguQNasWWnSpAlt27alTJkyet0QSWExMTEsXLiQGTNmcPToUfvHs2XLRt26dWnQoAHVq1fX69cTiouLY/369SxatIh169bZ99NB8iTU22+/zVtvvUW1atXIkSOHwaTyJFy2eLIsi+PHjxMWFkZYWBi7d++2l02Q/GamadOmtGrVihdffNFgUhERSe0SExPZtGkTixYt4ttvv+XevXv2P8ubNy/vvPMOtWvXJiAgQEvJxaklJSVx5MgRNm/ezHfffceuXbuIi4uz/3nu3Llp3bo1//znPylatKjBpI5l4sSJTJo0iXPnzv3m476+vtSvX58GDRpQpUoVTVmIGGBZFgcPHmT+/PksWbKE69ev2/8sffr0BAQEUKNGDf7xj39QpEgRlcJP4NatW6xfv55vvvmG9evX/6ZsB/D39ycgIIAKFSpQvnx5LSd3YC5VPN28eZMdO3awZcsWwsLC7Ds0HilVqhRNmjShcePG5MuXz0xIERGRPxEbG8uGDRtYvnw5a9eute9qAfD29qZ69eoEBQURGBhI0aJF9QZWHFpCQgLh4eHs3r2bPXv2sGfPHm7duvWbzylYsCANGjSgfv36lC1bVo9R/A8jR47k448/xtPTk4oVK9ovYkuWLKm/LxEHkpCQwM6dO1m9ejWrV6/+zTQzJD86XKlSJSpVqkTFihU1nfgE4uLi2LZtG5s3b2bLli0cOXLkd5/z0ksvUaZMGUqUKEGJEiXw9/cnV65c+jt2AE5dPP373/8mPDyc7du3s23bNg4fPsx//ud4eXkRGBhI7dq1qV27Nvnz5zeYVkRE5Mncv3+frVu32qd3Hy1ZfuTFF18kICCAwMBAKleuTNGiRXURKg4lNjaWzJkzEx8fb/9Y+vTpCQwMtJcnhQsX1kXBXzh37hzHjx+nSpUqZMiQwXQcEXkMSUlJREZG8t133/3PCc9cuXJx+fJlvf49pejoaL7//nt2797NDz/8wJEjR+y7Af9TtmzZKFq0KEWLFqVIkSIULlyYAgUKkC9fPk2RP0dOUzzFxsZy9OhR9u7dS0REBBEREZw5c+Z3n1ekSBECAwOpVasW1apVI3369AbSioiIPFuWZXHkyBHCwsL4/vvv2bNnDw8ePPjN52TKlIkyZcpQrlw5ypUrR6lSpciZM6ehxCLJ6tevj4eHh/0Ov5+fH56enqZjiYg8V3Fxcezfv98+Aerj48OXX35pOpbLuHv3Lvv27WP//v0cOnSIgwcPcuLEid+s2/lvvr6+5MuXjzx58pAnTx5eeuklfH198fHxIWfOnPj4+JApUyaVg8+AwxVPlmVx6dIloqKiOHLkCIcOHeLQoUOcOHHifzaYhQsXpmrVqgQEBBAQEKCTO0REJFWIi4tj7969bNu2je3bt/Ovf/3rN4/lPeLj44Ofnx/+/v74+/tTrFgxXn31VS07FREREZcWGxvLiRMnOHHiBFFRUURFRXHq1Cl+/vlnYmJiHuvfkSZNGrJly/abX5kzZyZLlixkyZKFTJkykTFjxt/8Spcu3W9+eXt7kzZt2lR9kJnx4un27dtMmzaNqKgo+xfFfy8NeyR79uyULl3afie3dOnSvPDCC885sYiIiONJSEjg6NGjREREsHfvXvbu3cvJkyf/550+Nzc38ufPbx89r1u3LhUrVjSQWhzFfy60FxERcWWWZXHjxg3Onj3LpUuXuHTpEleuXOHSpUtcvXqV6Ohorl279oe9xNNyd3cnbdq0eHl5kSZNGvs/PT098fDw+M3vPT09cXd3t//zP3+5ubn97vdubm64ublhs9l+83ubzWZfcJ8SHvcJM+PF07179373rLqHhwevvPIKr732mn0pmBaDiYiIPJn79+9z9OhR+/TwoUOHiIqK+t1y53HjxtG7d28zIcUh6P2ViIiIPKnHrZOMF08APXv2JGfOnPY7rwULFtSz/yIiIinAsiyuXbv2m7HzVq1aUaZMGdPRxCAVTyIiIvKknnnxpBFsEREReVI65MM56H2eiIiIPKln/qid7oSJiIjIk3KAwWoRERERMcjNdAAREREREREREXFNHo/7ic96o7uIiIiIiIiIiLg2h1guLiIiIiIiIiIirkeP2omIiIiIiIiISIpQ8SQiIiIiIiIiIilCxZOIiIiIiIiIiKQIFU8iIiIiIiIiIpIiVDyJiIiIiIiIiEiKUPEkIiIiIiIiIiIpQsWTiIiIiIiIiIikCBVPIiIiIiIiIiKSIlQ8iYiIiIiIiIhIilDxJCIiIiIiIiIiKULFk4iIiIiIiIiIpAgVTyIiIiIiIiIikiJUPImIiIiIiIiISIpQ8SQiIiIiIiIiIilCxZOIiIiIiIiIiKQIFU8iIiIiIiIiIpIiVDyJiIiIiIiIiEiKUPEkIiIiIiIiIiIpQsWTiIiIiIiIiIikCBVPIiIiIiIiIiKSIlQ8iYiIiIiIiIhIivh/6qFLiSi9dfQAAAAASUVORK5CYII=", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "ttesthyp2_fig" } }, "output_type": "display_data" } ], "source": [ "\n", "\n", "mu1 = 0\n", "sigma1 = 1\n", "mu2 = 2\n", "sigma2 = 1.45\n", "\n", "x1 = np.linspace(mu1 - 4*sigma, mu1 + 4*sigma1, 100)\n", "y1 = 100* stats.norm.pdf(x1, mu1, sigma1)\n", "x2 = np.linspace(mu2 - 4*sigma, mu2 + 4*sigma2, 100)\n", "y2 = 100* stats.norm.pdf(x2, mu2, sigma2)\n", "\n", "\n", "fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 5))\n", "\n", "\n", "sns.lineplot(x=x,y=y, color='black', ax = ax1)\n", "\n", "sns.lineplot(x=x,y=y, color='black', ax = ax2)\n", "sns.lineplot(x=x2,y=y2, color='black', ax = ax2)\n", "\n", "ax1.text(0, 47, 'null hypothesis', size=20, ha=\"center\")\n", "ax2.text(0, 47, 'alternative hypothesis', size=20, ha=\"center\")\n", "\n", "ax1.text(0, 41, r'$\\mu$', size=20, ha=\"center\")\n", "ax2.text(0, 41, r'$\\mu_1$', size=20, ha=\"center\")\n", "ax2.text(1.50, 30, r'$\\mu_2$', size=20, ha=\"left\")\n", "\n", "ax1.set_frame_on(False)\n", "ax2.set_frame_on(False)\n", "ax1.get_yaxis().set_visible(False)\n", "ax2.get_yaxis().set_visible(False)\n", "ax1.get_xaxis().set_visible(False)\n", "ax2.get_xaxis().set_visible(False)\n", "ax1.axhline(y=0, color='black')\n", "ax2.axhline(y=0, color='black')\n", "\n", "# show the figure in the book with caption\n", "plt.close()\n", "from myst_nb import glue\n", "glue(\"ttesthyp2_fig\", fig, display=False)\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "streaming-logan", "metadata": {}, "source": [ " ```{glue:figure} ttesthyp2_fig\n", ":figwidth: 600px\n", ":name: fig-ttesthyp2\n", "\n", "Graphical illustration of the null and alternative hypotheses assumed by the Welch $t$-test. Like the Student test we assume that both samples are drawn from a normal population; but the alternative hypothesis no longer requires the two populations to have equal variance.\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "phantom-purchase", "metadata": {}, "source": [ "### Doing the test in Python\n", "\n", "To run a Welch test in Python is pretty easy. All you have to do is not bother telling Python to assume equal variances. That is, the command is exactly the same as for the Student test, but where `correction = True`. In fact, in this case, you could just leave the `correction` argument off entirely, because Anasatasia and Bernadette had different numbers of students, and in cases where the $t$-test is comparing two unequal-sized samples, `pingouin` assumes unequal variance and does a Welch test by default, which is why we had to force it to do a Student test above by setting `correction` to `False`." ] }, { "cell_type": "code", "execution_count": 31, "id": "homeless-insight", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Tdofalternativep-valCI95%cohen-dBF10power
T-test2.03418723.024806two-sided0.05361[-0.09, 11.05]0.7395611.5560.53577
\n", "
" ], "text/plain": [ " T dof alternative p-val CI95% cohen-d \\\n", "T-test 2.034187 23.024806 two-sided 0.05361 [-0.09, 11.05] 0.739561 \n", "\n", " BF10 power \n", "T-test 1.556 0.53577 " ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from pingouin import ttest\n", "\n", "ttest(Harpo_wide['Anastasia'], Harpo_wide['Bernadette'], correction = True)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "precious-essex", "metadata": {}, "source": [ "Not too difficult, right? Not surprisingly, the output has exactly the same format as it did last time too: a test statistic $t$, and a $p$-value. So that's all pretty easy. \n", "\n", "Except, except... our result isn't significant anymore. When we ran the Student test, we did get a significant effect; but the Welch test on the same data set is not ($t(23.03) = 2.03$, $p = .054$). What does this mean? Should we panic? Is the sky burning? Probably not. The fact that one test is significant and the other isn't doesn't itself mean very much, especially since I kind of rigged the data so that this would happen. As a general rule, it's not a good idea to go out of your way to try to interpret or explain the difference between a $p$-value of .049 and a $p$-value of .051. If this sort of thing happens in real life, the *difference* in these $p$-values is almost certainly due to chance. What does matter is that you take a little bit of care in thinking about what test you use. The Student test and the Welch test have different strengths and weaknesses. If the two populations really do have equal variances, then the Student test is slightly more powerful (lower Type II error rate) than the Welch test. However, if they *don't* have the same variances, then the assumptions of the Student test are violated and you may not be able to trust it: you might end up with a higher Type I error rate. So it's a trade off. However, in real life, I tend to prefer the Welch test; because almost no-one *actually* believes that the population variances are identical.\n", "\n", "\n", "### Assumptions of the test\n", "\n", "The assumptions of the Welch test are very similar to those made by the [Student $t$-test](studentassumptions), except that the Welch test does not assume homogeneity of variance. This leaves only the assumption of normality, and the assumption of independence. The specifics of these assumptions are the same for the Welch test as for the Student test. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "liable-execution", "metadata": {}, "source": [ "(pairedsamplesttest)=\n", "## The paired-samples $t$-test\n", "\n", "Regardless of whether we're talking about the Student test or the Welch test, an independent samples $t$-test is intended to be used in a situation where you have two samples that are, well, independent of one another. This situation arises naturally when participants are assigned randomly to one of two experimental conditions, but it provides a very poor approximation to other sorts of research designs. In particular, a repeated measures design -- in which each participant is measured (with respect to the same outcome variable) in both experimental conditions -- is not suited for analysis using independent samples $t$-tests. For example, we might be interested in whether listening to music reduces people's working memory capacity. To that end, we could measure each person's working memory capacity in two conditions: with music, and without music. In an experimental design such as this one,[^note10] each participant appears in *both* groups. This requires us to approach the problem in a different way; by using the **_paired samples $t$-test_**. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "pharmaceutical-creator", "metadata": {}, "source": [ "### The data\n", "\n", "The data set that we'll use this time comes from Dr Chico's class.[^note11] In her class, students take two major tests, one early in the semester and one later in the semester. To hear her tell it, she runs a very hard class, one that most students find very challenging; but she argues that by setting hard assessments, students are encouraged to work harder. Her theory is that the first test is a bit of a \"wake up call\" for students: when they realise how hard her class really is, they'll work harder for the second test and get a better mark. Is she right? To test this, let's have a look at the `chico.csv` file: " ] }, { "cell_type": "code", "execution_count": 32, "id": "correct-hospital", "metadata": {}, "outputs": [], "source": [ "import pandas as pd\n", "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/chico.csv\")" ] }, { "attachments": {}, "cell_type": "markdown", "id": "romantic-laptop", "metadata": {}, "source": [ "The data frame `chico` contains three variables: an `id` variable that identifies each student in the class, the `grade_test1` variable that records the student grade for the first test, and the `grade_test2` variable that has the grades for the second test. Here's the first five students:" ] }, { "cell_type": "code", "execution_count": 33, "id": "bulgarian-botswana", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " id grade_test1 grade_test2\n", "0 student1 42.9 44.6\n", "1 student2 51.8 54.0\n", "2 student3 71.7 72.3\n", "3 student4 51.6 53.4\n", "4 student5 63.5 63.8" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.head()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "neutral-maine", "metadata": {}, "source": [ "At a glance, it does seem like the class is a hard one (most grades are between 50\\% and 60\\%), but it does look like there's an improvement from the first test to the second one. If we take a quick look at the descriptive statistics" ] }, { "cell_type": "code", "execution_count": 34, "id": "residential-criterion", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " grade_test1 grade_test2\n", "count 20.000000 20.000000\n", "mean 56.980000 58.385000\n", "std 6.616137 6.405612\n", "min 42.900000 44.600000\n", "25% 51.750000 53.100000\n", "50% 57.700000 59.700000\n", "75% 62.050000 63.050000\n", "max 71.700000 72.300000" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.describe()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "placed-feeding", "metadata": {}, "source": [ "we see that this impression seems to be supported. Across all 20 students[^note12] the mean grade for the first test is 57\\%, but this rises to 58\\% for the second test. Although, given that the standard deviations are 6.6\\% and 6.4\\% respectively, it's starting to feel like maybe the improvement is just illusory; maybe just random variation. This impression is reinforced when you see the means and confidence intervals plotted in {numref}`pairedta` panel A. If we were to rely on this plot alone, we'd come to the same conclusion that we got from looking at the descriptive statistics that the `describe()` method produced. Looking at how wide those confidence intervals are, we'd be tempted to think that the apparent improvement in student performance is pure chance." ] }, { "cell_type": "code", "execution_count": 35, "id": "casual-valve", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "pairedta_fig" } }, "output_type": "display_data" } ], "source": [ "fig, (ax1, ax2, ax3) = plt.subplots(1, 3, figsize=(15, 5))\n", "\n", "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/chico.csv\")\n", "\n", "sns.pointplot(data=df, ax = ax1)\n", "sns.scatterplot(data = df, x='grade_test1', y='grade_test2', ax = ax2)\n", "\n", "ax2.plot(ax2.get_xlim(), ax2.get_ylim(), ls=\"--\", c=\".3\")\n", "\n", "#ax1.set_ylim(40,80)\n", "#ax1.set_xlim(40,80)\n", "#ax1.set_ylim(40,80)\n", "\n", "\n", "df2 = df\n", "df2['improvement'] = df2['grade_test2']-df2['grade_test1']\n", "\n", "sns.histplot(data = df2, x='improvement')\n", "\n", "ax1.title.set_text('A')\n", "ax2.title.set_text('B')\n", "ax3.title.set_text('C')\n", "\n", "sns.despine()\n", "\n", "# show the figure in the book with caption\n", "plt.close()\n", "from myst_nb import glue\n", "glue(\"pairedta_fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "legislative-accessory", "metadata": {}, "source": [ "```{glue:figure} pairedta_fig\n", ":figwidth: 600px\n", ":name: fig-pairedta\n", "\n", "Mean grade for test 1 and test 2, with associated 95% confidence intervals (panel a). Scat- terplot showing the individual grades for test 1 and test 2 (panel b). Histogram showing the improvement made by each student in Dr Chico’s class (panel c). In panel c, notice that almost the entire distribution is above zero: the vast majority of students did improve their performance from the first test to the second one\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "inner-pathology", "metadata": {}, "source": [ "Nevertheless, this impression is wrong. To see why, take a look at the scatterplot of the grades for test 1 against the grades for test 2. shown in {numref}`fig-pairedta` panel B. \n", "\n", "\n", "\n", "In this plot, each dot corresponds to the two grades for a given student: if their grade for test 1 ($x$ co-ordinate) equals their grade for test 2 ($y$ co-ordinate), then the dot falls on the line. Points falling above the line are the students that performed better on the second test. Critically, almost all of the data points fall above the diagonal line: almost all of the students *do* seem to have improved their grade, if only by a small amount. This suggests that we should be looking at the *improvement* made by each student from one test to the next, and treating that as our raw data. To do this, we'll need to create a new variable for the `improvement` that each student makes, and add it to the data frame containing the `chico.csv` data. The easiest way to do this is as follows: " ] }, { "cell_type": "code", "execution_count": 36, "id": "silver-occasions", "metadata": {}, "outputs": [], "source": [ "df['improvement'] = df['grade_test2']-df['grade_test1']" ] }, { "attachments": {}, "cell_type": "markdown", "id": "orange-filter", "metadata": {}, "source": [ "Notice that I assigned the output to a variable called `df['improvement]`. That has the effect of creating a new column called `improvement` inside the `chico` data frame. Now that we've created and stored this `improvement` variable, we can draw a histogram showing the distribution of these improvement scores, shown in {numref}`fig-pairedta` panel C. \n", "\n", "\n", "When we look at histogram, it's very clear that there *is* a real improvement here. The vast majority of the students scored higher on the test 2 than on test 1, reflected in the fact that almost the entire histogram is above zero. In fact, if we use `scipy.stats.t.interval()` to compute a confidence interval for the population mean of this new variable, " ] }, { "cell_type": "code", "execution_count": 37, "id": "accessible-words", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(0.9508686092602991, 1.8591313907397005)" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import numpy as np\n", "from scipy import stats\n", "\n", "data = df['improvement']\n", "\n", "stats.t.interval(confidence=0.95, df=len(data)-1, loc=np.mean(data), scale=stats.sem(data))\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "interstate-bermuda", "metadata": {}, "source": [ "we see that it is 95\\% certain that the true (population-wide) average improvement would lie between 0.95\\% and 1.86\\%. So you can see, qualitatively, what's going on: there is a real \"within student\" improvement (everyone improves by about 1\\%), but it is very small when set against the quite large \"between student\" differences (student grades vary by about 20\\% or so). " ] }, { "attachments": {}, "cell_type": "markdown", "id": "selective-jungle", "metadata": {}, "source": [ "### What is the paired samples $t$-test?\n", "\n", "In light of the previous exploration, let's think about how to construct an appropriate $t$ test. One possibility would be to try to run an independent samples $t$-test using `grade_test1` and `grade_test2` as the variables of interest. However, this is clearly the wrong thing to do: the independent samples $t$-test assumes that there is no particular relationship between the two samples. Yet clearly that's not true in this case, because of the repeated measures structure to the data. To use the language that I introduced in the last section, if we were to try to do an independent samples $t$-test, we would be conflating the **_within subject_** differences (which is what we're interested in testing) with the **_between subject_** variability (which we are not). \n", "\n", "The solution to the problem is obvious, I hope, since we already did all the hard work in the previous section. Instead of running an independent samples $t$-test on `grade_test1` and `grade_test2`, we run a *one-sample* $t$-test on the within-subject difference variable, `improvement`. To formalise this slightly, if $X_{i1}$ is the score that the $i$-th participant obtained on the first variable, and $X_{i2}$ is the score that the same person obtained on the second one, then the difference score is:\n", "\n", "$$\n", "D_{i} = X_{i1} - X_{i2} \n", "$$\n", "\n", "Notice that the difference scores is *variable 1 minus variable 2* and not the other way around, so if we want improvement to correspond to a positive valued difference, we actually want \"test 2\" to be our \"variable 1\". Equally, we would say that $\\mu_D = \\mu_1 - \\mu_2$ is the population mean for this difference variable. So, to convert this to a hypothesis test, our null hypothesis is that this mean difference is zero; the alternative hypothesis is that it is not:\n", "\n", "$$\n", "\\begin{array}{ll}\n", "H_0: & \\mu_D = 0 \\\\\n", "H_1: & \\mu_D \\neq 0\n", "\\end{array}\n", "$$\n", "\n", "(this is assuming we're talking about a two-sided test here). This is more or less identical to the way we described the hypotheses for the one-sample $t$-test: the only difference is that the specific value that the null hypothesis predicts is 0. And so our $t$-statistic is defined in more or less the same way too. If we let $\\bar{D}$ denote the mean of the difference scores, then \n", "\n", "$$\n", "t = \\frac{\\bar{D}}{\\mbox{SE}({\\bar{D}})}\n", "$$\n", "\n", "which is \n", "\n", "$$\n", "t = \\frac{\\bar{D}}{\\hat\\sigma_D / \\sqrt{N}}\n", "$$\n", "\n", "where $\\hat\\sigma_D$ is the standard deviation of the difference scores. Since this is just an ordinary, one-sample $t$-test, with nothing special about it, the degrees of freedom are still $N-1$. And that's it: the paired samples $t$-test really isn't a new test at all: it's a one-sample $t$-test, but applied to the difference between two variables. It's actually very simple; the only reason it merits a discussion as long as the one we've just gone through is that you need to be able to recognise *when* a paired samples test is appropriate, and to understand *why* it's better than an independent samples $t$ test. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "cross-married", "metadata": {}, "source": [ "### Doing the test in Python \n", "\n", "How do you do a paired samples $t$-test in Python? One possibility is to follow the process I outlined above: create a \"difference\" variable and then run a one sample $t$-test on that, setting the population mean argument `popmean` to zero. Since we've already created a variable called `improvement`, let's do that:" ] }, { "cell_type": "code", "execution_count": 38, "id": "brazilian-raising", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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Tdofalternativep-valCI95%cohen-dBF10power
T-test6.47543619two-sided0.000003[0.95, 1.86]1.4479525991.5770.999984
\n", "
" ], "text/plain": [ " T dof alternative p-val CI95% cohen-d BF10 \\\n", "T-test 6.475436 19 two-sided 0.000003 [0.95, 1.86] 1.447952 5991.577 \n", "\n", " power \n", "T-test 0.999984 " ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from pingouin import ttest\n", "\n", "ttest(df['improvement'], 0)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "literary-parish", "metadata": {}, "source": [ "However, suppose you're lazy and you don't want to go to all the effort of creating a new variable. Or perhaps you just want to keep the difference between one-sample and paired-samples tests clear in your head. In that case, you can do:" ] }, { "cell_type": "code", "execution_count": 39, "id": "basic-cheat", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Tdofalternativep-valCI95%cohen-dBF10power
T-test6.47543619two-sided0.000003[0.95, 1.86]0.2157655991.5770.150446
\n", "
" ], "text/plain": [ " T dof alternative p-val CI95% cohen-d BF10 \\\n", "T-test 6.475436 19 two-sided 0.000003 [0.95, 1.86] 0.215765 5991.577 \n", "\n", " power \n", "T-test 0.150446 " ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from pingouin import ttest\n", "\n", "ttest(df['grade_test2'], df['grade_test1'], paired=True)\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "decreased-circumstances", "metadata": {}, "source": [ "Either way, you'll get the same $t$-value, and the same $p$-value, which is strangely comforting, actually. Not only that, but the result confirms our intuition. There’s an average improvement of 1.4% from test 1 to test 2, and this is significantly different from 0 ($t$(19) = 6.48, $p$ < .001). In fact, $p$ is quite a bit less than one, since the $p$-value has been given in scientific notation. The exact $p$-value is $3.32^{-06}$, that is, $p$ = 0.0000032." ] }, { "attachments": {}, "cell_type": "markdown", "id": "alive-variance", "metadata": {}, "source": [ "## One sided tests\n", "\n", "When introducing the theory of null hypothesis tests, I mentioned that there are some situations [when it's appropriate to specify a *one-sided* test](one-two-sided). So far, all of the $t$-tests have been two-sided tests. For instance, when we specified a one sample $t$-test for the grades in Dr Zeppo's class, the null hypothesis was that the true mean was 67.5\\%. The alternative hypothesis was that the true mean was greater than *or* less than 67.5\\%. Suppose we were only interested in finding out if the true mean is greater than 67.5\\%, and have no interest whatsoever in testing to find out if the true mean is lower than 67.5\\%. If so, our null hypothesis would be that the true mean is 67.5\\% or less, and the alternative hypothesis would be that the true mean is greater than 67.5\\%. The `ttest()` function lets you do this, by specifying the `alternative` argument. If you set `alternative = 'greater'`, it means that you're testing to see if the true mean is larger than `mu`. If you set `alternative = 'less'`, then you're testing to see if the true mean is smaller than `mu`. To see how it would work for Dr Zeppo's class, let's compare the results of the two-sided test we did before with the results of a one-sided test, where the alternative hypothesis is set to \"greater\":" ] }, { "cell_type": "code", "execution_count": 43, "id": "038d891c", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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Tdofalternativep-valCI95%cohen-dBF10power
T-test2.25471319two-sided0.036145[67.84, 76.76]0.5041691.7950.571446
\n", "
" ], "text/plain": [ " T dof alternative p-val CI95% cohen-d BF10 \\\n", "T-test 2.254713 19 two-sided 0.036145 [67.84, 76.76] 0.504169 1.795 \n", "\n", " power \n", "T-test 0.571446 " ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from pingouin import ttest\n", "\n", "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/zeppo.csv\")\n", "\n", "# two-sided test\n", "ttest(df['grades'], 67.5, alternative = 'two-sided')\n" ] }, { "cell_type": "code", "execution_count": 41, "id": "e71ad881", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Tdofalternativep-valCI95%cohen-dBF10power
T-test2.25471319greater0.018073[68.62, inf]0.5041693.590.701407
\n", "
" ], "text/plain": [ " T dof alternative p-val CI95% cohen-d BF10 \\\n", "T-test 2.254713 19 greater 0.018073 [68.62, inf] 0.504169 3.59 \n", "\n", " power \n", "T-test 0.701407 " ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# one-sided test\n", "ttest(df['grades'], 67.5, alternative = 'greater')" ] }, { "attachments": {}, "cell_type": "markdown", "id": "aware-tomato", "metadata": {}, "source": [ "The $t$-statistics are exactly the same, which makes sense, if you think about it, because the calculation of the $t$ is based on the mean and standard deviation, and these do not change. The $p$-value, on the other hand, is lower for the one-sided test. The only thing that changes between the two tests is the _expectation_ that we bring to data. The way that the $p$-value is calculated depends on those expectations, and they are the reason for choosing one test over the other. It should go without saying, but maybe is worth saying anyway, that our reasons for choosing one test over the other should be theoretical, and not based on which test is more likely to give us the $p$-value we want!\n", "\n", "\n", "So that's how to do a one-sided one sample $t$-test. However, all versions of the $t$-test can be one-sided. For an independent samples $t$ test, you could have a one-sided test if you're only interestd in testing to see if group A has *higher* scores than group B, but have no interest in finding out if group B has higher scores than group A. Let's suppose that, for Dr Harpo's class, you wanted to see if Anastasia's students had higher grades than Bernadette's. The `ttest_ind` function lets you do this, again by specifying the `alternative` argument. However, this time around the order that we enter the variables in the test makes a difference. If we expect that Anastasia's students have higher grades, and we want to conduct a one-sided test, we need to the data for Anastasia's students _first_. Otherwise, we end up testing the hypothesis that _Bernadette's_ students had the higher grade, which is the opposite of what we intended:" ] }, { "cell_type": "code", "execution_count": null, "id": "stretch-disclosure", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Two-sided: Ttest_indResult(statistic=array([2.11543239]), pvalue=array([0.04252949]))\n", "\n", "One-sided, Anastasia first: Ttest_indResult(statistic=array([2.11543239]), pvalue=array([0.02126474]))\n", "\n", "One-sided, Bernadette first: Ttest_indResult(statistic=array([-2.11543239]), pvalue=array([0.97873526]))\n" ] } ], "source": [ "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/harpo.csv\")\n", "\n", "# create two new variables for the grades from each tutor's students\n", "Anastasia = pd.DataFrame(df.loc[df['tutor'] == 'Anastasia']['grade'])\n", "Bernadette = pd.DataFrame(df.loc[df['tutor'] == 'Bernadette']['grade'])\n", "\n", "# run an independent samples t-test\n", "from scipy import stats\n", "\n", "print('Two-sided:', stats.ttest_ind(Anastasia, Bernadette, equal_var = True))\n", "print('')\n", "print('One-sided, Anastasia first:', stats.ttest_ind(Anastasia, Bernadette, equal_var = True, alternative = 'greater'))\n", "print('')\n", "print('One-sided, Bernadette first:', stats.ttest_ind(Bernadette, Anastasia, equal_var = True, alternative = 'greater'))" ] }, { "attachments": {}, "cell_type": "markdown", "id": "timely-suite", "metadata": {}, "source": [ "What about the paired samples $t$-test? Suppose we wanted to test the hypothesis that grades go *up* from test 1 to test 2 in Dr. Chico's class, and are not prepared to consider the idea that the grades go down. Again, we can use the `alternative` argument to specify the one-sided test, and it works the same way it does for the independent samples $t$-test. Since we are comparing test 1 to test 2 by substracting one from the other, it makes a difference whether we subract test 1 from test 2, or test 2 from test 1. So, to test the hypothesis that grades for test 2 are higher than test 2, we will need to enter the grades from test 2 first; otherwise we are testing the opposite hypothesis: " ] }, { "cell_type": "code", "execution_count": null, "id": "understood-arabic", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "test 2 - test 1: Ttest_relResult(statistic=6.475436088339379, pvalue=1.660335028063474e-06)\n", "\n", "test 1 - test 2: Ttest_relResult(statistic=-6.475436088339379, pvalue=0.9999983396649719)\n" ] } ], "source": [ "import pandas as pd\n", "from scipy.stats import ttest_rel\n", "\n", "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/chico.csv\")\n", "\n", "print('test 2 - test 1:', ttest_rel(df['grade_test2'], df['grade_test1'], alternative = 'greater'))\n", "print('')\n", "print('test 1 - test 2:', ttest_rel(df['grade_test1'], df['grade_test2'], alternative = 'greater')) " ] }, { "attachments": {}, "cell_type": "markdown", "id": "driven-basketball", "metadata": {}, "source": [ "(cohensd)=\n", "## Effect size\n", "\n", "The most commonly used measure of effect size for a $t$-test is **_Cohen's $d$_** {cite}`Cohen1988`. It's a very simple measure in principle, with quite a few wrinkles when you start digging into the details. Cohen himself defined it primarily in the context of an independent samples $t$-test, specifically the Student test. In that context, a natural way of defining the effect size is to divide the difference between the means by an estimate of the standard deviation. In other words, we're looking to calculate *something* along the lines of this:\n", "\n", "$$\n", "d = \\frac{\\mbox{(mean 1)} - \\mbox{(mean 2)}}{\\mbox{std dev}}\n", "$$\n", "\n", "and he suggested a rough guide for interpreting $d$ in the table below. You'd think that this would be pretty unambiguous, but it's not; largely because Cohen wasn't too specific on what he thought should be used as the measure of the standard deviation (in his defence, he was trying to make a broader point in his book, not nitpick about tiny details). As discussed by {cite}`McGrath2006`, there are several different version in common usage, and each author tends to adopt slightly different notation. For the sake of simplicity (as opposed to accuracy) I'll use $d$ to refer to any statistic that you calculate from the sample, and use $\\delta$ to refer to a theoretical population effect. Obviously, that does mean that there are several different things all called $d$.\n", "\n", "Although Cohen's $d$ is a very commonly-reported measure of effect size, there are not currently any modules that provide built-in tools to calculate it. At least, not any that I am aware of. Luckily, it isn't too hard to calculate \"by hand\"." ] }, { "attachments": {}, "cell_type": "markdown", "id": "instructional-climb", "metadata": {}, "source": [ "(dinterpretation)=\n", "\n", "The table below gives a (very) rough guide to interpreting Cohen's $d$. My personal recommendation is to not use these blindly. The $d$ statistic has a natural interpretation in and of itself: it redescribes the different in means as the number of standard deviations that separates those means. So it's generally a good idea to think about what that means in practical terms. In some contexts a \\\"small\\\" effect could be of big practical importance. In other situations a \\\"large\\\" effect may not be all that interesting." ] }, { "attachments": {}, "cell_type": "markdown", "id": "experimental-bailey", "metadata": {}, "source": [ "| d-value | rough interpretation |\n", "| :-------- | :------------------- |\n", "| about 0.2 | \"small\" effect |\n", "| about 0.5 | \"moderate\" effect |\n", "| about 0.8 | \"large\" effect |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "proof-xerox", "metadata": {}, "source": [ "### Cohen's $d$ from one sample\n", "\n", "The simplest situation to consider is the one corresponding to a one-sample $t$-test. In this case, the one sample mean $\\bar{X}$ and one (hypothesised) population mean $\\mu_o$ to compare it to. Not only that, there's really only one sensible way to estimate the population standard deviation: we just use our usual estimate $\\hat{\\sigma}$. Therefore, we end up with the following as the only way to calculate $d$, \n", "\n", "$$\n", "d = \\frac{\\bar{X} - \\mu_0}{\\hat{\\sigma}}\n", "$$" ] }, { "cell_type": "code", "execution_count": 45, "id": "78e1558a", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Cohen's d: 0.5041691240370937\n" ] } ], "source": [ "import pandas as pd\n", "import statistics\n", "\n", "# load Zeppo data\n", "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/zeppo.csv\")\n", "\n", "d = (df['grades'].mean() - 67.5) / df['grades'].std()\n", "print('Cohen\\'s d:', d)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "normal-import", "metadata": {}, "source": [ "What does this effect size mean? Overall, then, the psychology students in Dr Zeppo's class are achieving grades (mean = 72.3\\%) that are about .5 standard deviations higher than the level that you'd expect (67.5\\%) if they were performing at the same level as other students. Judged against Cohen's rough guide, this is a moderate effect size." ] }, { "attachments": {}, "cell_type": "markdown", "id": "intense-lithuania", "metadata": {}, "source": [ "### Cohen's $d$ from a Student $t$ test\n", "\n", "The majority of discussions of Cohen's $d$ focus on a situation that is analogous to Student's independent samples $t$ test, and it's in this context that the story becomes messier, since there are several different versions of $d$ that you might want to use in this situation. To understand why there are multiple versions of $d$, it helps to take the time to write down a formula that corresponds to the true population effect size $\\delta$. It's pretty straightforward, \n", "\n", "$$\n", "\\delta = \\frac{\\mu_1 - \\mu_2}{\\sigma}\n", "$$\n", "\n", "where, as usual, $\\mu_1$ and $\\mu_2$ are the population means corresponding to group 1 and group 2 respectively, and $\\sigma$ is the standard deviation (the same for both populations). The obvious way to estimate $\\delta$ is to do exactly the same thing that we did in the $t$-test itself: use the sample means as the top line, and a pooled standard deviation estimate for the bottom line:\n", "\n", "$$\n", "d = \\frac{\\bar{X}_1 - \\bar{X}_2}{\\hat{\\sigma}_p}\n", "$$\n", "\n", "where $\\hat\\sigma_p$ is the exact same pooled standard deviation measure that appears in the $t$-test. This is the most commonly used version of Cohen's $d$ when applied to the outcome of a Student $t$-test ,and is sometimes referred to as Hedges' $g$ statistic {cite}`Hedges1981`." ] }, { "attachments": {}, "cell_type": "markdown", "id": "greek-quantum", "metadata": {}, "source": [ "However, there are other possibilities, which I'll briefly describe. Firstly, you may have reason to want to use only one of the two groups as the basis for calculating the standard deviation. This approach (often called Glass' $\\Delta$) only makes most sense when you have good reason to treat one of the two groups as a purer reflection of \"natural variation\" than the other. This can happen if, for instance, one of the two groups is a control group. Secondly, recall that in the usual calculation of the pooled standard deviation we divide by $N-2$ to correct for the bias in the sample variance; in one version of Cohen's $d$ this correction is omitted. Instead, we divide by $N$. This version makes sense primarily when you're trying to calculate the effect size in the sample; rather than estimating an effect size in the population. Finally, there is a version based on {cite}`Hedges1985`, who point out there is a small bias in the usual (pooled) estimation for Cohen's $d$. Thus they introduce a small correction, by multiplying the usual value of $d$ by $(N-3)/(N-2.25)$. \n", "\n", "In any case, ignoring all those variations that you could make use of if you wanted, let's have a look at how to calculate the default version. In particular, suppose we look at the data from Dr Harpo's class." ] }, { "cell_type": "code", "execution_count": 47, "id": "9ac2aff1", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Cohen's d: 0.74\n" ] } ], "source": [ "import pandas as pd\n", "from numpy import sqrt, mean, std\n", "\n", "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/harpo.csv\")\n", "\n", "# create two new variables for the grades from each tutor's students\n", "tutor1 = pd.DataFrame(df.loc[df['tutor'] == 'Anastasia']['grade'])\n", "tutor2 = pd.DataFrame(df.loc[df['tutor'] == 'Bernadette']['grade'])\n", "\n", "# find number of student grades for each tutor\n", "n1 = len(tutor1)\n", "n2 = len(tutor2)\n", "\n", "# find mean student grade for each tutor\n", "u1 = tutor1['grade'].mean()\n", "u2 = tutor2['grade'].mean()\n", "\n", "# find variance in students' grades for each tutor\n", "v1 = tutor1['grade'].var()\n", "v2 = tutor2['grade'].var()\n", "\n", "\n", "# find pooled standard deviation (square root of weighted variation, \n", "# divided by total N, with a correction for bias of -2)\n", "\n", "sp = sqrt(((n1 - 1)*v1 + (n2-1)*v2) / (n1 + n2-2))\n", "\n", "# calculate Cohen's d\n", "d = (u1 - u2) / sp\n", "print('Cohen\\'s d:', round(d, 2))" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a56b01d6", "metadata": {}, "source": [ "But still, it's nice to know that there is nothing mysterious going on here. ``pinguoin`` isn't using any witchcraft to calculate Cohen's d, but it does save us a lot of time." ] }, { "attachments": {}, "cell_type": "markdown", "id": "breathing-version", "metadata": {}, "source": [ "### Cohen's $d$ from a Welch test\n", "\n", "Suppose the situation you're in is more like the Welch test: you still have two independent samples, but you no longer believe that the corresponding populations have equal variances. When this happens, we have to redefine what we mean by the population effect size. I'll refer to this new measure as $\\delta^\\prime$, so as to keep it distinct from the measure $\\delta$ which we defined previously. What {cite}`Cohen1988` suggests is that we could define our new population effect size by averaging the two population variances. What this means is that we get:\n", "\n", "$$\n", "\\delta^\\prime = \\frac{\\mu_1 - \\mu_2}{\\sigma^\\prime}\n", "$$\n", "\n", "where\n", "\n", "$$\n", "\\sigma^\\prime = \\sqrt{\\displaystyle{\\frac{ {\\sigma_1}^2 + {\\sigma_2}^2}{2}}}\n", "$$\n", "\n", "This seems quite reasonable, but notice that none of the measures that we've discussed so far are attempting to estimate this new quantity. It might just be my own ignorance of the topic, but I'm only aware of one version of Cohen's $d$ that actually estimates the unequal-variance effect size $\\delta^\\prime$ rather than the equal-variance effect size $\\delta$.\n", "All we do to calculate $d$ for this version is substitute the sample means $\\bar{X}_1$ and $\\bar{X}_2$ and the corrected sample standard deviations $\\hat{\\sigma}_1$ and $\\hat{\\sigma}_2$ into the equation for $\\delta^\\prime$. This gives us the following equation for $d$, \n", "\n", "$$\n", "d = \\frac{\\bar{X}_1 - \\bar{X}_2}{\\sqrt{\\displaystyle{\\frac{ {\\hat\\sigma_1}^2 + {\\hat\\sigma_2}^2}{2}}}}\n", "$$\n", "\n", "as our estimate of the effect size. " ] }, { "cell_type": "code", "execution_count": 48, "id": "96980d01", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Cohen's d: 0.72\n" ] } ], "source": [ "# find mean student grade for each tutor\n", "u1 = tutor1['grade'].mean()\n", "u2 = tutor2['grade'].mean()\n", "\n", "# find variance in students' grades for each tutor\n", "v1 = tutor1['grade'].var()\n", "v2 = tutor2['grade'].var()\n", "\n", "\n", "# find the mean variance of the two samples\n", "s = sqrt((v1 + v2)/2)\n", "\n", "# calculate Cohen's d\n", "d = (u1 - u2) / s\n", "print('Cohen\\'s d:', round(d, 2))" ] }, { "attachments": {}, "cell_type": "markdown", "id": "entire-pressure", "metadata": {}, "source": [ "### Cohen's $d$ from a paired-samples test\n", "\n", "\n", "Finally, what should we do for a paired samples $t$-test? In this case, the answer depends on what it is you're trying to do. *If* you want to measure your effect sizes relative to the distribution of difference scores, the measure of $d$ that you calculate is just \n", "\n", "$$\n", "d = \\frac{\\bar{D}}{\\hat{\\sigma}_D}\n", "$$\n", "\n", "where $\\hat{\\sigma}_D$ is the estimate of the standard deviation of the differences. The calculation here is pretty straightforward" ] }, { "cell_type": "code", "execution_count": 55, "id": "hispanic-level", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Cohen's d: 1.49\n" ] } ], "source": [ "from numpy import mean, std\n", "\n", "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/chico.csv\")\n", "\n", "difference = df['grade_test2'] - df['grade_test1']\n", "\n", "mean_diff = mean(difference)\n", "sd_diff = std(difference)\n", "\n", "d = mean_diff/sd_diff\n", "print('Cohen\\'s d:', round(d, 2))\n", "\n", "# translator's note: this version of Cohen's d is slightly different than one produced by the R function cohensD with \"method\" set to \"paired\".\n", "# the R function returns a value of 1.447952, and I have not yet figured out where this discrepancy comes from. Not a huge difference, but still.." ] }, { "attachments": {}, "cell_type": "markdown", "id": "fewer-complex", "metadata": {}, "source": [ "The only wrinkle is figuring out whether this is the measure you want or not. To the extent that you care about the practical consequences of your research, you often want to measure the effect size relative to the *original* variables, not the *difference* scores (e.g., the 1\\% improvement in Dr Chico's class is pretty small when measured against the amount of between-student variation in grades), in which case you use the same versions of Cohen's $d$ that you would use for a Student or Welch test. For instance, when we do that for Dr Chico's class, " ] }, { "cell_type": "code", "execution_count": 54, "id": "complex-antenna", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Cohen's d: 0.22\n" ] } ], "source": [ "# find mean student grade for each tutor\n", "u1 = df['grade_test1'].mean()\n", "u2 = df['grade_test2'].mean()\n", "\n", "# find variance in students' grades for each tutor\n", "v1 = df['grade_test1'].var()\n", "v2 = df['grade_test2'].var()\n", "\n", "\n", "# find the mean variance of the two samples\n", "s = sqrt((v1 + v2)/2)\n", "\n", "# calculate Cohen's d\n", "d = (u2 - u1) / s\n", "print('Cohen\\'s d:', round(d, 2))" ] }, { "attachments": {}, "cell_type": "markdown", "id": "dietary-illustration", "metadata": {}, "source": [ "what we see is that the overall effect size is quite small, when assessed on the scale of the original variables." ] }, { "attachments": {}, "cell_type": "markdown", "id": "painted-blend", "metadata": {}, "source": [ "(shapiro)=\n", "## Checking the normality of a sample\n", "\n", "All of the tests that we have discussed so far in this chapter have assumed that the data are normally distributed. This assumption is often quite reasonable, because the [central limit theorem](clt) does tend to ensure that many real world quantities are normally distributed: any time that you suspect that your variable is *actually* an average of lots of different things, there's a pretty good chance that it will be normally distributed; or at least close enough to normal that you can get away with using $t$-tests. However, life doesn't come with guarantees; and besides, there are lots of ways in which you can end up with variables that are highly non-normal. For example, any time you think that your variable is actually the minimum of lots of different things, there's a very good chance it will end up quite skewed. In psychology, response time (RT) data is a good example of this. If you suppose that there are lots of things that could trigger a response from a human participant, then the actual response will occur the first time one of these trigger events occurs.[^note13] This means that RT data are systematically non-normal. Okay, so if normality is assumed by all the tests, and is mostly but not always satisfied (at least approximately) by real world data, how can we check the normality of a sample? In this section I discuss two methods: QQ plots, and the Shapiro-Wilk test." ] }, { "attachments": {}, "cell_type": "markdown", "id": "light-speed", "metadata": {}, "source": [ "(qq)=\n", "### QQ plots\n", "\n", "One way to check whether a sample violates the normality assumption is to draw a **_\"quantile-quantile\" plot_** (QQ plot). This allows you to visually check whether you're seeing any systematic violations. In a QQ plot, each observation is plotted as a single dot. The x co-ordinate is the theoretical quantile that the observation should fall in, if the data were normally distributed (with mean and variance estimated from the sample) and on the y co-ordinate is the actual quantile of the data within the sample. If the data are normal, the dots should form a straight line. For instance, lets see what happens if we generate data by sampling from a normal distribution, and then drawing a QQ plot, using `probplot` from `scipy`. We can compare this with a histogram of the data as well:" ] }, { "cell_type": "code", "execution_count": 64, "id": "opposite-bachelor", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/text/plain": "" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "qq_fig" } }, "output_type": "display_data" }, { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import seaborn as sns\n", "from matplotlib import pyplot as plt\n", "from scipy.stats import probplot\n", "\n", "np.random.seed(42)\n", "normal_data = np.random.normal(size=100)\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 5))\n", "\n", "probplot(normal_data, dist=\"norm\", plot = axes[0]);\n", "sns.histplot(normal_data, axes=axes[1]);\n", "\n", "# format the figures\n", "titles = ['QQ plot', 'Histogram']\n", "for i, ax in enumerate(axes):\n", " ax.spines[['right', 'top']].set_visible(False)\n", " ax.set_title(titles[i])\n", "\n", "\n", "# show the figure in the book with caption\n", "\n", "from myst_nb import glue\n", "glue(\"qq_fig\", ax, display=False)\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "helpful-positive", "metadata": {}, "source": [ " ```{glue:figure} qq_fig\n", ":figwidth: 600px\n", ":name: fig-qq\n", "\n", "QQ plot (left) and histogram (right) of `normal_data`, a normally distributed sample with 100 observations. The Shapiro-Wilk statistic associated with these data is $W = .99$, indicating that no significant departures from normality were detected ($p = .73$).\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "muslim-adoption", "metadata": {}, "source": [ "And the results are shown in {numref}`fig-qq`, above.\n", "\n", "As you can see, these data form a pretty straight line; which is no surprise given that we sampled them from a normal distribution! In contrast, have a look at the data shown in {numref}`fig-qqskew` and {numref}`fig-qqheavy`, which show the histogram and a QQ plot for a data sets that are highly skewed and have a heavy tail (i.e., high kurtosis), respectively." ] }, { "cell_type": "code", "execution_count": null, "id": "limited-renaissance", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/skewed_data.csv\")\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 5))\n", "\n", "probplot(df['data'], dist=\"norm\", plot = axes[0])\n", "sns.histplot(df['data'], axes=axes[1])\n", "\n", "# format the figures\n", "titles = ['QQ plot', 'Histogram']\n", "for i, ax in enumerate(axes):\n", " ax.spines[['right', 'top']].set_visible(False)\n", " ax.set_title(titles[i])\n", "\n", "glue(\"qqskew_fig\", ax, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "addressed-residence", "metadata": {}, "source": [ " ```{glue:figure} qqskew_fig\n", ":figwidth: 600px\n", ":name: fig-qqskew\n", "\n", "The skewness of these data of 100 observations is 1.94, and is reflected in a QQ plot that curves upwards. As a consequence, the Shapiro-Wilk statistic is $W=.80$, reflecting a significant departure from normality ($p<.001$). \n", "\n", "```" ] }, { "cell_type": "code", "execution_count": null, "id": "hundred-pharmaceutical", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/text/plain": "" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "qqheavy_fig" } }, "output_type": "display_data" }, { "data": { "image/png": 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/heavy_tailed_data.csv\")\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 5))\n", "\n", "qq = probplot(df['data'], dist=\"norm\", plot = axes[0])\n", "hist = sns.histplot(df['data'], axes=axes[1])\n", "\n", "# format the figures\n", "titles = ['QQ plot', 'Histogram']\n", "for i, ax in enumerate(axes):\n", " ax.spines[['right', 'top']].set_visible(False)\n", " ax.set_title(titles[i])\n", "\n", "glue(\"qqheavy_fig\", ax, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "complete-bacon", "metadata": {}, "source": [ " ```{glue:figure} qqheavy_fig\n", ":figwidth: 600px\n", ":name: fig-qqheavy\n", "\n", "Plots for a heavy tailed data set, again consisting of 100 observations. In this case, the heavy tails in the data produce a high kurtosis (2.80), and cause the QQ plot to flatten in the middle, and curve away sharply on either side. The resulting Shapiro-Wilk statistic is $W = .93$, again reflecting significant non-normality ($p < .001$).\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "tribal-diary", "metadata": {}, "source": [ "(shapiro-wilk)=\n", "\n", "### Shapiro-Wilk tests\n", "\n", "Although QQ plots provide a nice way to informally check the normality of your data, sometimes you'll want to do something a bit more formal. And when that moment comes, the **_Shapiro-Wilk test_** {cite}`Shapiro1965` is probably what you're looking for.[^note14] As you'd expect, the null hypothesis being tested is that a set of $N$ observations is normally distributed. The test statistic that it calculates is conventionally denoted as $W$, and it's calculated as follows. First, we sort the observations in order of increasing size, and let $X_1$ be the smallest value in the sample, $X_2$ be the second smallest and so on. Then the value of $W$ is given by\n", "\n", "$$\n", "W = \\frac{ \\left( \\sum_{i = 1}^N a_i X_i \\right)^2 }{ \\sum_{i = 1}^N (X_i - \\bar{X})^2}\n", "$$\n", "\n", "where $\\bar{X}$ is the mean of the observations, and the $a_i$ values are ... mumble, mumble ... something complicated that is a bit beyond the scope of an introductory text. \n", "\n", "Because it's a little hard to explain the maths behind the $W$ statistic, a better idea is to give a broad brush description of how it behaves. Unlike most of the test statistics that we'll encounter in this book, it's actually *small* values of $W$ that indicated departure from normality. The $W$ statistic has a maximum value of 1, which arises when the data look \"perfectly normal\". The smaller the value of $W$, the less normal the data are. However, the sampling distribution for $W$ -- which is not one of the standard ones that I discussed in chapter on [probability](probability) and is in fact a complete pain in the arse to work with -- does depend on the sample size $N$. To give you a feel for what these sampling distributions look like, I've plotted three of them in {numref}`fig-swdist`. Notice that, as the sample size starts to get large, the sampling distribution becomes very tightly clumped up near $W=1$, and as a consequence, for larger samples $W$ doesn't have to be very much smaller than 1 in order for the test to be significant. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "sought-highlight", "metadata": {}, "source": [ "```{figure} ../img/ttest2/shapirowilkdist.png\n", ":name: fig-swdist\n", ":width: 600px\n", ":align: center\n", "\n", "Sampling distribution of the Shapiro-Wilk $W$ statistic, under the null hypothesis that the data are normally distributed, for samples of size 10, 20 and 50. Note that *small* values of $W$ indicate departure from normality.\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "demonstrated-wholesale", "metadata": {}, "source": [ "To run the test in Python, we use the `scipy.stats.shapiro` method. It has only a single argument `x`, which is a numeric vector containing the data whose normality needs to be tested. For example, when we apply this function to our `normal_data`, we get the following:" ] }, { "cell_type": "code", "execution_count": null, "id": "raised-suspect", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "ShapiroResult(statistic=0.9898834228515625, pvalue=0.6551706790924072)" ] }, "execution_count": 708, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from scipy.stats import shapiro\n", "\n", "shapiro(normal_data)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "liquid-departure", "metadata": {}, "source": [ "So, not surprisingly, we have no evidence that these data depart from normality. When reporting the results for a Shapiro-Wilk test, you should (as usual) make sure to include the test statistic $W$ and the $p$ value, though given that the sampling distribution depends so heavily on $N$ it would probably be a politeness to include $N$ as well." ] }, { "attachments": {}, "cell_type": "markdown", "id": "direct-theorem", "metadata": {}, "source": [ "(wilcox)=\n", "## Testing non-normal data with Wilcoxon tests\n", "\n", "Okay, suppose your data turn out to be pretty substantially non-normal, but you still want to run something like a $t$-test? This situation occurs a lot in real life: for the AFL winning margins data, for instance, the Shapiro-Wilk test made it very clear that the normality assumption is violated. This is the situation where you want to use Wilcoxon tests. \n", "\n", "Like the $t$-test, the Wilcoxon test comes in two forms, one-sample and two-sample, and they're used in more or less the exact same situations as the corresponding $t$-tests. Unlike the $t$-test, the Wilcoxon test doesn't assume normality, which is nice. In fact, they don't make any assumptions about what kind of distribution is involved: in statistical jargon, this makes them **_nonparametric tests_**. While avoiding the normality assumption is nice, there's a drawback: the Wilcoxon test is usually less powerful than the $t$-test (i.e., higher Type II error rate). I won't discuss the Wilcoxon tests in as much detail as the $t$-tests, but I'll give you a brief overview." ] }, { "attachments": {}, "cell_type": "markdown", "id": "conservative-freeze", "metadata": {}, "source": [ "### Two sample Wilcoxon test\n", "\n", "I'll start by describing the **_two sample Wilcoxon test_** (also known as the Mann-Whitney test), since it's actually simpler than the one sample version. Suppose we're looking at the scores of 10 people on some test. Since my imagination has now failed me completely, let's pretend it's a \"test of awesomeness\", and there are two groups of people, \"A\" and \"B\". I'm curious to know which group is more awesome. The data are included in the file `awesome.Rdata`, and like many of the data sets I've been using, it contains only a single data frame, in this case called `awesome`. Here's the data:" ] }, { "cell_type": "code", "execution_count": null, "id": "broken-picking", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " score_A score_B\n", "0 6.4 14.5\n", "1 10.7 10.4\n", "2 11.9 12.9\n", "3 7.3 11.7\n", "4 10.0 13.0" ] }, "execution_count": 740, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/awesome2.csv\")\n", "df" ] }, { "attachments": {}, "cell_type": "markdown", "id": "healthy-lancaster", "metadata": {}, "source": [ "As long as there are no ties (i.e., people with the exact same awesomeness score), then the test that we want to do is surprisingly simple. All we have to do is construct a table that compares every observation in group $A$ against every observation in group $B$. Whenever the group $A$ datum is larger, we place a check mark in the table:" ] }, { "attachments": {}, "cell_type": "markdown", "id": "likely-jaguar", "metadata": {}, "source": [ "\n", "| | | | group B | | | |\n", "|---------|------|------|---------|------|------|----|\n", "| | | 14.5 | 10.4 | 12.4 | 11.7 | 13 |\n", "| | 6.4 | . | . | . | . | . |\n", "| group A | 10.7 | . | ✓ | . | . | . |\n", "| | 11.9 | . | ✓ | . | ✓ | . |\n", "| | 7.3 | . | . | . | . | . |\n", "| | 10 | . | . | . | . | . |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "perceived-cross", "metadata": {}, "source": [ "We then count up the number of checkmarks. This is our test statistic, $W$.[^note15] The actual sampling distribution for $W$ is somewhat complicated, and I'll skip the details. For our purposes, it's sufficient to note that the interpretation of $W$ is qualitatively the same as the interpretation of $t$ or $z$. That is, if we want a two-sided test, then we reject the null hypothesis when $W$ is very large or very small; but if we have a directional (i.e., one-sided) hypothesis, then we only use one or the other. " ] }, { "cell_type": "code", "execution_count": null, "id": "romantic-algebra", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(1.0, 0.125)" ] }, "execution_count": 741, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from scipy.stats import wilcoxon\n", " \n", "w,p = wilcoxon(df['score_A'], df['score_B'] )\n", "w,p" ] }, { "attachments": {}, "cell_type": "markdown", "id": "soviet-alloy", "metadata": {}, "source": [ "### One sample Wilcoxon test\n", "\n", "\n", "What about the **_one sample Wilcoxon test_** (or equivalently, the paired samples Wilcoxon test)? Suppose I'm interested in finding out whether taking a statistics class has any effect on the happiness of students. Here's my data:" ] }, { "cell_type": "code", "execution_count": null, "id": "closing-conspiracy", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " before after change\n", "0 30 6 -24\n", "1 43 29 -14\n", "2 21 11 -10\n", "3 24 31 7\n", "4 23 17 -6\n", "5 40 2 -38\n", "6 29 31 2\n", "7 56 21 -35\n", "8 38 8 -30\n", "9 16 21 5" ] }, "execution_count": 743, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/happiness.csv\")\n", "df" ] }, { "attachments": {}, "cell_type": "markdown", "id": "interpreted-remove", "metadata": {}, "source": [ "What I've measured here is the happiness of each student `before` taking the class and `after` taking the class; the `change` score is the difference between the two. Just like we saw with the $t$-test, there's no fundamental difference between doing a paired-samples test using `before` and `after`, versus doing a one-sample test using the `change` scores. As before, the simplest way to think about the test is to construct a tabulation. The way to do it this time is to take those change scores that are positive valued, and tabulate them against all the complete sample. What you end up with is a table that looks like this:" ] }, { "attachments": {}, "cell_type": "markdown", "id": "athletic-obligation", "metadata": {}, "source": [ "| | all differences | | | | | | | | | | |\n", "|----------------------|---|-----|-----|-----|---|-----------------|-----|---|-----|-----|---|\n", "| | | -24 | -14 | -10 | 7 | -6 | -38 | 2 | -35 | -30 | 5 |\n", "| positive differences | 7 | . | . | . | ✓ | ✓ | . | ✓ | . | | ✓ |\n", "| | 2 | . | . | . | . | . | . | ✓ | . | . | |\n", "| | 5 | . | . | . | . | . | . | ✓ | . | . | ✓ |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "loaded-foundation", "metadata": {}, "source": [ "Counting up the tick marks this time, we get a test statistic of $V = 7$. As before, if our test is two sided, then we reject the null hypothesis when $V$ is very large or very small. As far of running it with Python goes, it's pretty much what you'd expect. For the one-sample version, the command you would use is" ] }, { "cell_type": "code", "execution_count": null, "id": "streaming-tuition", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "(7.0, 0.037109375)" ] }, "execution_count": 745, "metadata": {}, "output_type": "execute_result" } ], "source": [ "w,p = wilcoxon(df['change'])\n", "w,p" ] }, { "attachments": {}, "cell_type": "markdown", "id": "indoor-governor", "metadata": {}, "source": [ "As this shows, we have a significant effect. Evidently, taking a statistics class does have an effect on your happiness.\n", "\n", "## Summary\n", "\n", "- A [one sample $t$-test](ttest_onesample) is used to compare a single sample mean against a hypothesised value for the population mean.\n", "- An independent samples $t$-test is used to compare the means of two groups, and tests the null hypothesis that they have the same mean. It comes in two forms: the [Student test](studentttest) assumes that the groups have the same standard deviation, the [Welch test](welchttest) does not.\n", "- A [paired samples $t$-test](pairedsamplesttest) is used when you have two scores from each person, and you want to test the null hypothesis that the two scores have the same mean. It is equivalent to taking the difference between the two scores for each person, and then running a one sample $t$-test on the difference scores.\n", "- [Effect size calculations](cohensd) for the difference between means can be calculated via the Cohen's $d$ statistic..\n", "- You can check the normality of a sample using [QQ plots](qq) and the [Shapiro-Wilk test](shapiro-wilk).\n", "- If your data are non-normal, you can use [Wilcoxon tests](wilcox) instead of $t$-tests." ] }, { "attachments": {}, "cell_type": "markdown", "id": "former-token", "metadata": {}, "source": [ "[^note1]: We won't cover multiple predictors until the chapter on [regression](regression).\n", "\n", "[^note2]: Informal experimentation in my garden suggests that yes, it does. Australian natives are adapted to low phosphorus levels relative to everywhere else on Earth, apparently, so if you've bought a house with a bunch of exotics and you want to plant natives, don't follow my example: keep them separate. Nutrients to European plants are poison to Australian ones. There's probably a joke in that, but I can't figure out what it is.\n", "\n", "[^note3]: Actually this is too strong. Strictly speaking the $z$ test only requires that the sampling distribution of the mean be normally distributed; if the population is normal then it necessarily follows that the sampling distribution of the mean is also normal. However, as we saw when talking about the central limit theorem, it's quite possible (even commonplace) for the sampling distribution to be normal even if the population distribution itself is non-normal. However, in light of the sheer ridiculousness of the assumption that the true standard deviation is known, there really isn't much point in going into details on this front!\n", "\n", "[^note4]: Well, sort of. As I understand the history, Gosset only provided a partial solution: the general solution to the problem was provided by Sir Ronald Fisher.\n", "\n", "[^note5]: More seriously, I tend to think the reverse is true: I get very suspicious of technical reports that fill their results sections with nothing except the numbers. It might just be that I'm an arrogant jerk, but I often feel like an author that makes no attempt to explain and interpret their analysis to the reader either doesn't understand it themselves, or is being a bit lazy. Your readers are smart, but not infinitely patient. Don't annoy them if you can help it.\n", "\n", "[^note6]: Although it is the simplest, which is why I started with it.\n", "\n", "[^note7]: A funny question almost always pops up at this point: what the heck *is* the population being referred to in this case? Is it the set of students actually taking Dr Harpo's class (all 33 of them)? The set of people who might take the class (an unknown number) of them? Or something else? Does it matter which of these we pick? It's traditional in an introductory behavioural stats class to mumble a lot at this point, but since I get asked this question every year by my students, I'll give a brief answer. Technically yes, it does matter: if you change your definition of what the \"real world\" population actually is, then the sampling distribution of your observed mean $\\bar{X}$ changes too. The $t$-test relies on an assumption that the observations are sampled at random from an infinitely large population; and to the extent that real life isn't like that, then the $t$-test can be wrong. In practice, however, this isn't usually a big deal: even though the assumption is almost always wrong, it doesn't lead to a lot of pathological behaviour from the test, so we tend to just ignore it.\n", "\n", "[^note8]: Yes, I have a \"favourite\" way of thinking about pooled standard deviation estimates. So what?\n", "\n", "[^note9]: Well, I guess you can average apples and oranges, and what you end up with is a delicious fruit smoothie. But no one really thinks that a fruit smoothie is a very good way to describe the original fruits, do they?\n", "\n", "[^note10]: This design is very similar to the one in [](chisquare) that motivated the McNemar test. This should be no surprise. Both are standard repeated measures designs involving two measurements. The only difference is that this time our outcome variable is interval scale (working memory capacity) rather than a binary, nominal scale variable (a yes-or-no question).\n", "\n", "[^note11]: At this point we have Drs Harpo, Chico and Zeppo. No prizes for guessing who Dr Groucho is.\n", "\n", "[^note12]: This is obviously a class being taught at a very small or very expensive university, or else is a postgraduate class. *I've* never taught an intro stats class with less than 350 students.\n", "\n", "[^note13]: This is a massive oversimplification.\n", "\n", "[^note14]: Either that, or the Kolmogorov-Smirnov test, which is probably more traditional than the Shapiro-Wilk, though most things I've read seem to suggest Shapiro-Wilk is the better test of normality, although Kolomogorov-Smirnov is a general purpose test of distributional equivalence, so it can be adapted to handle other kinds of distribution tests.\n", "\n", "[^note15]: Actually, there are two different versions of the test statistic; they differ from each other by a constant value. I'll just describe one of them here." ] } ], "metadata": { "celltoolbar": "Tags", "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.3" } }, "nbformat": 4, "nbformat_minor": 5 } (/hdKa0K k LqxԈL9d'K} U 333]۵TUvpܠc45im3:4?%O ?k]խukڍa%BH)DD\x&_t XcZʘ]`UGhLn%_ylv+odl12Z+:oSlklJ3}խ}>3}hS¤p^94Խlb-ఎ#q_MfŠ b :P! @6A@L@D@B:"DARq`FFާi_Dx Ig$EZw jp.!5m=龮|3BZXmR/vƧTvr^/tH9e̒gO.{cjy!uEFn + *QiueHI6,yD۫cv1"Z)dd @V03Dr؇2aC.5Z•-pPɡb'uJ75tJ-fovJ-}eap[[v8A/fZc: 39M1yM1hiBiIrݍ>M209> ACZ+Ju<ž\}ruMim(kyjpn[}J:իڭg@G;4̰] u_ A@\.'btk*,$dJnC7PDt>/w*ȗ#cͅ{4tԝÚɭFW`n. **,To~h0fD8tQB@[S1#aH>l;!㡢#|]J;AUP( "y_IX97߰1~cCY?c46 ի14swT?94VURZ9*)@ YeV)3iV9zи c/2ʦRZ@ސM9;W6oi[!Cg~Zr{lSKH.痶^c9HRvt926;6qw](1$Zq^Gy_&+#{öm*/]s9lDEk~А5%فO+j?3`}bFk]?I3ro::́k{3ߐ rb8k_SGZ{ΕA3|[ 6n b{Yoc:oKkCotǖpy˜u@$ oc&l\^r2Z86fSָ/M܃ *<ʄ‘^UL&$@e"WiH\8ET8QI (0@fs@D͘ Mz pO -cʳ66PV"c 3c8"(_?8 5hZA.utStA+R##`KH`U#DcREq B1 !  *w*hX6-ke1!w+vH @,q?./Kn . 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The basic technique was developed by Sir Ronald Fisher in the early 20th century, and it is to him that we owe the rather unfortunate terminology. The term ANOVA is a little misleading, in two respects. Firstly, although the name of the technique refers to variances, ANOVA is concerned with investigating differences in means. Secondly, there are several different things out there that are all referred to as ANOVAs, some of which have only a very tenuous connection to one another. Third (and this is Ethan speaking now) _everything_ in statistics is an analysis of variance: analyzing variance is the entire purpose of statistics, so it seems a little unfair to give that name to any particular test. I challenge you to show me any statistical procedure that is not, in some sense, an analysis of variance. Still, Fisher got there first, so what can you do? Later on in the book we'll encounter a range of different ANOVA methods that apply in quite different situations, but for the purposes of this chapter we'll only consider the simplest form of ANOVA, in which we have several different groups of observations, and we're interested in finding out whether those groups differ in terms of some outcome variable of interest. This is the question that is addressed by a **_one-way ANOVA_**. \n", "\n", "The structure of this chapter is as follows: In [](anxifree) I'll introduce a fictitious data set that we'll use as a running example throughout the chapter. After introducing the data, I'll describe the mechanics of [how a one-way ANOVA actually works](anovaintro) and then focus on [how you can run one in Python](introduceaov). These two sections are the core of the chapter. The remainder of the chapter discusses a range of important topics that inevitably arise when running an ANOVA, namely [how to calculate effect sizes](etasquared), [post hoc tests and corrections for multiple comparisons](posthoc) and the [assumptions](anovaassumptions) that ANOVA relies upon. We'll also talk about how to check those assumptions and some of the things you can do if the assumptions are violated in the sections from [](levene) to [](kruskalwallis). At the end of the chapter we'll talk a little about the [relationship](anovaandt) between ANOVA and other statistical tools. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "dbeb4731", "metadata": {}, "source": [ "(anxifree)=\n", "## An illustrative data set\n", "\n", "Suppose you've become involved in a clinical trial in which you are testing a new antidepressant drug called *Joyzepam*. In order to construct a fair test of the drug's effectiveness, the study involves three separate drugs to be administered. One is a placebo, and the other is an existing antidepressant / anti-anxiety drug called *Anxifree*. A collection of 18 participants with moderate to severe depression are recruited for your initial testing. Because the drugs are sometimes administered in conjunction with psychological therapy, your study includes 9 people undergoing cognitive behavioural therapy (CBT) and 9 who are not. Participants are randomly assigned (doubly blinded, of course) a treatment, such that there are 3 CBT people and 3 no-therapy people assigned to each of the 3 drugs. A psychologist assesses the mood of each person after a 3 month run with each drug: and the overall *improvement* in each person's mood is assessed on a scale ranging from $-5$ to $+5$. \n", "\n", "With that as the study design, let's now look at what we've got in the data file:" ] }, { "cell_type": "code", "execution_count": 1, "id": "3c54e65d", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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drugtherapymood_gain
0placebono.therapy0.5
1placebono.therapy0.3
2placebono.therapy0.1
3anxifreeno.therapy0.6
4anxifreeno.therapy0.4
5anxifreeno.therapy0.2
6joyzepamno.therapy1.4
7joyzepamno.therapy1.7
8joyzepamno.therapy1.3
9placeboCBT0.6
10placeboCBT0.9
11placeboCBT0.3
12anxifreeCBT1.1
13anxifreeCBT0.8
14anxifreeCBT1.2
15joyzepamCBT1.8
16joyzepamCBT1.3
17joyzepamCBT1.4
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" ], "text/plain": [ " drug therapy mood_gain\n", "0 placebo no.therapy 0.5\n", "1 placebo no.therapy 0.3\n", "2 placebo no.therapy 0.1\n", "3 anxifree no.therapy 0.6\n", "4 anxifree no.therapy 0.4\n", "5 anxifree no.therapy 0.2\n", "6 joyzepam no.therapy 1.4\n", "7 joyzepam no.therapy 1.7\n", "8 joyzepam no.therapy 1.3\n", "9 placebo CBT 0.6\n", "10 placebo CBT 0.9\n", "11 placebo CBT 0.3\n", "12 anxifree CBT 1.1\n", "13 anxifree CBT 0.8\n", "14 anxifree CBT 1.2\n", "15 joyzepam CBT 1.8\n", "16 joyzepam CBT 1.3\n", "17 joyzepam CBT 1.4" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/clintrial.csv\")\n", "df" ] }, { "attachments": {}, "cell_type": "markdown", "id": "d5aa7346", "metadata": {}, "source": [ "So we have a single data frame called `clin.trial`, containing three variables; `drug`, `therapy` and `mood_gain`." ] }, { "attachments": {}, "cell_type": "markdown", "id": "30d9a6ab", "metadata": {}, "source": [ "For the purposes of this chapter, what we're really interested in is the effect of `drug` on `mood_gain`. The first thing to do is calculate some descriptive statistics and draw some graphs. In the chapters on [descriptive statistics](descriptives) and [data-wrangling](datawrangling) we discussed a variety of different functions that can be used for this purpose. For instance, we can use the `pd.crosstab()` function to see how many people we have in each group:" ] }, { "cell_type": "code", "execution_count": 2, "id": "5d1c335f", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
therapyCBTno.therapy
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placebo33
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" ], "text/plain": [ "therapy CBT no.therapy\n", "drug \n", "anxifree 3 3\n", "joyzepam 3 3\n", "placebo 3 3" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pd.crosstab(df['drug'], df['therapy'])" ] }, { "attachments": {}, "cell_type": "markdown", "id": "76588ebe", "metadata": {}, "source": [ "Similarly, we can use the `aggregate()` function to calculate means and standard deviations for the `mood_gain` variable broken down by which `drug` was administered:" ] }, { "cell_type": "code", "execution_count": 3, "id": "979991d0", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
meanstd
drug
anxifree0.7166670.392003
joyzepam1.4833330.213698
placebo0.4500000.281069
\n", "
" ], "text/plain": [ " mean std\n", "drug \n", "anxifree 0.716667 0.392003\n", "joyzepam 1.483333 0.213698\n", "placebo 0.450000 0.281069" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.groupby('drug')['mood_gain'].agg(['mean', 'std'])" ] }, { "attachments": {}, "cell_type": "markdown", "id": "dbd10fbb", "metadata": {}, "source": [ "Finally, we can use `pointplot()` from the `seaborn` package to produce a pretty picture." ] }, { "cell_type": "code", "execution_count": 4, "id": "e68da6dc", "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "\n", "fig = sns.pointplot(x='drug', y = 'mood_gain', data = df)\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "7aeb0f1f", "metadata": {}, "source": [ " ```{glue:figure} moodgain_fig\n", ":figwidth: 600px\n", ":name: fig-moodgain\n", "\n", "Average mood gain as a function of drug administered. Error bars depict 95% confidence intervals associated with each of the group means.\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "053f1e4e", "metadata": {}, "source": [ "\n", "The results are shown in the figure above, which plots the average mood gain for all three conditions; error bars show 95\\% confidence intervals. As the plot makes clear, there is a larger improvement in mood for participants in the Joyzepam group than for either the Anxifree group or the placebo group. The Anxifree group shows a larger mood gain than the control group, but the difference isn't as large. \n", "\n", "The question that we want to answer is: are these difference \"real\", or are they just due to chance?\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "f8438ac7", "metadata": {}, "source": [ "(anovaintro)=\n", "## How ANOVA works\n", "\n", "In order to answer the question posed by our clinical trial data, we're going to run a one-way ANOVA. As usual, I'm going to start by showing you how to do it the hard way, building the statistical tool from the ground up and showing you how you could do it in Python if you didn't have access to any of the cool ANOVA functions that kind people have made for us. And, as always, I hope you'll read it carefully, try to do it the long way once or twice to make sure you really understand how ANOVA works, and then -- once you've grasped the concept -- never *ever* do it this way again.\n", "\n", "The experimental design that I described in the previous section strongly suggests that we're interested in comparing the average mood change for the three different drugs. In that sense, we're talking about an analysis similar to the [$t$-test](ttest), but involving more than two groups. If we let $\\mu_P$ denote the population mean for the mood change induced by the placebo, and let $\\mu_A$ and $\\mu_J$ denote the corresponding means for our two drugs, Anxifree and Joyzepam, then the (somewhat pessimistic) null hypothesis that we want to test is that all three population means are identical: that is, *neither* of the two drugs is any more effective than a placebo. Mathematically, we write this null hypothesis like this:\n", "\n", "$$\n", "\\begin{array}{rcl}\n", "H_0 &:& \\mbox{it is true that } \\mu_P = \\mu_A = \\mu_J\n", "\\end{array}\n", "$$\n", "\n", "As a consequence, our alternative hypothesis is that at least one of the three different treatments is different from the others. It's a little trickier to write this mathematically, because (as we'll discuss) there are quite a few different ways in which the null hypothesis can be false. So for now we'll just write the alternative hypothesis like this:\n", "\n", "$$\n", "\\begin{array}{rcl}\n", "H_1 &:& \\mbox{it is *not* true that } \\mu_P = \\mu_A = \\mu_J\n", "\\end{array}\n", "$$\n", "\n", "This null hypothesis is a lot trickier to test than any of the ones we've seen previously. How shall we do it? A sensible guess would be to \"do an ANOVA\", since that's the title of the chapter, but it's not particularly clear why an \"analysis of *variances*\" will help us learn anything useful about the *means*. In fact, this is one of the biggest conceptual difficulties that people have when first encountering ANOVA. To see how this works, I find it most helpful to start by talking about variances. In fact, what I'm going to do is start by playing some mathematical games with the formula that describes the variance. That is, we'll start out by playing around with variances, and it will turn out that this gives us a useful tool for investigating means." ] }, { "attachments": {}, "cell_type": "markdown", "id": "9032c226", "metadata": {}, "source": [ "### Two formulas for the variance of $Y$\n", "\n", "Firstly, let's start by introducing some notation. We'll use $G$ to refer to the total number of groups. For our data set, there are three drugs, so there are $G=3$ groups. Next, we'll use $N$ to refer to the total sample size: there are a total of $N=18$ people in our data set. Similarly, let's use $N_k$ to denote the number of people in the $k$-th group. In our fake clinical trial, the sample size is $N_k = 6$ for all three groups.[^note1] Finally, we'll use $Y$ to denote the outcome variable: in our case, $Y$ refers to mood change. Specifically, we'll use $Y_{ik}$ to refer to the mood change experienced by the $i$-th member of the $k$-th group. Similarly, we'll use $\\bar{Y}$ to be the average mood change, taken across all 18 people in the experiment, and $\\bar{Y}_k$ to refer to the average mood change experienced by the 6 people in group $k$. \n", "\n", "Excellent. Now that we've got our notation sorted out, we can start writing down formulas. To start with, let's recall the formula for the variance that we used in, way back in those [kinder days](var) when we were just doing descriptive statistics. The sample variance of $Y$ is defined as follows:\n", "\n", "$$\n", "\\mbox{Var}(Y) = \\frac{1}{N} \\sum_{k=1}^G \\sum_{i=1}^{N_k} \\left(Y_{ik} - \\bar{Y} \\right)^2\n", "$$\n", "\n", "This formula looks pretty much identical to the formula for the variance that we [used earlier](var) in the chapter on descriptive statistics. The only difference is that this time around I've got two summations here: I'm summing over groups (i.e., values for $k$) and over the people within the groups (i.e., values for $i$). This is purely a cosmetic detail: if I'd instead used the notation $Y_p$ to refer to the value of the outcome variable for person $p$ in the sample, then I'd only have a single summation. The only reason that we have a double summation here is that I've classified people into groups, and then assigned numbers to people within groups. \n", "\n", "A concrete example might be useful here. Let's consider this table, in which we have a total of $N=5$ people sorted into $G=2$ groups. Arbitrarily, let's say that the \"cool\" people are group 1, and the \"uncool\" people are group 2, and it turns out that we have three cool people ($N_1 = 3$) and two uncool people ($N_2 = 2$)." ] }, { "attachments": {}, "cell_type": "markdown", "id": "162d74d6", "metadata": {}, "source": [ "|name |person ($p$) |group |group num ($k$) |index in group ($i$) |grumpiness ($Y_{ik}$ or $Y_p$) |\n", "|:----|:------------|:------|:---------------|:--------------------|:------------------------------|\n", "|Ann |1 |cool |1 |1 |20 |\n", "|Ben |2 |cool |1 |2 |55 |\n", "|Cat |3 |cool |1 |3 |21 |\n", "|Dan |4 |uncool |2 |1 |91 |\n", "|Egg |5 |uncool |2 |2 |22 |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "6148d7f7", "metadata": {}, "source": [ "Notice that I've constructed two different labelling schemes here. We have a \"person\" variable $p$, so it would be perfectly sensible to refer to $Y_p$ as the grumpiness of the $p$-th person in the sample. For instance, the table shows that Dan is the fourth person, so we'd say $p = 4$. So, when talking about the grumpiness $Y$ of this \"Dan\" person, whoever they might be, we could refer to Dan's grumpiness by saying that $Y_p = 91$, for person $p = 4$ that is. However, that's not the only way we could refer to Dan. As an alternative, we could note that Dan belongs to the \"uncool\" group ($k = 2$), and is in fact the first person listed in the uncool group ($i = 1$). So it's equally valid to refer to Dan's grumpiness by saying that $Y_{ik} = 91$, where $k = 2$ and $i = 1$. In other words, each person $p$ corresponds to a unique $ik$ combination, and so the formula that I gave above is actually identical to our original formula for the variance, which would be\n", "\n", "$$\n", "\\mbox{Var}(Y) = \\frac{1}{N} \\sum_{p=1}^N \\left(Y_{p} - \\bar{Y} \\right)^2\n", "$$\n", "\n", "In both formulas, all we're doing is summing over all of the observations in the sample. Most of the time we would just use the simpler $Y_p$ notation: the equation using $Y_p$ is clearly the simpler of the two. However, when doing an ANOVA it's important to keep track of which participants belong in which groups, and we need to use the $Y_{ik}$ notation to do this. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "67523957", "metadata": {}, "source": [ "### From variances to sums of squares\n", "\n", "Okay, now that we've got a good grasp on how the variance is calculated, let's define something called the **_total sum of squares_**, which is denoted SS$_{tot}$. This is very simple: instead of averaging the squared deviations, which is what we do when calculating the variance, we just add them up. So the formula for the total sum of squares is almost identical to the formula for the variance:\n", "\n", "$$\n", "\\mbox{SS}_{tot} = \\sum_{k=1}^G \\sum_{i=1}^{N_k} \\left(Y_{ik} - \\bar{Y} \\right)^2\n", "$$ \n", "\n", "When we talk about analysing variances in the context of ANOVA, what we're really doing is working with the total sums of squares rather than the actual variance. One very nice thing about the total sum of squares is that we can break it up into two different kinds of variation. Firstly, we can talk about the **_within-group sum of squares_**, in which we look to see how different each individual person is from their own group mean:\n", "\n", "$$\n", "\\mbox{SS}_w = \\sum_{k=1}^G \\sum_{i=1}^{N_k} \\left( Y_{ik} - \\bar{Y}_k \\right)^2\n", "$$\n", "\n", "where $\\bar{Y}_k$ is a group mean. In our example, $\\bar{Y}_k$ would be the average mood change experienced by those people given the $k$-th drug. So, instead of comparing individuals to the average of *all* people in the experiment, we're only comparing them to those people in the the same group. As a consequence, you'd expect the value of $\\mbox{SS}_w$ to be smaller than the total sum of squares, because it's completely ignoring any group differences -- that is, the fact that the drugs (if they work) will have different effects on people's moods.\n", " \n", "Next, we can define a third notion of variation which captures *only* the differences between groups. We do this by looking at the differences between the group means $\\bar{Y}_k$ and grand mean $\\bar{Y}$. In order to quantify the extent of this variation, what we do is calculate the **_between-group sum of squares_**:\n", "\n", "$$\n", "\\begin{array}{rcl}\n", "\\mbox{SS}_{b} &=& \\sum_{k=1}^G \\sum_{i=1}^{N_k} \\left( \\bar{Y}_k - \\bar{Y} \\right)^2\n", " \\\\\n", "&=& \\sum_{k=1}^G N_k \\left( \\bar{Y}_k - \\bar{Y} \\right)^2\n", "\\end{array}\n", "$$\n", "\n", "It's not too difficult to show that the total variation among people in the experiment $\\mbox{SS}_{tot}$ is actually the sum of the differences between the groups $\\mbox{SS}_b$ and the variation inside the groups $\\mbox{SS}_w$. That is:\n", "\n", "$$\n", "\\mbox{SS}_w + \\mbox{SS}_{b} = \\mbox{SS}_{tot}\n", "$$\n", "\n", "Yay." ] }, { "cell_type": "code", "execution_count": 5, "id": "d62b8ec9", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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", 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" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "anovavar_fig" } }, "output_type": "display_data" } ], "source": [ "\n", "import numpy as np\n", "from scipy import stats\n", "from matplotlib import pyplot as plt\n", "\n", "mu1 = -4\n", "mu2 = -.25\n", "mu3 = 3.5\n", "sigma = 2\n", "\n", "\n", "x1 = np.linspace(mu1 - 4*sigma, mu1 + 4*sigma, 100)\n", "y1 = 100* stats.norm.pdf(x1, mu1, sigma)\n", "x2 = np.linspace(mu2 - 4*sigma, mu2 + 4*sigma, 100)\n", "y2 = 100* stats.norm.pdf(x2, mu2, sigma)\n", "x3 = np.linspace(mu3 - 4*sigma, mu3 + 4*sigma, 100)\n", "y3 = 100* stats.norm.pdf(x3, mu3, sigma)\n", "\n", "\n", "fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 5))\n", "\n", "\n", "sns.lineplot(x=x1,y=y1, color='black', ax = ax1)\n", "sns.lineplot(x=x2,y=y2, color='black', ax = ax1)\n", "sns.lineplot(x=x3,y=y3, color='black', ax = ax1)\n", "\n", "sns.lineplot(x=x1,y=y1, color='black', ax = ax2)\n", "sns.lineplot(x=x2,y=y2, color='black', ax = ax2)\n", "sns.lineplot(x=x3,y=y3, color='black', ax = ax2)\n", "\n", "ax1.text(0, 24, 'Between−group variation', size=20, ha=\"center\")\n", "ax1.text(0, 22, '(i.e., differences among group means))', size=20, ha=\"center\")\n", "\n", "ax2.text(0, 24, 'Within−group variation', size=20, ha=\"center\")\n", "ax2.text(0, 22, '(i.e., deviations from group means)', size=20, ha=\"center\")\n", "\n", "\n", "ax1.annotate(text = '', xy = (mu1,18), xytext = (mu2,18), arrowprops = dict(arrowstyle='<->'))\n", "ax1.annotate(text = '', xy = (mu2,18), xytext = (mu3,18), arrowprops = dict(arrowstyle='<->'))\n", "ax1.annotate(text = '', xy = (mu1,16), xytext = (mu3,16), arrowprops = dict(arrowstyle='<->'))\n", "\n", "ax2.annotate(text = '', xy = (mu1-(sigma/2),18), xytext = (mu1+(sigma/2),18), arrowprops = dict(arrowstyle='<->'))\n", "ax2.annotate(text = '', xy = (mu2-(sigma/2),18), xytext = (mu2+(sigma/2),18), arrowprops = dict(arrowstyle='<->'))\n", "ax2.annotate(text = '', xy = (mu3-(sigma/2),18), xytext = (mu3+(sigma/2),18), arrowprops = dict(arrowstyle='<->'))\n", "\n", "\n", "ax1.annotate(text = 'A', xy = (-12,16), size = 20)\n", "ax2.annotate(text = 'B', xy = (-12,16), size = 20)\n", "\n", "\n", "ax1.set_frame_on(False)\n", "ax2.set_frame_on(False)\n", "ax1.get_yaxis().set_visible(False)\n", "ax2.get_yaxis().set_visible(False)\n", "ax1.get_xaxis().set_visible(False)\n", "ax2.get_xaxis().set_visible(False)\n", "ax1.axhline(y=0, color='black')\n", "ax2.axhline(y=0, color='black')\n", "\n", "# show the figure in the book with caption\n", "plt.close()\n", "from myst_nb import glue\n", "glue(\"anovavar_fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "3168985d", "metadata": {}, "source": [ " ```{glue:figure} anovavar_fig\n", ":figwidth: 600px\n", ":name: fig-anovavar\n", "\n", "Graphical illustration of “between groups” variation (panel A) and “within groups” variation (panel B). On the left, the arrows show the differences in the group means; on the right, the arrows highlight the variability within each group.\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "46266e85", "metadata": {}, "source": [ "Okay, so what have we found out? We've discovered that the total variability associated with the outcome variable ($\\mbox{SS}_{tot}$) can be mathematically carved up into the sum of \"the variation due to the differences in the sample means for the different groups\" ($\\mbox{SS}_{b}$) plus \"all the rest of the variation\" ($\\mbox{SS}_{w}$). How does that help me find out whether the groups have different population means? Um. Wait. Hold on a second... now that I think about it, this is *exactly* what we were looking for. If the null hypothesis is true, then you'd expect all the sample means to be pretty similar to each other, right? And that would imply that you'd expect $\\mbox{SS}_{b}$ to be really small, or at least you'd expect it to be a lot smaller than the \"the variation associated with everything else\", $\\mbox{SS}_{w}$. Hm. I detect a hypothesis test coming on...\n", "\n", "### From sums of squares to the $F$-test\n", "\n", "As we saw in the last section, the *qualitative* idea behind ANOVA is to compare the two sums of squares values $\\mbox{SS}_b$ and $\\mbox{SS}_w$ to each other: if the between-group variation is $\\mbox{SS}_b$ is large relative to the within-group variation $\\mbox{SS}_w$ then we have reason to suspect that the population means for the different groups aren't identical to each other. In order to convert this into a workable hypothesis test, there's a little bit of \"fiddling around\" needed. What I'll do is first show you *what* we do to calculate our test statistic -- which is called an **_$F$ ratio_** -- and then try to give you a feel for *why* we do it this way. \n", "\n", "In order to convert our SS values into an $F$-ratio, the first thing we need to calculate is the **_degrees of freedom_** associated with the SS$_b$ and SS$_w$ values. As usual, the degrees of freedom corresponds to the number of unique \"data points\" that contribute to a particular calculation, minus the number of \"constraints\" that they need to satisfy. For the within-groups variability, what we're calculating is the variation of the individual observations ($N$ data points) around the group means ($G$ constraints). In contrast, for the between groups variability, we're interested in the variation of the group means ($G$ data points) around the grand mean (1 constraint). Therefore, the degrees of freedom here are:\n", "\n", "$$\n", "\\begin{array}{lcl}\n", "\\mbox{df}_b &=& G-1 \\\\\n", "\\mbox{df}_w &=& N-G \\\\\n", "\\end{array}\n", "$$\n", "\n", "Okay, that seems simple enough. What we do next is convert our summed squares value into a \"mean squares\" value, which we do by dividing by the degrees of freedom: \n", "\n", "$$\n", "\\begin{array}{lcl}\n", "\\mbox{MS}_b &=& \\displaystyle\\frac{\\mbox{SS}_b }{ \\mbox{df}_b} \\\\\n", "\\mbox{MS}_w &=& \\displaystyle\\frac{\\mbox{SS}_w }{ \\mbox{df}_w} \n", "\\end{array}\n", "$$\n", "\n", "Finally, we calculate the $F$-ratio by dividing the between-groups MS by the within-groups MS:\n", "\n", "$$\n", "F = \\frac{\\mbox{MS}_b }{ \\mbox{MS}_w } \n", "$$\n", "\n", "At a very general level, the intuition behind the $F$ statistic is straightforward: bigger values of $F$ means that the between-groups variation is large, relative to the within-groups variation. As a consequence, the larger the value of $F$, the more evidence we have against the null hypothesis. But how large does $F$ have to be in order to actually *reject* $H_0$? In order to understand this, you need a slightly deeper understanding of what ANOVA is and what the mean squares values actually are. \n", "\n", "The next section discusses that in a bit of detail, but for readers that aren't interested in the details of what the test is actually measuring, I'll cut to the chase. In order to complete our hypothesis test, we need to know the sampling distribution for $F$ if the null hypothesis is true. Not surprisingly, the sampling distribution for the $F$ *statistic* under the null hypothesis is an $F$ *distribution*. If you recall back to our discussion of the $F$ distribution in [](probability), the $F$ distribution has two parameters, corresponding to the two degrees of freedom involved: the first one df$_1$ is the between groups degrees of freedom df$_b$, and the second one df$_2$ is the within groups degrees of freedom df$_w$. \n", "\n", "A summary of all the key quantities involved in a one-way ANOVA, including the formulas showing how they are calculated, is shown in the table below:" ] }, { "attachments": {}, "cell_type": "markdown", "id": "228a9725", "metadata": {}, "source": [ "| |df |sum of squares |mean squares |$F$ statistic |$p$ value |\n", "|:--------------|:-------------------|:------------------------------------------------------------------|:------------------------------------------------|:----------------------------------------|:-------------|\n", "|between groups |$\\mbox{df}_b = G-1$ |SS$_b = \\displaystyle\\sum_{k=1}^G N_k (\\bar{Y}_k - \\bar{Y})^2$ |$\\mbox{MS}_b = \\frac{\\mbox{SS}_b}{\\mbox{df}_b}$ |$F = \\frac{\\mbox{MS}_b }{ \\mbox{MS}_w }$ |[complicated] |\n", "|within groups |$\\mbox{df}_w = N-G$ |SS$_w = \\sum_{k=1}^G \\sum_{i = 1}^{N_k} ( {Y}_{ik} - \\bar{Y}_k)^2$ |$\\mbox{MS}_w = \\frac{\\mbox{SS}_w}{\\mbox{df}_w}$ |- |- |\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "20868a2c", "metadata": {}, "source": [ "(anovamodel)=\n", "### The model for the data and the meaning of $F$ (advanced)\n", "\n", "At a fundamental level, ANOVA is a competition between two different statistical models, $H_0$ and $H_1$. When I described the null and alternative hypotheses at the start of the section, I was a little imprecise about what these models actually are. I'll remedy that now, though you probably won't like me for doing so. If you recall, our null hypothesis was that all of the group means are identical to one another. If so, then a natural way to think about the outcome variable $Y_{ik}$ is to describe individual scores in terms of a single population mean $\\mu$, plus the deviation from that population mean. This deviation is usually denoted $\\epsilon_{ik}$ and is traditionally called the *error* or **_residual_** associated with that observation. Be careful though: just like we saw with the word \"significant\", the word \"error\" has a technical meaning in statistics that isn't quite the same as its everyday English definition. In everyday language, \"error\" implies a mistake of some kind; in statistics, it doesn't (or at least, not necessarily). With that in mind, the word \"residual\" is a better term than the word \"error\". In statistics, both words mean \"leftover variability\": that is, \"stuff\" that the model can't explain. In any case, here's what the null hypothesis looks like when we write it as a statistical model:\n", "\n", "$$\n", "Y_{ik} = \\mu + \\epsilon_{ik}\n", "$$\n", "\n", "where we make the *assumption* (discussed later) that the residual values $\\epsilon_{ik}$ are normally distributed, with mean 0 and a standard deviation $\\sigma$ that is the same for all groups. To use the notation that we introduced in the [chapter on probability](probability) we would write this assumption like this:\n", "\n", "$$\n", "\\epsilon_{ik} \\sim \\mbox{Normal}(0, \\sigma^2)\n", "$$\n", "\n", "What about the alternative hypothesis, $H_1$? The only difference between the null hypothesis and the alternative hypothesis is that we allow each group to have a different population mean. So, if we let $\\mu_k$ denote the population mean for the $k$-th group in our experiment, then the statistical model corresponding to $H_1$ is:\n", "\n", "$$\n", "Y_{ik} = \\mu_k + \\epsilon_{ik}\n", "$$\n", "\n", "where, once again, we assume that the error terms are normally distributed with mean 0 and standard deviation $\\sigma$. That is, the alternative hypothesis also assumes that \n", "\n", "$$\n", "\\epsilon_{ik} \\sim \\mbox{Normal}(0, \\sigma^2)\n", "$$\n", "\n", "Okay, now that we've described the statistical models underpinning $H_0$ and $H_1$ in more detail, it's now pretty straightforward to say what the mean square values are measuring, and what this means for the interpretation of $F$. I won't bore you with the proof of this, but it turns out that the within-groups mean square, MS$_w$, can be viewed as an estimator (in the technical sense: [estimation](estimation) of the error variance $\\sigma^2$. The between-groups mean square MS$_b$ is also an estimator; but what it estimates is the error variance *plus* a quantity that depends on the true differences among the group means. If we call this quantity $Q$, then we can see that the $F$-statistic is basically [^note2]\n", "\n", "$$\n", "F = \\frac{\\hat{Q} + \\hat\\sigma^2}{\\hat\\sigma^2}\n", "$$\n", "\n", "where the true value $Q=0$ if the null hypothesis is true, and $Q > 0$ if the alternative hypothesis is true (if you must know more, than check out chapter 10 of {cite}`Hays1994`). Therefore, at a bare minimum *the $F$ value must be larger than 1* to have any chance of rejecting the null hypothesis. Note that this *doesn't* mean that it's impossible to get an $F$-value less than 1. What it means is that, if the null hypothesis is true, then the sampling distribution of the $F$ ratio has a mean of 1,[^note3] and so we need to see $F$-values larger than 1 in order to safely reject the null.\n", "\n", "To be a bit more precise about the sampling distribution, notice that if the null hypothesis is true, both MS$_b$ and MS$_w$ are estimators of the variance of the residuals $\\epsilon_{ik}$. If those residuals are normally distributed, then you might suspect that the estimate of the variance of $\\epsilon_{ik}$ is chi-square distributed... because (as discussed [much earlier](otherdists)) that's what a chi-square distribution *is*: it's what you get when you square a bunch of normally-distributed things and add them up. And since the $F$ distribution is (again, by definition) what you get when you take the ratio between two things that are $\\chi^2$ distributed... we have our sampling distribution. Obviously, I'm glossing over a whole lot of stuff when I say this, but in broad terms, this really is where our sampling distribution comes from." ] }, { "attachments": {}, "cell_type": "markdown", "id": "42758f34", "metadata": {}, "source": [ "(anovacalc)=\n", "### A worked example\n", "\n", "The previous discussion was fairly abstract, and a little on the technical side, so I think that at this point it might be useful to see a worked example. For that, let's go back to the clinical trial data that I introduced at the start of the chapter. The descriptive statistics that we calculated at the beginning tell us our group means: an average mood gain of 0.45 for the placebo, 0.72 for Anxifree, and 1.48 for Joyzepam. With that in mind, let's party like it's 1899 [^note4] and start doing some pencil and paper calculations. I'll only do this for the first 5 observations, because it's not bloody 1899 and I'm very lazy. Let's start by calculating $\\mbox{SS}_{w}$, the within-group sums of squares. First, let's draw up a nice table to help us with our calculations... " ] }, { "attachments": {}, "cell_type": "markdown", "id": "ac05a567", "metadata": {}, "source": [ "\n", "|group ($k$) |outcome ($Y_{ik}$) |\n", "|:-----------|:------------------|\n", "|placebo |0.5 |\n", "|placebo |0.3 |\n", "|placebo |0.1 |\n", "|anxifree |0.6 |\n", "|anxifree |0.4 |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "6f6ec067", "metadata": {}, "source": [ "At this stage, the only thing I've included in the table is the raw data itself: that is, the grouping variable (i.e., `drug`) and outcome variable (i.e. `mood_gain`) for each person. Note that the outcome variable here corresponds to the $Y_{ik}$ value in our equation previously. The next step in the calculation is to write down, for each person in the study, the corresponding group mean; that is, $\\bar{Y}_k$. This is slightly repetitive, but not particularly difficult since we already calculated those group means when doing our descriptive statistics:" ] }, { "attachments": {}, "cell_type": "markdown", "id": "5e5d842c", "metadata": {}, "source": [ "\n", "|group ($k$) |outcome ($Y_{ik}$) |**group mean** ($\\bar{Y}_k$) |\n", "|:-----------|:------------------|:----------------------------|\n", "|placebo |0.5 |**0.45** |\n", "|placebo |0.3 |**0.45** |\n", "|placebo |0.1 |**0.45** |\n", "|anxifree |0.6 |**0.72** |\n", "|anxifree |0.4 |**0.72** |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "08ab43c5", "metadata": {}, "source": [ "\n", "\n", "Now that we've written those down, we need to calculate -- again for every person -- the deviation from the corresponding group mean. That is, we want to subtract $Y_{ik} - \\bar{Y}_k$. After we've done that, we need to square everything. When we do that, here's what we get:" ] }, { "attachments": {}, "cell_type": "markdown", "id": "0b55d8a3", "metadata": {}, "source": [ "\n", "|group ($k$) |outcome ($Y_{ik}$) |group mean ($\\bar{Y}_k$) |**dev. from group mean** ($Y_{ik} - \\bar{Y}_{k}$) |**squared deviation** ($(Y_{ik} - \\bar{Y}_{k})^2$) |\n", "|:-----------|:------------------|:------------------------|:--------------------------------------------------|:--------------------------------------------------|\n", "|placebo |0.5 |0.45 |**0.05** |**0.0025** |\n", "|placebo |0.3 |0.45 |**-0.15** |**0.0225** |\n", "|placebo |0.1 |0.45 |**-0.35** |**0.1225** |\n", "|anxifree |0.6 |0.72 |**-0.12** |**0.0136** |\n", "|anxifree |0.4 |0.72 |**-0.32** |**0.1003** |\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "1ec36aa0", "metadata": {}, "source": [ "The last step is equally straightforward. In order to calculate the within-group sum of squares, we just add up the squared deviations across all observations:\n", "\n", "$$\n", "\\begin{array}{rcl}\n", "\\mbox{SS}_w &=& 0.0025 + 0.0225 + 0.1225 + 0.0136 + 0.1003 \\\\\n", "&=& 0.2614\n", "\\end{array}\n", "$$\n", "\n", "Of course, if we actually wanted to get the *right* answer, we'd need to do this for all 18 observations in the data set, not just the first five. We could continue with the pencil and paper calculations if we wanted to, but it's pretty tedious. Alternatively, click the button to see how we can get Python to do this tedious work for us [^tedium]:\n", "\n", "[^tedium]: Ok, I suppose you could make a pretty solid case that writing all this code to do a bit of multiplication and substraction is also tedious, but I would counter your argument by pointing out that coding your way to these solutions has the benefit of being reproducible (if you later discover a mistake, you can trace it back and fix it) and resusable (next time you want to do solve a similar problem, you can take your old code and modify it, because now you already know how to do that sort of thing). Plus, doing it by hand is maybe ok for 18 participants, but what if you had a study with five hundred or a thousand participants, hmmm...?" ] }, { "cell_type": "code", "execution_count": 6, "id": "83ea7d72", "metadata": { "tags": [ "hide-input" ] }, "outputs": [], "source": [ "# select the columns from the dataframe with the group and outcome measure data\n", "# this is not strictly necessary, but it makes it clearer what we are doing when we plug these data back into the new dataframe Y\n", "group = df['drug']\n", "outcome = df['mood_gain']\n", "\n", "# make a new dataframe called grouped, in which our data are associated by which group (drug treatment) the participant was in\n", "grouped = df.groupby('drug')\n", "\n", "# make a dataframe with the mean values for each group\n", "gp_means = grouped.mean(numeric_only=True)\n", "\n", "# flatten the indices of the new dataframe gp_means, because multiindices can be pretty annoying\n", "gp_means.reset_index(inplace=True)\n", "\n", "# make a list (grouped_means) that the is the length of the number of participants (which we get from the length of the variable \"group\",\n", "# which contained all the rows of the original dataframe)\n", "grouped_means = [0]*len(group)\n", "\n", "# go through every row in the dataframe \"gp_means\", which contains the names of the three drugs,\n", "# then check in every row of the variable group (which contains the drug name for each participant)\n", "# and if the drug names match, put the mean mood gain in the corresponding row of \"grouped_means\"\n", "for s, val in enumerate(gp_means['drug']):\n", " for x, drug in enumerate(group):\n", " if val == drug:\n", " grouped_means[x] = round(gp_means['mood_gain'][s],2)\n", "\n", "# build a new dataframe Y with a row for each participant, with columns for drug name (group), each participants' outcome (mood gain),\n", "# and the average (mean) mood gain for all particpants who were in the same treatment group.\n", "Y = pd.DataFrame(\n", " {'group': group,\n", " 'outcome': outcome,\n", " 'group_means': grouped_means\n", " }) \n", "\n", "# add a column to the dataframe Y showing the distance (residual) of each individual from their group's mean\n", "Y['dev_from_group_means'] = Y['outcome'] - Y['group_means']\n", "\n", "# add a column to the dataframe Y with the same values as the column before (residuals), but squared (multiplied by themeselves)\n", "Y['squared_devs'] = Y['dev_from_group_means']**2\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "94da7078", "metadata": {}, "source": [ "This is where the basic Python [skills](getting-started-with-python), programming [skills](programming), and data-wrangling [skills](datawrangling) you learned way back at the beginning of this book really start to pay off. It might not be obvious from inspection what these commands are doing: as a general rule, the human brain seems to just shut down when faced with a big block of programming. However, I strongly suggest that -- if you're like me and tend to find that the mere sight of this code makes you want to look away and see if there's any beer left in the fridge or a game of footy on the telly -- you take a moment and look closely at these commands, and the many comments I have written, one at a time. Every single one of these commands is something you've seen before somewhere else in the book. There's nothing novel about them (though I admit that `reset_index` has appeared only once, and there aren't many nested `for` statements in this book), so if you're not quite sure how these commands work, this might be a good time to try playing around with them yourself, to try to get a sense of what's happening. On the other hand, if this does seem to make sense, then you won't be all that surprised at what happens when we look at the output of all this code, `Y`, and we see..." ] }, { "cell_type": "code", "execution_count": 7, "id": "844010dc", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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groupoutcomegroup_meansdev_from_group_meanssquared_devs
0placebo0.50.450.050.0025
1placebo0.30.45-0.150.0225
2placebo0.10.45-0.350.1225
3anxifree0.60.72-0.120.0144
4anxifree0.40.72-0.320.1024
5anxifree0.20.72-0.520.2704
6joyzepam1.41.48-0.080.0064
7joyzepam1.71.480.220.0484
8joyzepam1.31.48-0.180.0324
9placebo0.60.450.150.0225
10placebo0.90.450.450.2025
11placebo0.30.45-0.150.0225
12anxifree1.10.720.380.1444
13anxifree0.80.720.080.0064
14anxifree1.20.720.480.2304
15joyzepam1.81.480.320.1024
16joyzepam1.31.48-0.180.0324
17joyzepam1.41.48-0.080.0064
\n", "
" ], "text/plain": [ " group outcome group_means dev_from_group_means squared_devs\n", "0 placebo 0.5 0.45 0.05 0.0025\n", "1 placebo 0.3 0.45 -0.15 0.0225\n", "2 placebo 0.1 0.45 -0.35 0.1225\n", "3 anxifree 0.6 0.72 -0.12 0.0144\n", "4 anxifree 0.4 0.72 -0.32 0.1024\n", "5 anxifree 0.2 0.72 -0.52 0.2704\n", "6 joyzepam 1.4 1.48 -0.08 0.0064\n", "7 joyzepam 1.7 1.48 0.22 0.0484\n", "8 joyzepam 1.3 1.48 -0.18 0.0324\n", "9 placebo 0.6 0.45 0.15 0.0225\n", "10 placebo 0.9 0.45 0.45 0.2025\n", "11 placebo 0.3 0.45 -0.15 0.0225\n", "12 anxifree 1.1 0.72 0.38 0.1444\n", "13 anxifree 0.8 0.72 0.08 0.0064\n", "14 anxifree 1.2 0.72 0.48 0.2304\n", "15 joyzepam 1.8 1.48 0.32 0.1024\n", "16 joyzepam 1.3 1.48 -0.18 0.0324\n", "17 joyzepam 1.4 1.48 -0.08 0.0064" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Y" ] }, { "attachments": {}, "cell_type": "markdown", "id": "0ecddfba", "metadata": {}, "source": [ "If you compare this output to the contents of the table I've been constructing by hand, you can see that Python has done exactly the same calculations that I was doing (albeit with a few miniscule differences due to rounding), and much faster too. So, if we want to finish the calculations of the within-group sum of squares in Python, we just ask for the `sum()` of the `squared_devs` variable:" ] }, { "cell_type": "code", "execution_count": 8, "id": "80a7f44e", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "1.3918000000000001" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SSw = Y['squared_devs'].sum()\n", "SSw" ] }, { "attachments": {}, "cell_type": "markdown", "id": "e2353437", "metadata": {}, "source": [ "Okay. Now that we've calculated the within groups variation, $\\mbox{SS}_w$, it's time to turn our attention to the between-group sum of squares, $\\mbox{SS}_b$. The calculations for this case are very similar. The main difference is that, instead of calculating the differences between an observation $Y_{ik}$ and a group mean $\\bar{Y}_k$ for all of the observations, we calculate the differences between the group means $\\bar{Y}_k$ and the grand mean $\\bar{Y}$ (in this case 0.88) for all of the groups..." ] }, { "attachments": {}, "cell_type": "markdown", "id": "037aee26", "metadata": {}, "source": [ "\n", "|group ($k$) |group mean ($\\bar{Y}_k$) |grand mean ($\\bar{Y}$) |deviation ($\\bar{Y}_{k} - \\bar{Y}$) |squared deviations ($(\\bar{Y}_{k} - \\bar{Y})^2$) |\n", "|:-----------|:------------------------|:----------------------|:-----------------------------------|:------------------------------------------------|\n", "|placebo |0.45 |0.88 |-0.43 |0.18 |\n", "|anxifree |0.72 |0.88 |-0.16 |0.03 |\n", "|joyzepam |1.48 |0.88 |0.60 |0.36 |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "3fc9ea68", "metadata": {}, "source": [ "However, for the between group calculations we need to multiply each of these squared deviations by $N_k$, the number of observations in the group. We do this because every *observation* in the group (all $N_k$ of them) is associated with a between group difference. So if there are six people in the placebo group, and the placebo group mean differs from the grand mean by 0.19, then the *total* between group variation associated with these six people is $6 \\times 0.16 = 1.14$. So we have to extend our little table of calculations..." ] }, { "attachments": {}, "cell_type": "markdown", "id": "b2527b75", "metadata": {}, "source": [ "\n", "|group ($k$) |squared deviations ($(\\bar{Y}_{k} - \\bar{Y})^2$) |sample size ($N_k$) |weighted squared dev ($N_k (\\bar{Y}_{k} - \\bar{Y})^2$) |\n", "|:-----------|:------------------------------------------------|:-------------------|:------------------------------------------------------|\n", "|placebo |0.18 |6 |1.11 |\n", "|anxifree |0.03 |6 |0.16 |\n", "|joyzepam |0.36 |6 |2.18 |\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "0da12255", "metadata": {}, "source": [ "And so now our between group sum of squares is obtained by summing these \"weighted squared deviations\" over all three groups in the study:\n", "\n", "$$\n", "\\begin{array}{rcl}\n", "\\mbox{SS}_{b} &=& 1.11 + 0.16 + 2.18 \\\\\n", "&=& 3.45\n", "\\end{array}\n", "$$\n", "\n", "\n", "As you can see, the between group calculations are a lot shorter, so you probably wouldn't usually want to bother using Python as your calculator. However, if you *did* decide to do so, here's one way you could do it:" ] }, { "cell_type": "code", "execution_count": 9, "id": "443ce436", "metadata": {}, "outputs": [], "source": [ "grouped = df.groupby('drug')\n", "gp_means = grouped.mean(numeric_only=True)\n", "gp_means.reset_index(inplace=True)\n", "group = list(gp_means['drug'])\n", "gp_means = list(gp_means['mood_gain'])\n", "\n", "grand_mean = round(df['mood_gain'].mean(),2)\n", "grand_mean = [grand_mean]*3\n", "\n", "\n", "\n", "Y = pd.DataFrame(\n", " {'group': group,\n", " 'gp_means': gp_means,\n", " 'grand_mean': grand_mean \n", " }) \n", "\n", "Y['dev_from_grandmean'] = Y['gp_means'] - Y['grand_mean']\n", "Y['squared_devs'] = Y['dev_from_grandmean']**2\n", "\n", "xtab = pd.crosstab(index = df[\"drug\"], columns = \"count\")\n", "xtab.reset_index(inplace=True)\n", "\n", "Y['group_sizes'] = xtab['count']\n", "Y['weighted_squared_devs'] = Y['group_sizes'] * Y['squared_devs']\n", "\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "d2651792", "metadata": {}, "source": [ "I won't actually try to explain this code line by line this time, but -- just like last time -- there's nothing in there that we haven't seen in several places elsewhere in the book, so I'll leave it as an exercise for you to make sure you understand it. Once again, we can dump all our variables into a data frame so that we can print it out as a nice table:" ] }, { "cell_type": "code", "execution_count": 10, "id": "3f6bc1f9", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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groupgp_meansgrand_meandev_from_grandmeansquared_devsgroup_sizesweighted_squared_devs
0anxifree0.7166670.88-0.1633330.02667860.160067
1joyzepam1.4833330.880.6033330.36401162.184067
2placebo0.4500000.88-0.4300000.18490061.109400
\n", "
" ], "text/plain": [ " group gp_means grand_mean dev_from_grandmean squared_devs \\\n", "0 anxifree 0.716667 0.88 -0.163333 0.026678 \n", "1 joyzepam 1.483333 0.88 0.603333 0.364011 \n", "2 placebo 0.450000 0.88 -0.430000 0.184900 \n", "\n", " group_sizes weighted_squared_devs \n", "0 6 0.160067 \n", "1 6 2.184067 \n", "2 6 1.109400 " ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Y" ] }, { "attachments": {}, "cell_type": "markdown", "id": "dc2855f0", "metadata": {}, "source": [ "Clearly, these are basically the same numbers that we got before. There are a few tiny differences, but that's only because the hand-calculated versions have some small errors caused by the fact that I rounded all my numbers to 2 decimal places at each step in the calculations, whereas Python only does it at the end (obviously, Python's version is more accurate). Anyway, here's the Python command showing the final step:" ] }, { "cell_type": "code", "execution_count": 11, "id": "24ccbfe8", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3.453533333333334" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SSb = sum(Y['weighted_squared_devs'])\n", "SSb" ] }, { "attachments": {}, "cell_type": "markdown", "id": "c14f8756", "metadata": {}, "source": [ "which is (ignoring the slight differences due to rounding error) the same answer that I got when doing things by hand.\n", "\n", "Now that we've calculated our sums of squares values, $\\mbox{SS}_b$ and $\\mbox{SS}_w$, the rest of the ANOVA is pretty painless. The next step is to calculate the degrees of freedom. Since we have $G = 3$ groups and $N = 18$ observations in total, our degrees of freedom can be calculated by simple subtraction:\n", "\n", "$$\n", "\\begin{array}{lclcl}\n", "\\mbox{df}_b &=& G - 1 &=& 2 \\\\\n", "\\mbox{df}_w &=& N - G &=& 15 \n", "\\end{array}\n", "$$\n", "\n", "Next, since we've now calculated the values for the sums of squares and the degrees of freedom, for both the within-groups variability and the between-groups variability, we can obtain the mean square values by dividing one by the other:\n", "\n", "$$\n", "\\begin{array}{lclclcl}\n", "\\mbox{MS}_b &=& \\displaystyle\\frac{\\mbox{SS}_b }{ \\mbox{df}_b } &=& \\displaystyle\\frac{3.45}{ 2} &=& 1.73\n", "\\end{array}\n", "$$\n", "\n", "$$\n", "\\begin{array}{lclclcl}\n", "\\mbox{MS}_w &=& \\displaystyle\\frac{\\mbox{SS}_w }{ \\mbox{df}_w } &=& \\displaystyle\\frac{1.39}{15} &=& 0.09\n", "\\end{array}\n", "$$\n", "\n", "We're almost done. The mean square values can be used to calculate the $F$-value, which is the test statistic that we're interested in. We do this by dividing the between-groups MS value by the and within-groups MS value.\n", "\n", "$$\n", "F \\ = \\ \\frac{\\mbox{MS}_b }{ \\mbox{MS}_w } \\ = \\ \\frac{1.73}{0.09} \\ = \\ 18.6\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "d5560864", "metadata": {}, "source": [ "Woohooo! This is terribly exciting, yes? Now that we have our test statistic, the last step is to find out whether the test itself gives us a significant result. As discussed back when we talked about [hypothesis testing](hypothesistesting), what we really *ought* to do is choose an $\\alpha$ level (i.e., acceptable Type I error rate) ahead of time, construct our rejection region, etc etc. But in practice it's just easier to directly calculate the $p$-value. Back in the \"old days\", what we'd do is open up a statistics textbook or something and flick to the back section which would actually have a huge lookup table... that's how we'd \"compute\" our $p$-value, because it's too much effort to do it any other way. However, since we have access to Python, I'll use the `stats.f.cdf()` method from `scipy` to do it instead.[^eveneasier]\n", "\n", "[^eveneasier]: In fact, in practice we won't do any of this stuff. As we'll see in a minute, there are already modules for Python that will just take care of all of these steps for us, including calculating the $p$-value. But we're still doing things \"by hand\" here, remember?" ] }, { "cell_type": "code", "execution_count": 12, "id": "6e94ff10", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "8.672726890401883e-05" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from scipy import stats\n", "p = 1-stats.f.cdf(18.6, 2, 15)\n", "p" ] }, { "attachments": {}, "cell_type": "markdown", "id": "f51f40d5", "metadata": {}, "source": [ "(onewayanovatable)=\n", "Therefore, our $p$-value comes to 0.0000867, or $8.67 \\times 10^{-5}$ in scientific notation. So, unless we're being *extremely* conservative about our Type I error rate, we're pretty much guaranteed to reject the null hypothesis. \n", "\n", "At this point, we're basically done. Having completed our calculations, it's traditional to organise all these numbers into an ANOVA table like the one below:" ] }, { "attachments": {}, "cell_type": "markdown", "id": "e2412737", "metadata": {}, "source": [ "\n", "| |df |sum of squares |mean squares |$F$-statistic |$p$-value |\n", "|:--------------|:--|:--------------|:------------|:-------------|:---------------------|\n", "|between groups |2 |3.45 |1.73 |18.6 |$8.67 \\times 10^{-5}$ |\n", "|within groups |15 |1.39 |0.09 |- |- |\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "22e9fa55", "metadata": {}, "source": [ "These days, you'll probably never have much reason to want to construct one of these tables yourself, but you *will* find that almost all statistical software tends to organise the output of an ANOVA into a table like this, so it's a good idea to get used to reading them. However, although the software will output a full ANOVA table, there's almost never a good reason to include the whole table in your write up. A pretty standard way of reporting this result would be to write something like this:\n", "\n", "> \"One-way ANOVA showed a significant effect of drug on mood gain ($F(2,15) = 18.6, p<.001$).\"\n", "\n", "Sigh. So much work for one short sentence." ] }, { "attachments": {}, "cell_type": "markdown", "id": "b899a069", "metadata": {}, "source": [ "(introduceaov)=\n", "## Running an ANOVA in Python\n", "\n", "I'm pretty sure I know what you're thinking after reading the last section, *especially* if you followed my advice and tried typing all the commands in yourself.... doing the ANOVA calculations yourself *sucks*. There's quite a lot of calculations that we needed to do along the way, and it would be tedious to have to do this over and over again every time you wanted to do an ANOVA. One possible solution to the problem would be to take all these calculations and turn them into some Python functions yourself. You'd still have to do a lot of typing, but at least you'd only have to do it the one time: once you've created the functions, you can reuse them over and over again. However, writing your own functions is a lot of work, so this is kind of a last resort. Besides, it's much better if someone else does all the work for you..." ] }, { "attachments": {}, "cell_type": "markdown", "id": "9f9c0be7", "metadata": {}, "source": [ "### Getting Python to calculate ANOVA for you\n", "\n", "Now, as you can imagine, there are several Python modules available that provided tools for taking the pain out calculating ANOVA's. `scipy`, for example, has `stats.f_oneway` which, in the typical, laconic fashion of `scipy`, spits out an $F$-value, a $p$-value, and not much more. We have already used `scipy.stats` above to find our $p$-value by hand. I am going to reccomend an entirely different package, though: `statsmodels`. These things go in and out of fashion, but for my money, `statsmodels` is quickly becoming the go-to Python package for anthing beyond a humble $t$-test. `statsmodels` builds on a more modern approach to statistics which goes beyond what we can deal with here, but suffice to say, that among other things, it is an easy way to get your ANOVA work done. \n", "\n", "The key to working with `statsmodels` is the `formula`, so let's take a look at that. A `statsmodels` formula (which will instantly feel familiar to anyone who has worked with the programming language R before) begins with the _dependent variable_ or _outcome measure_: basically the thing we are trying to undersand. The _outcome measure_ goes on the left side of the `~`character, which I read in my head as \"predicted by\". To the right of the `~` comes, unsurprisingly, the _predictor variables_, or _independent variables_ if you prefer. So, in our case, we are investigating whether or not there are significant differences between the three drug types: placebo, anxifree, and joyzepam. In other words, we want to know if variation in the variable `mood_gain` that can be accounted for by these three groups is greater than the variation within the groups. Or, in still other words, whether `mood_gain` can be _predicted by_ `drug`. So, our formula is `'mood_gain ~ drug'`.\n", "\n", "Before we run the ANOVA, let's just remember what the data look like:" ] }, { "cell_type": "code", "execution_count": 13, "id": "4f8258aa", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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drugtherapymood_gain
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" ], "text/plain": [ " drug therapy mood_gain\n", "0 placebo no.therapy 0.5\n", "1 placebo no.therapy 0.3\n", "2 placebo no.therapy 0.1\n", "3 anxifree no.therapy 0.6\n", "4 anxifree no.therapy 0.4" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.head()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "680928af", "metadata": {}, "source": [ "Now, let's let `statsmodels` do its magic. To do the same ANOVA that I laboriously calculated in the previous section, I’d use a few simple commands like this:" ] }, { "cell_type": "code", "execution_count": 14, "id": "1fda74ff", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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sum_sqdfFPR(>F)
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" ], "text/plain": [ " sum_sq df F PR(>F)\n", "drug 3.453333 2.0 18.610778 0.000086\n", "Residual 1.391667 15.0 NaN NaN" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import statsmodels.api as sm\n", "from statsmodels.formula.api import ols\n", "\n", "formula = 'mood_gain ~ drug'\n", "\n", "model = ols(formula, data=df).fit()\n", "\n", "aov_table = sm.stats.anova_lm(model, typ=2)\n", "aov_table\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "7eb49a99", "metadata": {}, "source": [ "Now, isn't that better? We got to the same place, but with a lot less pain and suffering!\n", "\n", "Once we get done celebrating, we might take a little closer look at this wonder function. The function is called `anova_lm`. The ANOVA part makes sense, sure, but what is this `lm` thing all about? `lm` stands for \"linear model\", and later on, we'll see that this reflects a pretty deep statistical relationship between ANOVA and [linear regression](regression), and this has some important implications for what can be done with both ANOVAs and linear regressions... but I'm getting ahead of myself. For now, let's just bask in the glory of the anova table that `statsmodels` gives us. We get the sums of squares, the degrees of freedom, the $F$-statistic, and the $p$-value itself. These are all identical to the numbers that we calculated ourselves when doing it the long and tedious way, and it's even organised into a nice table, like the [one that we filled out by hand](onewayanovatable). You might be wondering what the ``NaN``s in the last two cells are. If you compare this table to the [one that we filled out by hand](onewayanovatable), you will see that these are simply empty cells; it doesn't make sense to calculate an $F$-value or $p$-value for the residuals. ``NaN`` just means \"not a number\". It would be nicer if ``statsmodels`` just left these cells empty, but after all the work it has just done for us, I'm willing to overlook this.\n", "\n", "So, `statsmodels` provides a pretty easy way to get ANOVAs done, but back when we were doing t-tests, I was all excited about `pingouin` and said it made things so much easier. So couldn't we use `pingouin` to do ANOVA's? Sure, of course we can!" ] }, { "cell_type": "code", "execution_count": 15, "id": "4d0428ec", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " Source SS DF MS F p-unc np2\n", "0 drug 3.453333 2 1.726667 18.610778 0.000086 0.712762\n", "1 Within 1.391667 15 0.092778 NaN NaN NaN" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" }, { "name": "stderr", "output_type": "stream", "text": [ "/Users/ethan/opt/miniconda3/envs/pythonbook3/lib/python3.11/site-packages/outdated/utils.py:14: OutdatedPackageWarning: The package pingouin is out of date. Your version is 0.5.3, the latest is 0.5.4.\n", "Set the environment variable OUTDATED_IGNORE=1 to disable these warnings.\n", " return warn(\n" ] } ], "source": [ "import pingouin as pg\n", "\n", "pg.anova(dv='mood_gain', \n", " between='drug', \n", " data=df,\n", " detailed=True)\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "18420ed4", "metadata": {}, "source": [ "If you check, you'll see we get the same answer either way. So that's _two_ easy ways to do ANOVA in Python! Time for beer and football!\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "f843e6a0", "metadata": {}, "source": [ "(etasquared)=\n", "## Effect size\n", "\n", "\n", "There's a few different ways you could measure the effect size in an ANOVA, but the most commonly used measures are $\\eta^2$ (**_eta squared_**) and partial $\\eta^2$. For a one way analysis of variance they're identical to each other, so for the moment I'll just explain $\\eta^2$. The definition of $\\eta^2$ is actually really simple:\n", "\n", "$$\n", "\\eta^2 = \\frac{\\mbox{SS}_b}{\\mbox{SS}_{tot}}\n", "$$\n", "\n", "That's all it is. So when I look at the ANOVA table above, I see that $\\mbox{SS}_b = 3.45$ and $\\mbox{SS}_{tot} = 3.45 + 1.39 = 4.84$. Thus we get an $\\eta^2$ value of \n", "\n", "$$\n", "\\eta^2 = \\frac{3.45}{4.84} = 0.71\n", "$$\n", "\n", "The interpretation of $\\eta^2$ is equally straightforward: it refers to the proportion of the variability in the outcome variable (`mood_gain`) that can be explained in terms of the predictor (`drug`). A value of $\\eta^2 = 0$ means that there is no relationship at all between the two, whereas a value of $\\eta^2 = 1$ means that the relationship is perfect. Better yet, the $\\eta^2$ value is very closely related to a squared correlation (i.e., $r^2$). So, if you're trying to figure out whether a particular value of $\\eta^2$ is big or small, it's sometimes useful to remember that \n", "\n", "$$\n", "\\eta= \\sqrt{\\frac{\\mbox{SS}_b}{\\mbox{SS}_{tot}}}\n", "$$\n", "\n", "can be interpreted as if it referred to the *magnitude* of a Pearson correlation. So in our drugs example, the $\\eta^2$ value of .71 corresponds to an $\\eta$ value of $\\sqrt{.71} = .84$. If we think about this as being equivalent to a correlation of about .84, we'd conclude that the relationship between `drug` and `mood_gain` is strong. \n", "\n", "It's pretty straightforward to calculate it directly from the numbers in the ANOVA table. In fact, since I've already got the `SSw` and `SSb` variables lying around from my earlier calculations, I can do this:" ] }, { "cell_type": "code", "execution_count": 16, "id": "d7d782f2", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.7127545404512933" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SStot = SSb + SSw # total sums of squares\n", "eta_squared = SSb / SStot # eta-squared value\n", "eta_squared" ] }, { "attachments": {}, "cell_type": "markdown", "id": "5eb57e4b", "metadata": {}, "source": [ "And, in fact, if you look at the last column in the `pingouin` output above, you'll see that `pingouin` has already calculated $\\eta^2$ for us." ] }, { "attachments": {}, "cell_type": "markdown", "id": "4f5b3176", "metadata": {}, "source": [ "(posthoc)=\n", "## Multiple comparisons and post hoc tests\n", "\n", "Any time you run an ANOVA with more than two groups, and you end up with a significant effect, the first thing you'll probably want to ask is which groups are actually different from one another. In our drugs example, our null hypothesis was that all three drugs (placebo, Anxifree and Joyzepam) have the exact same effect on mood. But if you think about it, the null hypothesis is actually claiming *three* different things all at once here. Specifically, it claims that:\n", "\n", "- Your competitor's drug (Anxifree) is no better than a placebo (i.e., $\\mu_A = \\mu_P$)\n", "- Your drug (Joyzepam) is no better than a placebo (i.e., $\\mu_J = \\mu_P$)\n", "- Anxifree and Joyzepam are equally effective (i.e., $\\mu_J = \\mu_A$)\n", "\n", "If any one of those three claims is false, then the null hypothesis is also false. So, now that we've rejected our null hypothesis, we're thinking that *at least* one of those things isn't true. But which ones? All three of these propositions are of interest: you certainly want to know if your new drug Joyzepam is better than a placebo, and it would be nice to know how well it stacks up against an existing commercial alternative (i.e., Anxifree). It would even be useful to check the performance of Anxifree against the placebo: even if Anxifree has already been extensively tested against placebos by other researchers, it can still be very useful to check that your study is producing similar results to earlier work.\n", "\n", "When we characterise the null hypothesis in terms of these three distinct propositions, it becomes clear that there are eight possible \"states of the world\" that we need to distinguish between:" ] }, { "attachments": {}, "cell_type": "markdown", "id": "0c9dfa98", "metadata": {}, "source": [ "\n", "|possibility: |is $\\mu_P = \\mu_A$? |is $\\mu_P = \\mu_J$? |is $\\mu_A = \\mu_J$? |which hypothesis? |\n", "|:------------|:-------------------|:-------------------|:-------------------|:-----------------|\n", "|1 |$\\checkmark$ |$\\checkmark$ |$\\checkmark$ |null |\n", "|2 |$\\checkmark$ |$\\checkmark$ | |alternative |\n", "|3 |$\\checkmark$ | |$\\checkmark$ |alternative |\n", "|4 |$\\checkmark$ | | |alternative |\n", "|5 | |$\\checkmark$ |$\\checkmark$ |alternative |\n", "|6 | |$\\checkmark$ | |alternative |\n", "|7 | | |$\\checkmark$ |alternative |\n", "|8 | | | |alternative |\n", "> " ] }, { "attachments": {}, "cell_type": "markdown", "id": "fc08804a", "metadata": {}, "source": [ "By rejecting the null hypothesis, we've decided that we *don't* believe that \\#1 is the true state of the world. The next question to ask is, which of the other seven possibilities *do* we think is right? When faced with this situation, its usually helps to look at the data. For instance, remember the figure we made at the beginning of this chapter, that looked like this?" ] }, { "cell_type": "code", "execution_count": 17, "id": "6d643270", "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "\n", "fig = sns.pointplot(x='drug', y = 'mood_gain', data = df)\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "15fd0fcf", "metadata": {}, "source": [ "Looking at the figure, it's tempting to conclude that Joyzepam is better than the placebo and better than Anxifree, but there's no real difference between Anxifree and the placebo. However, if we want to get a clearer answer about this, it might help to run some tests. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "25ef7856", "metadata": {}, "source": [ "### Running \"pairwise\" $t$-tests\n", "\n", "How might we go about solving our problem? Given that we've got three separate pairs of means (placebo versus Anxifree, placebo versus Joyzepam, and Anxifree versus Joyzepam) to compare, what we could do is run three separate $t$-tests and see what happens. There's a couple of ways that we could do this. One method would be to construct new variables corresponding the groups you want to compare (e.g., `anxifree`, `placebo` and `joyzepam`), and then run a $t$-test on these new variables." ] }, { "attachments": {}, "cell_type": "markdown", "id": "5861f526", "metadata": {}, "source": [ "(fwer)=\n", "This \"lots of $t$-tests idea\" isn't a bad strategy, but there is a problem with just running lots and lots of $t$-tests. The concern is that when running these analyses, what we're doing is going on a \"fishing expedition\": we're running lots and lots of tests without much theoretical guidance, in the hope that some of them come up significant. This kind of theory-free search for group differences is referred to as **_post hoc analysis_** (\"post hoc\" being Latin for \"after this\").[^plannedcomparisons] \n", "\n", "It's okay to run post hoc analyses, but a lot of care is required, because post-hoc t-tests can be pretty dangerous: each *individual* $t$-test is designed to have a 5\\% Type I error rate (i.e., $\\alpha = .05$), but imagine if I ran three of these tests. Intuitively, you may be able to feel that if every test I do has a 5\\% chance of committing a Type I error, then the more tests I do on the same set of data, the more likely that at least _some_ of them will lead me to incorrectly reject the null hypothesis. \n", "\n", "In fact, we can calculate a statistic that tells us just how risky it is to conduct many post-hoc $t$-tests on our data. It's called the _family-wise error rate_, and looks like this:\n", "\n", "$$FWER = 1-(1-\\alpha)^m$$\n", "\n", "where $m$ is the number of tests I do. So, while we might feel fine about choosing an $\\alpha$ of 0.05 for one $t$-test, which gives us a 5\\% chance of committing a Type I error, we might not feel so hot about running three post-hoc $t$-tests on our data, since \n", "\n", "$$1-(1-0.05)^3=0.142625$$\n", "\n", "Now the family-wise error rate is 0.14 -- in other words, we have now gone from a 5\\% chance of incorrectly rejecting a null hypothesis to a nearly 15\\% chance of incorrectly rejecting a null hypothesis, and maybe those odds don't feel so good anymore. If we did 100 $t$-tests on our data, each with an $\\alpha$ of 0.05, now the FWER goes up to\n", "\n", "$$1-(1-0.05)^{100}=0.99407947$$\n", "\n", "That is, it is nearly certain that we will be incorrectly rejecting some null hypotheses, but which ones? We have no idea.\n", "\n", "As we saw when we talked about [hypothesis testing](hypothesistesting), the central organising principle behind null hypothesis testing is that we seek to control our Type I error rate, but once I start running lots of $t$-tests at once, in order to determine the source of my ANOVA results, my actual Type I error rate across this whole *family* of tests has gotten completely out of control. \n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a7736a39", "metadata": {}, "source": [ "(correctionformultiplecomarisons)=\n", "\n", "The usual solution to this problem is to introduce an adjustment to the $p$-value, which aims to control the total error rate across the family of tests (see {cite}`Shaffer1995`). An adjustment of this form, which is usually (but not always) applied because one is doing post hoc analysis, is often referred to as a **_correction for multiple comparisons_**, though it is sometimes referred to as \"simultaneous inference\". In any case, there are quite a few different ways of doing this adjustment. I'll discuss a few of them in this section and in [a later section](posthoc2), but you should be aware that there are many other methods out there (see, e.g., {cite}`Hsu1996`). \n", "\n", "### Bonferroni corrections\n", "\n", "\n", "The simplest of these adjustments is called the **_Bonferroni correction_** {cite}`Dunn1961`, and it's very very simple indeed. Suppose that my post hoc analysis consists of $m$ separate tests, and I want to ensure that the total probability of making *any* Type I errors at all is at most $\\alpha$.[^bonf] If so, then the Bonferroni correction just says \"multiply all your raw $p$-values by $m$\". If we let $p$ denote the original $p$-value, and let $p^\\prime_j$ be the corrected value, then the Bonferroni correction tells that:\n", "\n", "$$\n", "p^\\prime = m \\times p\n", "$$\n", "\n", "And therefore, if you're using the Bonferroni correction, you would reject the null hypothesis if $p^\\prime < \\alpha$. The logic behind this correction is very straightforward. We're doing $m$ different tests; so if we arrange it so that each test has a Type I error rate of at most $\\alpha / m$, then the *total* Type I error rate across these tests cannot be larger than $\\alpha$. That's pretty simple, so much so that in the original paper, the author writes:\n", "\n", "> The method given here is so simple and so general that I am sure it must have been used before this. I do not find it, however, so can only conclude that perhaps its very simplicity has kept statisticians from realizing that it is a very good method in some situations (pp 52-53 {cite}`Dunn1961`)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "4f1c223e", "metadata": {}, "source": [ "### Holm corrections\n", "\n", "Although the Bonferroni correction is the simplest adjustment out there, it's not usually the best one to use. One method that is often used instead is the **_Holm correction_** {cite}`Holm1979`. The idea behind the Holm correction is to pretend that you're doing the tests sequentially; starting with the smallest (raw) $p$-value and moving onto the largest one. For the $j$-th largest of the $p$-values, the adjustment is *either*\n", "\n", "$$\n", "p^\\prime_j = j \\times p_j \n", "$$\n", "\n", "(i.e., the biggest $p$-value remains unchanged, the second biggest $p$-value is doubled, the third biggest $p$-value is tripled, and so on), *or*\n", "\n", "$$\n", "p^\\prime_j = p^\\prime_{j+1}\n", "$$\n", "\n", "whichever one is *larger*. This might sound a little confusing, so let's go through it a little more slowly. Here's what the Holm correction does. First, you sort all of your $p$-values in order, from smallest to largest. For the smallest $p$-value all you do is multiply it by $m$, and you're done. However, for all the other ones it's a two-stage process. For instance, when you move to the second smallest $p$ value, you first multiply it by $m-1$. If this produces a number that is bigger than the adjusted $p$-value that you got last time, then you keep it. But if it's smaller than the last one, then you copy the last $p$-value. To illustrate how this works, consider the table below, which shows the calculations of a Holm correction for a collection of five $p$-values:\n", "\n", "\n", "|raw $p$ |rank $j$ |$p \\times j$ |Holm $p$ |\n", "|:-------|:--------|:------------|:--------|\n", "|.001 |5 |.005 |.005 |\n", "|.005 |4 |.020 |.020 |\n", "|.019 |3 |.057 |.057 |\n", "|.022 |2 |.044 |.057 |\n", "|.103 |1 |.103 |.103 |\n", "\n", "\n", "Hopefully that makes things clear. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "74d534e4", "metadata": {}, "source": [ "Although it's a little harder to calculate, the Holm correction has some very nice properties: it's more powerful than Bonferroni (i.e., it has a lower Type II error rate), but -- counterintuitive as it might seem -- it has the *same* Type I error rate. As a consequence, in practice there's never any reason to use the simpler Bonferroni correction, since it is always outperformed by the slightly more elaborate Holm correction. Running pairwise t-tests with different correction methods is pretty easy with `pingouin`: we just use the `padjust` argument to say what kind of correction we want to use[^notebonf]:\n", "\n", "[^notebonf]: If you really want to use a Bonferrroni correction anyway, you can write `bonf` instead of `holm` for `padjust`." ] }, { "cell_type": "code", "execution_count": 18, "id": "5e875e91", "metadata": { "tags": [ "remove-cell" ] }, "outputs": [ { "data": { "text/plain": [ "{'divide': 'warn', 'over': 'warn', 'under': 'ignore', 'invalid': 'warn'}" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import numpy as np\n", "np.seterr(all=\"ignore\")" ] }, { "cell_type": "code", "execution_count": 19, "id": "0c5d91b0", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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ContrastABPairedParametricTdofalternativep-uncp-corrp-adjustBF10hedges
0druganxifreejoyzepamFalseTrue-4.20622210.0two-sided0.0018110.003621holm17.947-2.241659
1druganxifreeplaceboFalseTrue1.35418310.0two-sided0.2054860.205486holm0.8140.721696
2drugjoyzepamplaceboFalseTrue7.16870810.0two-sided0.0000300.000091holm475.2313.820482
\n", "
" ], "text/plain": [ " Contrast A B Paired Parametric T dof \\\n", "0 drug anxifree joyzepam False True -4.206222 10.0 \n", "1 drug anxifree placebo False True 1.354183 10.0 \n", "2 drug joyzepam placebo False True 7.168708 10.0 \n", "\n", " alternative p-unc p-corr p-adjust BF10 hedges \n", "0 two-sided 0.001811 0.003621 holm 17.947 -2.241659 \n", "1 two-sided 0.205486 0.205486 holm 0.814 0.721696 \n", "2 two-sided 0.000030 0.000091 holm 475.231 3.820482 " ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pingouin as pg\n", "\n", "# pairwise t-tests with Holm correction\n", "pg.pairwise_tests(dv='mood_gain', \n", " between='drug', \n", " padjust='holm', \n", " data=df)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a11aef0f", "metadata": {}, "source": [ "\n", "As you can see, the biggest $p$-value (corresponding to the comparison between Anxifree and the placebo) is unaltered: it is exactly the same as the value in the ``p-unc`` (\"uncorrected\" $p$-value) column as in the ``p-corr`` (\"corrected\" $p$-value) column. In contrast, the smallest $p$-value (Joyzepam versus placebo) has been multiplied by three.\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "7fa74b40", "metadata": {}, "source": [ "(madness)=\n", "### Statistical madness?\n", "\n", "In my experience, it is usually at around this point that students start to get antsy when I talk about these things in class. They look a little skeptical when I introduce the idea of the null hypothesis, and they shift uncomfortably in their chairs when we talk about $p$-values, but when I start talking about Bonferroni or Holm corrections there are typically one or two who simply can't take it any more, raise their hands, and ask me if I'm not just playing games with numbers at this point.\n", "\n", "In truth, they're not wrong. In a sense, we are playing games with numbers, but it's a serious game: we are trying to use math to help us understand the universe, and all of the infrastructure that has been erected over the years ($p$-values, effect sizes, degrees of freedom, etc.) has been put in place in an attempt to do so in a principled way. It is usually at this point that, in an effort to regain their trust, that I point to the BF10 column in the `pingouin` output. This column shows the \"bayes factor\" associated with the test, and for many (including myself) it is a more intuitively satisfying and sensible way to think about statistical results. There is a lot of [great material](http://xcelab.net/rm/statistical-rethinking/) out there on Bayesian statistics, and while it is beyond the scope of this book in its current form, I will have something to say on the subject [towards the end](bayes) of this book.\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a41e7677", "metadata": {}, "source": [ "(writingupposthoctests)=\n", "### Writing up the post hoc test\n", "\n", "Finally, having run the post hoc analysis to determine which groups are significantly different to one another, you might write up the result like this:\n", "\n", "> Post hoc tests (using the Holm correction to adjust $p$) indicated that Joyzepam produced a significantly larger mood change than both Anxifree ($p = .001$) and the placebo ($p = 9.1 \\times 10^{-5}$). We found no evidence that Anxifree performed better than the placebo ($p = .15$).\n", "\n", "Or, if you don't like the idea of reporting exact $p$-values, then you'd change those numbers to $p<.01$, $p<.001$ and $p > .05$ respectively. Either way, the key thing is that you indicate that you used Holm's correction to adjust the $p$-values. And of course, I'm assuming that elsewhere in the write up you've included the relevant descriptive statistics (i.e., the group means and standard deviations), since these $p$-values on their own aren't terribly informative. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "8201b24d", "metadata": {}, "source": [ "(anovaassumptions)=\n", "## Assumptions of one-way ANOVA\n", "\n", "Like any statistical test, analysis of variance relies on some assumptions about the data. There are three key assumptions that you need to be aware of: *normality*, *homogeneity of variance* and *independence*. If you remember back to [this section](anovamodel) -- which I hope you at least skimmed even if you didn't read the whole thing -- I described the statistical models underpinning ANOVA, which I wrote down like this:\n", "\n", "$$\n", "\\begin{array}{lrcl}\n", "H_0: & Y_{ik} &=& \\mu + \\epsilon_{ik} \\\\\n", "H_1: & Y_{ik} &=& \\mu_k + \\epsilon_{ik} \n", "\\end{array}\n", "$$\n", "\n", "In these equations $\\mu$ refers to a single, grand population mean which is the same for all groups, and $\\mu_k$ is the population mean for the $k$-th group. Up to this point we've been mostly interested in whether our data are best described in terms of a single grand mean (the null hypothesis) or in terms of different group-specific means (the alternative hypothesis). This makes sense, of course: that's actually the important research question! However, all of our testing procedures have -- implicitly -- relied on a specific assumption about the residuals, $\\epsilon_{ik}$, namely that\n", "\n", "$$\n", "\\epsilon_{ik} \\sim \\mbox{Normal}(0, \\sigma^2)\n", "$$\n", "\n", "None of the maths works properly without this bit. Or, to be precise, you can still do all the calculations, and you'll end up with an $F$-statistic, but you have no guarantee that this $F$-statistic actually measures what you think it's measuring, and so any conclusions that you might draw on the basis of the $F$ test might be wrong. \n", "\n", "So, how do we check whether this assumption about the residuals is accurate? Well, as I indicated above, there are three distinct claims buried in this one statement, and we'll consider them separately.\n", "\n", "- **_Normality_**. The residuals are assumed to be normally distributed. [As we saw](shapiro), we can assess this by looking at QQ plots or running a Shapiro-Wilk test. I'll talk about this in an ANOVA context [below](anovanormality). \n", "- **_Homogeneity of variance_**. Notice that we've only got the one value for the population standard deviation (i.e., $\\sigma$), rather than allowing each group to have it's own value (i.e., $\\sigma_k$). This is referred to as the homogeneity of variance (sometimes called homoscedasticity) assumption. ANOVA assumes that the population standard deviation is the same for all groups. We'll talk about this [extensively](levene). \n", "- **_Independence_**. The independence assumption is a little trickier. What it basically means is that, knowing one residual tells you nothing about any other residual. All of the $\\epsilon_{ik}$ values are assumed to have been generated without any \"regard for\" or \"relationship to\" any of the other ones. There's not an obvious or simple way to test for this, but there are some situations that are clear violations of this: for instance, if you have a repeated-measures design, where each participant in your study appears in more than one condition, then independence doesn't hold; there's a special relationship between some observations... namely those that correspond to the same person! When that happens, you need to use something like repeated measures ANOVA. I don't currently talk about repeated measures ANOVA in this book, but it will be included in later versions, if I get [around to it](https://en.wiktionary.org/wiki/round_tuit).\n", "\n", "\n", "### How robust is ANOVA?\n", "\n", "One question that people often want to know the answer to is the extent to which you can trust the results of an ANOVA if the assumptions are violated. Or, to use the technical language, how **_robust_** is ANOVA to violations of the assumptions? Again, I'm running out of time here, and won't be able to address this in the current version of the book. Maybe [someday](https://en.wikipedia.org/wiki/Hofstadter%27s_law)?" ] }, { "attachments": {}, "cell_type": "markdown", "id": "df6b367a", "metadata": {}, "source": [ "(levene)=\n", "## Checking the homogeneity of variance assumption\n", "\n", "There's more than one way to skin a cat, as the saying goes, and more than one way to test the homogeneity of variance assumption, too (though for some reason no-one made a saying out of that). The most commonly used test for this that I've seen in the literature is the **_Levene test_** {cite}`Levene1960`, and the closely related **_Brown-Forsythe test_** {cite}`BrownForsythe1974`, both of which I'll describe here. \n", "\n", "Levene's test is shockingly simple. Suppose we have our outcome variable $Y_{ik}$. All we do is define a new variable, which I'll call $Z_{ik}$, corresponding to the absolute deviation from the group mean:\n", "\n", "$$\n", "Z_{ik} = \\left| Y_{ik} - \\bar{Y}_k \\right|\n", "$$\n", "\n", "Okay, what good does this do us? Well, let's take a moment to think about what $Z_{ik}$ actually is, and what we're trying to test. The value of $Z_{ik}$ is a measure of how the $i$-th observation in the $k$-th group deviates from its group mean. And our null hypothesis is that all groups have the same variance; that is, the same overall deviations from the group means! So, the null hypothesis in a Levene's test is that the population means of $Z$ are identical for all groups. Hm. So what we need now is a statistical test of the null hypothesis that all group means are identical. Where have we seen that before? Oh right, that's what ANOVA is... and so all that the Levene's test does is run an ANOVA on the new variable $Z_{ik}$. Done.\n", "\n", "What about the Brown-Forsythe test? Does that do anything particularly different? Nope. The only change from the Levene's test is that it constructs the transformed variable $Z$ in a slightly different way, using deviations from the group *medians* rather than deviations from the group *means*. That is, for the Brown-Forsythe test, \n", "\n", "$$\n", "Z_{ik} = \\left| Y_{ik} - \\mbox{median}_k(Y) \\right|\n", "$$\n", "\n", "where $\\mbox{median}_k(Y)$ is the median for group $k$. Regardless of whether you're doing the standard Levene test or the Brown-Forsythe test, the test statistic -- which is sometimes denoted $F$, but sometimes written as $W$ -- is calculated in exactly the same way that the $F$-statistic for the regular ANOVA is calculated, just using a $Z_{ik}$ rather than $Y_{ik}$. With that in mind, let's just move on and look at how to run the test in Python." ] }, { "attachments": {}, "cell_type": "markdown", "id": "a551e429", "metadata": {}, "source": [ "### Running the Levene's test with Python\n", "\n", "Okay, so how do we run the Levene test? Obviously, since the Levene test is just an ANOVA, it would be easy enough to manually create the transformed variable $Z_{ik}$ and then run an ANOVA on that. However, that's the tedious way to do it. Much simpler would be to just get `pingouin` to do it for us. Maybe I should take a drink every time I mention `pingouin` in this book. Then again, maybe I shouldn't!" ] }, { "cell_type": "code", "execution_count": 20, "id": "f834ee63", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Wpvalequal_var
levene1.470.26True
\n", "
" ], "text/plain": [ " W pval equal_var\n", "levene 1.47 0.26 True" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pingouin as pg\n", "\n", "pg.homoscedasticity(data=df, \n", " dv=\"mood_gain\", \n", " group=\"drug\").round(2)\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "d394b012", "metadata": {}, "source": [ "If we look at the output, we see that the test is non-significant $(F_{2,15} = 1.47, p = .26)$, so it looks like the homogeneity of variance assumption is fine. By default, the `pingouin`'s `homoscedasticity` function actually does the Brown-Forsythe test. If you want to use the mean instead, then you need to explicitly set the `center` argument, like this:" ] }, { "cell_type": "code", "execution_count": 21, "id": "46be870d", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Wpvalequal_var
levene1.450.27True
\n", "
" ], "text/plain": [ " W pval equal_var\n", "levene 1.45 0.27 True" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Original Levene test\n", "pg.homoscedasticity(data=df, \n", " dv=\"mood_gain\", \n", " group=\"drug\",\n", " center = \"mean\").round(2)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a8b44f95", "metadata": {}, "source": [ "That being said, in most cases it's probably best to stick to the default value, since the Brown-Forsythe test is a bit more robust than the original Levene test." ] }, { "attachments": {}, "cell_type": "markdown", "id": "5435351e", "metadata": {}, "source": [ "(welchoneway)=\n", "## Removing the homogeneity of variance assumption\n", "\n", "In our example, the homogeneity of variance assumption turned out to be a pretty safe one: the Levene test came back non-significant, so we probably don't need to worry. However, in real life we aren't always that lucky. How do we save our ANOVA when the homogeneity of variance assumption is violated? If you recall from our discussion of $t$-tests, we've seen this problem before. The Student $t$-test assumes equal variances, so the solution was to use the Welch $t$-test, which does not. In fact, Welch {cite}`Welch1951` also showed how we can solve this problem for ANOVA too (the **_Welch one-way test_**). It's implemented in `pingouin` using the `welch_anova` function:" ] }, { "cell_type": "code", "execution_count": 22, "id": "cbb9896e", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Sourceddof1ddof2Fp-uncnp2
0drug29.4926.320.00.71
\n", "
" ], "text/plain": [ " Source ddof1 ddof2 F p-unc np2\n", "0 drug 2 9.49 26.32 0.0 0.71" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pingouin as pg\n", "\n", "pg.welch_anova(dv='mood_gain', \n", " between='drug', \n", " data=df).round(2)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "103e77a0", "metadata": {}, "source": [ "To understand what's happening here, let's compare these numbers to what we got [earlier](introduceaov) when we ran our original ANOVA. To save you the trouble of flicking back, here are those numbers again:" ] }, { "cell_type": "code", "execution_count": 23, "id": "ea395d3e", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Sourceddof1ddof2Fp-uncnp2
0drug21518.610.00.71
\n", "
" ], "text/plain": [ " Source ddof1 ddof2 F p-unc np2\n", "0 drug 2 15 18.61 0.0 0.71" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pg.anova(dv='mood_gain', \n", " between='drug', \n", " data=df).round(2)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "033772fb", "metadata": {}, "source": [ "Okay, so originally our ANOVA gave us the result $F(2,15) = 18.6$, whereas the Welch one-way test gave us $F(2,9.49) = 26.32$. In other words, the Welch test has reduced the within-groups degrees of freedom from 15 to 9.49, and the $F$-value has increased from 18.6 to 26.32. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "c25265d4", "metadata": {}, "source": [ "(anovanormality)=\n", "## Checking the normality assumption\n", "\n", "Testing the normality assumption is relatively straightforward. We have already covered [most of what you need to know](shapiro). The only thing we really need to know how to do is pull out the residuals (i.e., the $\\epsilon_{ik}$ values) so that we can draw our QQ plot and run our Shapiro-Wilk test. Now, I really hate to say this, but as of the time of writing (Monday, the 11th of April, 2022), `pingouin` does not currently have a way to give you the residuals from the ANOVA calculation. They expect to add this soon, but in the meantime, we can use `statsmodels` to give us what we need:" ] }, { "cell_type": "code", "execution_count": 24, "id": "894b5a45", "metadata": {}, "outputs": [], "source": [ "import statsmodels.api as sm\n", "from statsmodels.formula.api import ols\n", "\n", "formula = 'mood_gain ~ drug'\n", "\n", "model = ols(formula, data=df).fit()\n", "res = model.resid\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "69eac270", "metadata": {}, "source": [ "We can print them out too, though it's not exactly an edifying experience. In fact, given that I'm on the verge of putting *myself* to sleep just typing this, it might be a good idea to skip that step. Instead, let's draw some pictures and run ourselves a hypothesis test: " ] }, { "cell_type": "code", "execution_count": 25, "id": "a3fa6e42", "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# QQ plot\n", "ax = pg.qqplot(res, dist='norm')\n", "sns.despine()" ] }, { "cell_type": "code", "execution_count": 26, "id": "9d4ebbd7", "metadata": {}, "outputs": [ { "data": { "image/png": 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UKrXsF4vF7hoPAMhB7m+mXbduXXzwgx+MQqEQV199ddxzzz0xcuTINs+tra2NysrKlq2qqqqbpwUAulPuoTJ8+PBYu3ZtrFixIr7+9a/HpEmT4umnn27z3OnTp0djY2PLtmnTpm6eFgDoTrm/9NO3b9/46Ec/GhERY8aMiZUrV8aPfvSj+Pd///e9zi0UClEoFLp7RAAgJ7lfUXm3LMtavQ8FADh45XpF5Zvf/GZMnDgxqqqqYvv27TF//vxYunRpLFq0KM+xAIBE5Boqr7zyStTU1MTmzZujsrIyRo0aFYsWLYrx48fnORYAkIhcQ+WOO+7I8+kBgMQl9x4VAIA9hAoAkCyhAgAkS6gAAMkSKgBAsoQKAJAsoQIAJEuoAADJEioAQLKECgCQLKECACRLqAAAyRIqAECyhAoAkCyhAgAkS6gAAMkSKgBAsoQKAJAsoQIAJEuoAADJEioAQLKECgCQLKECACRLqAAAyRIqAECyhAoAkCyhAgAkS6gAAMkSKgBAsoQKAJAsoQIAJEuoAADJEioAQLKECgCQLKECACRLqAAAyRIqAECyhAoAkCyhAgAkS6gAAMnKNVRqa2vj9NNPj/79+8egQYPiC1/4Qjz77LN5jgQAJCTXUHn00Udj8uTJsWLFili8eHG8/fbbMWHChHjjjTfyHAsASER5nk++aNGiVvvz5s2LQYMGRV1dXfz5n/95TlMBAKnINVTerbGxMSIiBgwY0ObtpVIpSqVSy36xWOyWudi/+vr6aGhoyHuM5G3YsCHvEXoca3ZgSqVSFAqFvMdInp+nnieZUMmyLKZOnRp/9md/FieffHKb59TW1sbMmTO7eTL2p76+PkaMODF27nwz71F6jN2lXXmPkLydja9GRFlcfvnleY/SM5SVRWRZ3lP0GP4O9hzJhMo111wTTz75ZPz3f//3Ps+ZPn16TJ06tWW/WCxGVVVVd4zHe2hoaIidO9+MM66cERWDq/MeJ2mb1y2P9ffNjbfffjvvUZK3+83tEZHFqZfeEEcPG5H3OEnb83NlrfbP38GeJ4lQufbaa+O+++6Lxx57LI477rh9nlcoFFzaTFjF4OoYMGR43mMkrbj5xbxH6HE+OGiIn6v92PNzZa32z9/BnifXUMmyLK699tq45557YunSpTFs2LA8xwEAEpNrqEyePDnuuuuu+M///M/o379/bNmyJSIiKisr49BDD81zNAAgAbl+jsqcOXOisbExPv3pT8fgwYNbtgULFuQ5FgCQiNxf+gEA2Bff9QMAJEuoAADJEioAQLKECgCQLKECACRLqAAAyRIqAECyhAoAkCyhAgAkS6gAAMkSKgBAsoQKAJAsoQIAJEuoAADJEioAQLKECgCQLKECACRLqAAAyRIqAECyhAoAkCyhAgAkS6gAAMkSKgBAsoQKAJAsoQIAJEuoAADJEioAQLKECgCQLKECACRLqAAAyRIqAECyhAoAkCyhAgAkS6gAAMkSKgBAsoQKAJAsoQIAJEuoAADJEioAQLJyDZXHHnssLr744jj22GOjrKws7r333jzHAQASk2uovPHGG/Hxj388br311jzHAAASVZ7nk0+cODEmTpyY5wgAQMJyDZX2KpVKUSqVWvaLxWKXPl99fX00NDR06XP0Bhs2bMh7BAB6qR4VKrW1tTFz5sxuea76+voYMeLE2LnzzW55vt5gd2lX3iMA0Mv0qFCZPn16TJ06tWW/WCxGVVVVlzxXQ0ND7Nz5Zpxx5YyoGFzdJc/RW2xetzzW3zc33n777bxHAaCX6VGhUigUolAodOtzVgyujgFDhnfrc/Y0xc0v5j0CAL2Uz1EBAJKV6xWVHTt2xPPPP9+yv3Hjxli7dm0MGDAghgwZkuNkAEAKcg2VVatWxbhx41r297z/ZNKkSXHnnXfmNBUAkIpcQ+XTn/50ZFmW5wgAQMK8RwUASJZQAQCSJVQAgGQJFQAgWUIFAEiWUAEAkiVUAIBkCRUAIFlCBQBIllABAJIlVACAZAkVACBZQgUASJZQAQCSJVQAgGQJFQAgWUIFAEiWUAEAkiVUAIBkCRUAIFlCBQBIllABAJIlVACAZAkVACBZQgUASJZQAQCSJVQAgGQJFQAgWUIFAEiWUAEAkiVUAIBkCRUAIFlCBQBIllABAJIlVACAZAkVACBZQgUASJZQAQCSJVQAgGTlHiqzZ8+OYcOGRb9+/WL06NGxbNmyvEcCABKRa6gsWLAgrrvuurjxxhtjzZo1cfbZZ8fEiROjvr4+z7EAgETkGiq33HJLfPWrX42vfe1rceKJJ8asWbOiqqoq5syZk+dYAEAiyvN64l27dkVdXV1Mmzat1fEJEybE448/3uZ9SqVSlEqllv3GxsaIiCgWi50+344dOyIi4rWXno23Szs7/fF7k+LmlyIiovHl56JPeVnO06TNWh04a3XgrNWBs1btU9zyv69w7Nixo0v+re3fv3+Ule3nv4csJy+//HIWEdlvfvObVse/853vZCeccEKb95kxY0YWETabzWaz2XrB1tjYuN9eyO2Kyh7vLqksy/ZZV9OnT4+pU6e27Dc3N8drr70WRx111P6L7CBXLBajqqoqNm3aFBUVFXmP02tZ5+5jrbuHde4eB+s69+/ff7/n5BYqAwcOjEMOOSS2bNnS6vjWrVvjmGOOafM+hUIhCoVCq2NHHHFEV43YK1VUVBxUfwnyYp27j7XuHta5e1jnveX2Ztq+ffvG6NGjY/Hixa2OL168OM4666ycpgIAUpLrSz9Tp06NmpqaGDNmTIwdOzbmzp0b9fX1cfXVV+c5FgCQiFxD5ctf/nK8+uqrcfPNN8fmzZvj5JNPjvvvvz+GDh2a51i9UqFQiBkzZuz10hmdyzp3H2vdPaxz97DO+1aWZVmW9xAAAG3J/SP0AQD2RagAAMkSKgBAsoQKAJAsodKLbdu2LWpqaqKysjIqKyujpqYmXn/99QO+/1VXXRVlZWUxa9asLpuxN2jvOu/evTtuuOGGOOWUU+Lwww+PY489Nq644or44x//2H1D9wCzZ8+OYcOGRb9+/WL06NGxbNmy9zz/0UcfjdGjR0e/fv3i+OOPjx//+MfdNGnP1561XrhwYYwfPz6OPvroqKioiLFjx8YDDzzQjdP2XO39md7jN7/5TZSXl8epp57atQMmSqj0YpdeemmsXbs2Fi1aFIsWLYq1a9dGTU3NAd333nvvjSeeeCKOPfbYLp6y52vvOr/55puxevXq+Pa3vx2rV6+OhQsXxu9+97v43Oc+141Tp23BggVx3XXXxY033hhr1qyJs88+OyZOnBj19fVtnr9x48b47Gc/G2effXasWbMmvvnNb8aUKVPiP/7jP7p58p6nvWv92GOPxfjx4+P++++Purq6GDduXFx88cWxZs2abp68Z2nvOu/R2NgYV1xxRZx33nndNGmC3v/XC5Kip59+OouIbMWKFS3Hli9fnkVE9swzz7znff/whz9kH/7wh7P169dnQ4cOzX74wx928bQ91/tZ53f67W9/m0VE9tJLL3XFmD3OJz/5yezqq69udWzEiBHZtGnT2jz/n/7pn7IRI0a0OnbVVVdlZ555ZpfN2Fu0d63bMnLkyGzmzJmdPVqv0tF1/vKXv5x961vfymbMmJF9/OMf78IJ0+WKSi+1fPnyqKysjDPOOKPl2JlnnhmVlZXx+OOP7/N+zc3NUVNTE//4j/8YJ510UneM2qN1dJ3frbGxMcrKynx3VUTs2rUr6urqYsKECa2OT5gwYZ9runz58r3Ov+CCC2LVqlWxe/fuLpu1p+vIWr9bc3NzbN++PQYMGNAVI/YKHV3nefPmxe9///uYMWNGV4+YtNy/PZmusWXLlhg0aNBexwcNGrTXF0G+0/e///0oLy+PKVOmdOV4vUZH1/md3nrrrZg2bVpceumlvowsIhoaGqKpqWmvLyc95phj9rmmW7ZsafP8t99+OxoaGmLw4MFdNm9P1pG1frcf/OAH8cYbb8SXvvSlrhixV+jIOj/33HMxbdq0WLZsWZSXH9z/VLui0sPcdNNNUVZW9p7bqlWrIiKirKxsr/tnWdbm8YiIurq6+NGPfhR33nnnPs85WHTlOr/T7t2745JLLonm5uaYPXt2p/85erJ3r9/+1rSt89s6zt7au9Z73H333XHTTTfFggUL2gx2WjvQdW5qaopLL700Zs6cGSeccEJ3jZesgzvTeqBrrrkmLrnkkvc8p7q6Op588sl45ZVX9rrtT3/6015Vv8eyZcti69atMWTIkJZjTU1N8fd///cxa9asePHFF9/X7D1JV67zHrt3744vfelLsXHjxnj44YddTfk/AwcOjEMOOWSv/6e5devWfa7phz70oTbPLy8vj6OOOqrLZu3pOrLWeyxYsCC++tWvxi9+8Ys4//zzu3LMHq+967x9+/ZYtWpVrFmzJq655pqI+N+X2LIsi/Ly8njwwQfj3HPP7ZbZUyBUepiBAwfGwIED93ve2LFjo7GxMX7729/GJz/5yYiIeOKJJ6KxsTHOOuusNu9TU1Oz1//gXHDBBVFTUxNf+cpX3v/wPUhXrnPE/4+U5557Lh555BH/mL5D3759Y/To0bF48eL4i7/4i5bjixcvjs9//vNt3mfs2LHxq1/9qtWxBx98MMaMGRN9+vTp0nl7so6sdcT/Xkm58sor4+67744LL7ywO0bt0dq7zhUVFbFu3bpWx2bPnh0PP/xw/PKXv4xhw4Z1+cxJyfGNvHSxz3zmM9moUaOy5cuXZ8uXL89OOeWU7KKLLmp1zvDhw7OFCxfu8zH81s/+tXedd+/enX3uc5/LjjvuuGzt2rXZ5s2bW7ZSqZTHHyE58+fPz/r06ZPdcccd2dNPP51dd9112eGHH569+OKLWZZl2bRp07KampqW81944YXssMMOy66//vrs6aefzu64446sT58+2S9/+cu8/gg9RnvX+q677srKy8uz2267rdXP7uuvv57XH6FHaO86v9vB/Fs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"text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# histogram of residuals\n", "\n", "import seaborn as sns\n", "\n", "\n", "ax = sns.histplot(res)\n", "sns.despine()" ] }, { "cell_type": "code", "execution_count": 27, "id": "f99b8d77", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Wpvalnormal
00.960190.605309True
\n", "
" ], "text/plain": [ " W pval normal\n", "0 0.96019 0.605309 True" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#Shapiro-Wilk test with Pingouin\n", "\n", "pg.normality(res)" ] }, { "cell_type": "code", "execution_count": 28, "id": "5d945ef6", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "ShapiroResult(statistic=0.9601902365684509, pvalue=0.6053088307380676)" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#Shapiro-Wilk test with scipy\n", "\n", "from scipy.stats import shapiro\n", "\n", "shapiro(res)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "06fb0310", "metadata": {}, "source": [ "\n", "The histogram and QQ plot are both look pretty normal to me. Not perfect, of course. The histogram in particuar you might need to squint at a bit, to see a normal distringution. But the results of our Shapiro-Wilk test ($W = .96$, $p = .61$) finds no indication that normality is violated, so even if these residuals are not _perfectly_ normally distributed, they seem to be well within the range of the acceptable." ] }, { "attachments": {}, "cell_type": "markdown", "id": "a584eee0", "metadata": {}, "source": [ "(kruskalwallis)=\n", "## Removing the normality assumption\n", "\n", "Now that we've seen how to check for normality, we are led naturally to ask what we can do to address violations of normality. In the context of a one-way ANOVA, the easiest solution is probably to switch to a non-parametric test (i.e., one that doesn't rely on any particular assumption about the kind of distribution involved). We've seen non-parametric tests [before](wilcox): when you only have two groups, the Wilcoxon test provides the non-parametric alternative that you need. When you've got three or more groups, you can use the **_Kruskal-Wallis rank sum test_** {cite}`KruskalWallis1952`. So that's the test we'll talk about next.\n", "\n", "### The logic behind the Kruskal-Wallis test\n", "\n", "The Kruskal-Wallis test is surprisingly similar to ANOVA, in some ways. In ANOVA, we started with $Y_{ik}$, the value of the outcome variable for the $i$th person in the $k$th group. For the Kruskal-Wallis test, what we'll do is rank order all of these $Y_{ik}$ values, and conduct our analysis on the ranked data. So let's let $R_{ik}$ refer to the ranking given to the $i$th member of the $k$th group. Now, let's calculate $\\bar{R}_k$, the average rank given to observations in the $k$th group:\n", "\n", "$$\n", "\\bar{R}_k = \\frac{1}{N_K} \\sum_{i} R_{ik}\n", "$$\n", "and let's also calculate $\\bar{R}$, the grand mean rank:\n", "$$\n", "\\bar{R} = \\frac{1}{N} \\sum_{i} \\sum_{k} R_{ik}\n", "$$\n", "\n", "Now that we've done this, we can calculate the squared deviations from the grand mean rank $\\bar{R}$. When we do this for the individual scores -- i.e., if we calculate $(R_{ik} - \\bar{R})^2$ -- what we have is a \"nonparametric\" measure of how far the $ik$-th observation deviates from the grand mean rank. When we calculate the squared deviation of the group means from the grand means -- i.e., if we calculate $(\\bar{R}_k - \\bar{R} )^2$ -- then what we have is a nonparametric measure of how much the *group* deviates from the grand mean rank. With this in mind, let's follow the same logic that we did with ANOVA, and define our *ranked* sums of squares measures in much the same way that we did earlier. First, we have our \"total ranked sums of squares\":\n", "\n", "$$\n", "\\mbox{RSS}_{tot} = \\sum_k \\sum_i ( R_{ik} - \\bar{R} )^2\n", "$$\n", "\n", "and we can define the \"between groups ranked sums of squares\" like this:\n", "\n", "$$\n", "\\begin{array}{rcl}\n", "\\mbox{RSS}_{b} &=& \\sum_k \\sum_i ( \\bar{R}_k - \\bar{R} )^2 \\\\\n", "&=& \\sum_k N_k ( \\bar{R}_k - \\bar{R} )^2 \n", "\\end{array}\n", "$$\n", "\n", "So, if the null hypothesis is true and there are no true group differences at all, you'd expect the between group rank sums $\\mbox{RSS}_{b}$ to be very small, much smaller than the total rank sums $\\mbox{RSS}_{tot}$. Qualitatively this is very much the same as what we found when we went about constructing the ANOVA $F$-statistic; but for technical reasons the Kruskal-Wallis test statistic, usually denoted $K$, is constructed in a slightly different way: \n", "\n", "$$\n", "K = (N - 1) \\times \\frac{\\mbox{RSS}_b}{\\mbox{RSS}_{tot}}\n", "$$\n", "\n", "and, if the null hypothesis is true, then the sampling distribution of $K$ is *approximately* chi-square with $G-1$ degrees of freedom (where $G$ is the number of groups). The larger the value of $K$, the less consistent the data are with null hypothesis, so this is a one-sided test: we reject $H_0$ when $K$ is sufficiently large." ] }, { "attachments": {}, "cell_type": "markdown", "id": "3d4a0807", "metadata": {}, "source": [ "### Additional details\n", "\n", "The description in the previous section illustrates the logic behind the Kruskal-Wallis test. At a conceptual level, this is the right way to think about how the test works. However, from a purely mathematical perspective it's needlessly complicated. I won't show you the derivation, but you can use a bit of algebraic jiggery-pokery[^notejig] to show that the equation for $K$ can be rewritten as \n", "\n", "$$\n", "K = \\frac{12}{N(N-1)} \\sum_k N_k {\\bar{R}_k}^2 - 3(N+1)\n", "$$\n", "\n", "It's this last equation that you sometimes see given for $K$. This is way easier to calculate than the version I described in the previous section, it's just that it's totally meaningless to actual humans. It's probably best to think of $K$ the way I described it earlier... as an analogue of ANOVA based on ranks. But keep in mind that the test statistic that gets calculated ends up with a rather different look to it than the one we used for our original ANOVA.\n", "\n", "But wait, there's more! Dear lord, why is there always *more*? The story I've told so far is only actually true when there are no ties in the raw data. That is, if there are no two observations that have exactly the same value. If there *are* ties, then we have to introduce a correction factor to these calculations. At this point I'm assuming that even the most diligent reader has stopped caring (or at least formed the opinion that the tie-correction factor is something that doesn't require their immediate attention). So I'll very quickly tell you how it's calculated, and omit the tedious details about *why* it's done this way. Suppose we construct a frequency table for the raw data, and let $f_j$ be the number of observations that have the $j$-th unique value. This might sound a bit abstract, so here's the Python code showing a concrete example:" ] }, { "cell_type": "code", "execution_count": 29, "id": "a14090f4", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " unique_values counts\n", "5 0.1 1\n", "7 0.2 1\n", "0 0.3 2\n", "6 0.4 1\n", "4 0.5 1\n", "1 0.6 2\n", "11 0.8 1\n", "9 0.9 1\n", "10 1.1 1\n", "12 1.2 1\n", "3 1.3 2\n", "2 1.4 2\n", "8 1.7 1\n", "13 1.8 1" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Make a frequency table of the counts of unique values (mood gain)\n", "\n", "f = df['mood_gain'].value_counts().rename_axis('unique_values').reset_index(name='counts').sort_values('unique_values')\n", "f" ] }, { "attachments": {}, "cell_type": "markdown", "id": "8a9b9baa", "metadata": {}, "source": [ "Looking at these frequencies, notice that the third entry in the frequency table has a value of $2$. Since this corresponds to a `mood_gain` of 0.3, this table is telling us that two people's mood increased by 0.3. More to the point, note that we can say that `f['counts'][2]` has a value of `2`. Or, in the mathematical notation I introduced above, this is telling us that $f_3 = 2$. Yay. So, now that we know this, the tie correction factor (TCF) is:\n", "\n", "$$\n", "\\mbox{TCF} = 1 - \\frac{\\sum_j {f_j}^3 - f_j}{N^3 - N} \n", "$$\n", "\n", "The tie-corrected value of the Kruskal-Wallis statistic obtained by dividing the value of $K$ by this quantity: it is this tie-corrected version that Python calculates. And at long last, we're actually finished with the theory of the Kruskal-Wallis test. I'm sure you're all terribly relieved that I've cured you of the existential anxiety that naturally arises when you realise that you *don't* know how to calculate the tie-correction factor for the Kruskal-Wallis test. Right?\n", "\n", "\n", "[^notejig]: A technical term." ] }, { "attachments": {}, "cell_type": "markdown", "id": "11100d3d", "metadata": {}, "source": [ "### How to run the Kruskal-Wallis test in Python\n", "\n", "Despite the horror that we've gone through in trying to understand what the Kruskal-Wallis test actually does, it turns out that running the test is pretty painless, since `pingouin` has a function called `kruskal()`:" ] }, { "cell_type": "code", "execution_count": 30, "id": "e2fb5eb0", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Sourceddof1Hp-unc
Kruskaldrug212.0760.002
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" ], "text/plain": [ " Source ddof1 H p-unc\n", "Kruskal drug 2 12.076 0.002" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pingouin as pg\n", "\n", "pg.kruskal(data=df, \n", " dv='mood_gain', \n", " between='drug').round(3)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "c03f300d", "metadata": {}, "source": [ "(anovaandt)=\n", "\n", "## On the relationship between ANOVA and the Student $t$ test\n", "\n", "There's one last thing I want to point out before finishing. It's something that a lot of people find kind of surprising, but it's worth knowing about: an ANOVA with two groups is identical to the Student $t$-test. No, really. It's not just that they are similar, but they are actually equivalent in every meaningful way. I won't try to prove that this is always true, but I will show you a single concrete demonstration. Suppose that, instead of running an ANOVA on our `mood_gain` predicted by `drug` model, let's instead do it using `therapy` as the predictor. If we run this ANOVA, here's what we get`\n" ] }, { "cell_type": "code", "execution_count": 31, "id": "587df80e", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Sourceddof1ddof2Fp-uncnp2
0therapy1161.7080.210.096
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" ], "text/plain": [ " Source ddof1 ddof2 F p-unc np2\n", "0 therapy 1 16 1.708 0.21 0.096" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pingouin as pg\n", "pg.anova(dv='mood_gain', \n", " between='therapy', \n", " data=df).round(3)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "b49c28fc", "metadata": {}, "source": [ "Overall, it looks like there's no significant effect here at all but, as we'll [soon see](anova2) this is actually a misleading answer! In any case, it's irrelevant to our current goals: our interest here is in the $F$-statistic, which is $F(1,16) = 1.71$, and the $p$-value, which is .21. Since we only have two groups, I didn't actually need to resort to an ANOVA, I could have just decided to run a Student $t$-test. So let's see what happens when I do that. First, I'll just re-arrange the data in to wide format:" ] }, { "cell_type": "code", "execution_count": 32, "id": "cf8c9b68", "metadata": {}, "outputs": [], "source": [ "df_wide = pd.DataFrame(\n", " {'no_therapy': df.loc[df['therapy'] == 'no.therapy']['mood_gain'],\n", " 'CBT': df.loc[df['therapy'] == 'CBT']['mood_gain']})\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a2cb639d", "metadata": {}, "source": [ "Then we can run the test" ] }, { "cell_type": "code", "execution_count": 33, "id": "31057011", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Tdofalternativep-valCI95%cohen-dBF10power
T-test-1.30716two-sided0.21[-0.84, 0.2]0.6160.7320.233
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" ], "text/plain": [ " T dof alternative p-val CI95% cohen-d BF10 power\n", "T-test -1.307 16 two-sided 0.21 [-0.84, 0.2] 0.616 0.732 0.233" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pingouin as pg\n", "\n", "pg.ttest(df_wide['no_therapy'], \n", " df_wide['CBT']).round(3)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "52a61d11", "metadata": {}, "source": [ "Curiously, the $p$-values are identical: once again we obtain a value of $p = .21$. But what about the test statistic? Having run a $t$-test instead of an ANOVA, we get a somewhat different answer, namely $t(16) = -1.3068$. However, there is a fairly straightforward relationship here. If we square the $t$-statistic" ] }, { "cell_type": "code", "execution_count": 34, "id": "4b244729", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "1.708" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "round((-1.307)**2,3)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "quarterly-designer", "metadata": {}, "source": [ "\n", "we get the $F$-statistic from before. Kinda cool, innit?" ] }, { "attachments": {}, "cell_type": "markdown", "id": "28f00f01", "metadata": {}, "source": [ "## Summary\n", "\n", "There's a fair bit covered in this chapter, but there's still a lot missing. Most obviously, I haven't yet discussed any analog of the paired samples $t$-test for more than two groups. There is a way of doing this, known as *repeated measures ANOVA*, which I am happy to report, ``pingouin`` [handles](https://pingouin-stats.org/build/html/generated/pingouin.rm_anova.html#pingouin.rm_anova) with [aplomb](https://en.wikipedia.org/wiki/Aplomb). I also haven't discussed how to run an ANOVA when you are interested in more than one grouping variable, but that will be discussed in a lot of detail in a [later chapter](anova2). In terms of what we have discussed, the key topics were:\n", "\n", "\n", "- The basic logic behind [how ANOVA works](anovaintro) and [how to run one in Python](introduceaov).\n", "- How to compute an [effect size for an ANOVA](etasquared).\n", "- [Post hoc analysis and corrections for multiple testing](posthoc).\n", "- The [assumptions](anovaassumptions) made by ANOVA.\n", "- [How to check the homogeneity of variance assumption](levene) and [what to do](welchoneway) if it is violated.\n", "- [How to check the normality assumption](anovanormality) and [what to do](kruskalwallis) if it is violated.\n", "\n", "\n", "As with all of the chapters in this book, there are quite a few different sources that I've relied upon, but the one stand-out text that I've been most heavily influenced by is {cite}`Sahai2000`. It's not a good book for beginners, but it's an excellent book for more advanced readers who are interested in understanding the mathematics behind ANOVA." ] }, { "attachments": {}, "cell_type": "markdown", "id": "378f5b48", "metadata": {}, "source": [ "[^note1]: When all groups have the same number of observations, the experimental design is said to be \"balanced\". Balance isn't such a big deal for one-way ANOVA, which is the topic of this chapter. It becomes more important when you start doing more complicated ANOVAs.\n", "\n", "[^note2]: In a later versions I'm intending to expand on this. But because I'm writing in a rush, and am already over my deadlines, I'll just briefly note that if you read ahead to the chapter on [factorial anovas](anova2) and look at how the \"treatment effect\" at level $k$ of a factor is defined in terms of the $\\alpha_k$ values (check out the section on [interactions](interactions)), it turns out that $Q$ refers to a weighted mean of the squared treatment effects, $Q=(\\sum_{k=1}^G N_k\\alpha_k^2)/(G-1)$.\n", "\n", "[^note3]: Or, if we want to be sticklers for accuracy, $1 + \\frac{2}{df_2 - 2}$.\n", "\n", "[^note4]: Or, to be precise, party like \"it's 1899 and we've got no friends and nothing better to do with our time than do some calculations that wouldn't have made any sense in 1899 because ANOVA didn't exist until about the 1920s\".\n", "\n", "[^plannedcomparisons]: If you *do* have some theoretical basis for wanting to investigate some comparisons but not others, it's a different story. In those circumstances you're not really running \"post hoc\" analyses at all: you're making \"planned comparisons\". I do talk about this situation [later in the book](plannedcomparisons)), but for now I want to keep things simple.\n", "\n", "[^bonf]: It's worth noting in passing that not all adjustment methods try to do this. What I've described here is an approach for controlling \"family wise Type I error rate\". However, there are other post hoc tests seek to control the \"false discovery rate\", which is a somewhat different thing." ] }, { "cell_type": "code", "execution_count": null, "id": "8706dbf8", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "celltoolbar": "Tags", "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.3" } }, "nbformat": 4, "nbformat_minor": 5 } { "cells": [ { "attachments": {}, "cell_type": "markdown", "id": "constant-nation", "metadata": {}, "source": [ "(regression)=\n", "# Linear regression" ] }, { "attachments": {}, "cell_type": "markdown", "id": "34ea410f", "metadata": {}, "source": [ "\n", "\n", "The goal in this chapter is to introduce **_linear regression_**. Stripped to its bare essentials, linear regression models are basically a slightly fancier version of the [Pearson correlation](correl), though as we'll see, regression models are much more powerful tools." ] }, { "attachments": {}, "cell_type": "markdown", "id": "e7971302", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "Since the basic ideas in regression are closely tied to correlation, we'll return to the `parenthood.csv` file that we were using to illustrate how correlations work. Recall that, in this data set, we were trying to find out why Dan is so very grumpy all the time, and our working hypothesis was that I'm not getting enough sleep. " ] }, { "cell_type": "code", "execution_count": 58, "id": "56714750", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " dan_sleep baby_sleep dan_grump day\n", "0 7.59 10.18 56 1\n", "1 7.91 11.66 60 2\n", "2 5.14 7.92 82 3\n", "3 7.71 9.61 55 4\n", "4 6.68 9.75 67 5" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "file = 'https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/parenthood.csv'\n", "df = pd.read_csv(file)\n", "\n", "df.head()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "d75e9ab0", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "We drew some scatterplots to help us examine the relationship between the amount of sleep I get, and my grumpiness the following day. " ] }, { "cell_type": "code", "execution_count": 59, "id": "b8cd9bbd", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "sleepycorrelation-fig" } }, "output_type": "display_data" } ], "source": [ "from myst_nb import glue\n", "from matplotlib import pyplot as plt\n", "import seaborn as sns\n", "\n", "fig = plt.figure()\n", "\n", "ax = sns.scatterplot(data = df,\n", " x = 'dan_sleep', \n", " y = 'dan_grump')\n", "ax.set(title = 'Grumpiness and sleep', ylabel = 'My grumpiness (0-100)', xlabel='My sleep (hours)')\n", "sns.despine()\n", "\n", "plt.close()\n", "\n", "glue(\"sleepycorrelation-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "eea7dea4", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " ```{glue:figure} sleepycorrelation-fig\n", ":figwidth: 600px\n", ":name: fig-sleepycorrelation\n", "\n", "Scatterplot showing grumpiness as a function of hours slept.\n", "\n", "```\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "75808a56", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "The actual scatterplot that we draw is the one shown in {numref}`fig-sleepycorrelation`, and as we saw previously this corresponds to a correlation of $r=-.90$, but what we find ourselves secretly imagining is something that looks closer to the left panel in {numref}`fig-sleep_regressions_1`. That is, we mentally draw a straight line through the middle of the data. In statistics, this line that we're drawing is called a **_regression line_**. Notice that -- since we're not idiots -- the regression line goes through the middle of the data. We don't find ourselves imagining anything like the rather silly plot shown in the right panel in {numref}`fig-sleep_regressions_1`. " ] }, { "cell_type": "code", "execution_count": 60, "id": "74508823", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "sleep_regressions_1-fig" } }, "output_type": "display_data" } ], "source": [ "import numpy as np \n", "import matplotlib.pyplot as plt\n", "import statsmodels.formula.api as smf\n", "\n", "# find the regression coefficients to allow manually plotting the line\n", "model = smf.ols(formula=\"dan_grump ~ dan_sleep\", data=df).fit()\n", "intercept = model.params.Intercept\n", "slope = model.params.dan_sleep\n", "\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(10, 5), sharey=True)\n", "\n", "x = np.linspace(4,10)\n", "\n", "\n", "sns.scatterplot(data = df, x = 'dan_sleep', y = 'dan_grump', ax = axes[0])\n", "fig.axes[0].set_title(\"The best-fitting regression line\")\n", "fig.axes[0].set_xlabel(\"My sleep (hours)\")\n", "fig.axes[0].set_ylabel(\"My grumpiness (0-10)\")\n", "fig.axes[0].plot(x,slope*x+intercept)\n", "\n", "sns.scatterplot(data = df, x = 'dan_sleep', y = 'dan_grump', ax = axes[1])\n", "fig.axes[1].set_title(\"Not the best-fitting regression line!\")\n", "fig.axes[1].set_xlabel(\"My sleep (hours)\")\n", "fig.axes[1].set_ylabel(\"My grumpiness (0-10)\")\n", "fig.axes[1].plot(x,-3*x+80)\n", "\n", "sns.despine()\n", "\n", "plt.close()\n", "\n", "glue(\"sleep_regressions_1-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "2ee1dcd1", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " ```{glue:figure} sleep_regressions_1-fig\n", ":figwidth: 600px\n", ":name: fig-sleep_regressions_1\n", "\n", "The panel to the left shows the sleep-grumpiness scatterplot from {numref}`fig-sleepycorrelation` with the best fitting regression line drawn over the top. Not surprisingly, the line goes through the middle of the data. In contrast, the panel to the right shows the same data, but with a very poor choice of regression line drawn over the top.\n", "\n", "```\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "69a143d7", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "This is not highly surprising: the line that I've drawn in panel to the right doesn't \"fit\" the data very well, so it doesn't make a lot of sense to propose it as a way of summarising the data, right? This is a very simple observation to make, but it turns out to be very powerful when we start trying to wrap just a little bit of maths around it. To do so, let's start with a refresher of some high school maths. The formula for a straight line is usually written like this:\n", "\n", "$$\n", "y = mx + c\n", "$$ \n", "\n", "\n", "Or, at least, that's what it was when I went to high school all those years ago.[^americanhighschool] The two *variables* are $x$ and $y$, and we have two *coefficients*, $m$ and $c$. The coefficient $m$ represents the *slope* of the line, and the coefficient $c$ represents the *$y$-intercept* of the line. Digging further back into our decaying memories of high school (sorry, for some of us high school was a long time ago), we remember that the intercept is interpreted as \"the value of $y$ that you get when $x=0$\". Similarly, a slope of $m$ means that if you increase the $x$-value by 1 unit, then the $y$-value goes up by $m$ units; a negative slope means that the $y$-value would go down rather than up. Ah yes, it's all coming back to me now. \n", "\n", "Now that we've remembered that, it should come as no surprise to discover that we use the exact same formula to describe a regression line. If $Y$ is the outcome variable (the DV) and $X$ is the predictor variable (the IV), then the formula that describes our regression is written like this:\n", "\n", "$$\n", "\\hat{Y_i} = b_1 X_i + b_0\n", "$$\n", "\n", "Hm. Looks like the same formula, but there's some extra frilly bits in this version. Let's make sure we understand them. Firstly, notice that I've written $X_i$ and $Y_i$ rather than just plain old $X$ and $Y$. This is because we want to remember that we're dealing with actual data. In this equation, $X_i$ is the value of predictor variable for the $i$th observation (i.e., the number of hours of sleep that I got on day $i$ of my little study), and $Y_i$ is the corresponding value of the outcome variable (i.e., my grumpiness on that day). And although I haven't said so explicitly in the equation, what we're assuming is that this formula works for all observations in the data set (i.e., for all $i$). Secondly, notice that I wrote $\\hat{Y}_i$ and not $Y_i$. This is because we want to make the distinction between the *actual data* $Y_i$, and the *estimate* $\\hat{Y}_i$ (i.e., the prediction that our regression line is making). Thirdly, I changed the letters used to describe the coefficients from $m$ and $c$ to $b_1$ and $b_0$. That's just the way that statisticians like to refer to the coefficients in a regression model. I've no idea why they chose $b$, but that's what they did. In any case $b_0$ always refers to the intercept term, and $b_1$ refers to the slope.\n", "\n", "\n", "Excellent, excellent. Next, I can't help but notice that -- regardless of whether we're talking about the good regression line or the bad one -- the data don't fall perfectly on the line. Or, to say it another way, the data $Y_i$ are not identical to the predictions of the regression model $\\hat{Y_i}$. Since statisticians love to attach letters, names and numbers to everything, let's refer to the difference between the model prediction and that actual data point as a *residual*, and we'll refer to it as $\\epsilon_i$.[^noteepsilon] Written using mathematics, the residuals are defined as:\n", "\n", "$$\n", "\\epsilon_i = Y_i - \\hat{Y}_i\n", "$$\n", "\n", "which in turn means that we can write down the complete linear regression model as:\n", "\n", "$$\n", "Y_i = b_1 X_i + b_0 + \\epsilon_i\n", "$$\n", "\n", "[^noteepsilon]: The $\\epsilon$ symbol is the Greek letter epsilon. It's traditional to use $\\epsilon_i$ or $e_i$ to denote a residual.\n", "\n", "[^americanhighschool]: Translator's note: and when _I_ went to high school in the United States ca. [1990](https://en.wikipedia.org/wiki/1990) [CE](https://en.wikipedia.org/wiki/Common_Era) we learned this as $y=ax+b$. Why? Who decides whether it is $a$ or $m$? $c$ or $b$? I truly have no idea, but either way, it represents the same idea." ] }, { "attachments": {}, "cell_type": "markdown", "id": "c247ec1d", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(regressionestimation)=\n", "## Estimating a linear regression model\n", "\n", "\n", "Okay, now let's redraw our pictures, but this time I'll add some lines to show the size of the residual for all observations. When the regression line is good, our residuals (the lengths of the solid black lines) all look pretty small, as shown in the left panel of {numref}`fig-sleep_regressions_2`, but when the regression line is a bad one, the residuals are a lot larger, as you can see from looking at the right panel of {numref}`fig-sleep_regressions_2`. Hm. Maybe what we \"want\" in a regression model is *small* residuals. Yes, that does seem to make sense. In fact, I think I'll go so far as to say that the \"best fitting\" regression line is the one that has the smallest residuals. Or, better yet, since statisticians seem to like to take squares of everything why not say that ...\n", "\n", "> The estimated regression coefficients, $\\hat{b}_0$ and $\\hat{b}_1$ are those that minimise the sum of the squared residuals, which we could either write as $\\sum_i (Y_i - \\hat{Y}_i)^2$ or as $\\sum_i {\\epsilon_i}^2$.\n", "\n", "Yes, yes that sounds even better. And since I've indented it like that, it probably means that this is the right answer. And since this is the right answer, it's probably worth making a note of the fact that our regression coefficients are *estimates* (we're trying to guess the parameters that describe a population!), which is why I've added the little hats, so that we get $\\hat{b}_0$ and $\\hat{b}_1$ rather than $b_0$ and $b_1$. Finally, I should also note that -- since there's actually more than one way to estimate a regression model -- the more technical name for this estimation process is **_ordinary least squares (OLS) regression_**. \n", "\n", "At this point, we now have a concrete definition for what counts as our \"best\" choice of regression coefficients, $\\hat{b}_0$ and $\\hat{b}_1$. The natural question to ask next is, if our optimal regression coefficients are those that minimise the sum squared residuals, how do we *find* these wonderful numbers? The actual answer to this question is complicated, and it doesn't help you understand the logic of regression.[^notekungfu] As a result, this time I'm going to let you off the hook. Instead of showing you how to do it the long and tedious way first, and then \"revealing\" the wonderful shortcut that Python provides you with, let's cut straight to the chase... and use Python to do all the heavy lifting. \n", "\n", "\n", "[^notekungfu]: Or at least, I'm assuming that it doesn't help most people. But on the off chance that someone reading this is a proper kung fu master of linear algebra (and to be fair, I always have a few of these people in my intro stats class), it *will* help *you* to know that the solution to the estimation problem turns out to be $\\hat{b} = (X^TX)^{-1} X^T y$, where $\\hat{b}$ is a vector containing the estimated regression coefficients, $X$ is the \"design matrix\" that contains the predictor variables (plus an additional column containing all ones; strictly $X$ is a matrix of the regressors, but I haven't discussed the distinction yet), and $y$ is a vector containing the outcome variable. For everyone else, this isn't exactly helpful, and can be downright scary. However, since quite a few things in linear regression can be written in linear algebra terms, you'll see a bunch of footnotes like this one in this chapter. If you can follow the maths in them, great. If not, ignore it." ] }, { "cell_type": "code", "execution_count": 61, "id": "ffd9dca1", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "sleep_regressions_2-fig" } }, "output_type": "display_data" } ], "source": [ "import numpy, scipy, matplotlib\n", "import matplotlib.pyplot as plt\n", "from scipy.optimize import curve_fit\n", "\n", "xData = df['dan_sleep']\n", "yData = numpy.array(df['dan_grump'])\n", "\n", "# (the solution to this figure stolen shamelessly from this stack-overflow answer by James Phillips:\n", "# https://stackoverflow.com/questions/53779773/python-linear-regression-best-fit-line-with-residuals)\n", "\n", "# fit linear regression model and save parameters\n", "def func(x, a, b):\n", " return a * x + b\n", "\n", "initialParameters = numpy.array([1.0, 1.0])\n", "\n", "fittedParameters, pcov = curve_fit(func, xData, yData, initialParameters)\n", "\n", "modelPredictions = func(xData, *fittedParameters) \n", "\n", "data = pd.DataFrame({'x': xData,\n", " 'y': yData})\n", "\n", "# plot data points\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 5), sharey=True)\n", "sns.scatterplot(data = data, x = 'x', y = 'y', ax = axes[0])\n", "fig.axes[0].set_title(\"The best-fitting regression line!\")\n", "fig.axes[0].set_xlabel(\"My sleep (hours)\")\n", "fig.axes[0].set_ylabel(\"My grumpiness (0-10)\")\n", "\n", "# add regression line\n", "xModel = numpy.linspace(min(xData), max(xData))\n", "yModel = func(xModel, *fittedParameters)\n", "\n", "axes[0].plot(xModel, yModel)\n", "\n", "# add drop lines\n", "for i in range(len(xData)):\n", " lineXdata = (xData[i], xData[i]) # same X\n", " lineYdata = (yData[i], modelPredictions[i]) # different Y\n", " axes[0].plot(lineXdata, lineYdata)\n", "\n", " \n", "#####\n", "\n", "# create poor-fitting model\n", "badParameters = np.array([-3, 80])\n", "badPredictions = func(xData, *badParameters) \n", "\n", "bad_xModel = numpy.linspace(min(xData), max(xData))\n", "bad_yModel = func(bad_xModel, *badParameters)\n", "\n", "# plot data with poor-fitting model\n", "sns.scatterplot(data = data, x = 'x', y = 'y', ax = axes[1])\n", "fig.axes[1].set_title(\"Not the best-fitting regression line!\")\n", "fig.axes[1].set_xlabel(\"My sleep (hours)\")\n", "fig.axes[1].set_ylabel(\"My grumpiness (0-10)\")\n", "fig.axes[1].plot(bad_xModel, bad_yModel) \n", "\n", "# add drop lines\n", "for i in range(len(xData)):\n", " lineXdata = (xData[i], xData[i]) \n", " lineYdata = (yData[i], badPredictions[i]) \n", " axes[1].plot(lineXdata, lineYdata)\n", " \n", " \n", "sns.despine()\n", "\n", "plt.close()\n", "glue(\"sleep_regressions_2-fig\", fig, display=False)\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "72ceba5b", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " ```{glue:figure} sleep_regressions_2-fig\n", ":figwidth: 600px\n", ":name: fig-sleep_regressions_2\n", "\n", "A depiction of the residuals associated with the best fitting regression line (left panel), and the residuals associated with a poor regression line (right panel). The residuals are much smaller for the good regression line. Again, this is no surprise given that the good line is the one that goes right through the middle of the data.\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "17e9c720", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(pingouinregression)=\n", "## Linear Regression with Python\n", "\n", "As always, there are several different ways we could go about calculating a linear regression in Python, but we'll stick with `pingouin`[^drink], which for my money is one of the simplest and easiest packages to use. The `pingouin` command for linear regression is, well, `linear_regression`, so that couldn't be much more straightforward. After that, we just need to tell `pinguoin` which variable we want to use as a predictor variable (independent variable), and which one we want to use as the outcome variable (dependent variable). `pingouin` wants the predictor variable first, so, since we want to model my grumpiness as a function of my sleep, we write:\n", "\n", "[^drink]: You saw that coming, didn't you? If not, then I bet you haven't read the sections on [t-tests](ttest) or [ANOVA](ANOVA) yet, have you? I'm such a shill for `pingouin`. " ] }, { "cell_type": "code", "execution_count": 62, "id": "solved-yesterday", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [ "import pingouin as pg\n", "\n", "mod1 = pg.linear_regression(df['dan_sleep'], df['dan_grump'])" ] }, { "cell_type": "code", "execution_count": 63, "id": "lined-recommendation", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/html": [ "
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namescoefseTpvalr2adj_r2CI[2.5%]CI[97.5%]
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" ], "text/plain": [ " names coef se T pval r2 adj_r2 CI[2.5%] CI[97.5%]\n", "0 Intercept 125.96 3.02 41.76 0.0 0.82 0.81 119.97 131.94\n", "1 dan_sleep -8.94 0.43 -20.85 0.0 0.82 0.81 -9.79 -8.09" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Display results, rounded to two decimal places.\n", "mod1.round(2)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "09bbc1ca", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "As is its way, `pingouin` gives us a nice simple table, with a lot of information. Most importantly for now, we can see that `pingouin` has caclulated the intercept $\\hat{b}_0 = 125.96$ and the slope $\\hat{b}_1 = -8.94$. In other words, the best-fitting regression line that I plotted in {numref}`fig-sleep_regressions_1` has this formula: \n", "\n", "$$\n", "\\hat{Y}_i = -8.94 \\ X_i + 125.96\n", "$$ " ] }, { "attachments": {}, "cell_type": "markdown", "id": "7fdd35c3", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "### Warning!!!\n", "\n", "Remember, it's critical that you put the variables in the right order. If you reverse the predictor and outcome variables, `pinguoin` will happily calculate a result for you, but it will not be the one you are looking for. If instead, we had written `pg.linear_regression(df['dan_grump'], df['dan_sleep'])`, we would get the following:" ] }, { "cell_type": "code", "execution_count": 64, "id": "f27c3f32", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/html": [ "
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namescoefseTpvalr2adj_r2CI[2.5%]CI[97.5%]
0Intercept12.780.2845.270.00.820.8112.2213.34
1dan_grump-0.090.00-20.850.00.820.81-0.10-0.08
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" ], "text/plain": [ " names coef se T pval r2 adj_r2 CI[2.5%] CI[97.5%]\n", "0 Intercept 12.78 0.28 45.27 0.0 0.82 0.81 12.22 13.34\n", "1 dan_grump -0.09 0.00 -20.85 0.0 0.82 0.81 -0.10 -0.08" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "modx = pg.linear_regression(df['dan_grump'], df['dan_sleep'])\n", "modx.round(2)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "ccc72aef", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "The output looks valid enough on the face of it, and it is even statistically significant. But in this model, we just predicted my son's sleepiness as a function of my grumpiness, which is madness! Reversing the direction of causality would make a great scifi movie[^noteNolan], but it's no good in statistics. So remember, predictor first, outcome second[^noteformula]\n", "\n", "[^noteNolan]: Christopher Nolan, have your people call my people if you're interested, we'll do lunch!\n", "[^noteformula]: This is extra confusing if you happen to have come from the world of R, where this sort of model is usually defined with a formula, in which the outcome measure comes first, followed by the predictor(s), or even if you have used `statsmodels`, which also preserves the R-style formula notation." ] }, { "attachments": {}, "cell_type": "markdown", "id": "31c60fc0", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "### Interpreting the estimated model\n", "\n", "The most important thing to be able to understand is how to interpret these coefficients. Let's start with $\\hat{b}_1$, the slope. If we remember the definition of the slope, a regression coefficient of $\\hat{b}_1 = -8.94$ means that if I increase $X_i$ by 1, then I'm decreasing $Y_i$ by 8.94. That is, each additional hour of sleep that I gain will improve my mood, reducing my grumpiness by 8.94 grumpiness points. What about the intercept? Well, since $\\hat{b}_0$ corresponds to \"the expected value of $Y_i$ when $X_i$ equals 0\", it's pretty straightforward. It implies that if I get zero hours of sleep ($X_i =0$) then my grumpiness will go off the scale, to an insane value of ($Y_i = 125.96$). Best to be avoided, I think.\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "ef122d96", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(multipleregression)=\n", "## Multiple linear regression\n", "\n", "The simple linear regression model that we've discussed up to this point assumes that there's a single predictor variable that you're interested in, in this case `dan_sleep`. In fact, up to this point, *every* statistical tool that we've talked about has assumed that your analysis uses one predictor variable and one outcome variable. However, in many (perhaps most) research projects you actually have multiple predictors that you want to examine. If so, it would be nice to be able to extend the linear regression framework to be able to include multiple predictors. Perhaps some kind of **_multiple regression_** model would be in order?\n", "\n", "Multiple regression is conceptually very simple. All we do is add more terms to our regression equation. Let's suppose that we've got two variables that we're interested in; perhaps we want to use both `dan_sleep` and `baby_sleep` to predict the `dan_grump` variable. As before, we let $Y_i$ refer to my grumpiness on the $i$-th day. But now we have two $X$ variables: the first corresponding to the amount of sleep I got and the second corresponding to the amount of sleep my son got. So we'll let $X_{i1}$ refer to the hours I slept on the $i$-th day, and $X_{i2}$ refers to the hours that the baby slept on that day. If so, then we can write our regression model like this:\n", "\n", "$$\n", "Y_i = b_2 X_{i2} + b_1 X_{i1} + b_0 + \\epsilon_i\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "8d98bead", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "As before, $\\epsilon_i$ is the residual associated with the $i$-th observation, $\\epsilon_i = {Y}_i - \\hat{Y}_i$. In this model, we now have three coefficients that need to be estimated: $b_0$ is the intercept, $b_1$ is the coefficient associated with my sleep, and $b_2$ is the coefficient associated with my son's sleep. However, although the number of coefficients that need to be estimated has changed, the basic idea of how the estimation works is unchanged: our estimated coefficients $\\hat{b}_0$, $\\hat{b}_1$ and $\\hat{b}_2$ are those that minimise the sum squared residuals. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "ad3ad281", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(pingouinmultiplelinearregression)=\n", "## Multiple Linear Regression in Python\n", "\n", "Doing mulitiple linear regression in `pingouin` is just as easy as adding some more predictor variables, like this:" ] }, { "cell_type": "code", "execution_count": 65, "id": "f9c9eddf", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [ " mod2 = pg.linear_regression(df[['dan_sleep', 'baby_sleep']], df['dan_grump'])" ] }, { "attachments": {}, "cell_type": "markdown", "id": "8a117839", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "Still, there is one thing to watch out for. If you look carefully at the command above, you will notice that not only have we added a new predictor (`baby_sleep`), we have also added some extra brackets. While before our predictor variable was `['dan_sleep']`, now we have `[['dan_sleep', 'baby_sleep']]`. Why the extra set of `[]`?\n", "\n", "This is because we are using the brackets in two different ways. When we wrote `['dan_sleep']`, the square brackets meant \"select the column in the `pandas` dataframe with the header 'dan_sleep'\". But now we are giving `pingouin` a _list_ of columns to select, and `list` objects are _also_ defined by square brackets in Python. To keep things clear, another way to achieve the same result would be to define the list of predictor variables outside the call to `pingouin`:" ] }, { "cell_type": "code", "execution_count": 66, "id": "49eff9c6", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [ "predictors = ['dan_sleep', 'baby_sleep']\n", "outcome = 'dan_grump'\n", "\n", "mod2 = pg.linear_regression(df[predictors], df[outcome])" ] }, { "attachments": {}, "cell_type": "markdown", "id": "fa7a4ce3", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "You could even do all the work outside of `pinguoin`, like this:" ] }, { "cell_type": "code", "execution_count": 67, "id": "b3dddf41", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [ "predictors = df[['dan_sleep', 'baby_sleep']]\n", "outcome = df['dan_grump']\n", "\n", "mod2 = pg.linear_regression(predictors, outcome)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "d6650a32", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "All three of these will give the same result, so it's up to you choose what makes most sense to you. But now it's time to take a look at the results:" ] }, { "cell_type": "code", "execution_count": 68, "id": "3acaa6b8", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/html": [ "
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namescoefseTpvalr2adj_r2CI[2.5%]CI[97.5%]
0Intercept125.973.0441.420.000.820.81119.93132.00
1dan_sleep-8.950.55-16.170.000.820.81-10.05-7.85
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" ], "text/plain": [ " names coef se T pval r2 adj_r2 CI[2.5%] CI[97.5%]\n", "0 Intercept 125.97 3.04 41.42 0.00 0.82 0.81 119.93 132.00\n", "1 dan_sleep -8.95 0.55 -16.17 0.00 0.82 0.81 -10.05 -7.85\n", "2 baby_sleep 0.01 0.27 0.04 0.97 0.82 0.81 -0.53 0.55" ] }, "execution_count": 68, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mod2.round(2)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "5d1e86a2", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "The coefficient associated with dan_sleep is quite large, suggesting that every hour of sleep I lose makes me a lot grumpier. However, the coefficient for baby_sleep is very small, suggesting that it doesn’t really matter how much sleep my son gets, not really. What matters as far as my grumpiness goes is how much sleep _I_ get. Although conceptually similar, multiple linear regressions are much harder to visualize than a simple linear regression with only one predictor. To get a sense of what this multiple regression model with two predictors looks like, {numref}`fig-sleep_regressions_3d` shows a 3D plot that plots all three variables, along with the regression model itself." ] }, { "cell_type": "code", "execution_count": 69, "id": "4757a6d2", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "sleep_regressions_3d-fig" } }, "output_type": "display_data" } ], "source": [ "import matplotlib.pyplot as plt\n", "import pandas as pd\n", "import seaborn as sns\n", "from mpl_toolkits.mplot3d import Axes3D\n", "\n", "# style the plot\n", "sns.set_style(\"whitegrid\")\n", "\n", "# construct 3d plot space\n", "fig = plt.figure(figsize=(25, 10)) \n", "ax = fig.add_subplot(111, projection = '3d')\n", "\n", "# define axes\n", "x = df['dan_sleep']\n", "y = df['baby_sleep']\n", "z = df['dan_grump']\n", "\n", "# set axis labels\n", "ax.set_xlabel(\"dan_sleep\")\n", "ax.set_ylabel(\"baby_sleep\")\n", "ax.set_zlabel(\"dan_grump\")\n", "\n", "\n", "# get intercept and regression coefficients from the lmm model\n", "coefs = list(mod2['coef'][1:])\n", "intercept = mod2['coef'][0]\n", "\n", "# create a 3d plane representation of the lmm predictions\n", "xs = np.tile(np.arange(12), (12,1))\n", "ys = np.tile(np.arange(12), (12,1)).T\n", "zs = xs*coefs[0]+ys*coefs[1]+intercept\n", "ax.plot_surface(xs,ys,zs, alpha=0.5)\n", "\n", "# plot the data and plane\n", "ax.plot_surface(xs,ys,zs, alpha=0.01)\n", "ax.scatter(x, y, z, color = 'blue')\n", "\n", "# adjust the viewing angle\n", "ax.view_init(11,97)\n", "\n", "\n", "\n", "# plot the figure in the book with the caption (and no duplicate figure)\n", "plt.close()\n", "glue(\"sleep_regressions_3d-fig\", fig, display=False)\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "db5e9ad1", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " ```{glue:figure} sleep_regressions_3d-fig\n", ":figwidth: 600px\n", ":name: fig-sleep_regressions_3d\n", "\n", "A 3D visualisation of a multiple regression model. There are two predictors in the model, `dan_sleep` and `baby_sleep`; the outcome variable is `dan.grump`. Together, these three variables form a 3D space: each observation (blue dots) is a point in this space. In much the same way that a simple linear regression model forms a line in 2D space, this multiple regression model forms a plane in 3D space. When we estimate the regression coefficients, what we're trying to do is find a plane that is as close to all the blue dots as possible.\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "67bddbf7", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "### Formula for the general case\n", "\n", "The equation that I gave above shows you what a multiple regression model looks like when you include two predictors. Not surprisingly, then, if you want more than two predictors, all you have to do is add more $X$ terms and more $b$ coefficients. In other words, if you have $K$ predictor variables in the model then the regression equation looks like this:\n", "\n", "$$\n", "Y_i = \\left( \\sum_{k=1}^K b_{k} X_{ik} \\right) + b_0 + \\epsilon_i\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "3689e030", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(r2)=\n", "## Quantifying the fit of the regression model\n", "\n", "So we now know how to estimate the coefficients of a linear regression model. The problem is, we don't yet know if this regression model is any good. For example, our multiple linear regression model `mod2` *claims* that every hour of sleep will improve my mood by quite a lot, but it might just be rubbish. Remember, the regression model only produces a prediction $\\hat{Y}_i$ about what my mood is like: my actual mood is $Y_i$. If these two are very close, then the regression model has done a good job. If they are very different, then it has done a bad job. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "c4d35773", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "### The $R^2$ value\n", "\n", "Once again, let's wrap a little bit of mathematics around this. First, we've got the sum of the squared residuals:\n", "\n", "$$\n", "\\mbox{SS}_{res} = \\sum_i (Y_i - \\hat{Y}_i)^2\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "f19b18b6", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "which we would hope to be pretty small. Specifically, what we'd like is for it to be very small in comparison to the total variability in the outcome variable, \n", "\n", "$$\n", "\\mbox{SS}_{tot} = \\sum_i (Y_i - \\bar{Y})^2\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "2ad3aed6", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "While we're here, let's calculate these values in Python. Just to make my Python commands look a bit more similar to the mathematical equations, I'll create variables `X` and `Y`:" ] }, { "cell_type": "code", "execution_count": 70, "id": "4d5fe885", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [ "X = df['dan_sleep'] # the predictor\n", "Y = df['dan_grump'] # the outcome" ] }, { "attachments": {}, "cell_type": "markdown", "id": "c6aa9b62", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "First, lets just examine the output for the simple model that uses only a single predictor:" ] }, { "cell_type": "code", "execution_count": 71, "id": "3c11290e", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/html": [ "
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namescoefseTpvalr2adj_r2CI[2.5%]CI[97.5%]
0Intercept125.963.0241.760.00.820.81119.97131.94
1dan_sleep-8.940.43-20.850.00.820.81-9.79-8.09
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" ], "text/plain": [ " names coef se T pval r2 adj_r2 CI[2.5%] CI[97.5%]\n", "0 Intercept 125.96 3.02 41.76 0.0 0.82 0.81 119.97 131.94\n", "1 dan_sleep -8.94 0.43 -20.85 0.0 0.82 0.81 -9.79 -8.09" ] }, "execution_count": 71, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mod1 = pg.linear_regression(X, Y)\n", "mod1.round(2)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "fe76ae05", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "In this output, we can see that Python has calculated an intercept of 125.96 and a regression coefficient ($beta$) of -8.94. So for every hour of sleep I get, the model estimates that this will correspond to a decrease in grumpiness of about 9 on my incredibly scientific grumpiness scale. We can use this information to calculate $\\hat{Y}$, that is, the values that the model _predicts_ for the outcome measure, as opposed to $Y$, which are the actual data we observed. So, for each value of the predictor variable X, we multiply that value by the regression coefficient -8.84, and add the intercept 125.97:" ] }, { "cell_type": "code", "execution_count": 72, "id": "6f66d0de", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [ "\n", "Y_pred = -8.94 * X + 125.97" ] }, { "attachments": {}, "cell_type": "markdown", "id": "654af8d9", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "Okay, now that we've got a variable which stores the regression model predictions for how grumpy I will be on any given day, let's calculate our sum of squared residuals. We would do that using the following command:" ] }, { "cell_type": "code", "execution_count": 73, "id": "bcce8ffe", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/plain": [ "1838.7224883200004" ] }, "execution_count": 73, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SS_resid = sum( (Y - Y_pred)**2 )\n", "SS_resid" ] }, { "attachments": {}, "cell_type": "markdown", "id": "1f8403ba", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "Wonderful. A big number that doesn't mean very much. Still, let's forge boldly onwards anyway, and calculate the total sum of squares as well. That's also pretty simple:" ] }, { "cell_type": "code", "execution_count": 74, "id": "738f942e", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/plain": [ "9998.590000000002" ] }, "execution_count": 74, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import numpy as np\n", "SS_tot = sum( (Y - np.mean(Y))**2 )\n", "SS_tot" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a4671c01", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "\n", "Hm. Well, it's a much bigger number than the last one, so this does suggest that our regression model was making good predictions. But it's not very interpretable. \n", "\n", "Perhaps we can fix this. What we'd like to do is to convert these two fairly meaningless numbers into one number. A nice, interpretable number, which for no particular reason we'll call $R^2$. What we would like is for the value of $R^2$ to be equal to 1 if the regression model makes no errors in predicting the data. In other words, if it turns out that the residual errors are zero, that is, if $\\mbox{SS}_{res} = 0$, then we expect $R^2 = 1$. Similarly, if the model is completely useless, we would like $R^2$ to be equal to 0. What do I mean by \"useless\"? Tempting as it is demand that the regression model move out of the house, cut its hair and get a real job, I'm probably going to have to pick a more practical definition: in this case, all I mean is that the residual sum of squares is no smaller than the total sum of squares, $\\mbox{SS}_{res} = \\mbox{SS}_{tot}$. Wait, why don't we do exactly that? In fact, the formula that provides us with our $R^2$ value is pretty simple to write down,\n", "\n", "$$\n", "R^2 = 1 - \\frac{\\mbox{SS}_{res}}{\\mbox{SS}_{tot}}\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "cc472da0", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "and equally simple to calculate in Python:" ] }, { "cell_type": "code", "execution_count": 75, "id": "a0ddbf1a", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/plain": [ "0.816101821524835" ] }, "execution_count": 75, "metadata": {}, "output_type": "execute_result" } ], "source": [ "R2 = 1- (SS_resid / SS_tot)\n", "R2" ] }, { "attachments": {}, "cell_type": "markdown", "id": "f83a76ce", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "The $R^2$ value, sometimes called the **_coefficient of determination_**[^notenever] has a simple interpretation: it is the *proportion* of the variance in the outcome variable that can be accounted for by the predictor. So in this case, the fact that we have obtained $R^2 = .816$ means that the predictor (`my_sleep`) explains 81.6\\% of the variance in the outcome (`my_grump`). \n", "\n", "Naturally, you don't actually need to type in all these commands yourself if you want to obtain the $R^2$ value for your regression model. And as you have probably already noticed, `pingouin` calculates $R^2$ for us without even being asked to. But there's another property of $R^2$ that I want to point out. \n", "\n", "[^notenever]: And by \"sometimes\" I mean \"almost never\". In practice everyone just calls it \"$R$-squared\"." ] }, { "attachments": {}, "cell_type": "markdown", "id": "12c7087b", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "### The relationship between regression and correlation\n", "\n", "At this point we can revisit my earlier claim that regression, in this very simple form that I've discussed so far, is basically the same thing as a correlation. Previously, we used the symbol $r$ to denote a Pearson correlation. Might there be some relationship between the value of the correlation coefficient $r$ and the $R^2$ value from linear regression? Of course there is: the squared correlation $r^2$ is identical to the $R^2$ value for a linear regression with only a single predictor. To illustrate this, here's the squared correlation:" ] }, { "cell_type": "code", "execution_count": 76, "id": "ce50df54", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/plain": [ "0.8161027191478786" ] }, "execution_count": 76, "metadata": {}, "output_type": "execute_result" } ], "source": [ "r = X.corr(Y) # calculate the correlation\n", "r**2 # print the squared correlation" ] }, { "attachments": {}, "cell_type": "markdown", "id": "2805acd4", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "\n", "Yep, same number. In other words, running a Pearson correlation is more or less equivalent to running a linear regression model that uses only one predictor variable.\n", "\n", "### The adjusted $R^2$ value\n", "\n", "One final thing to point out before moving on. It's quite common for people to report a slightly different measure of model performance, known as \"adjusted $R^2$\". The motivation behind calculating the adjusted $R^2$ value is the observation that adding more predictors into the model will *always* cause the $R^2$ value to increase (or at least not decrease). The adjusted $R^2$ value introduces a slight change to the calculation, as follows. For a regression model with $K$ predictors, fit to a data set containing $N$ observations, the adjusted $R^2$ is:\n", "\n", "$$\n", "\\mbox{adj. } R^2 = 1 - \\left(\\frac{\\mbox{SS}_{res}}{\\mbox{SS}_{tot}} \\times \\frac{N-1}{N-K-1} \\right)\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "b1291af5", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "This adjustment is an attempt to take the degrees of freedom into account. The big advantage of the adjusted $R^2$ value is that when you add more predictors to the model, the adjusted $R^2$ value will only increase if the new variables improve the model performance more than you'd expect by chance. The big disadvantage is that the adjusted $R^2$ value *can't* be interpreted in the elegant way that $R^2$ can. $R^2$ has a simple interpretation as the proportion of variance in the outcome variable that is explained by the regression model; to my knowledge, no equivalent interpretation exists for adjusted $R^2$. \n", "\n", "An obvious question then, is whether you should report $R^2$ or adjusted $R^2$. This is probably a matter of personal preference. If you care more about interpretability, then $R^2$ is better. If you care more about correcting for bias, then adjusted $R^2$ is probably better. Speaking just for myself, I prefer $R^2$: my feeling is that it's more important to be able to interpret your measure of model performance. Besides, as we'll soon see in the upcoming section on hypothesis tests for regression models, which I am going to link to even though it is [literally the next section in this book](regressiontests), if you're worried that the improvement in $R^2$ that you get by adding a predictor is just due to chance and not because it's a better model, well, we've got hypothesis tests for that. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "fd5caab6", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(regressiontests)=\n", "## Hypothesis tests for regression models\n", "\n", "So far we've talked about what a regression model is, how the coefficients of a regression model are estimated, and how we quantify the performance of the model (the last of these, incidentally, is basically our measure of effect size). The next thing we need to talk about is hypothesis tests. There are two different (but related) kinds of hypothesis tests that we need to talk about: those in which we test whether the regression model as a whole is performing significantly better than a null model; and those in which we test whether a particular regression coefficient is significantly different from zero. \n", "\n", "At this point, you're probably groaning internally, thinking that I'm going to introduce a whole new collection of tests. You're probably sick of hypothesis tests by now, and don't want to learn any new ones. Me too. I'm so sick of hypothesis tests that I'm going to shamelessly reuse the $F$-test from the [chapter on ANOVAs](anova) and the $t$-test from [the chapter on t-tests](ttest). In fact, all I'm going to do in this section is show you how those tests are imported wholesale into the regression framework. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "6c435d23", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "### Testing the model as a whole\n", "\n", "Okay, suppose you've estimated your regression model. The first hypothesis test you might want to try is one in which the null hypothesis that there is *no relationship* between the predictors and the outcome, and the alternative hypothesis is that *the data are distributed in exactly the way that the regression model predicts*. Formally, our \"null model\" corresponds to the fairly trivial \"regression\" model in which we include 0 predictors, and only include the intercept term $b_0$\n", "\n", "$$\n", "H_0: Y_i = b_0 + \\epsilon_i\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "5c500ec4", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "If our regression model has $K$ predictors, the \"alternative model\" is described using the usual formula for a multiple regression model:\n", "\n", "$$\n", "H_1: Y_i = \\left( \\sum_{k=1}^K b_{k} X_{ik} \\right) + b_0 + \\epsilon_i\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "53f575d7", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "How can we test these two hypotheses against each other? The trick is to understand that just like we did with ANOVA, it's possible to divide up the total variance $\\mbox{SS}_{tot}$ into the sum of the residual variance $\\mbox{SS}_{res}$ and the regression model variance $\\mbox{SS}_{mod}$. I'll skip over the technicalities, since we covered most of them in the [ANOVA chapter](anova), and just note that:\n", "\n", "$$\n", "\\mbox{SS}_{mod} = \\mbox{SS}_{tot} - \\mbox{SS}_{res}\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "e1306fd7", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "And, just like we did with the ANOVA, we can convert the sums of squares into mean squares by dividing by the degrees of freedom. \n", "\n", "$$\n", "\\begin{array}{rcl}\n", "\\mbox{MS}_{mod} &=& \\displaystyle\\frac{\\mbox{SS}_{mod} }{df_{mod}} \\\\ \\\\\n", "\\mbox{MS}_{res} &=& \\displaystyle\\frac{\\mbox{SS}_{res} }{df_{res} }\n", "\\end{array}\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "db91e296", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "So, how many degrees of freedom do we have? As you might expect, the $df$ associated with the model is closely tied to the number of predictors that we've included. In fact, it turns out that $df_{mod} = K$. For the residuals, the total degrees of freedom is $df_{res} = N -K - 1$. \n", "\n", "Now that we've got our mean square values, you're probably going to be entirely unsurprised (possibly even bored) to discover that we can calculate an $F$-statistic like this:\n", "\n", "$$\n", "F = \\frac{\\mbox{MS}_{mod}}{\\mbox{MS}_{res}}\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "63692822", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "and the degrees of freedom associated with this are $K$ and $N-K-1$. This $F$ statistic has exactly the same interpretation as the one we introduced [when learning about ANOVAs](anova). Large $F$ values indicate that the null hypothesis is performing poorly in comparison to the alternative hypothesis." ] }, { "attachments": {}, "cell_type": "markdown", "id": "9e9a64e9", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "\"Ok, this is fine\", I hear you say, \"but now show me the easy way! Show me how easy it is to get an $F$ statistic from `pingouin`! `pingouin` makes everything so much easier! Surely `pingouin` does this for me as well?\"\n", "\n", "Yeah. About that... actually, as of the time of writing (Tuesday the 17th of May, 2022), `pingouin` does _not_ automatically calculate the $F$ statistic for the model for you. This seems like kind of a strange omission to me, since it is pretty normal to report overall $F$ and $p$ values for a model, and `pingouin` seems to be all about making the normal things easy. So, I can only assume this will get added at some point, but for now, sadly, we are left to ourselves on this one. \n", "\n", "I should mention that there are other statistics packages for Python that will do this for you. [statsmodels](https://www.statsmodels.org/stable/regression.html) comes to mind, for instance. But this is opening a whole new can of worms that I'd rather avoid for now, so instead I provide you with code to calculate the $F$ statistic and $p$-value for the model \"manually\" below:" ] }, { "cell_type": "code", "execution_count": 77, "id": "eb455782", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "F= 215.2382865368443 p= 2.1457300163209325e-36\n" ] } ], "source": [ "import numpy as np\n", "from scipy import stats as st\n", "\n", "# your predictor and outcome variables (aka, \"the data\")\n", "predictors = df[['dan_sleep', 'baby_sleep']]\n", "outcome = df['dan_grump']\n", "\n", "# model the data, and store the model information in a variable called \"mod\"\n", "mod = pg.linear_regression(predictors, outcome)\n", "\n", "\n", "# call the outcome data \"Y\", just for the sake of generalizability\n", "Y = outcome\n", "\n", "# get the model residuals from the model object\n", "res = mod.residuals_\n", "\n", "# calculate the residual, the model, and the total sums of squares\n", "SS_res = np.sum(np.square(res))\n", "SS_tot = sum( (Y - np.mean(Y))**2 )\n", "SS_mod = SS_tot - SS_res\n", "\n", "# get the degrees of freedom for the model and the residuals\n", "df_mod = mod.df_model_\n", "df_res = mod.df_resid_\n", "\n", "# caluculate the mean squares for the model and the residuals\n", "MS_mod = SS_mod / df_mod\n", "MS_res = SS_res / df_res\n", "\n", "# calculate the F-statistic\n", "F = MS_mod / MS_res\n", "\n", "# estimate the p-value\n", "p = st.f.sf(F, df_mod, df_res)\n", "\n", "# display the results\n", "print(\"F=\",F, \"p=\", p)\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "e29bbba8", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(ftestfunction)=\n", "### An F-test function\n", "\n", "A more compact way to do this would be to take everything I have done above and put it inside a [function](functions). I've done this below, not least so that I will be able to copy/paste from it myself at some later date. Here is a function called `regression_f` that takes as its arguments a list of predictors, and an outcome variable, and spits out the $F$ and $p$ values." ] }, { "cell_type": "code", "execution_count": 78, "id": "387b7ea4", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [ "def regression_f(predictors, outcome):\n", " mod = pg.linear_regression(predictors, outcome)\n", " Y = outcome\n", " res = mod.residuals_\n", " SS_res = np.sum(np.square(res))\n", " SS_tot = sum( (Y - np.mean(Y))**2 )\n", " SS_mod = SS_tot - SS_res\n", " df_mod = mod.df_model_\n", " df_res = mod.df_resid_\n", " MS_mod = SS_mod / df_mod\n", " MS_res = SS_res / df_res\n", " F = MS_mod / MS_res\n", " p = st.f.sf(F, df_mod, df_res)\n", " return(F, p)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "9120a43b", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "Once we have run the function, all we need to do is plug in our values, and `regression_f` does the rest:" ] }, { "cell_type": "code", "execution_count": 79, "id": "50ffea08", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/plain": [ "(215.2382865368443, 2.1457300163209325e-36)" ] }, "execution_count": 79, "metadata": {}, "output_type": "execute_result" } ], "source": [ "predictors = df[['dan_sleep', 'baby_sleep']]\n", "outcome = df['dan_grump']\n", "\n", "regression_f(predictors, outcome)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "6c23c6a4", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "### Tests for individual coefficients\n", "\n", "The $F$-test that we've just introduced is useful for checking that the model as a whole is performing better than chance. This is important: if your regression model doesn't produce a significant result for the $F$-test then you probably don't have a very good regression model (or, quite possibly, you don't have very good data). However, while failing this test is a pretty strong indicator that the model has problems, *passing* the test (i.e., rejecting the null) doesn't imply that the model is good! Why is that, you might be wondering? The answer to that can be found by looking at the coefficients for the multiple linear regression model we calculated earlier:" ] }, { "cell_type": "code", "execution_count": 80, "id": "5fc70431", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/html": [ "
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namescoefseTpvalr2adj_r2CI[2.5%]CI[97.5%]
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" ], "text/plain": [ " names coef se T pval r2 adj_r2 CI[2.5%] CI[97.5%]\n", "0 Intercept 125.97 3.04 41.42 0.00 0.82 0.81 119.93 132.00\n", "1 dan_sleep -8.95 0.55 -16.17 0.00 0.82 0.81 -10.05 -7.85\n", "2 baby_sleep 0.01 0.27 0.04 0.97 0.82 0.81 -0.53 0.55" ] }, "execution_count": 80, "metadata": {}, "output_type": "execute_result" } ], "source": [ "predictors = df[['dan_sleep', 'baby_sleep']]\n", "outcome = df['dan_grump']\n", "\n", "mod2 = pg.linear_regression(predictors, outcome)\n", "mod2.round(2)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "67c75cc1", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "\n", "I can't help but notice that the estimated regression coefficient for the `baby_sleep` variable is tiny (0.01), relative to the value that we get for `dan_sleep` (-8.95). Given that these two variables are absolutely on the same scale (they're both measured in \"hours slept\"), I find this suspicious. In fact, I'm beginning to suspect that it's really only the amount of sleep that *I* get that matters in order to predict my grumpiness.\n", "\n", "Once again, we can reuse a hypothesis test that we discussed earlier, this time the $t$-test. The test that we're interested has a null hypothesis that the true regression coefficient is zero ($b = 0$), which is to be tested against the alternative hypothesis that it isn't ($b \\neq 0$). That is:\n", "\n", "$$\n", "\\begin{array}{rl}\n", "H_0: & b = 0 \\\\\n", "H_1: & b \\neq 0 \n", "\\end{array}\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "59650808", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "How can we test this? Well, if the [central limit theorem](clt) is kind to us, we might be able to guess that the sampling distribution of $\\hat{b}$, the estimated regression coefficient, is a normal distribution with mean centred on $b$. What that would mean is that if the null hypothesis were true, then the sampling distribution of $\\hat{b}$ has mean zero and unknown standard deviation. Assuming that we can come up with a good estimate for the [standard error](clt) of the regression coefficient, $\\mbox{SE}({\\hat{b}})$, then we're in luck. That's *exactly* the situation for which we introduced the one-sample $t$ way back in [the chapter on t-tests](ttest). So let's define a $t$-statistic like this,\n", "\n", "$$\n", "t = \\frac{\\hat{b}}{\\mbox{SE}({\\hat{b})}}\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "829f64ab", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "I'll skip over the reasons why, but our degrees of freedom in this case are $df = N- K- 1$. Irritatingly, the estimate of the standard error of the regression coefficient, $\\mbox{SE}({\\hat{b}})$, is not as easy to calculate as the standard error of the mean that we used for the simpler $t$-tests [earlier](ttest). In fact, the formula is somewhat ugly, and not terribly helpful to look at. For our purposes it's sufficient to point out that the standard error of the estimated regression coefficient depends on both the predictor and outcome variables, and is somewhat sensitive to violations of the homogeneity of variance assumption (discussed shortly). \n", "\n", "In any case, this $t$-statistic can be interpreted in the same way as the $t$-statistics that we discussed [earlier](ttest). Assuming that you have a two-sided alternative (i.e., you don't really care if $b >0$ or $b < 0$), then it's the extreme values of $t$ (i.e., a lot less than zero or a lot greater than zero) that suggest that you should reject the null hypothesis. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "3c306575", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "Now we are in a position to understand all the values in the multiple regression table provided by `pingouin`:" ] }, { "cell_type": "code", "execution_count": 81, "id": "782d33f1", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/html": [ "
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namescoefseTpvalr2adj_r2CI[2.5%]CI[97.5%]
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2baby_sleep0.010.270.040.970.820.81-0.530.55
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" ], "text/plain": [ " names coef se T pval r2 adj_r2 CI[2.5%] CI[97.5%]\n", "0 Intercept 125.97 3.04 41.42 0.00 0.82 0.81 119.93 132.00\n", "1 dan_sleep -8.95 0.55 -16.17 0.00 0.82 0.81 -10.05 -7.85\n", "2 baby_sleep 0.01 0.27 0.04 0.97 0.82 0.81 -0.53 0.55" ] }, "execution_count": 81, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mod2.round(2)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "21e8af68", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "Each row in this table refers to one of the coefficients in the regression model. The first row is the intercept term, and the later ones look at each of the predictors. The columns give you all of the relevant information. The `coef` column is the actual estimate of $b$ (e.g., 125.96 for the intercept, -8.9 for the `dan_sleep` predictor, and 0.01 for the `baby_sleep` predictor). The `se` column is the standard error estimate $\\hat\\sigma_b$. The `T` column gives you the $t$-statistic, and it's worth noticing that in this table $t= \\hat{b}/\\mbox{SE}({\\hat{b}})$ every time. The `pval` column gives you the actual $p$ value for each of these tests.[^notecorrection] The `r2`and àdj_r2` columns give the $R^2$ value and the adjusted $R^2$ for the model, and the last two columns give us the upper and lower [confidence interval](ci) bounds for each estimate.\n", "\n", "[^notecorrection]: Note that, although `pingouin` has done multiple tests here, it hasn't done a Bonferroni correction or anything. These are standard one-sample $t$-tests with a two-sided alternative. If you want to make corrections for multiple tests, you need to do that yourself." ] }, { "attachments": {}, "cell_type": "markdown", "id": "ac9e50c0", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "If we add our [F-test results](ftestfunction) to the mix:" ] }, { "cell_type": "code", "execution_count": 82, "id": "1e7e0a09", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "F: 215.24 p: 2.1457300163209325e-36\n" ] } ], "source": [ "f = regression_f(predictors, outcome)\n", "print(\"F:\", f[0].round(2), \"p:\", f[1])" ] }, { "attachments": {}, "cell_type": "markdown", "id": "50523e3b", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "we have everything we need to evaluate our model. In this case, the model performs significantly better than you'd expect by chance ($F(2,97) = 215.2$, $p<.001$), which isn't all that surprising: the $R^2 = .812$ value indicate that the regression model accounts for 81.2\\% of the variability in the outcome measure. However, when we look back up at the $t$-tests for each of the individual coefficients, we have pretty strong evidence that the `baby_sleep` variable has no significant effect; all the work is being done by the `dan_sleep` variable. Taken together, these results suggest that `mod2` is actually the wrong model for the data: you'd probably be better off dropping the `baby_sleep` predictor entirely. In other words, the `mod1` model that we started with is the better model. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "cc471f67", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(corrhyp)=\n", "## Testing the significance of a correlation\n", "\n", "\n", "### Hypothesis tests for a single correlation\n", "\n", "I don't want to spend too much time on this, but it's worth very briefly returning to the point I made earlier, that Pearson correlations are basically the same thing as linear regressions with only a single predictor added to the model. What this means is that the hypothesis tests that I just described in a regression context can also be applied to correlation coefficients. To see this, let's just revist our `mod1` model:" ] }, { "cell_type": "code", "execution_count": 83, "id": "dbac25b8", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/html": [ "
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namescoefseTpvalr2adj_r2CI[2.5%]CI[97.5%]
0Intercept125.963.0241.760.00.820.81119.97131.94
1dan_sleep-8.940.43-20.850.00.820.81-9.79-8.09
\n", "
" ], "text/plain": [ " names coef se T pval r2 adj_r2 CI[2.5%] CI[97.5%]\n", "0 Intercept 125.96 3.02 41.76 0.0 0.82 0.81 119.97 131.94\n", "1 dan_sleep -8.94 0.43 -20.85 0.0 0.82 0.81 -9.79 -8.09" ] }, "execution_count": 83, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X = df['dan_sleep'] # the predictor\n", "Y = df['dan_grump'] # the outcome\n", "mod1 = pg.linear_regression(X, Y)\n", "mod1.round(2)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "e0987de1", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "The important thing to note here is the $t$ test associated with the predictor, in which we get a result of $t(98) = -20.85$, $p<.001$. Now let's compare this to the output of the `corr` function from `pinguoin`, which runs a hypothesis test to see if the observed correlation between two variables is significantly different from 0. " ] }, { "cell_type": "code", "execution_count": 84, "id": "77497fa4", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
nrCI95%p-valBF10power
pearson100-0.903384[-0.93, -0.86]8.176426e-382.591e+341.0
\n", "
" ], "text/plain": [ " n r CI95% p-val BF10 power\n", "pearson 100 -0.903384 [-0.93, -0.86] 8.176426e-38 2.591e+34 1.0" ] }, "execution_count": 84, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pg.corr(X,Y)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "466ae212", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "Now, just like the $F$-test from earlier, `pingouin` unfortunately doesn't calculate a $t$-statistic for us automatically when running a correlation. But the formula for the $t$-statistic of a Pearson correlation is just\n", "\n", "\n", "$$\n", "t = r\\sqrt{\\frac{n-2}{1-r^2}}\n", "$$\n", "\n", "so, with the output from `pg.corr(X,Y)` above, it's not too difficult to find $t$:" ] }, { "cell_type": "code", "execution_count": 85, "id": "d4567e48", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/plain": [ "-20.85439996017091" ] }, "execution_count": 85, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from math import sqrt\n", "\n", "t = -0.903384*sqrt((100-2) / (1-(-0.903384)**2))\n", "t" ] }, { "attachments": {}, "cell_type": "markdown", "id": "cd2e26bc", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "Look familiar? -20.85 was the same $t$-value that we got when we ran the regression model. That's because the test for the significance of a correlation is identical to the $t$ test that we run on a coefficient in a regression model. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "319fae53", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(corrhyp2)=\n", "### Hypothesis tests for all pairwise correlations\n", "\n", "Okay, one more digression before I return to regression properly. In the previous section I talked about you can run a hypothesis test on a single correlation. But we aren't restricted to computing a single correlation: you can compute *all* pairwise correlations among the variables in your data set. This leads people to the natural question: can we also run hypothesis tests on all of the pairwise correlations in our data using `pg.corr`?\n", "\n", "The answer is no, and there's a very good reason for this. Testing a single correlation is fine: if you've got some reason to be asking \"is A related to B?\", then you should absolutely run a test to see if there's a significant correlation. But if you've got variables A, B, C, D and E and you're thinking about testing the correlations among all possible pairs of these, a statistician would want to ask: what's your hypothesis? If you're in the position of wanting to test all possible pairs of variables, then you're pretty clearly on a fishing expedition, hunting around in search of significant effects when you don't actually have a clear research hypothesis in mind. This is *dangerous*, and perhaps the authors of the `corr` function didn't want to endorse this sort of behavior. `corr` does have the nice feature that you can call it as an attribute of your dataframe, so for our parenthood data, if we want to see all the parwise correlations in the data, you can simply write" ] }, { "cell_type": "code", "execution_count": 86, "id": "03a9c5b8", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/html": [ "
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dan_sleepbaby_sleepdan_grumpday
dan_sleep1.0000000.627949-0.903384-0.098408
baby_sleep0.6279491.000000-0.565964-0.010434
dan_grump-0.903384-0.5659641.0000000.076479
day-0.098408-0.0104340.0764791.000000
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" ], "text/plain": [ " dan_sleep baby_sleep dan_grump day\n", "dan_sleep 1.000000 0.627949 -0.903384 -0.098408\n", "baby_sleep 0.627949 1.000000 -0.565964 -0.010434\n", "dan_grump -0.903384 -0.565964 1.000000 0.076479\n", "day -0.098408 -0.010434 0.076479 1.000000" ] }, "execution_count": 86, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.corr()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "06550355", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "and you get a nice correlation matrix, but no p-values.\n", "\n", "On the other hand... a somewhat less hardline view might be to argue we've encountered this situation before, back when we talked about *post hoc tests* in ANOVA. When running post hoc tests, we didn't have any specific comparisons in mind, so what we did was apply a correction (e.g., Bonferroni, Holm, etc) in order to avoid the possibility of an inflated Type I error rate. From this perspective, it's okay to run hypothesis tests on all your pairwise correlations, but you must treat them as post hoc analyses, and if so you need to apply a correction for multiple comparisons. `rcorr`, also from `pingouin`, lets you do this. You can use the `padjust` argument to specify what kind of correction you would like to apply; here I have chosen a Bonferroni correction:" ] }, { "cell_type": "code", "execution_count": 87, "id": "c1d4b47d", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/html": [ "
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dan_sleepbaby_sleepdan_grumpday
dan_sleep-******
baby_sleep0.628-***
dan_grump-0.903-0.566-
day-0.098-0.010.076-
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" ], "text/plain": [ " dan_sleep baby_sleep dan_grump day\n", "dan_sleep - *** *** \n", "baby_sleep 0.628 - *** \n", "dan_grump -0.903 -0.566 - \n", "day -0.098 -0.01 0.076 -" ] }, "execution_count": 87, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.rcorr(padjust = 'bonf')" ] }, { "attachments": {}, "cell_type": "markdown", "id": "7e7bcec5", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "The little stars indicate the \"significance level\": one star for $p<0.05$, two stars for $p<0.01$, and three stars for $p<0.001$.\n", "\n", "So there you have it. If you really desperately want to do pairwise hypothesis tests on your correlations, the `rcorr` function will let you do it. But please, **please** be careful. I can't count the number of times I've had a student panicking in my office because they've run these pairwise correlation tests, and they get one or two significant results that don't make any sense. For some reason, the moment people see those little significance stars appear, they feel compelled to throw away all common sense and assume that the results must correspond to something real that requires an explanation. In most such cases, my experience has been that the right answer is \"it's a Type I error\". Remember when we talked about the [family-wise error rate](fwer)? The more tests you do on the same data, the greater your chances of finding statistically significant results just by, uh, chance." ] }, { "attachments": {}, "cell_type": "markdown", "id": "1329623c", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(regressioncoefs)=\n", "### Calculating standardised regression coefficients\n", "\n", "One more thing that you might want to do is to calculate \"standardised\" regression coefficients, often denoted $\\beta$. The rationale behind standardised coefficients goes like this. In a lot of situations, your variables are on fundamentally different scales. Suppose, for example, my regression model aims to predict people's IQ scores, using their educational attainment (number of years of education) and their income as predictors. Obviously, educational attainment and income are not on the same scales: the number of years of schooling can only vary by 10s of years, whereas income would vary by 10,000s of dollars (or more). The units of measurement have a big influence on the regression coefficients: the $b$ coefficients only make sense when interpreted in light of the units, both of the predictor variables and the outcome variable. This makes it very difficult to compare the coefficients of different predictors. Yet there are situations where you really do want to make comparisons between different coefficients. Specifically, you might want some kind of standard measure of which predictors have the strongest relationship to the outcome. This is what **_standardised coefficients_** aim to do. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "ccd35c64", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "The basic idea is quite simple: the standardised coefficients are the coefficients that you would have obtained if you'd converted all the variables to [standard scores](zscores) $z$-scores before running the regression.[^noteregressors] The idea here is that, by converting all the predictors to $z$-scores, they all go into the regression on the same scale, thereby removing the problem of having variables on different scales. Regardless of what the original variables were, a $\\beta$ value of 1 means that an increase in the predictor of 1 standard deviation will produce a corresponding 1 standard deviation increase in the outcome variable. Therefore, if variable A has a larger absolute value of $\\beta$ than variable B, it is deemed to have a stronger relationship with the outcome. Or at least that's the idea: it's worth being a little cautious here, since this does rely very heavily on the assumption that \"a 1 standard deviation change\" is fundamentally the same kind of thing for all variables. It's not always obvious that this is true. \n", "\n", "[^noteregressors]: Strictly, you standardise all the *regressors*: that is, every \"thing\" that has a regression coefficient associated with it in the model. For the regression models that I've talked about so far, each predictor variable maps onto exactly one regressor, and vice versa. However, that's not actually true in general: we'll see some examples of this when we learn about [factorial ANOVA](anova2). But for now, we don't need to care too much about this distinction." ] }, { "attachments": {}, "cell_type": "markdown", "id": "229e5c52", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "Still, let's give it a try on the `parenthood` data.\n", "\n" ] }, { "cell_type": "code", "execution_count": 88, "id": "82d6568e", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/html": [ "
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namescoefseTpvalr2adj_r2CI[2.5%]CI[97.5%]
0Intercept-0.00000.0435-0.00001.00000.81610.8123-0.08640.0864
1dan_sleep_standard-0.90470.0559-16.17150.00000.81610.8123-1.0158-0.7937
2baby_sleep_standard0.00220.05590.03880.96910.81610.8123-0.10890.1132
\n", "
" ], "text/plain": [ " names coef se T pval r2 adj_r2 \\\n", "0 Intercept -0.0000 0.0435 -0.0000 1.0000 0.8161 0.8123 \n", "1 dan_sleep_standard -0.9047 0.0559 -16.1715 0.0000 0.8161 0.8123 \n", "2 baby_sleep_standard 0.0022 0.0559 0.0388 0.9691 0.8161 0.8123 \n", "\n", " CI[2.5%] CI[97.5%] \n", "0 -0.0864 0.0864 \n", "1 -1.0158 -0.7937 \n", "2 -0.1089 0.1132 " ] }, "execution_count": 88, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from scipy import stats\n", "import pingouin as pg\n", "\n", "df['dan_sleep_standard'] = stats.zscore(df['dan_sleep'])\n", "df['baby_sleep_standard'] = stats.zscore(df['baby_sleep'])\n", "df['dan_grump'] = stats.zscore(df['dan_grump'])\n", "\n", "\n", "predictors = df[['dan_sleep_standard', 'baby_sleep_standard']]\n", "outcome = df['dan_grump']\n", "\n", "mod3 = pg.linear_regression(predictors, outcome)\n", "mod3.round(4)\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "63cc2cde", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "This clearly shows that the `dan_sleep` variable has a much stronger effect than the `baby_sleep` variable. However, this is a perfect example of a situation where it would probably make sense to use the original coefficients $b$ rather than the standardised coefficients $\\beta$. After all, my sleep and the baby's sleep are *already* on the same scale: number of hours slept. Why complicate matters by converting these to $z$-scores?" ] }, { "attachments": {}, "cell_type": "markdown", "id": "ab779f4e", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(regressionassumptions)=\n", "\n", "## Assumptions of regression\n", "\n", "The linear regression model that I've been discussing relies on several assumptions. In the section on [regression diagnostics](regressiondiagnostics) we'll talk a lot more about how to check that these assumptions are being met, but first, let's have a look at each of them.\n", "\n", "\n", "- *Normality*. Like half the models in statistics, standard linear regression relies on an assumption of normality. Specifically, it assumes that the *residuals* are normally distributed. It's actually okay if the predictors $X$ and the outcome $Y$ are non-normal, so long as the residuals $\\epsilon$ are normal. See [](regressionnormality).\n", "- *Linearity*. A pretty fundamental assumption of the linear regression model is that relationship between $X$ and $Y$ actually be linear! Regardless of whether it's a simple regression or a multiple regression, we assume that the relatiships involved are linear. See [](regressionlinearity).\n", "- *Homogeneity of variance*. Strictly speaking, the regression model assumes that each residual $\\epsilon_i$ is generated from a normal distribution with mean 0, and (more importantly for the current purposes) with a standard deviation $\\sigma$ that is the same for every single residual. In practice, it's impossible to test the assumption that every residual is identically distributed. Instead, what we care about is that the standard deviation of the residual is the same for all values of $\\hat{Y}$, and (if we're being especially paranoid) all values of every predictor $X$ in the model. See [](regressionhomogeneity).\n", "- *Uncorrelated predictors*. The idea here is that, is a multiple regression model, you don't want your predictors to be too strongly correlated with each other. This isn't \"technically\" an assumption of the regression model, but in practice it's required. Predictors that are too strongly correlated with each other (referred to as \"collinearity\") can cause problems when evaluating the model. See [](regressioncollinearity)\n", "- *Residuals are independent of each other*. This is really just a \"catch all\" assumption, to the effect that \"there's nothing else funny going on in the residuals\". If there is something weird (e.g., the residuals all depend heavily on some other unmeasured variable) going on, it might screw things up.\n", "- *No \"bad\" outliers*. Again, not actually a technical assumption of the model (or rather, it's sort of implied by all the others), but there is an implicit assumption that your regression model isn't being too strongly influenced by one or two anomalous data points; since this raises questions about the adequacy of the model, and the trustworthiness of the data in some cases. See [](regressionoutliers).\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "00528a88", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(regressiondiagnostics)=\n", "## Model checking\n", "\n", "The main focus of this section is **_regression diagnostics_**, a term that refers to the art of checking that the assumptions of your regression model have been met, figuring out how to fix the model if the assumptions are violated, and generally to check that nothing \"funny\" is going on. I refer to this as the \"art\" of model checking with good reason: it's not easy, and while there are a lot of fairly standardised tools that you can use to diagnose and maybe even cure the problems that ail your model (if there are any, that is!), you really do need to exercise a certain amount of judgment when doing this. It's easy to get lost in all the details of checking this thing or that thing, and it's quite exhausting to try to remember what all the different things are. This has the very nasty side effect that a lot of people get frustrated when trying to learn *all* the tools, so instead they decide not to do *any* model checking. This is a bit of a worry! \n", "\n", "In this section, I describe several different things you can do to check that your regression model is doing what it's supposed to. It doesn't cover the full range of things you could do, but it's still much more detailed than what I see a lot of people doing in practice; and I don't usually cover all of this in my intro stats class myself. However, I do think it's important that you get a sense of what tools are at your disposal, so I'll try to introduce a bunch of them here. \n", "\n", "\n", "### Three kinds of residuals\n", "\n", "The majority of regression diagnostics revolve around looking at the residuals, and by now you've probably formed a sufficiently pessimistic theory of statistics to be able to guess that -- precisely *because* of the fact that we care a lot about the residuals -- there are several different kinds of residual that we might consider. In particular, the following three kinds of residual are referred to in this section: \"ordinary residuals\", \"standardised residuals\", and \"Studentised residuals\". There is a fourth kind that you'll see referred to in some of the figures, and that's the \"Pearson residual\": however, for the models that we're talking about in this chapter, the Pearson residual is identical to the ordinary residual. \n", "\n", "The first and simplest kind of residuals that we care about are **_ordinary residuals_**. These are the actual, raw residuals that I've been talking about throughout this chapter. The ordinary residual is just the difference between the fitted value $\\hat{Y}_i$ and the observed value $Y_i$. I've been using the notation $\\epsilon_i$ to refer to the $i$-th ordinary residual, and by gum I'm going to stick to it. With this in mind, we have the very simple equation\n", "\n", "$$\n", "\\epsilon_i = Y_i - \\hat{Y}_i\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "42eb1e52", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "This is of course what we saw earlier, and unless I specifically refer to some other kind of residual, this is the one I'm talking about. So there's nothing new here: I just wanted to repeat myself. In any case, if you have run your regression model using `pingouin`, you can access the residuals from your model (in our case, our `mod2`)like this:" ] }, { "cell_type": "code", "execution_count": 89, "id": "3742285c", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [ "res = mod2.residuals_" ] }, { "attachments": {}, "cell_type": "markdown", "id": "7a19efa9", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "One drawback to using ordinary residuals is that they're always on a different scale, depending on what the outcome variable is and how good the regression model is. That is, unless you've decided to run a regression model without an intercept term, the ordinary residuals will have mean 0; but the variance is different for every regression. In a lot of contexts, especially where you're only interested in the *pattern* of the residuals and not their actual values, it's convenient to estimate the **_standardised residuals_**, which are normalised in such a way as to have standard deviation 1. The way we calculate these is to divide the ordinary residual by an estimate of the (population) standard deviation of these residuals. For technical reasons, mumble mumble, the formula for this is:\n", "\n", "$$\n", "\\epsilon_{i}^\\prime = \\frac{\\epsilon_i}{\\hat{\\sigma} \\sqrt{1-h_i}}\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "efea5e30", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "where $\\hat\\sigma$ in this context is the estimated population standard deviation of the ordinary residuals, and $h_i$ is the \"hat value\" of the $i$th observation. I haven't explained hat values to you yet (but have no fear,[^notehope] it's coming shortly), so this won't make a lot of sense. For now, it's enough to interpret the standardised residuals as if we'd converted the ordinary residuals to $z$-scores. In fact, that is more or less the truth, it's just that we're being a bit fancier. Now, unfortunately, `pingouin` does not provide standardized residuals, so if we want to inspect these, the best option is probably `statsmodels`:\n", "\n", "[^notehope]: Or have no hope, as the case may be." ] }, { "cell_type": "code", "execution_count": 90, "id": "930915eb", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [ "import statsmodels.api as sm\n", "\n", "predictors = df[['dan_sleep', 'baby_sleep']]\n", "outcome = df['dan_grump']\n", "\n", "## fit regression model\n", "predictors = sm.add_constant(predictors)\n", "mod = sm.OLS(outcome, predictors)\n", "est = mod.fit()\n", "\n", "#obtain standardized residuals\n", "influence = est.get_influence()\n", "res_standard = influence.resid_studentized_internal\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "d8724979", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "\n", "The third kind of residuals are **_Studentised residuals_** (also called \"jackknifed residuals\") and they're even fancier than standardised residuals. Again, the idea is to take the ordinary residual and divide it by some quantity in order to estimate some standardised notion of the residual, but the formula for doing the calculations this time is subtly different:\n", "\n", "$$\n", "\\epsilon_{i}^* = \\frac{\\epsilon_i}{\\hat{\\sigma}_{(-i)} \\sqrt{1-h_i}}\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "cb00bfc5", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "Notice that our estimate of the standard deviation here is written $\\hat{\\sigma}_{(-i)}$. What this corresponds to is the estimate of the residual standard deviation that you *would have obtained*, if you just deleted the $i$th observation from the data set. This sounds like the sort of thing that would be a nightmare to calculate, since it seems to be saying that you have to run $N$ new regression models (even a modern computer might grumble a bit at that, especially if you've got a large data set). Fortunately, some terribly clever person has shown that this standard deviation estimate is actually given by the following equation:\n", "\n", "$$\n", "\\hat\\sigma_{(-i)} = \\hat{\\sigma} \\ \\sqrt{\\frac{N-K-1 - {\\epsilon_{i}^\\prime}^2}{N-K-2}}\n", "$$\n", "\n", "Isn't that a pip?\n", "\n", "If you ever need to calculate studentised residuals yourself, this is also possible using `statsmodels`. Since we have already used `statmodels` to estimate our model above, when we calculated the standardized residuals, we can just re-use our model estimate `est` from before, and the first column of the resulting dataframe gives us our studentized residuals:" ] }, { "cell_type": "code", "execution_count": 91, "id": "e1e357d9", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/html": [ "
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student_residunadj_pbonf(p)
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" ], "text/plain": [ " student_resid unadj_p bonf(p)\n", "0 -0.494821 0.621857 1.0\n", "1 1.105570 0.271676 1.0\n", "2 0.461729 0.645321 1.0\n", "3 -0.475346 0.635620 1.0\n", "4 0.166721 0.867940 1.0" ] }, "execution_count": 91, "metadata": {}, "output_type": "execute_result" } ], "source": [ "stud_res = est.outlier_test()\n", "stud_res.head()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "fde761fe", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "Before moving on, I should point out that you don't often need to manually extract these residuals yourself, even though they are at the heart of almost all regression diagnostics. Most of the time the various functions that run the diagnostics will take care of these calculations for you." ] }, { "attachments": {}, "cell_type": "markdown", "id": "183d7642", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(regressionoutliers)=\n", "\n", "### Three kinds of anomalous data\n", "One danger that you can run into with linear regression models is that your analysis might be disproportionately sensitive to a smallish number of \"unusual\" or \"anomalous\" observations. In the context of linear regression, there are three conceptually distinct ways in which an observation might be called \"anomalous\". All three are interesting, but they have rather different implications for your analysis.\n", "\n", "The first kind of unusual observation is an **_outlier_**. The definition of an outlier (in this context) is an observation that is very different from what the regression model predicts. An example is shown in {numref}`fig-outlier`. In practice, we operationalise this concept by saying that an outlier is an observation that has a very large Studentised residual, $\\epsilon_i^*$. Outliers are interesting: a big outlier *might* correspond to junk data -- e.g., the variables might have been entered incorrectly, or some other defect may be detectable. Note that you shouldn't throw an observation away just because it's an outlier. But the fact that it's an outlier is often a cue to look more closely at that case, and try to find out why it's so different." ] }, { "cell_type": "code", "execution_count": 92, "id": "dcc141dc", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "outlier-fig" } }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from scipy.optimize import curve_fit\n", "import pandas as pd\n", "import seaborn as sns\n", "\n", "\n", "df = pd.DataFrame(\n", " {'x': [1, 1.3, 1.8, 1.9, 2.4, 2.3, 2.4, 2.6, 2.8, 3.6, 4],\n", " 'y': [1.5, 1.4, 1.9, 1.7, 2.3, 2.1, 2.6, 2.8, 2.4, 2.6, 2.8],\n", " 'y2': [1.5, 1.4, 1.9, 1.7, 4, 2.1, 2.6, 2.8, 2.4, 2.6, 2.8]\n", " })\n", "\n", "\n", "\n", "# fit linear regression model and save parameters\n", "def func(x, a, b):\n", " return a * x + b\n", "\n", "initialParameters = np.array([1.0, 1.0])\n", "\n", "fittedParameters, pcov = curve_fit(func, df['x'], df['y'], initialParameters)\n", "\n", "modelPredictions = func(df['x'], *fittedParameters) \n", "\n", "xModel = np.linspace(min(df['x']), max(df['x']))\n", "yModel = func(xModel, *fittedParameters)\n", "\n", "\n", "# plot data\n", "fig = plt.figure() \n", "ax = fig.add_subplot()\n", "\n", "\n", "sns.scatterplot(data = df, x='x', y='y')\n", "\n", "# add regression line\n", "ax.plot(xModel, yModel)\n", "\n", "\n", "initialParameters = np.array([1.0, 1.0])\n", "\n", "fittedParameters, pcov = curve_fit(func, df['x'], df['y2'], initialParameters)\n", "\n", "modelPredictions = func(df['x'], *fittedParameters) \n", "\n", "xModel = np.linspace(min(df['x']), max(df['x']))\n", "yModel = func(xModel, *fittedParameters)\n", "\n", "ax.plot(xModel, yModel)\n", "ax.plot(2.4, 4, 'ro')\n", "ax.plot([2.4, 2.4], [2.4 ,4], linestyle='dashed')\n", "ax.grid(False)\n", "\n", "sns.despine()\n", "plt.close()\n", "glue(\"outlier-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "358631a1", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " ```{glue:figure} outlier-fig\n", ":figwidth: 600px\n", ":name: fig-outlier\n", "\n", "An illustration of outliers. The orange line plots the regression line estimated when the anomalous (red) data point is included, and the dotted line shows the residual for the outlier. The blue line shows the regression line that would have been estimated without the anomalous observation included. The outlier has an unusual value on the outcome (y axis location) but not the predictor (x axis location), and lies a long way from the regression line.\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "c72b4d9e", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "The second way in which an observation can be unusual is if it has high **_leverage_**: this happens when the observation is very different from all the other observations. This doesn't necessarily have to correspond to a large residual: if the observation happens to be unusual on all variables in precisely the same way, it can actually lie very close to the regression line. An example of this is shown in panel A of {numref}`fig-leverage-influence`. The leverage of an observation is operationalised in terms of its *hat value*, usually written $h_i$. The formula for the hat value is rather complicated[^notehatmatrix] but its interpretation is not: $h_i$ is a measure of the extent to which the $i$-th observation is \"in control\" of where the regression line ends up going. We won't bother extracting the hat values here, but if you want to do this, you can use the `get_influence` method from the `statsmodels.api` package to inspect the relative influence of data points. For now, it is enough to get an intuitive, visual idea of what leverage can mean.\n", "\n", "[^notehatmatrix]: Again, for the linear algebra fanatics: the \"hat matrix\" is defined to be that matrix $H$ that converts the vector of observed values $y$ into a vector of fitted values $\\hat{y}$, such that $\\hat{y} = H y$. The name comes from the fact that this is the matrix that \"puts a hat on $y$\". The hat *value* of the $i$-th observation is the $i$-th diagonal element of this matrix (so technically I should be writing it as $h_{ii}$ rather than $h_{i}$). Oh, and in case you care, here's how it's calculated: $H = X(X^TX)^{-1} X^T$. Pretty, isn't it?" ] }, { "cell_type": "code", "execution_count": 93, "id": "ee9dcdf5", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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", 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" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "leverage-influence-fig" } }, "output_type": "display_data" } ], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from scipy.optimize import curve_fit\n", "import pandas as pd\n", "import seaborn as sns\n", "import string\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(10, 5), sharey=True)\n", "\n", "df = pd.DataFrame(\n", " {'x': [1, 1.3, 1.8, 1.9, 2.4, 2.3, 2.4, 2.6, 2.8, 3.6, 4, 8],\n", " 'y': [1.5, 1.4, 1.9, 1.7, 2.3, 2.1, 2.6, 2.8, 2.4, 2.6, 2.8, 7.8],\n", " 'y2': [1.5, 1.4, 1.9, 1.7, 4, 2.1, 2.6, 2.8, 2.4, 2.6, 2.8, 7.4]\n", " })\n", "\n", "\n", "\n", "# fit linear regression model and save parameters\n", "def func(x, a, b):\n", " return a * x + b\n", "\n", "initialParameters = np.array([1.0, 1.0])\n", "\n", "fittedParameters, pcov = curve_fit(func, df['x'], df['y'], initialParameters)\n", "\n", "modelPredictions = func(df['x'], *fittedParameters) \n", "\n", "xModel = np.linspace(min(df['x']), max(df['x']))\n", "yModel = func(xModel, *fittedParameters)\n", "\n", "\n", "\n", "sns.scatterplot(data = df, x='x', y='y', ax = axes[0])\n", "\n", "# add regression line\n", "axes[0].plot(xModel, yModel)\n", "\n", "\n", "initialParameters = np.array([1.0, 1.0])\n", "\n", "fittedParameters, pcov = curve_fit(func, df['x'], df['y2'], initialParameters)\n", "\n", "modelPredictions = func(df['x'], *fittedParameters) \n", "\n", "xModel = np.linspace(min(df['x']), max(df['x']))\n", "yModel = func(xModel, *fittedParameters)\n", "\n", "axes[0].plot(xModel, yModel)\n", "axes[0].plot(8, 7.4, 'ro')\n", "axes[0].plot([8, 8], [7.4 ,7.6], linestyle='dashed')\n", "axes[0].grid(False)\n", "\n", "# Plot 2\n", "\n", "df = pd.DataFrame(\n", " {'x': [1, 1.3, 1.8, 1.9, 2.4, 2.3, 2.4, 2.6, 2.8, 3.6, 4, 8],\n", " 'y': [1.5, 1.4, 1.9, 1.7, 2.3, 2.1, 2.6, 2.8, 2.4, 2.6, 2.8, 7.8],\n", " 'y2': [1.5, 1.4, 1.9, 1.7, 4, 2.1, 2.6, 2.8, 2.4, 2.6, 2.8, 5]\n", " })\n", "\n", "\n", "\n", "# fit linear regression model and save parameters\n", "def func(x, a, b):\n", " return a * x + b\n", "\n", "initialParameters = np.array([1.0, 1.0])\n", "\n", "fittedParameters, pcov = curve_fit(func, df['x'], df['y'], initialParameters)\n", "\n", "modelPredictions = func(df['x'], *fittedParameters) \n", "\n", "xModel = np.linspace(min(df['x']), max(df['x']))\n", "yModel = func(xModel, *fittedParameters)\n", "\n", "\n", "\n", "\n", "sns.scatterplot(data = df, x='x', y='y', ax = axes[1])\n", "\n", "# add regression line\n", "axes[1].plot(xModel, yModel)\n", "\n", "\n", "initialParameters = np.array([1.0, 1.0])\n", "\n", "fittedParameters, pcov = curve_fit(func, df['x'], df['y2'], initialParameters)\n", "\n", "modelPredictions = func(df['x'], *fittedParameters) \n", "\n", "xModel = np.linspace(min(df['x']), max(df['x']))\n", "yModel = func(xModel, *fittedParameters)\n", "\n", "axes[1].plot(xModel, yModel)\n", "axes[1].plot(8, 5, 'ro')\n", "axes[1].plot([8, 8], [5 ,7.3], linestyle='dashed')\n", "axes[1].grid(False)\n", "\n", "\n", "for n, ax in enumerate(axes): \n", " ax.text(-0.1, 1.1, string.ascii_uppercase[n], transform=ax.transAxes, \n", " size=20)\n", "\n", "\n", "sns.despine()\n", "plt.close()\n", "glue(\"leverage-influence-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "c279c3e6", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " ```{glue:figure} leverage-influence-fig\n", ":figwidth: 600px\n", ":name: fig-leverage-influence\n", "\n", "Outliers showing high leverage points (panel A) and high influence points (panel B).\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "f6587dc5", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "In general, if an observation lies far away from the other ones in terms of the predictor variables, it will have a large hat value (as a rough guide, high leverage is when the hat value is more than 2-3 times the average; and note that the sum of the hat values is constrained to be equal to $K+1$). High leverage points are also worth looking at in more detail, but they're much less likely to be a cause for concern unless they are also outliers.\n", "\n", "This brings us to our third measure of unusualness, the **_influence_** of an observation. A high influence observation is an outlier that has high leverage. That is, it is an observation that is very different to all the other ones in some respect, and also lies a long way from the regression line. This is illustrated in {numref}`fig-leverage-influence`, panel B. Notice the contrast to panel A, and to {numref}`fig-outlier`: outliers don't move the regression line much, and neither do high leverage points. But something that is an outlier and has high leverage... that has a big effect on the regression line." ] }, { "attachments": {}, "cell_type": "markdown", "id": "7c53757c", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "That's why we call these points high influence; and it's why they're the biggest worry. We operationalise influence in terms of a measure known as **_Cook's distance_**, \n", "\n", "$$\n", "D_i = \\frac{{\\epsilon_i^*}^2 }{K+1} \\times \\frac{h_i}{1-h_i}\n", "$$ \n", "\n", "Notice that this is a multiplication of something that measures the outlier-ness of the observation (the bit on the \n", "left), and something that measures the leverage of the observation (the bit on the right). In other words, in order to have a large Cook's distance, an observation must be a fairly substantial outlier *and* have high leverage." ] }, { "attachments": {}, "cell_type": "markdown", "id": "4b132160", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "Again, if you want to quantify Cook's distance, this can be done using using `statsmodels.api`. I won't go through this in detail, but if you are interested, you can click to show the code, and see how I got the Cook's distance values from `statmodels`." ] }, { "cell_type": "code", "execution_count": 94, "id": "fbb75951", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "cooks-fig" } }, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "import statsmodels.api as sm\n", "\n", "# Define a figure with two panels\n", "fig, axes = plt.subplots(1, 2, figsize=(8, 5))\n", "\n", "# Define our made-up data\n", "df = pd.DataFrame(\n", " {'x': [1, 1.3, 1.8, 1.9, 2.4, 2.3, 2.4, 2.6, 2.8, 3.6, 4, 8],\n", " 'y': [1.5, 1.4, 1.9, 1.7, 2.3, 2.1, 2.6, 2.8, 2.4, 2.6, 2.8, 7.8],\n", " 'y2': [1.5, 1.4, 1.9, 1.7, 4, 2.1, 2.6, 2.8, 2.4, 2.6, 2.8, 5]\n", " })\n", "\n", "# Get Cook's distance\n", "\n", "# model the data using statsmodels.api OLS (ordinary least squares)\n", "model = sm.OLS(df['y2'], df['x'])\n", "results = model.fit()\n", "\n", "# extract cook's distance\n", "influence = results.get_influence()\n", "cooks = influence.cooks_distance\n", "\n", "# for plotting, make a dataframe with the x data, and the corresponding cook's distances\n", "df_cooks = pd.DataFrame(\n", " {'x': df['x'],\n", " 'y': cooks[0]\n", " })\n", "\n", "# Panel A\n", "\n", "# plot Cook's distance against the x data points\n", "sns.scatterplot(data = df_cooks, x = 'x', y = 'y', ax = axes[0])\n", "axes[0].axhline(y=1, color='gray', linestyle='--')\n", "axes[0].axhline(y= 4/len(df['x']), color='gray', linestyle='--')\n", "\n", "\n", "\n", "# Panel B (same as before)\n", "\n", "df = pd.DataFrame(\n", " {'x': [1, 1.3, 1.8, 1.9, 2.4, 2.3, 2.4, 2.6, 2.8, 3.6, 4, 8],\n", " 'y': [1.5, 1.4, 1.9, 1.7, 2.3, 2.1, 2.6, 2.8, 2.4, 2.6, 2.8, 7.8],\n", " 'y2': [1.5, 1.4, 1.9, 1.7, 4, 2.1, 2.6, 2.8, 2.4, 2.6, 2.8, 5]\n", " })\n", "\n", "\n", "\n", "# fit linear regression model and save parameters\n", "def func(x, a, b):\n", " return a * x + b\n", "\n", "\n", "# first regression line\n", "initialParameters = np.array([1.0, 1.0])\n", "\n", "fittedParameters, pcov = curve_fit(func, df['x'], df['y'], initialParameters)\n", "\n", "modelPredictions = func(df['x'], *fittedParameters) \n", "\n", "xModel = np.linspace(min(df['x']), max(df['x']))\n", "yModel = func(xModel, *fittedParameters)\n", "\n", "\n", "# plot data points\n", "sns.scatterplot(data = df, x='x', y='y', ax = axes[1])\n", "\n", "# plot first regression line\n", "axes[1].plot(xModel, yModel)\n", "\n", "\n", "# second regression line\n", "\n", "initialParameters = np.array([1.0, 1.0])\n", "\n", "fittedParameters, pcov = curve_fit(func, df['x'], df['y2'], initialParameters)\n", "\n", "modelPredictions = func(df['x'], *fittedParameters) \n", "\n", "xModel = np.linspace(min(df['x']), max(df['x']))\n", "yModel = func(xModel, *fittedParameters)\n", "\n", "axes[1].plot(xModel, yModel)\n", "\n", "# add red point to show \"outlier\"\n", "axes[1].plot(8, 5, 'ro')\n", "\n", "# add dashed line to show change in residual\n", "axes[1].plot([8, 8], [5 ,7.3], linestyle='dashed')\n", "axes[1].grid(False)\n", "\n", "# add letter labels\n", "for n, ax in enumerate(axes): \n", " ax.text(-0.1, 1.1, string.ascii_uppercase[n], transform=ax.transAxes, \n", " size=20)\n", "\n", "# prettify\n", "sns.despine()\n", "for ax in axes:\n", " ax.grid(False)\n", "plt.close()\n", "glue(\"cooks-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "e2664025", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " ```{glue:figure} cooks-fig\n", ":figwidth: 600px\n", ":name: fig-cooks\n", "\n", "A visualization of Cook's distance as a means for identifying the influence of each data point. Panel B is the same as panel B in {numref}`fig-leverage-influence` above. The red dot show the effect on the entire regression line that sinking the final point from 7.4 to 5 has. Panel A shows Cook's distance for the data set in which the data point at x = 8 is 5 (the orange line). The dashed lines show two possible cutoff points for a \"large\" Cook's distance: either 1 or 4/N, in which N is the number of data points.\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "5c0c4ecb", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "As a rough guide, Cook's distance greater than 1 is often considered large (that's what I typically use as a quick and dirty rule), though a quick scan of the internet and a few papers suggests that $4/N$ has also been suggested as a possible rule of thumb. As the visualization in {numref}`fig-cooks` illustrates, the red data point at x = 8 has a large Cook's distance." ] }, { "attachments": {}, "cell_type": "markdown", "id": "e0ecd657", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "An obvious question to ask next is, if you do have large values of Cook's distance, what should you do? As always, there's no hard and fast rules. Probably the first thing to do is to try running the regression with that point excluded and see what happens to the model performance and to the regression coefficients. If they really are substantially different, it's time to start digging into your data set and your notes that you no doubt were scribbling as your ran your study; try to figure out *why* the point is so different. If you start to become convinced that this one data point is badly distorting your results, you might consider excluding it, but that's less than ideal unless you have a solid explanation for why this particular case is qualitatively different from the others and therefore deserves to be handled separately.[^noterobust] To give an example, let's delete the observation from day 64, the observation with the largest Cook's distance for the `mod2` model, where we predicted my grumpiness on the basis of my sleep, and my baby's sleep.\n", "\n", "[^noterobust]: An alternative is to run a \"robust regression\"; I might discuss robust regression in a later version of this book, but I don't promise." ] }, { "attachments": {}, "cell_type": "markdown", "id": "95401e5b", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "First, let's get back to the original sleep data, and remind ourselves of what our model coefficients looked like:" ] }, { "cell_type": "code", "execution_count": 95, "id": "cef42308", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/html": [ "
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namescoefseTpvalr2adj_r2CI[2.5%]CI[97.5%]
0Intercept125.973.0441.420.000.820.81119.93132.00
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" ], "text/plain": [ " names coef se T pval r2 adj_r2 CI[2.5%] CI[97.5%]\n", "0 Intercept 125.97 3.04 41.42 0.00 0.82 0.81 119.93 132.00\n", "1 dan_sleep -8.95 0.55 -16.17 0.00 0.82 0.81 -10.05 -7.85\n", "2 baby_sleep 0.01 0.27 0.04 0.97 0.82 0.81 -0.53 0.55" ] }, "execution_count": 95, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "import pingouin as pg\n", "\n", "file = 'https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/parenthood.csv'\n", "df = pd.read_csv(file)\n", "\n", "predictors = df[['dan_sleep', 'baby_sleep']]\n", "outcome = df['dan_grump']\n", "\n", "mod2 = pg.linear_regression(predictors, outcome)\n", "mod2.round(2)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "84576f7f", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "Then, we can remove the data from day 64, using the `drop()` method from `pandas`. Remember, as always, it's Python, and Python is zero-indexed, so it starts counting the days at 0 and not 1, and that means that day 64 is on row 63!" ] }, { "cell_type": "code", "execution_count": 96, "id": "41f92756", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [], "source": [ "df_2 = df.drop(63)" ] }, { "cell_type": "code", "execution_count": 97, "id": "d0b77749", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/html": [ "
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namescoefseTpvalr2adj_r2CI[2.5%]CI[97.5%]
0Intercept126.363.0042.190.000.820.82120.41132.30
1dan_sleep-8.830.55-16.130.000.820.82-9.91-7.74
2baby_sleep-0.130.28-0.480.630.820.82-0.680.41
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" ], "text/plain": [ " names coef se T pval r2 adj_r2 CI[2.5%] CI[97.5%]\n", "0 Intercept 126.36 3.00 42.19 0.00 0.82 0.82 120.41 132.30\n", "1 dan_sleep -8.83 0.55 -16.13 0.00 0.82 0.82 -9.91 -7.74\n", "2 baby_sleep -0.13 0.28 -0.48 0.63 0.82 0.82 -0.68 0.41" ] }, "execution_count": 97, "metadata": {}, "output_type": "execute_result" } ], "source": [ "predictors = df_2[['dan_sleep', 'baby_sleep']]\n", "outcome = df_2['dan_grump']\n", "\n", "mod3 = pg.linear_regression(predictors, outcome)\n", "mod3.round(2)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "d8025450", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "As you can see, those regression coefficients have barely changed in comparison to the values we got earlier. In other words, we really don't have any problem as far as anomalous data are concerned." ] }, { "attachments": {}, "cell_type": "markdown", "id": "5e354965", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(regressionnormality)=\n", "### Checking the normality of the residuals\n", "\n", "Like many of the statistical tools we've discussed in this book, regression models rely on a normality assumption. In this case, we assume that the residuals are normally distributed. The tools for testing this aren't fundamentally different to those that we discussed [earlier](shapiro). First, I firmly believe that it never hurts to draw a good old-fashioned histogram:" ] }, { "cell_type": "code", "execution_count": 98, "id": "09e7adf8", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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", 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" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "res-hist-fig" } }, "output_type": "display_data" } ], "source": [ "import pandas as pd\n", "import pingouin as pg\n", "import seaborn as sns\n", "\n", "predictors = df[['dan_sleep', 'baby_sleep']]\n", "outcome = df['dan_grump']\n", "\n", "mod2 = pg.linear_regression(predictors, outcome)\n", "\n", "res = mod2.residuals_\n", "res = pd.DataFrame(res, columns = ['Residuals'])\n", "\n", "fig = plt.figure()\n", "\n", "ax = sns.histplot(data = res, x = 'Residuals')\n", "ax.grid(False)\n", "sns.despine()\n", "plt.close()\n", "glue(\"res-hist-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "2b6d8383", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " ```{glue:figure} res-hist-fig\n", ":figwidth: 600px\n", ":name: fig-res-hist\n", "\n", "A histogram of the (ordinary) residuals in the `mod2` model. These residuals look very close to being normally distributed, much moreso than is typically seen with real data. This shouldn’t surprise you... they aren’t real data, and they aren’t real residuals!\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "2f56983c", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "The resulting plot is shown in {numref}`fig-res-hist`, and as you can see the plot looks pretty damn close to normal, almost unnaturally so. I could also run a Shapiro-Wilk test to check, using the `normality` function from `pingouin`:" ] }, { "cell_type": "code", "execution_count": 99, "id": "b566420f", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/html": [ "
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Wpvalnormal
Residuals0.990.84True
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" ], "text/plain": [ " W pval normal\n", "Residuals 0.99 0.84 True" ] }, "execution_count": 99, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sw = pg.normality(res, method = 'shapiro')\n", "sw.round(2)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a9593ad0", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "The W value of .99, at this sample size, is non-significant ($p$ = .82), again suggesting that the normality assumption isn’t in any danger here. As a third measure, we might also want to draw a QQ-plot. This can be most easily done using the `qqplot` function from `statsmodels.api`:" ] }, { "cell_type": "code", "execution_count": 100, "id": "a75c505c", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "import statsmodels.api as sm\n", "\n", "\n", "sm.qqplot(res['Residuals'], line = 's')\n", "sns.despine()\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "f94bc8ba", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " ```{glue:figure} qq-fig\n", ":figwidth: 600px\n", ":name: fig-qq\n", "\n", "A Q-Q (quantile-quantile) plot, the theoretical quantiles according to the model, against the quantiles of the standardised residuals.\n", "\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "2077e62d", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "If our residuals were _perfectly_ normally distributed, then they would lie right on the red line. These residuals veer off a little at the ends, but again, pretty damn close, so the Q-Q plot further confirms our conviction based on the histogram and the Shapiro-Wilk test, that it is appropriate to model these data with a linear regression.\n", "\n", "A little note: Q-Q plots are often created by plotting the quantiles of the _standardized_ residuals against the theoretical quantiles. Since using Q-Q plots for assessing normality is basically a matter of squinting at the plot at getting a _feeling_, based on your great experience of squinting at plots, for whether the data _seem_ normal enough, it probably doesn't matter so much which way you do it. Below you can see a side-by-side comparison of a Q-Q plot of the ordinary (left) and standardized (right) residuals. Honestly, they look about the same to me, and you can get the ordinary residuals straight from `pingouin`, without needing to invoke `statsmodels`." ] }, { "cell_type": "code", "execution_count": 101, "id": "20d8719d", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Define a figure with two panels\n", "import matplotlib.pyplot as plt\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(15, 5))\n", "\n", "res_standard = pd.DataFrame(res_standard, columns = ['Standardized_Residuals'])\n", "\n", "sm.qqplot(res['Residuals'], line = 's', ax = axes[0])\n", "sm.qqplot(res_standard['Standardized_Residuals'], line = 's', ax = axes[1])\n", "\n", "axes[0].set_title('Ordinary residuals')\n", "axes[1].set_title('Standardized residuals')\n", "\n", "for ax in axes:\n", " ax.grid(False)\n", "\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "49829d8d", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(regressionlinearity)=\n", "### Checking the linearity of the relationship\n", "\n", "The third thing we might want to test is the linearity of the relationships between the predictors and\n", "the outcomes. There’s a few different things that you might want to do in order to check this. Firstly, it\n", "never hurts to just plot the relationship between the fitted values $\\hat{Y}_i$ and the observed values $Y_i$ for the\n", "outcome variable, as illustrated in {numref}`fig-fitted_v_observed`." ] }, { "cell_type": "code", "execution_count": 102, "id": "55be1b48", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "fitted_v_observed-fig" } }, "output_type": "display_data" } ], "source": [ "predictors = df[['dan_sleep', 'baby_sleep']]\n", "outcome = df['dan_grump']\n", "\n", "mod2 = pg.linear_regression(predictors, outcome, as_dataframe = False)\n", "\n", "df_res_pred = pd.DataFrame(\n", " {'observed': outcome,\n", " 'fitted': mod2['pred']\n", " })\n", "\n", "fig = plt.figure()\n", "\n", "ax=sns.scatterplot(data = df_res_pred, x = 'fitted', y = 'observed')\n", "ax.grid(False)\n", "sns.despine()\n", "plt.close()\n", "glue(\"fitted_v_observed-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "cd7ce4ef", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " ```{glue:figure} fitted_v_observed-fig\n", ":figwidth: 600px\n", ":name: fig-fitted_v_observed\n", "\n", "Plot of the fitted values against the observed values of the outcome variable. A straight line is what we’re hoping to see here. This looks pretty good, suggesting that there’s nothing grossly wrong, but there could be hidden subtle issues.\n", "\n", "```\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "ab1b5312", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "\n", "One of the reasons I like to draw these plots is that they give you a kind of “big picture view”. If this plot looks approximately linear, then we’re probably not doing too badly (though that’s not to say that there aren’t problems). However, if you can see big departures from linearity here, then it strongly suggests that you need to make some changes.\n", "\n", "In any case, in order to get a more detailed picture it’s often more informative to look at the relationship between the fitted values and the residuals themselves. Let's take a look. Here, we use the `residplot` function from `seaborn`, and in addition to drawing a straight line through all the points, we can ask Python to compute a \"locally-weighed linear linear\" regression, by setting the argument `lowess` to `True`. This gives us a sort of moving window, as Python fits a series of little regressions as it moves through the data. With a little work, we can also use ``regplot`` to produce the same figure for the multiple linear regression model with both ``dan_sleep`` and ``baby_sleep`` as predictors (click the show button to see how). If the relationship in our data is truly linear, then we should see a straight, perfectly horizontal line. There’s some hint of curvature here, but it’s not clear whether or not we be concerned. " ] }, { "cell_type": "code", "execution_count": 103, "id": "e2586aee", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "resid1-fig" } }, "output_type": "display_data" } ], "source": [ "# Define a figure with two panels\n", "import matplotlib.pyplot as plt\n", "\n", "fig, axes = plt.subplots(3, 1, figsize=(8, 5))\n", "\n", "# Fit linear regressions for both predictors, and plot residuals\n", "\n", "p1=sns.residplot(data=df, x='dan_sleep', y='dan_grump', lowess=True, line_kws=dict(color=\"r\"), ax=axes[0])\n", "p2=p3=sns.residplot(data=df, x='baby_sleep', y='dan_grump', lowess=True, line_kws=dict(color=\"r\"), ax=axes[1])\n", "\n", "\n", "# Add panel for multiple linear regression\n", "\n", "df_res_pred['residuals'] = mod2['residuals']\n", "\n", "\n", "p3=sns.regplot(data = df_res_pred, x = 'fitted', y = 'residuals', lowess = True, line_kws=dict(color=\"r\"), ax=axes[2])\n", "\n", "plt.axhline(y=0, color='b', linestyle='dotted')\n", "\n", "# Make figures look nicer\n", "for n, ax in enumerate(axes):\n", " ax.text(-0.1, 1.1, string.ascii_uppercase[n], transform=ax.transAxes, \n", " size=20)\n", " ax.grid(False)\n", " sns.despine()\n", "\n", "subplots=[p1,p2,p3]\n", "for p in subplots:\n", " p.tick_params(axis='x', which='both', bottom=False, top=False, labelbottom=False)\n", "\n", "\n", "# Plot figure in book, with caption\n", "plt.close(fig)\n", "glue(\"resid1-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "dff2beef", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "\n", " ```{glue:figure} resid1-fig\n", ":figwidth: 600px\n", ":name: fig-resid1\n", "\n", "Scatterplots of the residuals when a linear regression is fit separately for either ``dan_sleep`` as a predictor (panel A) or ``baby_sleep`` as a predictor (panel B). Panel C plots the residuals for the overall fit of the model including both predictors. The dotted horizontal line represents the fitted regression model. If a linear model is appropriate for our data, then the points should lie evenly above and below the dotted line, the whole away along the x-axis. The red line shows a locally-weighted smoothing of the regressionn line. Too much curvature in this line would suggest that maybe a straight line is not the best model for our data. But how much curvature is too much?\n", "\n", "```\n", "\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "f7e8e9bb", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "One method for testing the degree of curvature is known as Tukey's test for nonadditivity {cite:`tukey1949one, fox2018r`}. The test is quite simple: you just take the fitted (predicted) values from your original model, square them, and then include these squared fitted values as a predictor when you re-run the model. If the $t$-test for the new predictor comes up significant, this implies that there is some nonlinear relationship between the variable and the residuals. Let's give this a try:" ] }, { "cell_type": "code", "execution_count": 104, "id": "10bf9e26", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/html": [ "
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0Intercept1.54057.6420.0270.9790.8250.819-112.878115.958
1dan_sleep2.8675.4940.5220.6030.8250.819-8.03913.773
2baby_sleep0.0110.2660.0430.9660.8250.819-0.5170.540
3Tukey0.0100.0052.1620.0330.8250.8190.0010.020
\n", "
" ], "text/plain": [ " names coef se T pval r2 adj_r2 CI[2.5%] CI[97.5%]\n", "0 Intercept 1.540 57.642 0.027 0.979 0.825 0.819 -112.878 115.958\n", "1 dan_sleep 2.867 5.494 0.522 0.603 0.825 0.819 -8.039 13.773\n", "2 baby_sleep 0.011 0.266 0.043 0.966 0.825 0.819 -0.517 0.540\n", "3 Tukey 0.010 0.005 2.162 0.033 0.825 0.819 0.001 0.020" ] }, "execution_count": 104, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# square the fitted values for our model, and add them to our dataframe\n", "df['Tukey'] = mod2['pred']**2\n", "\n", "#re-run the model, with squared fitted values added as predictor (here I have called them \"Tukey\")\n", "mod_curve_check = pg.linear_regression(df[['dan_sleep', 'baby_sleep', 'Tukey']], df['dan_grump'])\n", "mod_curve_check.round(3)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "df18df5a", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "Since the $t$-test on the coefficients for new ``Tukey`` predictor is significant ($p$ < 0.05), this suggests that the curvature we see in {numref}`fig-resid1` is genuine, although it still bears remembering that the pattern in {numref}`fig-fitted_v_observed` is pretty damn straight: in other words the deviations from linearity are fairly small, and probably not worth worrying about." ] }, { "attachments": {}, "cell_type": "markdown", "id": "10695d53", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "In a lot of cases, the solution to this problem (and many others) is to transform one or more of the variables. We have already discussed the [basics of variable transformation](transform), but I do want to make special note of one additional possibility that I didn't mention earlier: the Box-Cox transform. The Box-Cox function is a fairly simple one, but it's very widely used \n", "\n", "$$\n", "f(x,\\lambda) = \\frac{x^\\lambda - 1}{\\lambda}\n", "$$\n", "\n", "for all values of $\\lambda$ except $\\lambda = 0$. When $\\lambda = 0$ we just take the natural logarithm (i.e., $\\ln(x)$). You can calculate it using the ``boxcox`` function from ``scipy.stats``, which will automatically calculate the optimal value of $\\lambda$. In the case of our ``dan_sleep`` data though, the data were already pretty normally distributed (panel A), so applying the transformation (panel B) really doesn't do much. Now, if only the _quality_ of my sleep could be transformed, and not just the _distribution_, ah, then we would really be on to something!" ] }, { "cell_type": "code", "execution_count": 105, "id": "7e60bcee", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "hide-input" ] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Optimal value for lambda: 1.7912772737277842\n" ] }, { "data": { "application/papermill.record/image/png": 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", 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" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "boxcox-fig" } }, "output_type": "display_data" } ], "source": [ "from scipy.stats import boxcox\n", "\n", "df['dan_sleep_transformed'], best_lambda = boxcox(df['dan_sleep'])\n", "\n", "# Define a figure with two panels\n", "import matplotlib.pyplot as plt\n", "\n", "fig, axes = plt.subplots(1, 2, figsize=(8, 5))\n", "\n", "# Fit linear regressions for both predictors, and plot residuals\n", "\n", "p1=sns.kdeplot(df['dan_sleep'], ax=axes[0])\n", "p2=sns.kdeplot(df['dan_sleep_transformed'], ax=axes[1])\n", "\n", "# Make figures look nicer\n", "for n, ax in enumerate(axes):\n", " ax.text(-0.1, 1.1, string.ascii_uppercase[n], transform=ax.transAxes, \n", " size=20)\n", " ax.grid(False)\n", " sns.despine()\n", "\n", "subplots=[p1,p2,p3]\n", "for p in subplots:\n", " p.tick_params(axis='x', which='both', bottom=False, top=False, labelbottom=False)\n", "\n", "print('Optimal value for lambda:', best_lambda)\n", "\n", "\n", "# Plot figure in book, with caption\n", "plt.close(fig)\n", "glue(\"boxcox-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "692b6be8", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " ```{glue:figure} boxcox-fig\n", ":figwidth: 600px\n", ":name: fig-boxcox\n", "\n", "Distribution for the raw ``dan_sleep`` data (panel A) and the distribution same data after a Box-Cox transformation (panel B). In this case, the transformation has not had much of an effect, as the data were reasonably normally-distributed to begin with.\n", "\n", "```\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "8787af7f", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "(regressionhomogeneity)=\n", "### Checking the homogeneity of variance\n", "\n", "The regression models that we've talked about all make a homogeneity of variance assumption: the variance of the residuals is assumed to be constant. One way to inspect this visually is to use a \"Scale-Location Plot\", which is just a variation on the plots we made in {numref}`fig-resid1`. In a Scale-Location Plot, instead of plotting the fitted values against the raw residuals, we plot them against the square root of the absolute value of the standardized residuals, which can make it easier to visually assess the homogeneity of the variance.\n", "\n", "As far as I know (and admittedly, that's not so far), there is no ready-made package in Python that will produce a Scale-Location Plot, but if you have made it this far in the book, you already have the knowledge to build one of these yourself. It could go something like this:" ] }, { "cell_type": "code", "execution_count": 106, "id": "554b5a3c", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [ "hide-input" ] }, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "\n", "predictors = df[['dan_sleep', 'baby_sleep']]\n", "outcome = df['dan_grump']\n", "\n", "mod2 = pg.linear_regression(predictors, outcome, as_dataframe = False)\n", "\n", "\n", "# Thanks to Ivan Savov (https://github.com/ivanistheone) for contributing this solution for getting the standardized residuals\n", "SS_resid = np.sum(mod2[\"residuals\"]**2)\n", "n = len(mod2['residuals'])\n", "p = len(list(predictors))\n", "sigmahat = np.sqrt(abs( SS_resid/(n-p-1) ))\n", "stand_res = mod2['residuals'] / sigmahat\n", "df_slplot = pd.DataFrame(\n", " {'fitted': mod2['pred'],\n", " 'sqrt_abs_stand_res': np.sqrt(np.abs(stand_res))\n", " })\n", "\n", "\n", "fig = plt.figure()\n", "\n", "ax=sns.regplot(data = df_slplot, x = 'fitted', y = 'sqrt_abs_stand_res', lowess = True, line_kws=dict(color=\"r\"))\n", "\n", "# Make figures look nicer\n", "ax.set_ylabel(r'$\\sqrt{|standardized residuals|}$')\n", "ax.set_xlabel('Fitted Values')\n", "ax.tick_params(axis='x', which='both', bottom=False, top=False, labelbottom=False)\n", "ax.grid(False)\n", "sns.despine()\n", "\n", "\n", "\n", "# Plot figure in book, with caption\n", "\n", "#glue(\"sl-plot-fig\", fig, display=False)\n", "#plt.close(fig)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "af65c95d", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ " ```{glue:figure} sl-plot-fig\n", ":figwidth: 600px\n", ":name: fig_sl-plot\n", "\n", "Plot of the fitted values (model predictions) against the square root of the abs standardised residuals. This plot is used to diagnose violations of homogeneity of variance. If the variance is really constant, then the line through the middle should be horizontal and flat.\n", "\n", "```\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "e3689558", "metadata": {}, "source": [ "A slightly more formal approach is to run a hypothesis test such as the Breusch–Pagan test {cite}`breusch1979simple`. Unfortunately, to run the test, we'll have to leave the cozy world of `pingouin` and use `statsmodels` instead, but luckily the code is not complicated. Basically, we just run our regression in `statsmodels` instead of `pinguoin`, and then run the Breusch-Pagan test on the output (see below). I won't go into how it works, other than to say that after we fit our regression model, we then fit _another_ regression model in which we use our predicted values to predict our residuals. We then use something called the \"Lagrange multiplier statistic\" (similar to a $x^2$ test) to check the significance of our new model. If the new model is not significant, this can be used to support the assumption that we are indeed dealing with a relationship that can be modelled linearly." ] }, { "cell_type": "code", "execution_count": 107, "id": "1b4b0ee0", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Lagrange multiplier statistic: 0.4886831614554388\n", "p: 0.7832200556890354\n" ] } ], "source": [ "import statsmodels.formula.api as smf\n", "import statsmodels.stats.api as sms\n", "\n", "# fit our regression model using statsmodels\n", "fit = smf.ols('dan_grump ~ dan_sleep+baby_sleep', data=df).fit()\n", "\n", "bptest=sms.diagnostic.het_breuschpagan(fit.resid, fit.model.exog)\n", "\n", "print('Lagrange multiplier statistic:', bptest[0])\n", "print('p:', bptest[1])\n", "\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "cafb5c9d", "metadata": {}, "source": [ "We see that our original impression was right: there’s no violations of homogeneity of variance in this data.\n", "\n", "It’s a bit beyond the scope of this chapter to talk too much about how to deal with violations of homogeneity of variance, but I’ll give you a quick sense of what you need to consider. The main thing to worry about, if homogeneity of variance is violated, is that the standard error estimates associated with the regression coefficients are no longer entirely reliable, and so your $t$-tests for the coefficients aren’t quite right either. A simple fix to the problem is to make use of a “heteroscedasticity corrected covariance matrix” when estimating the standard errors. \n", "\n", "Again, this goes beyond what we can do with `pingouin`, but `statmodels` can get us there. Before I show you how, let's just re-assure ourselves that both `pingouin` and `statsmodels` are doing the same basic calculations when we fit our regression models. We'll start by re-running our mulitiple regression with `pingouin`, but I'll bump up the number of decimal places that we round our answer to, so that it is easier to compare with the `statsmodels` output:" ] }, { "cell_type": "code", "execution_count": 108, "id": "7f78b86a", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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namescoefseTpvalr2adj_r2CI[2.5%]CI[97.5%]
0Intercept125.96563.040941.42310.00000.81610.8123119.9301132.0010
1dan_sleep-8.95020.5535-16.17150.00000.81610.8123-10.0487-7.8518
2baby_sleep0.01050.27110.03880.96910.81610.8123-0.52750.5485
\n", "
" ], "text/plain": [ " names coef se T pval r2 adj_r2 CI[2.5%] \\\n", "0 Intercept 125.9656 3.0409 41.4231 0.0000 0.8161 0.8123 119.9301 \n", "1 dan_sleep -8.9502 0.5535 -16.1715 0.0000 0.8161 0.8123 -10.0487 \n", "2 baby_sleep 0.0105 0.2711 0.0388 0.9691 0.8161 0.8123 -0.5275 \n", "\n", " CI[97.5%] \n", "0 132.0010 \n", "1 -7.8518 \n", "2 0.5485 " ] }, "execution_count": 108, "metadata": {}, "output_type": "execute_result" } ], "source": [ "predictors = df[['dan_sleep', 'baby_sleep']]\n", "outcome = df['dan_grump']\n", "\n", "mod2 = pg.linear_regression(predictors, outcome)\n", "mod2.round(4)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "0ad3c960", "metadata": {}, "source": [ "Next, we'll fit the same model, but using `statsmodels` instead:" ] }, { "cell_type": "code", "execution_count": 109, "id": "6cf22983", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
OLS Regression Results
Dep. Variable: dan_grump R-squared: 0.816
Model: OLS Adj. R-squared: 0.812
Method: Least Squares F-statistic: 215.2
Date: Thu, 02 May 2024 Prob (F-statistic): 2.15e-36
Time: 11:12:57 Log-Likelihood: -287.48
No. Observations: 100 AIC: 581.0
Df Residuals: 97 BIC: 588.8
Df Model: 2
Covariance Type: nonrobust
\n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
coef std err t P>|t| [0.025 0.975]
Intercept 125.9656 3.041 41.423 0.000 119.930 132.001
dan_sleep -8.9502 0.553 -16.172 0.000 -10.049 -7.852
baby_sleep 0.0105 0.271 0.039 0.969 -0.527 0.549
\n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
Omnibus: 0.593 Durbin-Watson: 2.120
Prob(Omnibus): 0.743 Jarque-Bera (JB): 0.218
Skew: -0.053 Prob(JB): 0.897
Kurtosis: 3.203 Cond. No. 76.9


Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified." ], "text/latex": [ "\\begin{center}\n", "\\begin{tabular}{lclc}\n", "\\toprule\n", "\\textbf{Dep. Variable:} & dan\\_grump & \\textbf{ R-squared: } & 0.816 \\\\\n", "\\textbf{Model:} & OLS & \\textbf{ Adj. R-squared: } & 0.812 \\\\\n", "\\textbf{Method:} & Least Squares & \\textbf{ F-statistic: } & 215.2 \\\\\n", "\\textbf{Date:} & Thu, 02 May 2024 & \\textbf{ Prob (F-statistic):} & 2.15e-36 \\\\\n", "\\textbf{Time:} & 11:12:57 & \\textbf{ Log-Likelihood: } & -287.48 \\\\\n", "\\textbf{No. Observations:} & 100 & \\textbf{ AIC: } & 581.0 \\\\\n", "\\textbf{Df Residuals:} & 97 & \\textbf{ BIC: } & 588.8 \\\\\n", "\\textbf{Df Model:} & 2 & \\textbf{ } & \\\\\n", "\\textbf{Covariance Type:} & nonrobust & \\textbf{ } & \\\\\n", "\\bottomrule\n", "\\end{tabular}\n", "\\begin{tabular}{lcccccc}\n", " & \\textbf{coef} & \\textbf{std err} & \\textbf{t} & \\textbf{P$> |$t$|$} & \\textbf{[0.025} & \\textbf{0.975]} \\\\\n", "\\midrule\n", "\\textbf{Intercept} & 125.9656 & 3.041 & 41.423 & 0.000 & 119.930 & 132.001 \\\\\n", "\\textbf{dan\\_sleep} & -8.9502 & 0.553 & -16.172 & 0.000 & -10.049 & -7.852 \\\\\n", "\\textbf{baby\\_sleep} & 0.0105 & 0.271 & 0.039 & 0.969 & -0.527 & 0.549 \\\\\n", "\\bottomrule\n", "\\end{tabular}\n", "\\begin{tabular}{lclc}\n", "\\textbf{Omnibus:} & 0.593 & \\textbf{ Durbin-Watson: } & 2.120 \\\\\n", "\\textbf{Prob(Omnibus):} & 0.743 & \\textbf{ Jarque-Bera (JB): } & 0.218 \\\\\n", "\\textbf{Skew:} & -0.053 & \\textbf{ Prob(JB): } & 0.897 \\\\\n", "\\textbf{Kurtosis:} & 3.203 & \\textbf{ Cond. No. } & 76.9 \\\\\n", "\\bottomrule\n", "\\end{tabular}\n", "%\\caption{OLS Regression Results}\n", "\\end{center}\n", "\n", "Notes: \\newline\n", " [1] Standard Errors assume that the covariance matrix of the errors is correctly specified." ], "text/plain": [ "\n", "\"\"\"\n", " OLS Regression Results \n", "==============================================================================\n", "Dep. Variable: dan_grump R-squared: 0.816\n", "Model: OLS Adj. R-squared: 0.812\n", "Method: Least Squares F-statistic: 215.2\n", "Date: Thu, 02 May 2024 Prob (F-statistic): 2.15e-36\n", "Time: 11:12:57 Log-Likelihood: -287.48\n", "No. Observations: 100 AIC: 581.0\n", "Df Residuals: 97 BIC: 588.8\n", "Df Model: 2 \n", "Covariance Type: nonrobust \n", "==============================================================================\n", " coef std err t P>|t| [0.025 0.975]\n", "------------------------------------------------------------------------------\n", "Intercept 125.9656 3.041 41.423 0.000 119.930 132.001\n", "dan_sleep -8.9502 0.553 -16.172 0.000 -10.049 -7.852\n", "baby_sleep 0.0105 0.271 0.039 0.969 -0.527 0.549\n", "==============================================================================\n", "Omnibus: 0.593 Durbin-Watson: 2.120\n", "Prob(Omnibus): 0.743 Jarque-Bera (JB): 0.218\n", "Skew: -0.053 Prob(JB): 0.897\n", "Kurtosis: 3.203 Cond. No. 76.9\n", "==============================================================================\n", "\n", "Notes:\n", "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n", "\"\"\"" ] }, "execution_count": 109, "metadata": {}, "output_type": "execute_result" } ], "source": [ "smf.ols('dan_grump ~ dan_sleep+baby_sleep', data=df).fit().summary()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "1fb72643", "metadata": {}, "source": [ "Of course, `statsmodels` gives us way more information than `pingouin`, but if you hunt around in the output for the key things we are used to talking about (the coefficients for `dan_sleep` and `baby_sleep`, the t-values, the $R^2$ and adjusted $R^2$, etc.) it's all pretty much the same, right?\n", "\n", "Good. Now, we'll re-fit our regression model, but ask `statmodels` to take the potential non-homogeneity (aka the heteroskedasticity) of variance in our model into account by specifying a ``cov_type``, that is, the type of \"robust sandwich estimator\" we want to use. Yes. It really is called a \"robust sandwich estimator\". I'm not even kidding.[^sandwich]\n", "\n", "[^sandwich]: Again, a footnote that should be read only by the two readers of this book that love linear algebra (mmmm... I love the smell of matrix computations in the morning; smells like... nerd). In these estimators, the covariance matrix for $b$ is given by $(X^T X)^{-1}\\ X^T \\Sigma X \\ (X^T X)^{-1}$. See, it's a \"sandwich\"? Assuming you think that $(X^T X)^{-1} = \\mbox{\"bread\"}$ and $X^T \\Sigma X = \\mbox{\"filling\"}$, that is. Which of course everyone does, right? In any case, the usual estimator is what you get when you set $\\Sigma = \\hat\\sigma^2 I$. But there are others. For instance, the corrected version that I learned originally uses $\\Sigma = \\mbox{diag}(\\epsilon_i^2)$ {cite}`White1980`. `statsmodels` gives you a choice of four different sandwich estimators you can use, but not (at least that I can find today on June 21 2023) a lot of information on what they actually are, or references for where they come from, so if you _really_ need this information, you'll have to do your own digging. Sorry." ] }, { "cell_type": "code", "execution_count": 110, "id": "3588a5e1", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
OLS Regression Results
Dep. Variable: dan_grump R-squared: 0.816
Model: OLS Adj. R-squared: 0.812
Method: Least Squares F-statistic: 201.7
Date: Thu, 02 May 2024 Prob (F-statistic): 2.78e-35
Time: 11:12:57 Log-Likelihood: -287.48
No. Observations: 100 AIC: 581.0
Df Residuals: 97 BIC: 588.8
Df Model: 2
Covariance Type: HC1
\n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
coef std err z P>|z| [0.025 0.975]
Intercept 125.9656 3.149 39.996 0.000 119.793 132.138
dan_sleep -8.9502 0.597 -14.987 0.000 -10.121 -7.780
baby_sleep 0.0105 0.282 0.037 0.970 -0.541 0.563
\n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
Omnibus: 0.593 Durbin-Watson: 2.120
Prob(Omnibus): 0.743 Jarque-Bera (JB): 0.218
Skew: -0.053 Prob(JB): 0.897
Kurtosis: 3.203 Cond. No. 76.9


Notes:
[1] Standard Errors are heteroscedasticity robust (HC1)" ], "text/latex": [ "\\begin{center}\n", "\\begin{tabular}{lclc}\n", "\\toprule\n", "\\textbf{Dep. Variable:} & dan\\_grump & \\textbf{ R-squared: } & 0.816 \\\\\n", "\\textbf{Model:} & OLS & \\textbf{ Adj. R-squared: } & 0.812 \\\\\n", "\\textbf{Method:} & Least Squares & \\textbf{ F-statistic: } & 201.7 \\\\\n", "\\textbf{Date:} & Thu, 02 May 2024 & \\textbf{ Prob (F-statistic):} & 2.78e-35 \\\\\n", "\\textbf{Time:} & 11:12:57 & \\textbf{ Log-Likelihood: } & -287.48 \\\\\n", "\\textbf{No. Observations:} & 100 & \\textbf{ AIC: } & 581.0 \\\\\n", "\\textbf{Df Residuals:} & 97 & \\textbf{ BIC: } & 588.8 \\\\\n", "\\textbf{Df Model:} & 2 & \\textbf{ } & \\\\\n", "\\textbf{Covariance Type:} & HC1 & \\textbf{ } & \\\\\n", "\\bottomrule\n", "\\end{tabular}\n", "\\begin{tabular}{lcccccc}\n", " & \\textbf{coef} & \\textbf{std err} & \\textbf{z} & \\textbf{P$> |$z$|$} & \\textbf{[0.025} & \\textbf{0.975]} \\\\\n", "\\midrule\n", "\\textbf{Intercept} & 125.9656 & 3.149 & 39.996 & 0.000 & 119.793 & 132.138 \\\\\n", "\\textbf{dan\\_sleep} & -8.9502 & 0.597 & -14.987 & 0.000 & -10.121 & -7.780 \\\\\n", "\\textbf{baby\\_sleep} & 0.0105 & 0.282 & 0.037 & 0.970 & -0.541 & 0.563 \\\\\n", "\\bottomrule\n", "\\end{tabular}\n", "\\begin{tabular}{lclc}\n", "\\textbf{Omnibus:} & 0.593 & \\textbf{ Durbin-Watson: } & 2.120 \\\\\n", "\\textbf{Prob(Omnibus):} & 0.743 & \\textbf{ Jarque-Bera (JB): } & 0.218 \\\\\n", "\\textbf{Skew:} & -0.053 & \\textbf{ Prob(JB): } & 0.897 \\\\\n", "\\textbf{Kurtosis:} & 3.203 & \\textbf{ Cond. No. } & 76.9 \\\\\n", "\\bottomrule\n", "\\end{tabular}\n", "%\\caption{OLS Regression Results}\n", "\\end{center}\n", "\n", "Notes: \\newline\n", " [1] Standard Errors are heteroscedasticity robust (HC1)" ], "text/plain": [ "\n", "\"\"\"\n", " OLS Regression Results \n", "==============================================================================\n", "Dep. Variable: dan_grump R-squared: 0.816\n", "Model: OLS Adj. R-squared: 0.812\n", "Method: Least Squares F-statistic: 201.7\n", "Date: Thu, 02 May 2024 Prob (F-statistic): 2.78e-35\n", "Time: 11:12:57 Log-Likelihood: -287.48\n", "No. Observations: 100 AIC: 581.0\n", "Df Residuals: 97 BIC: 588.8\n", "Df Model: 2 \n", "Covariance Type: HC1 \n", "==============================================================================\n", " coef std err z P>|z| [0.025 0.975]\n", "------------------------------------------------------------------------------\n", "Intercept 125.9656 3.149 39.996 0.000 119.793 132.138\n", "dan_sleep -8.9502 0.597 -14.987 0.000 -10.121 -7.780\n", "baby_sleep 0.0105 0.282 0.037 0.970 -0.541 0.563\n", "==============================================================================\n", "Omnibus: 0.593 Durbin-Watson: 2.120\n", "Prob(Omnibus): 0.743 Jarque-Bera (JB): 0.218\n", "Skew: -0.053 Prob(JB): 0.897\n", "Kurtosis: 3.203 Cond. No. 76.9\n", "==============================================================================\n", "\n", "Notes:\n", "[1] Standard Errors are heteroscedasticity robust (HC1)\n", "\"\"\"" ] }, "execution_count": 110, "metadata": {}, "output_type": "execute_result" } ], "source": [ "smf.ols('dan_grump ~ dan_sleep+baby_sleep', data=df).fit(cov_type='HC1').summary()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "860244c9", "metadata": {}, "source": [ "Not surprisingly, the $z$-values are pretty similar to the $t$-values that we saw when we fit our basic, non-robust regression, because the homogeneity of variance assumption wasn’t violated in our data anyway. But if it had been, we might have seen some more substantial differences." ] }, { "attachments": {}, "cell_type": "markdown", "id": "cbb97f5f", "metadata": {}, "source": [ "(regressioncollinearity)=\n", "### Checking for collinearity\n", "\n", "The last kind of regression diagnostic that I'm going to discuss in this chapter is the use of **_variance inflation factors_** (VIFs), which are useful for determining whether or not the predictors in your regression model are too highly correlated with each other. There is a variance inflation factor associated with each predictor $X_k$ in the model, and the formula for the $k$-th VIF is:\n", "\n", "$$\n", "\\mbox{VIF}_k = \\frac{1}{1-{R^2_{(-k)}}}\n", "$$\n", "\n", "where $R^2_{(-k)}$ refers to $R$-squared value you would get if you ran a regression using $X_k$ as the outcome variable, and all the other $X$ variables as the predictors. The idea here is that $R^2_{(-k)}$ is a very good measure of the extent to which $X_k$ is correlated with all the other variables in the model. Better yet, the square root of the VIF is pretty interpretable: it tells you how much wider the confidence interval for the corresponding coefficient $b_k$ is, relative to what you would have expected if the predictors are all nice and uncorrelated with one another. If you've only got two predictors, the VIF values are always going to be the same, as we can see if we use the ``variance_inflation_factor`` function from ``statsmodels``.\n", "\n", "First, we need a dataframe with our predictor variables as columns, plus a column with a constant value representing the model intercept:" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a5fdf7e5", "metadata": {}, "source": [] }, { "cell_type": "code", "execution_count": 111, "id": "d3ccbeda", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Interceptdan_sleepbaby_sleep
017.5910.18
117.9111.66
215.147.92
317.719.61
416.689.75
\n", "
" ], "text/plain": [ " Intercept dan_sleep baby_sleep\n", "0 1 7.59 10.18\n", "1 1 7.91 11.66\n", "2 1 5.14 7.92\n", "3 1 7.71 9.61\n", "4 1 6.68 9.75" ] }, "execution_count": 111, "metadata": {}, "output_type": "execute_result" } ], "source": [ "matrix = pd.DataFrame({'Intercept': [1]*len(df['dan_sleep']),\n", " 'dan_sleep': df['dan_sleep'], \n", " 'baby_sleep': df['baby_sleep']})\n", "matrix.head()\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "1857f79b", "metadata": {}, "source": [ "Then, we can calculate the VIF for each predictor and put the results in a dataframe:" ] }, { "cell_type": "code", "execution_count": 112, "id": "c433a0f4", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
VIFvariable
048.784569Intercept
11.651038dan_sleep
21.651038baby_sleep
\n", "
" ], "text/plain": [ " VIF variable\n", "0 48.784569 Intercept\n", "1 1.651038 dan_sleep\n", "2 1.651038 baby_sleep" ] }, "execution_count": 112, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from statsmodels.stats.outliers_influence import variance_inflation_factor\n", "\n", "vif = pd.DataFrame()\n", "vif['VIF'] = [variance_inflation_factor(matrix.values, i) for i in range(matrix.shape[1])]\n", "vif['variable'] = matrix.columns\n", "vif\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "8da33b9d", "metadata": {}, "source": [ "And since the square root of 1.65 is 1.28, which isn't really a huge change in the confidence intervals for our predictors, we see that the correlation between our two predictors isn’t causing much of a problem.\n", "\n", "To give a sense of how we could end up with a model that has bigger collinearity problems, suppose I were to run a much less interesting regression model, in which I tried to predict the day on which the data were collected, as a function of all the other variables in the data set. To see why this would be a bit of a problem, let’s have a look at the correlation matrix for all four variables:" ] }, { "cell_type": "code", "execution_count": 113, "id": "0d9d96d6", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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dan_sleepbaby_sleepdan_grumpdayTukeydan_sleep_transformed
dan_sleep1.0000000.627949-0.903384-0.098408-0.9970240.997866
baby_sleep0.6279491.000000-0.565964-0.010434-0.6261770.625397
dan_grump-0.903384-0.5659641.0000000.0764790.907817-0.895394
day-0.098408-0.0104340.0764791.0000000.103952-0.093393
Tukey-0.997024-0.6261770.9078170.1039521.000000-0.989864
dan_sleep_transformed0.9978660.625397-0.895394-0.093393-0.9898641.000000
\n", "
" ], "text/plain": [ " dan_sleep baby_sleep dan_grump day Tukey \\\n", "dan_sleep 1.000000 0.627949 -0.903384 -0.098408 -0.997024 \n", "baby_sleep 0.627949 1.000000 -0.565964 -0.010434 -0.626177 \n", "dan_grump -0.903384 -0.565964 1.000000 0.076479 0.907817 \n", "day -0.098408 -0.010434 0.076479 1.000000 0.103952 \n", "Tukey -0.997024 -0.626177 0.907817 0.103952 1.000000 \n", "dan_sleep_transformed 0.997866 0.625397 -0.895394 -0.093393 -0.989864 \n", "\n", " dan_sleep_transformed \n", "dan_sleep 0.997866 \n", "baby_sleep 0.625397 \n", "dan_grump -0.895394 \n", "day -0.093393 \n", "Tukey -0.989864 \n", "dan_sleep_transformed 1.000000 " ] }, "execution_count": 113, "metadata": {}, "output_type": "execute_result" } ], "source": [ "df.corr()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "98f6a09e", "metadata": {}, "source": [ "We have some fairly large correlations between some of our predictor variables! When we run the regression model and look at the VIF values, we see that the collinearity is causing a lot of uncertainty about the coefficients. First, we'll add the ``day`` column and the ``dan_grump`` column to our ``matrix`` dataframe..." ] }, { "cell_type": "code", "execution_count": 114, "id": "e0029a83", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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Interceptdan_sleepbaby_sleepdan_grumpday
017.5910.18561
117.9111.66602
215.147.92823
317.719.61554
416.689.75675
\n", "
" ], "text/plain": [ " Intercept dan_sleep baby_sleep dan_grump day\n", "0 1 7.59 10.18 56 1\n", "1 1 7.91 11.66 60 2\n", "2 1 5.14 7.92 82 3\n", "3 1 7.71 9.61 55 4\n", "4 1 6.68 9.75 67 5" ] }, "execution_count": 114, "metadata": {}, "output_type": "execute_result" } ], "source": [ "matrix['dan_grump'] = df['dan_grump']\n", "matrix['day'] = df['day']\n", "\n", "\n", "matrix.head()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "f92a066c", "metadata": {}, "source": [ "and then, we'll look at the VIFs..." ] }, { "cell_type": "code", "execution_count": 115, "id": "af44556e", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
VIFvariable
0922.960321Intercept
16.148528dan_sleep
21.658389baby_sleep
35.442617dan_grump
41.015119day
\n", "
" ], "text/plain": [ " VIF variable\n", "0 922.960321 Intercept\n", "1 6.148528 dan_sleep\n", "2 1.658389 baby_sleep\n", "3 5.442617 dan_grump\n", "4 1.015119 day" ] }, "execution_count": 115, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vif = pd.DataFrame()\n", "vif['VIF'] = [variance_inflation_factor(matrix.values, i) for i in range(matrix.shape[1])]\n", "vif['variable'] = matrix.columns\n", "vif" ] }, { "attachments": {}, "cell_type": "markdown", "id": "99660940", "metadata": {}, "source": [ "Yep, that’s some mighty fine collinearity you’ve got there. Of course, if you suspect that your model might have some degree of collinearity between predictors, you are still left with some questions to which there are no concrete answers (sorry!). How much collinearity is ok? What should I do about the collinearity in my model? There are some suggestions and guidelines out there (e.g., in the ``statsmodels`` documentation, a vif of 5 is suggested to be the limit, but [elsewhere](https://en.wikipedia.org/wiki/Variance_inflation_factor) you can find other suggested values as cutoff lines), but like so many things in statistics (and in life), I'm afraid you will need to exercise your own judgment." ] }, { "attachments": {}, "cell_type": "markdown", "id": "a33d12c5", "metadata": {}, "source": [ "(modelselreg)=\n", "## Model Selection \n", "\n", "One fairly major problem that remains is the problem of \"model selection\". That is, if we have a data set that contains several variables, which ones should we include as predictors, and which ones should we not include? In other words, we have a problem of **_variable selection_**. In general, model selection is a complex business, but it's made somewhat simpler if we restrict ourselves to the problem of choosing a subset of the variables that ought to be included in the model. Nevertheless, I'm not going to try covering even this reduced topic in a lot of detail. Instead, I'll talk about two broad principles that you need to think about; and then discuss one concrete tool to help you select a subset of variables to include in your model. Firstly, the two principles:\n", "\n", "- It's nice to have an actual substantive basis for your choices. That is, in a lot of situations you the researcher have good reasons to pick out a smallish number of possible regression models that are of theoretical interest; these models will have a sensible interpretation in the context of your field. Never discount the importance of this. Statistics serves the scientific process, not the other way around. \n", "- To the extent that your choices rely on statistical inference, there is a trade off between simplicity and goodness of fit. As you add more predictors to the model, you make it more complex; each predictor adds a new free parameter (i.e., a new regression coefficient), and each new parameter increases the model's capacity to \"absorb\" random variations. So the goodness of fit (e.g., $R^2$) continues to rise as you add more predictors no matter what. If you want your model to be able to generalise well to new observations, you need to avoid throwing in too many variables. \n", "\n", "This latter principle is often referred to as **_Ockham's razor_**, and is often summarised in terms of the following pithy saying: *do not multiply entities beyond necessity*. In this context, it means: don't chuck in a bunch of largely irrelevant predictors just to boost your $R^2$. Hm. Yeah, the original was better. \n", "\n", "In any case, what we need is an actual mathematical criterion that will implement the qualitative principle behind Ockham's razor in the context of selecting a regression model. As it turns out there are several possibilities. The one that I'll talk about is the **_Akaike information criterion_** {cite}``Akaike1974`` simply because it's a common one, and because ``statsmodels`` will calculate it for you automatically when you run a linear regression There are many others, including BIC (Bayesian Information Criterion), which ``statsmodels`` also gives you for free. In the context of a linear regression model (and ignoring terms that don't depend on the model in any way!), the AIC for a model that has $K$ predictor variables plus an intercept is:\n", "\n", "$$\n", "\\mbox{AIC} = \\displaystyle\\frac{\\mbox{SS}_{res}}{\\hat{\\sigma}}^2+ 2K\n", "$$\n", "\n", "\n", "The smaller the AIC value, the better the model performance is.[^AIC_calc] If we ignore the low level details, it's fairly obvious what the AIC does: on the left we have a term that increases as the model predictions get worse; on the right we have a term that increases as the model complexity increases. The best model is the one that fits the data well (low residuals; left hand side) using as few predictors as possible (low $K$; right hand side). In short, this is a simple implementation of Ockham's razor. \n", "\n", "\n", "\n", "[^AIC_calc]: Note, depending on exactly how AIC is calculated, the actual AIC value may be different. That is, if you calculate AIC using the formula above, a Python function, or an algorithm from some other statistics software. However, if you use the same method to calculate AIC for two different models, and take the differnce between them, then this should end up being the same, no matter what method was used. In practice, this is all you care about: the actual value of an AIC statistic isn't very informative, but the differences between two AIC values *are* useful, since these provide a measure of the extent to which one model outperforms another." ] }, { "attachments": {}, "cell_type": "markdown", "id": "b9c52570", "metadata": {}, "source": [ "### Backward elimination\n", "\n", "Okay, let’s have a look at how this works in practice. In this example I’ll keep it simple and use only the basic backward elimination approach. That is, start with the complete regression model, including all possible predictors. Then, at each “step” we try all possible ways of removing one of the variables, and whichever of these is best (in terms of lowest AIC value) is accepted. This becomes our new regression model; and we then try all possible deletions from the new model, again choosing the option with lowest AIC. This process continues until we end up with a model that has a lower AIC value than any of the other possible models that you could produce by deleting one of its predictors. Let’s see this in action. First, I need to define the model from which the process starts." ] }, { "cell_type": "code", "execution_count": 116, "id": "f5ac445f", "metadata": {}, "outputs": [], "source": [ "mod_full = smf.ols('dan_grump ~ dan_sleep + baby_sleep + day', data=df)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "8eee3272", "metadata": {}, "source": [ "We can get ``statsmodels`` to fit the model and return a summay, which will include the AIC value." ] }, { "cell_type": "code", "execution_count": 117, "id": "6ad334d1", "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
OLS Regression Results
Dep. Variable: dan_grump R-squared: 0.816
Model: OLS Adj. R-squared: 0.811
Method: Least Squares F-statistic: 142.2
Date: Thu, 02 May 2024 Prob (F-statistic): 3.42e-35
Time: 11:12:57 Log-Likelihood: -287.43
No. Observations: 100 AIC: 582.9
Df Residuals: 96 BIC: 593.3
Df Model: 3
Covariance Type: nonrobust
\n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
coef std err t P>|t| [0.025 0.975]
Intercept 126.2787 3.242 38.945 0.000 119.842 132.715
dan_sleep -8.9693 0.560 -16.016 0.000 -10.081 -7.858
baby_sleep 0.0157 0.273 0.058 0.954 -0.526 0.558
day -0.0044 0.015 -0.288 0.774 -0.035 0.026
\n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
Omnibus: 0.599 Durbin-Watson: 2.120
Prob(Omnibus): 0.741 Jarque-Bera (JB): 0.215
Skew: -0.037 Prob(JB): 0.898
Kurtosis: 3.215 Cond. No. 441.


Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified." ], "text/latex": [ "\\begin{center}\n", "\\begin{tabular}{lclc}\n", "\\toprule\n", "\\textbf{Dep. Variable:} & dan\\_grump & \\textbf{ R-squared: } & 0.816 \\\\\n", "\\textbf{Model:} & OLS & \\textbf{ Adj. R-squared: } & 0.811 \\\\\n", "\\textbf{Method:} & Least Squares & \\textbf{ F-statistic: } & 142.2 \\\\\n", "\\textbf{Date:} & Thu, 02 May 2024 & \\textbf{ Prob (F-statistic):} & 3.42e-35 \\\\\n", "\\textbf{Time:} & 11:12:57 & \\textbf{ Log-Likelihood: } & -287.43 \\\\\n", "\\textbf{No. Observations:} & 100 & \\textbf{ AIC: } & 582.9 \\\\\n", "\\textbf{Df Residuals:} & 96 & \\textbf{ BIC: } & 593.3 \\\\\n", "\\textbf{Df Model:} & 3 & \\textbf{ } & \\\\\n", "\\textbf{Covariance Type:} & nonrobust & \\textbf{ } & \\\\\n", "\\bottomrule\n", "\\end{tabular}\n", "\\begin{tabular}{lcccccc}\n", " & \\textbf{coef} & \\textbf{std err} & \\textbf{t} & \\textbf{P$> |$t$|$} & \\textbf{[0.025} & \\textbf{0.975]} \\\\\n", "\\midrule\n", "\\textbf{Intercept} & 126.2787 & 3.242 & 38.945 & 0.000 & 119.842 & 132.715 \\\\\n", "\\textbf{dan\\_sleep} & -8.9693 & 0.560 & -16.016 & 0.000 & -10.081 & -7.858 \\\\\n", "\\textbf{baby\\_sleep} & 0.0157 & 0.273 & 0.058 & 0.954 & -0.526 & 0.558 \\\\\n", "\\textbf{day} & -0.0044 & 0.015 & -0.288 & 0.774 & -0.035 & 0.026 \\\\\n", "\\bottomrule\n", "\\end{tabular}\n", "\\begin{tabular}{lclc}\n", "\\textbf{Omnibus:} & 0.599 & \\textbf{ Durbin-Watson: } & 2.120 \\\\\n", "\\textbf{Prob(Omnibus):} & 0.741 & \\textbf{ Jarque-Bera (JB): } & 0.215 \\\\\n", "\\textbf{Skew:} & -0.037 & \\textbf{ Prob(JB): } & 0.898 \\\\\n", "\\textbf{Kurtosis:} & 3.215 & \\textbf{ Cond. No. } & 441. \\\\\n", "\\bottomrule\n", "\\end{tabular}\n", "%\\caption{OLS Regression Results}\n", "\\end{center}\n", "\n", "Notes: \\newline\n", " [1] Standard Errors assume that the covariance matrix of the errors is correctly specified." ], "text/plain": [ "\n", "\"\"\"\n", " OLS Regression Results \n", "==============================================================================\n", "Dep. Variable: dan_grump R-squared: 0.816\n", "Model: OLS Adj. R-squared: 0.811\n", "Method: Least Squares F-statistic: 142.2\n", "Date: Thu, 02 May 2024 Prob (F-statistic): 3.42e-35\n", "Time: 11:12:57 Log-Likelihood: -287.43\n", "No. Observations: 100 AIC: 582.9\n", "Df Residuals: 96 BIC: 593.3\n", "Df Model: 3 \n", "Covariance Type: nonrobust \n", "==============================================================================\n", " coef std err t P>|t| [0.025 0.975]\n", "------------------------------------------------------------------------------\n", "Intercept 126.2787 3.242 38.945 0.000 119.842 132.715\n", "dan_sleep -8.9693 0.560 -16.016 0.000 -10.081 -7.858\n", "baby_sleep 0.0157 0.273 0.058 0.954 -0.526 0.558\n", "day -0.0044 0.015 -0.288 0.774 -0.035 0.026\n", "==============================================================================\n", "Omnibus: 0.599 Durbin-Watson: 2.120\n", "Prob(Omnibus): 0.741 Jarque-Bera (JB): 0.215\n", "Skew: -0.037 Prob(JB): 0.898\n", "Kurtosis: 3.215 Cond. No. 441.\n", "==============================================================================\n", "\n", "Notes:\n", "[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.\n", "\"\"\"" ] }, "execution_count": 117, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mod_full.fit().summary()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a1a648f3", "metadata": {}, "source": [ " This is great, but we will be doing this several times, and this is quite a lot of information to wade through just to get AIC value. So to reduce the output, we could write a function, like the one below, to return _only_ the AIC value." ] }, { "cell_type": "code", "execution_count": 118, "id": "66ce59c1", "metadata": {}, "outputs": [], "source": [ "# function to return only the AIC calculation for a statsmodels linear model\n", "def get_aic(model_name, model):\n", " aic = model.fit().summary().tables[0].as_text().split('\\n')[7].split('AIC:')[1].strip()\n", " output = model_name + ' AIC: ' + aic\n", " return output" ] }, { "attachments": {}, "cell_type": "markdown", "id": "ca81e771", "metadata": {}, "source": [ "The line in this function that defines the variable ``aic`` looks pretty hairly, but that is only because I have compressed it down to one line. If you break it down, and try running it bit by bit (adding the parts with a `.` in front of them and looking at the output), you will see that doesn't really involve anything new (other than maybe the ``strip()`` function, which just removes unnecessary whitespace from the output)\n", "\n", "Now we just give the function a name for the model, and the model itself, and it tells us the AIC:" ] }, { "cell_type": "code", "execution_count": 119, "id": "1ed5f89e", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'full model AIC: 582.9'" ] }, "execution_count": 119, "metadata": {}, "output_type": "execute_result" } ], "source": [ "get_aic('full model', mod_full)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "b48b816f", "metadata": {}, "source": [ "Now we can use our fancy new function to loop through all the possible models, each one removing one predictor from the full model:" ] }, { "cell_type": "code", "execution_count": 120, "id": "1541abd2", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/plain": [ "['Full Model AIC: 582.9',\n", " 'No Day AIC: 581.0',\n", " 'No Baby AIC: 580.9',\n", " 'No Dan AIC: 710.9']" ] }, "execution_count": 120, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mod_full = smf.ols('dan_grump ~ dan_sleep + baby_sleep + day', data=df)\n", "mod_no_day = smf.ols('dan_grump ~ dan_sleep + baby_sleep', data=df)\n", "mod_no_baby = smf.ols('dan_grump ~ dan_sleep + day', data=df)\n", "mod_no_dan = smf.ols('dan_grump ~ baby_sleep + day', data=df)\n", "\n", "models = [mod_full, mod_no_day, mod_no_baby, mod_no_dan]\n", "names = ['Full Model', 'No Day', 'No Baby', 'No Dan']\n", "\n", "aics = []\n", "for n, model in enumerate(models):\n", " aics.append(get_aic(names[n], model))\n", "\n", "aics" ] }, { "attachments": {}, "cell_type": "markdown", "id": "744f1267", "metadata": {}, "source": [ "Of these models, the one in which we removed ``baby_sleep`` as a predictor has the lowest AIC, so it seems like removing this predictor is probably a good idea. This shouldn't be surprising, really, given all the other checks we have already done on these data. But let's proceed. Now, we can do the same thing again, but comparing only models without ``baby_sleep``:" ] }, { "cell_type": "code", "execution_count": 121, "id": "b06eb416", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "outputs": [ { "data": { "text/plain": [ "['Full Model AIC: 580.9', 'No Day AIC: 579.0', 'No Dan AIC: 747.7']" ] }, "execution_count": 121, "metadata": {}, "output_type": "execute_result" } ], "source": [ "mod_full = smf.ols('dan_grump ~ dan_sleep + day', data=df)\n", "mod_no_day = smf.ols('dan_grump ~ dan_sleep', data=df)\n", "mod_no_dan = smf.ols('dan_grump ~ day', data=df)\n", "\n", "models = [mod_full, mod_no_day, mod_no_dan]\n", "names = ['Full Model', 'No Day', 'No Dan']\n", "\n", "aics = []\n", "for n, model in enumerate(models):\n", " aics.append(get_aic(names[n], model))\n", "\n", "aics" ] }, { "attachments": {}, "cell_type": "markdown", "id": "3e88ec56", "metadata": {}, "source": [ "Now, the model in which we removed ``day`` as a predictor performs best (has the lowest AIC), so the next step would be to remove ``day`` from our model. In this case, that only leaves us with one predictor: ``dan_sleep``. Which is (perhaps not all that surprisingly) the mod1 that we started with at the [beginning of the chapter](pingouinregression): \n", "\n", " mod1 = pg.linear_regression(df['dan_sleep'], df['dan_grump'])\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "2ac2fece", "metadata": {}, "source": [ "This process, as I have described it, is fairly straightforward, especially once you have a function to just grab the AIC from the ``statsmodels`` output. But if you find yourself doing this sort of thing often, you might want to automate it even further. It wouldn't take _that_ much work to expand on the code I have given you, and build a function that takes as input your full model, and then _automatically_ considers all possible variations, and continues eliminating predictors until it finds the optimal model. But this I leave as a programming exercise for you, if you feel so inclined." ] }, { "attachments": {}, "cell_type": "markdown", "id": "a2d73567", "metadata": {}, "source": [ "### A caveat\n", "\n", "Automated variable selection methods are seductive things. They provide an element of objectivity to your model selection, and that’s kind of nice. Unfortunately, they’re sometimes used as an excuse for thoughtlessness. No longer do you have to think carefully about which predictors to add to the model and what the theoretical basis for their inclusion might be... everything is solved by the magic of AIC. And if we start throwing around phrases like Ockham’s razor, well, it sounds like everything is wrapped up in a nice neat little package that no-one can argue with.\n", "\n", "Or, perhaps not. First of all, there’s very little agreement on what counts as an appropriate model selection criterion. When I was taught backward elimination as an undergraduate, we used F-tests to do it, because that was the default method used by the software. Here we are using AIC, and since this is an introductory text that’s the only method I’ve described, but the AIC is hardly the Word of the Gods of Statistics. It’s an approximation, derived under certain assumptions, and it’s guaranteed to work only for large samples when those assumptions are met. Alter those assumptions and you get a different criterion, like the BIC for instance. Take a different approach again and you get the NML criterion. Decide that you’re a Bayesian and you get model selection based on posterior odds ratios. Then there are a bunch of regression-specific tools that I haven’t mentioned. And so on. All of these different methods have strengths and weaknesses, and some are easier to calculate than others (AIC is probably the easiest of the lot, which might account for its popularity). Almost all of them produce the same answers when the answer is “obvious”, but there’s a fair amount of disagreement when the model selection problem becomes hard.\n", "\n", "What does this mean in practice? Well, you could go and spend several years teaching yourself the theory of model selection, learning all the ins and outs of it; so that you could finally decide on what you personally think the right thing to do is. Speaking as someone who actually did that, I wouldn’t recommend it: you’ll probably come out the other side even more confused than when you started. A better strategy is to show a bit of common sense... if you’re staring at the results of a stepwise AIC model comparison procedure, and the model that makes sense is close to having the smallest AIC, but is narrowly defeated by a model that doesn’t make any sense... trust your instincts. Statistical model selection is an inexact tool, and as I said at the beginning, interpretability matters." ] }, { "attachments": {}, "cell_type": "markdown", "id": "173a3198", "metadata": {}, "source": [ "### Comparing two regression models\n", "\n", "An alternative to using automated model selection procedures is for the researcher to explicitly select two or more regression models to compare to each other. You can do this in a few different ways, depending on what research question you're trying to answer. Suppose we want to know whether or not the amount of sleep that my son got has any relationship to my grumpiness, over and above what we might expect from the amount of sleep that I got. We also want to make sure that the day on which we took the measurement has no influence on the relationship. That is, we're interested in the relationship between `baby_sleep` and `dan_grump`, and from that perspective `dan_sleep` and `day` are nuisance variables or **_covariates_** that we want to control for. In this situation, what we would like to know is whether `dan_grump ~ dan_sleep + day + baby_sleep` (which I'll call Model 1, or `M1`) is a better regression model for these data than `dan_grump ~ dan_sleep + day` (which I'll call Model 0, or `M0`). There are two different ways we can compare these two models, one based on a model selection criterion like AIC, and the other based on an explicit hypothesis test. I'll show you the AIC-based approach first because it's simpler, and follows naturally from the method we used in the last section. The first thing I need to do is actually run the regressions. Since we want to calculate AIC, it will be easier to use `statsmodels` than `pigouin`. First we'll define the two models, then use our handy-dandy AIC function to get the AIC for each of them." ] }, { "cell_type": "code", "execution_count": 122, "id": "6ebd37cd", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "['M0 AIC: 580.9', 'M1 AIC: 582.9']" ] }, "execution_count": 122, "metadata": {}, "output_type": "execute_result" } ], "source": [ "M0 = smf.ols('dan_grump ~ dan_sleep + day', data=df)\n", "M1 = smf.ols('dan_grump ~ dan_sleep + day + baby_sleep', data=df)\n", "\n", "models = [M0, M1]\n", "names = ['M0', 'M1']\n", "\n", "aics = []\n", "for n, model in enumerate(models):\n", " aics.append(get_aic(names[n], model))\n", "\n", "aics" ] }, { "attachments": {}, "cell_type": "markdown", "id": "506f854d", "metadata": {}, "source": [ "Since Model 0 has the smaller AIC value, it is judged to be the better model for these data. \n", "\n", "By the way, I mentioned before that ``statsmodels`` gives us both AIC and BIC automatically. In fact, while I'm not particularly impressed with either AIC or BIC as model selection methods, if you do find yourself using one of these two, the empirical evidence suggests that BIC is the better criterion of the two. In most simulation studies that I've seen, BIC does a much better job of selecting the correct model. And indeed, it wouldn't be hard to modify our function ``get_aic`` from above to get both the AIC and BIC values from the ``statsmodels`` output, so why not just do that, like so [^exercise]:\n", "\n", "[^exercise]: I leave it as a \"fun\" programming exercise to you to adapt this function even further, so that it present the results in a nice table format, e.g. in a dataframe. Not necessary, but it would make it easier to compare models if the results were nicely lined up in columns, don't you think?" ] }, { "cell_type": "code", "execution_count": 123, "id": "e4b439ff", "metadata": {}, "outputs": [], "source": [ "def get_aic_bic(model_name, model):\n", " aic = model.fit().summary().tables[0].as_text().split('\\n')[7].split('AIC:')[1].strip()\n", " bic = model.fit().summary().tables[0].as_text().split('\\n')[8].split('BIC:')[1].strip()\n", " output = model_name + ' AIC: ' + aic + ' BIC: ' + bic\n", " return output" ] }, { "cell_type": "code", "execution_count": 124, "id": "8d456f8d", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "['M0 AIC: 580.9 BIC: 588.7', 'M1 AIC: 582.9 BIC: 593.3']" ] }, "execution_count": 124, "metadata": {}, "output_type": "execute_result" } ], "source": [ "models = [M0, M1]\n", "names = ['M0', 'M1']\n", "\n", "ics = []\n", "for n, model in enumerate(models):\n", " ics.append(get_aic_bic(names[n], model))\n", "\n", "ics" ] }, { "attachments": {}, "cell_type": "markdown", "id": "6f8efb5a", "metadata": {}, "source": [ "BIC is also smaller for `MO` than for `M1`, so based on both AIC and BIC, it looks like Model 0 is the better choice.\n", "\n", "A somewhat different approach to the problem comes out of the hypothesis testing framework. Suppose you have two regression models, where one of them (Model 0) contains a *subset* of the predictors from the other one (Model 1). That is, Model 1 contains all of the predictors included in Model 0, plus one or more additional predictors. When this happens we say that Model 0 is **_nested_** within Model 1, or possibly that Model 0 is a **_submodel_** of Model 1. Regardless of the terminology, what this means is that we can think of Model 0 as a null hypothesis and Model 1 as an alternative hypothesis. And in fact we can construct an $F$ test for this in a fairly straightforward fashion. We can fit both models to the data and obtain a residual sum of squares for both models. I'll denote these as SS$_{res}^{(0)}$ and SS$_{res}^{(1)}$ respectively. The superscripting here just indicates which model we're talking about. Then our $F$ statistic is\n", "\n", "$$\n", "F = \\frac{(\\mbox{SS}_{res}^{(0)} - \\mbox{SS}_{res}^{(1)})/k}{(\\mbox{SS}_{res}^{(1)})/(N-p-1)}\n", "$$\n", "\n", "where $N$ is the number of observations, $p$ is the number of predictors in the full model (not including the intercept), and $k$ is the difference in the number of parameters between the two models.^[It's worth noting in passing that this same $F$ statistic can be used to test a much broader range of hypotheses than those that I'm mentioning here. Very briefly: notice that the nested model M0 corresponds to the full model M1 when we constrain some of the regression coefficients to zero. It is sometimes useful to construct submodels by placing other kinds of constraints on the regression coefficients. For instance, maybe two different coefficients might have to sum to zero, or something like that. You can construct hypothesis tests for those kind of constraints too, but it is somewhat more complicated and the sampling distribution for $F$ can end up being something known as the non-central $F$ distribution, which is waaaaay beyond the scope of this book! All I want to do is alert you to this possibility.] The degrees of freedom here are $k$ and $N-p-1$. Note that it's often more convenient to think about the difference between those two SS values as a sum of squares in its own right. That is: \n", "\n", "$$\n", "\\mbox{SS}_\\Delta = \\mbox{SS}_{res}^{(0)} - \\mbox{SS}_{res}^{(1)}\n", "$$\n", "\n", "The reason why this his helpful is that we can express $\\mbox{SS}_\\Delta$ a measure of the extent to which the two models make different predictions about the the outcome variable. Specifically:\n", "$$\n", "\\mbox{SS}_\\Delta = \\sum_{i} \\left( \\hat{y}_i^{(1)} - \\hat{y}_i^{(0)} \\right)^2\n", "$$\n", "where $\\hat{y}_i^{(0)}$ is the fitted value for $y_i$ according to model $M_0$ and $\\hat{y}_i^{(1)}$ is the is the fitted value for $y_i$ according to model $M_1$. \n", "\n", "Okay, so that's the hypothesis test that we use to compare two regression models to one another. Now, how do we do it in Python? As it turns out, `statsmodels` has a function called `anova_lm` that will do just this. All we have to do is fit the two models that we want to compare, and then plug them in the function (null model first):\n" ] }, { "cell_type": "code", "execution_count": 125, "id": "b52f0730", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
df_residssrdf_diffss_diffFPr(>F)
097.01837.1559160.0NaNNaNNaN
196.01837.0922281.00.0636880.0033280.954116
\n", "
" ], "text/plain": [ " df_resid ssr df_diff ss_diff F Pr(>F)\n", "0 97.0 1837.155916 0.0 NaN NaN NaN\n", "1 96.0 1837.092228 1.0 0.063688 0.003328 0.954116" ] }, "execution_count": 125, "metadata": {}, "output_type": "execute_result" } ], "source": [ "from statsmodels.stats.anova import anova_lm\n", "\n", "M0 = M0.fit()\n", "M1 = M1.fit()\n", "\n", "anova_lm(M0,M1)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a66de911", "metadata": {}, "source": [ "Since we have $p>.05$ we retain the null hypothesis (`M0`). This approach to regression, in which we add all of our covariates into a null model, and then *add* the variables of interest into an alternative model, and then compare the two models in hypothesis testing framework, is often referred to as **_hierarchical regression_**." ] }, { "attachments": {}, "cell_type": "markdown", "id": "2ee5fadd", "metadata": { "editable": true, "slideshow": { "slide_type": "" }, "tags": [] }, "source": [ "## Summary\n", "\n", "\n", "- Basic ideas in [linear regression](regression) and how regression models are [estimated](regressionestimation).\n", "- [Multiple linear regression](multipleregression). \n", "- Measuring the [overall performance](r2) of a regression model using $R^2$\n", "- [Hypothesis tests for regression models](regressiontests)\n", "- [Calculating confidence intervals ](regressioncoefs) for regression coefficients, and standardised coefficients \n", "- The [assumptions of regression](regressionassumptions) and [how to check them](regressiondiagnostics)\n", "- [Selecting a regression model](modelselreg)" ] } ], "metadata": { "celltoolbar": "Tags", "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.3" } }, "nbformat": 4, "nbformat_minor": 5 } { "cells": [ { "attachments": {}, "cell_type": "markdown", "id": "insured-cedar", "metadata": {}, "source": [ "(anova2)=\n", "# Factorial ANOVA" ] }, { "attachments": {}, "cell_type": "markdown", "id": "d9a282ab", "metadata": {}, "source": [ "Over the course of the last few chapters you can probably detect a general trend. We started out looking at tools that you can use to compare two groups to one another, most notably the [$t$-test](ttest). Then, we introduced [analysis of variance](anova) (ANOVA) as a method for comparing more than two groups. The chapter on [regression](regression) covered a somewhat different topic, but in doing so it introduced a powerful new idea: building statistical models that have *multiple* predictor variables used to explain a single outcome variable. For instance, a regression model could be used to predict the number of errors a student makes in a reading comprehension test based on the number of hours they studied for the test, and their score on a standardised IQ test. The goal in this chapter is to import this idea into the ANOVA framework. For instance, suppose we were interested in using the reading comprehension test to measure student achievements in three different schools, and we suspect that girls and boys are developing at different rates (and so would be expected to have different performance on average). Each student is classified in two different ways: on the basis of their gender, and on the basis of their school. What we'd like to do is analyse the reading comprehension scores in terms of *both* of these grouping variables. The tool for doing so is generically referred to as **_factorial ANOVA_**. However, since we have two grouping variables, we sometimes refer to the analysis as a two-way ANOVA, in contrast to the one-way ANOVAs that we [ran earlier](anova)." ] }, { "attachments": {}, "cell_type": "markdown", "id": "19203557", "metadata": {}, "source": [ "(factorialanovasimple)=\n", "## Balanced designs, main effects\n", "\n", "When we discussed [analysis of variance](anova), we assumed a fairly simple experimental design: each person falls into one of several groups, and we want to know whether these groups have different means on some outcome variable. In this section, I'll discuss a broader class of experimental designs, known as **_factorial designs_**, in which we have more than one grouping variable. I gave one example of how this kind of design might arise above. Another example appears in the chapter on [one-way ANOVA](anova), in which we were looking at the effect of different drugs on the `mood_gain` experienced by each person. In that chapter we did find a significant effect of drug, but at the end of the chapter we also ran an analysis to see if there was an effect of therapy. We didn't find one, but there's something a bit worrying about trying to run two *separate* analyses trying to predict the same outcome. Maybe there actually *is* an effect of therapy on mood gain, but we couldn't find it because it was being \"hidden\" by the effect of drug? In other words, we're going to want to run a *single* analysis that includes *both* `drug` and `therapy` as predictors. For this analysis, each person is cross-classified by the drug they were given (a factor with 3 levels) and what therapy they received (a factor with 2 levels). We refer to this as a $3 \\times 2$ factorial design. If we cross-tabulate `drug` by `therapy` using the `crosstab()` in `pandas`, we get the following table:" ] }, { "cell_type": "code", "execution_count": 2, "id": "b38e94ae", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
therapyCBTno.therapy
drug
anxifree33
joyzepam33
placebo33
\n", "
" ], "text/plain": [ "therapy CBT no.therapy\n", "drug \n", "anxifree 3 3\n", "joyzepam 3 3\n", "placebo 3 3" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/clintrial.csv\")\n", "\n", "pd.crosstab(index=df[\"drug\"], columns=df[\"therapy\"],margins=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "2aee2f64", "metadata": {}, "source": [ "As you can see, not only do we have participants corresponding to all possible combinations of the two factors, indicating that our design is **_completely crossed_**, it turns out that there are an equal number of people in each group. In other words, we have a **_balanced_** design. In this section I'll talk about how to analyse data from balanced designs, since this is the simplest case. The story for unbalanced designs is quite tedious, so we'll put it to one side for the moment." ] }, { "attachments": {}, "cell_type": "markdown", "id": "cbf44bf3", "metadata": {}, "source": [ "(factanovahyp)=\n", "### What hypotheses are we testing?\n", "\n", "Like one-way ANOVA, factorial ANOVA is a tool for testing certain types of hypotheses about population means. So a sensible place to start would be to be explicit about what our hypotheses actually are. However, before we can even get to that point, it's really useful to have some clean and simple notation to describe the population means. Because of the fact that observations are cross-classified in terms of two different factors, there are quite a lot of different means that one might be interested. To see this, let's start by thinking about all the different sample means that we can calculate for this kind of design. Firstly, there's the obvious idea that we might be interested in this table of group means:" ] }, { "cell_type": "code", "execution_count": 3, "id": "1b4a176a", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
therapydrugmood_gain
0CBTanxifree1.03
1CBTjoyzepam1.50
2CBTplacebo0.60
3no.therapyanxifree0.40
4no.therapyjoyzepam1.47
5no.therapyplacebo0.30
\n", "
" ], "text/plain": [ " therapy drug mood_gain\n", "0 CBT anxifree 1.03\n", "1 CBT joyzepam 1.50\n", "2 CBT placebo 0.60\n", "3 no.therapy anxifree 0.40\n", "4 no.therapy joyzepam 1.47\n", "5 no.therapy placebo 0.30" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pd.DataFrame(round(df.groupby(by=['therapy', 'drug'])['mood_gain'].mean(),2)).reset_index()\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "baf64f4f", "metadata": {}, "source": [ "Now, this output shows a cross-tabulation of the group means for all possible combinations of the two factors (e.g., people who received the placebo and no therapy, people who received the placebo while getting CBT, etc). However, we can also construct tables that ignore one of the two factors. That gives us output that looks like this:" ] }, { "cell_type": "code", "execution_count": 4, "id": "8653859e", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
therapymood_gain
0CBT1.04
1no.therapy0.72
\n", "
" ], "text/plain": [ " therapy mood_gain\n", "0 CBT 1.04\n", "1 no.therapy 0.72" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pd.DataFrame(round(df.groupby(['therapy'])['mood_gain'].mean(),2)).reset_index()" ] }, { "cell_type": "code", "execution_count": 5, "id": "821fd280", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
drugmood_gain
0anxifree0.72
1joyzepam1.48
2placebo0.45
\n", "
" ], "text/plain": [ " drug mood_gain\n", "0 anxifree 0.72\n", "1 joyzepam 1.48\n", "2 placebo 0.45" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pd.DataFrame(round(df.groupby(['drug'])['mood_gain'].mean(),2)).reset_index()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a8db1851", "metadata": {}, "source": [ "But of course, if we can ignore one factor we can certainly ignore both. That is, we might also be interested in calculating the average mood gain across all 18 participants, regardless of what drug or psychological therapy they were given:" ] }, { "cell_type": "code", "execution_count": 6, "id": "b3a84461", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.88" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "round(df['mood_gain'].mean(),2)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "ee5175b3", "metadata": {}, "source": [ "At this point we have 12 different sample means to keep track of! It is helpful to organise all these numbers into a single table, which would look like this:\n", "\n", "| |no therapy |CBT |total |\n", "|:--------|:----------|:----|:-----|\n", "|placebo |0.30 |0.60 |0.45 |\n", "|anxifree |0.40 |1.03 |0.72 |\n", "|joyzepam |1.47 |1.50 |1.48 |\n", "|total |0.72 |1.04 |0.88 |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "2fec1291", "metadata": {}, "source": [ "Now, each of these different means is of course a sample statistic: it's a quantity that pertains to the specific observations that we've made during our study. What we want to make inferences about are the corresponding population parameters: that is, the true means as they exist within some broader population. Those population means can also be organised into a similar table, but we'll need a little mathematical notation to do so. As usual, I'll use the symbol $\\mu$ to denote a population mean. However, because there are lots of different means, I'll need to use subscripts to distinguish between them. \n", "\n", "Here's how the notation works. Our table is defined in terms of two factors: each row corresponds to a different level of Factor A (in this case `drug`), and each column corresponds to a different level of Factor B (in this case `therapy`). If we let $R$ denote the number of rows in the table, and $C$ denote the number of columns, we can refer to this as an $R \\times C$ factorial ANOVA. In this case $R=3$ and $C=2$. We'll use lowercase letters to refer to specific rows and columns, so $\\mu_{rc}$ refers to the population mean associated with the $r$th level of Factor A (i.e. row number $r$) and the $c$th level of Factor B (column number $c$).[^notesubscript] So the population means are now written like this:\n", "\n", "[^notesubscript]: The nice thing about the subscript notation is that generalises nicely: if our experiment had involved a third factor, then we could just add a third subscript. In principle, the notation extends to as many factors as you might care to include, but in this book we'll rarely consider analyses involving more than two factors, and never more than three. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "0f3fbcf3", "metadata": {}, "source": [ "| |no therapy |CBT |total |\n", "|:--------|:----------|:----------|:-----|\n", "|placebo |$\\mu_{11}$ |$\\mu_{12}$ | |\n", "|anxifree |$\\mu_{21}$ |$\\mu_{22}$ | |\n", "|joyzepam |$\\mu_{31}$ |$\\mu_{32}$ | |\n", "|total | | | |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "7eaf7b58", "metadata": {}, "source": [ "Okay, what about the remaining entries? For instance, how should we describe the average mood gain across the entire (hypothetical) population of people who might be given Joyzepam in an experiment like this, regardless of whether they were in CBT? We use the \"dot\" notation to express this. In the case of Joyzepam, notice that we're talking about the mean associated with the third row in the table. That is, we're averaging across two cell means (i.e., $\\mu_{31}$ and $\\mu_{32}$). The result of this averaging is referred to as a **_marginal mean_**, and would be denoted $\\mu_{3.}$ in this case. The marginal mean for CBT corresponds to the population mean associated with the second column in the table, so we use the notation $\\mu_{.2}$ to describe it. The grand mean is denoted $\\mu_{..}$ because it is the mean obtained by averaging (marginalising[^notemarginalising]) over both. So our full table of population means can be written down like this:\n", "\n", "[^notemarginalising]: Technically, marginalising isn't quite identical to a regular mean: it's a weighted average, where you take into account the frequency of the different events that you're averaging over. However, in a balanced design, all of our cell frequencies are equal by definition, so the two are equivalent. We'll discuss unbalanced designs later, and when we do so you'll see that all of our calculations become a real headache. But let's ignore this for now." ] }, { "attachments": {}, "cell_type": "markdown", "id": "8b8e1412", "metadata": {}, "source": [ "| |no therapy |CBT |total |\n", "|:--------|:----------|:----------|:----------|\n", "|placebo |$\\mu_{11}$ |$\\mu_{12}$ |$\\mu_{1.}$ |\n", "|anxifree |$\\mu_{21}$ |$\\mu_{22}$ |$\\mu_{2.}$ |\n", "|joyzepam |$\\mu_{31}$ |$\\mu_{32}$ |$\\mu_{3.}$ |\n", "|total |$\\mu_{.1}$ |$\\mu_{.2}$ |$\\mu_{..}$ |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "fad9faf3", "metadata": {}, "source": [ "Now that we have this notation, it is straightforward to formulate and express some hypotheses. Let's suppose that the goal is to find out two things: firstly, does the choice of drug have any effect on mood, and secondly, does CBT have any effect on mood? These aren't the only hypotheses that we could formulate of course, and we'll see a really important example of a different kind of hypothesis in the section on [interactions](interactions), but these are the two simplest hypotheses to test, and so we'll start there. Consider the first test. If drug has no effect, then we would expect all of the row means to be identical, right? So that's our null hypothesis. On the other hand, if the drug does matter then we should expect these row means to be different. Formally, we write down our null and alternative hypotheses in terms of the *equality of marginal means*:" ] }, { "attachments": {}, "cell_type": "markdown", "id": "4d946da2", "metadata": {}, "source": [ "| | |\n", "|:-----------------------------|:------------------------------------------------------------|\n", "|Null hypothesis $H_0$: |row means are the same i.e. $\\mu_{1.} = \\mu_{2.} = \\mu_{3.}$ |\n", "|Alternative hypothesis $H_1$: |at least one row mean is different. |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "3fb68110", "metadata": {}, "source": [ "It's worth noting that these are *exactly* the same statistical hypotheses that we formed when we ran a one-way ANOVA on these data [way back when](anova). Back then I used the notation $\\mu_P$ to refer to the mean mood gain for the placebo group, with $\\mu_A$ and $\\mu_J$ corresponding to the group means for the two drugs, and the null hypothesis was $\\mu_P = \\mu_A = \\mu_J$. So we're actually talking about the same hypothesis: it's just that the more complicated ANOVA requires more careful notation due to the presence of multiple grouping variables, so we're now referring to this hypothesis as $\\mu_{1.} = \\mu_{2.} = \\mu_{3.}$. However, as we'll see shortly, although the hypothesis is identical, the test of that hypothesis is subtly different due to the fact that we're now acknowledging the existence of the second grouping variable.\n", "\n", "Speaking of the other grouping variable, you won't be surprised to discover that our second hypothesis test is formulated the same way. However, since we're talking about the psychological therapy rather than drugs, our null hypothesis now corresponds to the equality of the column means:\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "6a4d1435", "metadata": {}, "source": [ "| | |\n", "|:-----------------------------|:---------------------------------------------------------|\n", "|Null hypothesis $H_0$: |column means are the same, i.e., $\\mu_{.1} = \\mu_{.2}$ |\n", "|Alternative hypothesis $H_1$: |column means are different, i.e., $\\mu_{.1} \\neq \\mu_{.2}$|" ] }, { "attachments": {}, "cell_type": "markdown", "id": "472ed7cd", "metadata": {}, "source": [ "### Running the analysis in Python\n", "\n", "If the data you're trying to analyse correspond to a balanced factorial design, then running your analysis of variance is easy. To see how easy it is, let's start by reproducing the original analysis from [earlier](anova). In case you've forgotten, for that analysis we were using only a single factor (i.e., `drug`) as our between-subjects variable to predict our outcome (dependent) variable (i.e., `mood_gain`), and so this was what we did:" ] }, { "cell_type": "code", "execution_count": 7, "id": "4d667428", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
SourceSSDFMSFp-uncnp2
0drug3.4521.7318.610.00.71
1Within1.39150.09NaNNaNNaN
\n", "
" ], "text/plain": [ " Source SS DF MS F p-unc np2\n", "0 drug 3.45 2 1.73 18.61 0.0 0.71\n", "1 Within 1.39 15 0.09 NaN NaN NaN" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pingouin as pg\n", "\n", "model1 = pg.anova(dv='mood_gain', between='drug', data=df, detailed=True)\n", "round(model1, 2)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "709da32e", "metadata": {}, "source": [ "Note that this time around I've used the name `model1` as the label for my output variable, since I'm planning on creating quite a few other models too. To start with, suppose I'm also curious to find out if `therapy` has a relationship to `mood_gain`. In light of what we've seen from our discussion of [multiple regression](regression), you probably won't be surprised that all we have to do is extend the formula: in other words, if we specify `dv=mood_gain, between=['drug', 'therapy']` as our model, we'll probably get what we're after:" ] }, { "cell_type": "code", "execution_count": 8, "id": "c10f48ee", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
SourceSSDFMSFp-uncnp2
0drug3.453321.726731.71430.00000.8409
1therapy0.467210.46728.58160.01260.4170
2drug * therapy0.271120.13562.48980.12460.2933
3Residual0.6533120.0544NaNNaNNaN
\n", "
" ], "text/plain": [ " Source SS DF MS F p-unc np2\n", "0 drug 3.4533 2 1.7267 31.7143 0.0000 0.8409\n", "1 therapy 0.4672 1 0.4672 8.5816 0.0126 0.4170\n", "2 drug * therapy 0.2711 2 0.1356 2.4898 0.1246 0.2933\n", "3 Residual 0.6533 12 0.0544 NaN NaN NaN" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "model2 = pg.anova(dv='mood_gain', between=['drug', 'therapy'], data=df, detailed=True)\n", "round(model2, 4)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "085e641f", "metadata": {}, "source": [ "Most of this output is pretty simple to read too: the first row of the table reports a between-group sum of squares (SS) value associated with the `drug` factor, along with a corresponding between-group $df$ value. It also calculates a mean square value (MS), and $F$-statistic, an (uncorrected) $p$-value, and an estimate of the effect size (`np2`, that is, partial eta-squared). There is also a row corresponding to the `therapy` factor, and a row corresponding to the residuals (i.e., the within groups variation). \n", "\n", "Now, the third row is a little trickier, so let's just save that one for [later](interactions), shall we? (Spoiler: this is the interaction of `drug` and `therapy`, but we'll get there soon).\n", "\n", "Not only are all of the individual quantities pretty familiar, the relationships between these different quantities has remained unchanged: just like we saw with the original one-way ANOVA, note that the mean square value is calculated by dividing SS by the corresponding $df$. That is, it's still true that\n", "\n", "$$\n", "\\mbox{MS} = \\frac{\\mbox{SS}}{df}\n", "$$\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "4badab0d", "metadata": {}, "source": [ "regardless of whether we're talking about `drug`, `therapy` or the residuals. To see this, let's not worry about how the sums of squares values are calculated: instead, let's take it on faith that Python has calculated the SS values correctly, and try to verify that all the rest of the numbers make sense. First, note that for the `drug` factor, we divide $3.45$ by $2$, and end up with a mean square value of $1.73$. For the `therapy` factor, there's only 1 degree of freedom, so our calculations are even simpler: dividing $0.47$ (the SS value) by 1 gives us an answer of $0.47$ (the MS value). \n", "\n", "Turning to the $F$ statistics and the $p$ values, notice that we have two of each: one corresponding to the `drug` factor and the other corresponding to the `therapy` factor. Regardless of which one we're talking about, the $F$ statistic is calculated by dividing the mean square value associated with the factor by the mean square value associated with the residuals. If we use \"A\" as shorthand notation to refer to the first factor (factor A; in this case `drug`) and \"R\" as shorthand notation to refer to the residuals, then the $F$ statistic associated with factor A is denoted $F_A$, and is calculated as follows:\n", "\n", "$$\n", "F_{A} = \\frac{\\mbox{MS}_{A}}{\\mbox{MS}_{R}}\n", "$$\n", "\n", "Note that this use of \"R\" to refer to residuals is a bit awkward, since we also used the letter R to refer to the number of rows in the table, but I'm only going to use \"R\" to mean residuals in the context of SS$_R$ and MS$_R$, so hopefully this shouldn't be confusing. Anyway, to apply this formula to the `drugs` factor, we take the mean square of $1.73$ and divide it by the residual mean square value of $0.07$, which gives us an $F$-statistic of $26.15$. The corresponding calculation for the `therapy` variable would be to divide $0.47$ by $0.07$ which gives $7.08$ as the $F$-statistic. Not surprisingly, of course, these are the same values that R has reported in the ANOVA table above.\n", "\n", "The last part of the ANOVA table is the calculation of the $p$ values. Once again, there is nothing new here: for each of our two factors, what we're trying to do is test the null hypothesis that there is no relationship between the factor and the outcome variable (I'll be a bit more precise about this later on). To that end, we've (apparently) followed a similar strategy that we did in the one way ANOVA, and have calculated an $F$-statistic for each of these hypotheses. To convert these to $p$ values, all we need to do is note that the that the sampling distribution for the $F$ *statistic* under the null hypothesis (that the factor in question is irrelevant) is an $F$ *distribution*: and that two degrees of freedom values are those corresponding to the factor, and those corresponding to the residuals. For the `drug` factor we're talking about an $F$ distribution with 2 and 14 degrees of freedom (I'll discuss degrees of freedom in more detail later). In contrast, for the `therapy` factor sampling distribution is $F$ with 1 and 14 degrees of freedom." ] }, { "attachments": {}, "cell_type": "markdown", "id": "c324cf91", "metadata": {}, "source": [ "At this point, I hope you can see that the ANOVA table for this more complicated analysis corresponding to `model2` should be read in much the same way as the ANOVA table for the simpler analysis for `model1`. In short, it's telling us that the factorial ANOVA for our $3 \\times 2$ design found a significant effect of drug ($F_{2,12} = 31.71, p < .001$) as well as a significant effect of therapy ($F_{1,12} = 8.58, p = .01$). Or, to use the more technically correct terminology, we would say that there are two **_main effects_** of drug and therapy. Why are these \"main\" effects? Well, because there could also be an *interaction* between the effect of `drug` and the effect of `therapy`. We'll explore the concept of [interactions](interactions) below." ] }, { "attachments": {}, "cell_type": "markdown", "id": "89c02ba5", "metadata": {}, "source": [ "In the previous section I had two goals: firstly, to show you that the Python commands needed to do factorial ANOVA are pretty much the same ones that we used for a one way ANOVA. The only difference is that we add to the number of predictors in the `between` argument of `pingouin`'s `anova()` function. Secondly, I wanted to show you what the ANOVA table looks like in this case, so that you can see from the outset that the basic logic and structure behind factorial ANOVA is the same as that which underpins one way ANOVA. Try to hold onto that feeling. It's genuinely true, insofar as factorial ANOVA is built in more or less the same way as the simpler one-way ANOVA model. It's just that this feeling of familiarity starts to evaporate once you start digging into the details. Traditionally, this comforting sensation is replaced by an urge to murder the the authors of statistics textbooks.\n", "\n", "Okay, let's start looking at some of those details. The explanation that I gave in the last section illustrates the fact that the hypothesis tests for the main effects (of drug and therapy in this case) are $F$-tests, but what it doesn't do is show you how the sum of squares (SS) values are calculated. Nor does it tell you explicitly how to calculate degrees of freedom ($df$ values) though that's a simple thing by comparison. Let's assume for now that we have only two predictor variables, Factor A and Factor B. If we use $Y$ to refer to the outcome variable, then we would use $Y_{rci}$ to refer to the outcome associated with the $i$-th member of group $rc$ (i.e., level/row $r$ for Factor A and level/column $c$ for Factor B). Thus, if we use $\\bar{Y}$ to refer to a sample mean, we can use the same notation as before to refer to group means, marginal means and grand means: that is, $\\bar{Y}_{rc}$ is the sample mean associated with the $r$th level of Factor A and the $c$th level of Factor B, $\\bar{Y}_{r.}$ would be the marginal mean for the $r$th level of Factor A, $\\bar{Y}_{.c}$ would be the marginal mean for the $c$th level of Factor B, and $\\bar{Y}_{..}$ is the grand mean. In other words, our sample means can be organised into the same table as the population means. For our clinical trial data, that table looks like this:" ] }, { "attachments": {}, "cell_type": "markdown", "id": "d76b792a", "metadata": {}, "source": [ "| |no therapy |CBT |total |\n", "|:--------|:--------------|:--------------|:--------------|\n", "|placebo |$\\bar{Y}_{11}$ |$\\bar{Y}_{12}$ |$\\bar{Y}_{1.}$ |\n", "|anxifree |$\\bar{Y}_{21}$ |$\\bar{Y}_{22}$ |$\\bar{Y}_{2.}$ |\n", "|joyzepam |$\\bar{Y}_{31}$ |$\\bar{Y}_{32}$ |$\\bar{Y}_{3.}$ |\n", "|total |$\\bar{Y}_{.1}$ |$\\bar{Y}_{.2}$ |$\\bar{Y}_{..}$ |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "417af7c0", "metadata": {}, "source": [ "And if we look at the sample means that I showed earlier, we have $\\bar{Y}_{11} = 0.30$, $\\bar{Y}_{12} = 0.60$ etc. In our clinical trial example, the `drugs` factor has 3 levels and the `therapy` factor has 2 levels, and so what we're trying to run is a $3 \\times 2$ factorial ANOVA. However, we'll be a little more general and say that Factor A (the row factor) has $R$ levels and Factor B (the column factor) has $C$ levels, and so what we're runnning here is an $R \\times C$ factorial ANOVA.\n", "\n", "Now that we've got our notation straight, we can compute the sum of squares values for each of the two factors in a relatively familiar way. For Factor A, our between group sum of squares is calculated by assessing the extent to which the (row) marginal means $\\bar{Y}_{1.}$, $\\bar{Y}_{2.}$ etc, are different from the grand mean $\\bar{Y}_{..}$. We do this in the same way that we did for one-way ANOVA: calculate the sum of squared difference between the $\\bar{Y}_{i.}$ values and the $\\bar{Y}_{..}$ values. Specifically, if there are $N$ people in each group, then we calculate this:\n", "\n", "$$\n", "\\mbox{SS}_{A} = (N \\times C) \\sum_{r=1}^R \\left( \\bar{Y}_{r.} - \\bar{Y}_{..} \\right)^2\n", "$$" ] }, { "attachments": {}, "cell_type": "markdown", "id": "3ce726e7", "metadata": {}, "source": [ "As with one-way ANOVA, the most interesting [^translation] part of this formula is the $\\left( \\bar{Y}_{r.} - \\bar{Y}_{..} \\right)^2$ bit, which corresponds to the squared deviation associated with level $r$. All that this formula does is calculate this squared deviation for all $R$ levels of the factor, add them up, and then multiply the result by $N \\times C$. The reason for this last part is that there are multiple cells in our design that have level $r$ on Factor A: in fact, there are $C$ of them, one corresponding to each possible level of Factor B! For instance, in our toy example, there are *two* different cells in the design corresponding to the `anxifree` drug: one for people with `no.therapy`, and one for the `CBT` group. Not only that, within each of these cells there are $N$ observations. So, if we want to convert our SS value into a quantity that calculates the between-groups sum of squares on a \"per observation\" basis, we have to multiply by by $N \\times C$. The formula for factor B is of course the same thing, just with some subscripts shuffled around:\n", "\n", "$$\n", "\\mbox{SS}_{B} = (N \\times R) \\sum_{c=1}^C \\left( \\bar{Y}_{.c} - \\bar{Y}_{..} \\right)^2\n", "$$\n", "\n", "[^translation]: English translation: \"least tedious\"." ] }, { "attachments": {}, "cell_type": "markdown", "id": "7ddfb08f", "metadata": {}, "source": [ "Now that we have these formulas, we can check them against the `pingouin` output from the earlier section. First, notice that we calculated all the marginal means (i.e., row marginal means $\\bar{Y}_{r.}$ and column marginal means $\\bar{Y}_{.c}$) earlier using `.mean()`, and we also calculated the grand mean. Let's repeat those calculations, but this time we'll save the results to variables so that we can use them in subsequent calculations:" ] }, { "cell_type": "code", "execution_count": 9, "id": "7694d6c2", "metadata": {}, "outputs": [], "source": [ "drug_means = df.groupby(['drug'])['mood_gain'].mean()\n", "therapy_means = df.groupby(['therapy'])['mood_gain'].mean()\n", "grand_mean = df['mood_gain'].mean()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "d15dc88c", "metadata": {}, "source": [ "Okay, now let’s calculate the sum of squares associated with the main effect of drug. There are a total of $N=3$ people in each group, and $C=2$ different types of therapy. Or, to put it another way, there are $3 \\times 2 = 6$ people who received any particular drug. So our calculations are:" ] }, { "cell_type": "code", "execution_count": 10, "id": "a76c801d", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "3.4533" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SS_drug = (3*2) * sum((drug_means - grand_mean)**2)\n", "round(SS_drug,4)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "cbec528b", "metadata": {}, "source": [ "Not surprisingly, this is the same number that you get when you look up the SS value for the drugs factor in the ANOVA table that I presented earlier. We can repeat the same kind of calculation for the effect of therapy. Again there are $N=3$ people in each group, but since there are $R=3$ different drugs, this time around we note that there are $3 \\times 3 = 9$ people who received CBT, and an additional 9 people who received the placebo. So our calculation is now:" ] }, { "cell_type": "code", "execution_count": 11, "id": "03d9109b", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.4672" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SS_therapy = (3*3) * sum((therapy_means - grand_mean)**2)\n", "round(SS_therapy,4)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "dd389570", "metadata": {}, "source": [ "and we are, once again, unsurprised to see that our calculations are identical to the ANOVA output.\n", "\n", "So that's how you calculate the SS values for the two main effects. These SS values are analogous to the between-group sum of squares values that we calculated when doing [one-way ANOVA](anova). However, it's not a good idea to think of them as between-groups SS values anymore, just because we have two different grouping variables and it's easy to get confused. In order to construct an $F$ test, however, we also need to calculate the within-groups sum of squares. In keeping with the terminology that we used in the [regression chapter](regression)) and the terminology that R uses when printing out the ANOVA table, I'll start referring to the within-groups SS value as the *residual* sum of squares SS$_R$. \n", "\n", "The easiest way to think about the residual SS values in this context, I think, is to think of it as the leftover variation in the outcome variable after you take into account the differences in the marginal means (i.e., after you remove SS$_A$ and SS$_B$). What I mean by that is we can start by calculating the total sum of squares, which I'll label SS$_T$. The formula for this is pretty much the same as it was for one-way ANOVA: we take the difference between each observation $Y_{rci}$ and the grand mean $\\bar{Y}_{..}$, square the differences, and add them all up\n", "\n", "$$\n", "\\mbox{SS}_T = \\sum_{r=1}^R \\sum_{c=1}^C \\sum_{i=1}^N \\left( Y_{rci} - \\bar{Y}_{..}\\right)^2\n", "$$\n", "\n", "The \"triple summation\" here looks more complicated than it is. In the first two summations, we're summing across all levels of Factor A (i.e., over all possible rows $r$ in our table), across all levels of Factor B (i.e., all possible columns $c$). Each $rc$ combination corresponds to a single group, and each group contains $N$ people: so we have to sum across all those people (i.e., all $i$ values) too. In other words, all we're doing here is summing across all observations in the data set (i.e., all possible $rci$ combinations). \n", "\n", "At this point, we know the total variability of the outcome variable SS$_T$, and we know how much of that variability can be attributed to Factor A (SS$_A$) and how much of it can be attributed to Factor B (SS$_B$). The residual sum of squares is thus defined to be the variability in $Y$ that *can't* be attributed to either of our two factors. In other words:\n", "\n", "$$\n", "\\mbox{SS}_R = \\mbox{SS}_T - (\\mbox{SS}_A + \\mbox{SS}_B)\n", "$$\n", "\n", "Of course, there is a formula that you can use to calculate the residual SS directly, but I think that it makes more conceptual sense to think of it like this. The whole point of calling it a residual is that it's the leftover variation, and the formula above makes that clear. I should also note that, in keeping with the terminology used in the regression chapter, it is commonplace to refer to $\\mbox{SS}_A + \\mbox{SS}_B$ as the variance attributable to the \"ANOVA model\", denoted SS$_M$, and so we often say that the total sum of squares is equal to the model sum of squares plus the residual sum of squares. Later on in this chapter we'll see that this isn't just a surface similarity: ANOVA and regression are actually the same thing under the hood. \n", "\n", "In any case, it's probably worth taking a moment to check that we can calculate SS$_R$ using this formula, and verify that we do obtain the same answer that `pingouin` produces in its ANOVA table. The calculations are pretty straightforward. First we calculate the total sum of squares:" ] }, { "cell_type": "code", "execution_count": 12, "id": "47f120a1", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "4.845" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SS_tot = sum((df['mood_gain'] - grand_mean)**2)\n", "round(SS_tot,4)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "e84ff2b3", "metadata": {}, "source": [ "and then we use it to calculate the residual sum of squares:" ] }, { "cell_type": "code", "execution_count": 13, "id": "74129b90", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.9244" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "SS_res = SS_tot - (SS_drug + SS_therapy)\n", "round(SS_res,4)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a385c1b7", "metadata": {}, "source": [ "Yet again, we get the same answer. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "b16684a6", "metadata": {}, "source": [ "### What are our degrees of freedom?\n", "\n", "The degrees of freedom are calculated in much the same way as for one-way ANOVA. For any given factor, the degrees of freedom is equal to the number of levels minus 1 (i.e., $R-1$ for the row variable, Factor A, and $C-1$ for the column variable, Factor B). So, for the `drugs` factor we obtain $df = 2$, and for the `therapy` factor we obtain $df=1$. Later on on, when we discuss the interpretation of [ANOVA as a regression model](anovalm)), I'll give a clearer statement of how we arrive at this number, but for the moment we can use the simple definition of degrees of freedom, namely that the degrees of freedom equals the number of quantities that are observed, minus the number of constraints. So, for the `drugs` factor, we observe 3 separate group means, but these are constrained by 1 grand mean; and therefore the degrees of freedom is 2. For the residuals, the logic is similar, but not quite the same. The total number of observations in our experiment is 18. The constraints correspond to the 1 grand mean, the 2 additional group means that the `drug` factor introduces, and the 1 additional group mean that the the `therapy` factor introduces, and so our degrees of freedom is 14. As a formula, this is $N-1 -(R-1)-(C-1)$, which simplifies to $N-R-C+1$." ] }, { "attachments": {}, "cell_type": "markdown", "id": "f2345149", "metadata": {}, "source": [ "### Factorial ANOVA versus one-way ANOVAs\n", "\n", "Now that we've seen *how* a factorial ANOVA works, it's worth taking a moment to compare it to the results of the one way analyses, because this will give us a really good sense of *why* it's a good idea to run the factorial ANOVA. In the [chapter on 1-way ANOVA](anova), I ran a one-way ANOVA that looked to see if there are any differences between drugs, and a second one-way ANOVA to see if there were any differences between therapies. As we saw in [previously](factanovahyp), the null and alternative hypotheses tested by the one-way ANOVAs are in fact identical to the hypotheses tested by the factorial ANOVA. Looking even more carefully at the ANOVA tables, we can see that the sum of squares associated with the factors are identical in the two different analyses (3.45 for `drug` and 0.92 for `therapy`), as are the degrees of freedom (2 for `drug`, 1 for `therapy`). But they don't give the same answers! Most notably, when we ran the [one-way ANOVA for `therapy`](anovaandt) we didn't find a significant effect (the $p$-value was 0.21). However, when we look at the main effect of `therapy` within the context of the two-way ANOVA, we do get a significant effect ($p=.019$). The two analyses are clearly not the same.\n", "\n", "Why does that happen? The answer lies in understanding how the *residuals* are calculated. Recall that the whole idea behind an $F$-test is to compare the variability that can be attributed to a particular factor with the variability that cannot be accounted for (the residuals). If you run a one-way ANOVA for `therapy`, and therefore ignore the effect of `drug`, the ANOVA will end up dumping all of the drug-induced variability into the residuals! This has the effect of making the data look more noisy than they really are, and the effect of `therapy` which is correctly found to be significant in the two-way ANOVA now becomes non-significant. If we ignore something that actually matters (e.g., `drug`) when trying to assess the contribution of something else (e.g., `therapy`) then our analysis will be distorted. Of course, it's perfectly okay to ignore variables that are genuinely irrelevant to the phenomenon of interest: if we had recorded the colour of the walls, and that turned out to be non-significant in a three-way ANOVA (i.e. `pg.anova(dv='mood_gain', between=['drug', 'therapy', 'wall_color']`), it would be perfectly okay to disregard it and just report the simpler two-way ANOVA that doesn't include this irrelevant factor. What you shouldn't do is drop variables that actually make a difference! " ] }, { "attachments": {}, "cell_type": "markdown", "id": "8a676948", "metadata": {}, "source": [ "### What kinds of outcomes does this analysis capture?\n", "\n", "The ANOVA model that we've been talking about so far covers a range of different patterns that we might observe in our data. For instance, in a two-way ANOVA design, there are four possibilities: (a) only Factor A matters, (b) only Factor B matters, (c) both A and B matter, and (d) neither A nor B matters. An example of each of these four possibilities is plotted in {numref}`fig-anovas-sans-interaction`. \n" ] }, { "cell_type": "code", "execution_count": 14, "id": "867640fb", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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", 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" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "anovas-sans-interaction-fig" } }, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "from matplotlib import pyplot as plt\n", "\n", "levels = ['Level 1', 'Level 1', 'Level 2', 'Level 2']\n", "factors = ['Factor B, Level 1', 'Factor B, Level 2', 'Factor B, Level 1', 'Factor B, Level 2']\n", "panel1 = [1,1.2,2,2.2]\n", "panel2 = [1.2, 1.9, 1.2, 1.9]\n", "panel3 = [0.6, 1.4, 1.5, 2.4]\n", "panel4 = [1.5, 1.6, 1.5, 1.6]\n", "\n", "d = pd.DataFrame({'Factor': factors,\n", " 'Factor A': levels,\n", " 'Panel 1': panel1,\n", " 'Panel 2': panel2,\n", " 'Panel 3': panel3,\n", " 'Panel 4': panel4})\n", "\n", "fig, axes = plt.subplots(2,2)\n", "\n", "\n", "p1 = sns.pointplot(data = d, x = 'Factor A', y = 'Panel 1', hue= 'Factor', errorbar=('ci', False), ax=axes[0,0])\n", "p2 = sns.pointplot(data = d, x = 'Factor A', y = 'Panel 2', hue= 'Factor', errorbar=('ci', False), ax=axes[0,1])\n", "p3 = sns.pointplot(data = d, x = 'Factor A', y = 'Panel 3', hue= 'Factor', errorbar=('ci', False), ax=axes[1,0])\n", "p4 = sns.pointplot(data = d, x = 'Factor A', y = 'Panel 4', hue= 'Factor', errorbar=('ci', False), ax=axes[1,1])\n", "\n", "ps = [p1, p2, p3, p4]\n", "panelID = ['A', 'B', 'C', 'D']\n", "titles = ['Only Factor A has an effect', 'Only Factor B has an effect', \n", " 'Both A and B have an effect', 'Neither A nor B has an effect']\n", "\n", "for n, p in enumerate(ps):\n", " p.set_ylim(0,3)\n", " p.set_ylabel('')\n", " p.set_xlabel('')\n", " p.get_legend().remove()\n", " p.text(0.1, 1, titles[n], horizontalalignment='left', verticalalignment='top', transform=p.transAxes)\n", " p.text(0, 1.1, panelID[n], horizontalalignment='left', verticalalignment='top', transform=p.transAxes)\n", " sns.despine()\n", "\n", "p1.set_xticklabels('')\n", "p2.set_xticklabels('')\n", "\n", "p3.set_xlabel('Factor A')\n", "p4.set_xlabel('Factor A')\n", "\n", "handles, labels = p1.get_legend_handles_labels()\n", "fig.legend(handles, ['Factor B, Level 1', 'Factor B, Level 2'], loc='upper center', ncol=2)\n", "\n", "\n", "# Plot figure in book, with caption\n", "from myst_nb import glue\n", "plt.close(fig)\n", "glue(\"anovas-sans-interaction-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "1486c14a", "metadata": {}, "source": [ " ```{glue:figure} anovas-sans-interaction-fig\n", ":figwidth: 600px\n", ":name: fig-anovas-sans-interaction\n", "\n", "The four different outcomes for a 2 x 2 ANOVA when no interactions are present. In Panel A we see a main effect of Factor A, and no effect of Factor B. Panel B shows a main effect of Factor B but no effect of Factor A. Panel C shows main effects of both Factor A and Factor B. Finally, Panel D shows no effect of either factor.\n", "\n", "```\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "bc0c6c1b", "metadata": {}, "source": [ "(interactions)=\n", "## Balanced designs, main effects and interactions\n", "\n", "The four patterns of data shown in {numref}`fig-anovas-sans-interaction` are all quite realistic: there are a great many data sets that produce exactly those patterns. However, they are not the whole story, and the ANOVA model that we have been talking about up to this point is not sufficient to fully account for a table of group means. Why not? Well, so far we have the ability to talk about the idea that drugs can influence mood, and therapy can influence mood, but no way of talking about the possibility of an **_interaction_** between the two. An interaction between A and B is said to occur whenever the effect of Factor A is *different*, depending on which level of Factor B we're talking about. Several examples of an interaction effect with the context of a 2 x 2 ANOVA are shown in {numref}`fig-anovas-avec-interaction`. To give a more concrete example, suppose that the operation of Anxifree and Joyzepam is governed quite different physiological mechanisms, and one consequence of this is that while Joyzepam has more or less the same effect on mood regardless of whether one is in therapy, Anxifree is actually much more effective when administered in conjunction with CBT. The ANOVA that we developed in the previous section does not capture this idea. To get some idea of whether an interaction is actually happening here, it helps to plot the various group means." ] }, { "cell_type": "code", "execution_count": 15, "id": "f30fd859", "metadata": { "tags": [ "hide-input" ] }, "outputs": [ { "data": { "application/papermill.record/image/png": 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", "application/papermill.record/text/plain": "
" }, "metadata": { "scrapbook": { "mime_prefix": "application/papermill.record/", "name": "anovas-avec-interaction-fig" } }, "output_type": "display_data" } ], "source": [ "import seaborn as sns\n", "from matplotlib import pyplot as plt\n", "\n", "levels = ['Level 1', 'Level 1', 'Level 2', 'Level 2']\n", "factors = ['Factor B, Level 1', 'Factor B, Level 2', 'Factor B, Level 1', 'Factor B, Level 2']\n", "panel1 = [1,2,2,1]\n", "panel2 = [1, 1.1, 1.4, 2]\n", "panel3 = [1, 1.1, 1, 2.4]\n", "panel4 = [1, 1.4, 1, 2.3]\n", "\n", "d = pd.DataFrame({'Factor': factors,\n", " 'Factor A': levels,\n", " 'Panel 1': panel1,\n", " 'Panel 2': panel2,\n", " 'Panel 3': panel3,\n", " 'Panel 4': panel4})\n", "\n", "fig, axes = plt.subplots(2,2)\n", "\n", "\n", "p1 = sns.pointplot(data = d, x = 'Factor A', y = 'Panel 1', hue= 'Factor', errorbar=('ci', False), ax=axes[0,0])\n", "p2 = sns.pointplot(data = d, x = 'Factor A', y = 'Panel 2', hue= 'Factor', errorbar=('ci', False), ax=axes[0,1])\n", "p3 = sns.pointplot(data = d, x = 'Factor A', y = 'Panel 3', hue= 'Factor', errorbar=('ci', False), ax=axes[1,0])\n", "p4 = sns.pointplot(data = d, x = 'Factor A', y = 'Panel 4', hue= 'Factor', errorbar=('ci', False), ax=axes[1,1])\n", "\n", "ps = [p1, p2, p3, p4]\n", "panelID = ['A', 'B', 'C', 'D']\n", "titles = ['Crossover interaction', 'Effect for one level of Factor A', \n", " 'One cell is different', 'Effect for one level of Factor B']\n", "\n", "for n, p in enumerate(ps):\n", " p.set_ylim(0,3)\n", " p.set_ylabel('')\n", " p.set_xlabel('')\n", " p.get_legend().remove()\n", " p.text(0.1, 1, titles[n], horizontalalignment='left', verticalalignment='top', transform=p.transAxes)\n", " p.text(0, 1.1, panelID[n], horizontalalignment='left', verticalalignment='top', transform=p.transAxes)\n", " sns.despine()\n", "\n", "p1.set_xticklabels('')\n", "p2.set_xticklabels('')\n", "\n", "p3.set_xlabel('Factor A')\n", "p4.set_xlabel('Factor A')\n", "\n", "handles, labels = p1.get_legend_handles_labels()\n", "fig.legend(handles, ['Factor B, Level 1', 'Factor B, Level 2'], loc='upper center', ncol=2)\n", "\n", "\n", "# Plot figure in book, with caption\n", "from myst_nb import glue\n", "plt.close(fig)\n", "glue(\"anovas-avec-interaction-fig\", fig, display=False)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "8455d86f", "metadata": {}, "source": [ " ```{glue:figure} anovas-avec-interaction-fig\n", ":figwidth: 600px\n", ":name: fig-anovas-avec-interaction\n", "\n", "Qualitatively different interactions for a 2 x 2 ANOVA\n", "\n", "```\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "26aaeed1", "metadata": {}, "source": [ "To give a more concrete example, suppose that the operation of Anxifree and Joyzepam is governed quite different physiological mechanisms, and one consequence of this is that while Joyzepam has more or less the same effect on mood regardless of whether one is in therapy, Anxifree is actually much more effective when administered in conjunction with CBT. The ANOVA that we developed in the previous section does not capture this idea. To get some idea of whether an interaction is actually happening here, it helps to plot the various group means. We can do this easily with `seaborn`:" ] }, { "cell_type": "code", "execution_count": 16, "id": "945c56fe", "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "sns.pointplot(data=df, x='drug', y='mood_gain', hue='therapy')\n", "sns.despine()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "6101b34f", "metadata": {}, "source": [ "Our main concern relates to the fact that the two lines aren't parallel. The effect of CBT (difference between solid line and dotted line) when the drug is Joyzepam (right side) appears to be near zero, even smaller than the effect of CBT when a placebo is used (left side). However, when Anxifree is administered, the effect of CBT is larger than the placebo (middle). Is this effect real, or is this just random variation due to chance? Our original ANOVA cannot answer this question, because we make no allowances for the idea that interactions even exist! In this section, we'll fix this problem.\n", "\n", "\n", "### What exactly *is* an interaction effect?\n", "\n", "The key idea that we're going to introduce in this section is that of an interaction effect. `pingouin` has already been calculating the for us, but to keep things simple, I have just ignored them. Now the time has come to look at that _other_ line in the ANOVA table, the one that about \"drug*therapy\".\n", "\n", "Intuitively, the idea behind an interaction effect is fairly simple: it just means that the effect of Factor A is different, depending on which level of Factor B we're talking about. But what does that actually mean in terms of our data? {numref}`fig-anovas-avec-interaction` depicts several different patterns that, although quite different to each other, would all count as an interaction effect. So it's not entirely straightforward to translate this qualitative idea into something mathematical that a statistician can work with. As a consequence, the way that the idea of an interaction effect is formalised in terms of null and alternative hypotheses is slightly difficult, and I'm guessing that a lot of readers of this book probably won't be all that interested. Even so, I'll try to give the basic idea here.\n", "\n", "To start with, we need to be a little more explicit about our main effects. Consider the main effect of Factor A (`drug` in our running example). We originally formulated this in terms of the null hypothesis that the two marginal means $\\mu_{r.}$ are all equal to each other. Obviously, if all of these are equal to each other, then they must also be equal to the grand mean $\\mu_{..}$ as well, right? So what we can do is define the *effect* of Factor A at level $r$ to be equal to the difference between the marginal mean $\\mu_{r.}$ and the grand mean $\\mu_{..}$. \n", "\n", "Let's denote this effect by $\\alpha_r$, and note that\n", "\n", "$$\n", "\\alpha_r = \\mu_{r.} - \\mu_{..} \n", "$$\n", "\n", "Now, by definition all of the $\\alpha_r$ values must sum to zero, for the same reason that the average of the marginal means $\\mu_{r.}$ must be the grand mean $\\mu_{..}$. We can similarly define the effect of Factor B at level $i$ to be the difference between the column marginal mean $\\mu_{.c}$ and the grand mean $\\mu_{..}$\n", "\n", "$$\n", "\\beta_c = \\mu_{.c} - \\mu_{..}\n", "$$\n", "\n", "and once again, these $\\beta_c$ values must sum to zero. The reason that statisticians sometimes like to talk about the main effects in terms of these $\\alpha_r$ and $\\beta_c$ values is that it allows them to be precise about what it means to say that there is no interaction effect. If there is no interaction at all, then these $\\alpha_r$ and $\\beta_c$ values will perfectly describe the group means $\\mu_{rc}$. Specifically, it means that\n", "\n", "$$\n", "\\mu_{rc} = \\mu_{..} + \\alpha_r + \\beta_c \n", "$$\n", "\n", "That is, there's nothing *special* about the group means that you couldn't predict perfectly by knowing all the marginal means. And that's our null hypothesis, right there. The alternative hypothesis is that\n", "\n", "$$\n", "\\mu_{rc} \\neq \\mu_{..} + \\alpha_r + \\beta_c \n", "$$\n", "\n", "for at least one group $rc$ in our table. However, statisticians often like to write this slightly differently. They'll usually define the specific interaction associated with group $rc$ to be some number, awkwardly referred to as $(\\alpha\\beta)_{rc}$, and then they will say that the alternative hypothesis is that \n", "\n", "$$\n", "\\mu_{rc} = \\mu_{..} + \\alpha_r + \\beta_c + (\\alpha\\beta)_{rc}\n", "$$\n", "\n", "where $(\\alpha\\beta)_{rc}$ is non-zero for at least one group. This notation is kind of ugly to look at, but it is handy as we'll see in the next section when discussing how to calculate the sum of squares.\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "3b137159", "metadata": {}, "source": [ "### Calculating sums of squares for the interaction\n", "\n", "How should we calculate the sum of squares for the interaction terms, SS$_{A:B}$? \n", "\n", "Well, first off, it helps to notice how the previous section defined the interaction effect in terms of the extent to which the actual group means differ from what you'd expect by just looking at the marginal means. Of course, all of those formulas refer to population parameters rather than sample statistics, so we don't actually know what they are. However, we can estimate them by using sample means in place of population means. So for Factor A, a good way to estimate the main effect at level $r$ as the difference between the *sample* marginal mean $\\bar{Y}_{rc}$ and the sample grand mean $\\bar{Y}_{..}$. That is, we would use this as our estimate of the effect:\n", "\n", "$$\n", "\\hat{\\alpha}_r = \\bar{Y}_{r.} - \\bar{Y}_{..}\n", "$$\n", "\n", "Similarly, our estimate of the main effect of Factor B at level $c$ can be defined as follows:\n", "\n", "$$\n", "\\hat{\\beta}_c = \\bar{Y}_{.c} - \\bar{Y}_{..}\n", "$$\n", "\n", "Now, if you go back to the formulas that I used to describe the SS values for the two main effects, you'll notice that these effect terms are exactly the quantities that we were squaring and summing! So what's the analog of this for interaction terms? The answer to this can be found by first rearranging the formula for the group means $\\mu_{rc}$ under the alternative hypothesis, so that we get this:\n", "\n", "$$\n", "\\begin{align}\n", "(\\alpha \\beta)_{rc} &= \\mu_{rc} - \\mu_{..} - \\alpha_r - \\beta_c \\\\\n", " &= \\mu_{rc} - \\mu_{..} - (\\mu_{r.} - \\mu_{..}) - (\\mu_{.c} - \\mu_{..}) \\\\\n", " & = \\mu_{rc} - \\mu_{r.} - \\mu_{.c} + \\mu_{..}\n", "\\end{align}\n", "$$\n", "\n", "\n", "\n", "So, once again, if we substitute our sample statistics in place of the population means, we get the following as our estimate of the interaction effect for group $rc$, which is\n", "\n", "$$\n", "\\hat{(\\alpha\\beta)}_{rc} = \\bar{Y}_{rc} - \\bar{Y}_{r.} - \\bar{Y}_{.c} + \\bar{Y}_{..}\n", "$$\n", "\n", "Now all we have to do is sum all of these estimates across all $R$ levels of Factor A and all $C$ levels of Factor B, and we obtain the following formula for the sum of squares associated with the interaction as a whole:\n", "\n", "$$\n", "\\mbox{SS}_{A:B} = N \\sum_{r=1}^R \\sum_{c=1}^C \\left( \\bar{Y}_{rc} - \\bar{Y}_{r.} - \\bar{Y}_{.c} + \\bar{Y}_{..} \\right)^2\n", "$$\n", "\n", "where, we multiply by $N$ because there are $N$ observations in each of the groups, and we want our SS values to reflect the variation among *observations* accounted for by the interaction, not the variation among groups. \n", "\n", "\n", "Now that we have a formula for calculating SS$_{A:B}$, it's important to recognise that the interaction term is part of the model (of course), so the total sum of squares associated with the model, SS$_M$ is now equal to the sum of the three relevant SS values, $\\mbox{SS}_A + \\mbox{SS}_B + \\mbox{SS}_{A:B}$. The residual sum of squares $\\mbox{SS}_R$ is still defined as the leftover variation, namely $\\mbox{SS}_T - \\mbox{SS}_M$, but now that we have the interaction term this becomes\n", "\n", "$$\n", "\\mbox{SS}_R = \\mbox{SS}_T - (\\mbox{SS}_A + \\mbox{SS}_B + \\mbox{SS}_{A:B})\n", "$$ \n", "\n", "\n", "As a consequence, the residual sum of squares SS$_R$ will be smaller than in our original ANOVA that didn't include interactions.\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "1f674b2f", "metadata": {}, "source": [ "### Degrees of freedom for the interaction\n", "\n", "Calculating the degrees of freedom for the interaction is, once again, slightly trickier than the corresponding calculation for the main effects. To start with, let's think about the ANOVA model as a whole. Once we include interaction effects in the model, we're allowing every single group to have a unique mean, $\\mu_{rc}$. For an $R \\times C$ factorial ANOVA, this means that there are $R \\times C$ quantities of interest in the model, and only the one constraint: all of the group means need to average out to the grand mean. So the model as a whole needs to have $(R\\times C) - 1$ degrees of freedom. But the main effect of Factor A has $R-1$ degrees of freedom, and the main effect of Factor B has $C-1$ degrees of freedom. Which means that the degrees of freedom associated with the interaction is \n", "\n", "$$\n", "\\begin{align}\n", "df_{A:B} &= (R\\times C - 1) - (R - 1) - (C -1 ) \\\\\n", " &= RC - R - C + 1 \\\\\n", " &= (R-1)(C-1)\n", "\\end{align}\n", "$$\n", "\n", "which is just the product of the degrees of freedom associated with the row factor and the column factor.\n", "\n", "What about the residual degrees of freedom? Because we've added interaction terms, which absorb some degrees of freedom, there are fewer residual degrees of freedom left over. Specifically, note that if the model with interaction has a total of $(R\\times C) - 1$, and there are $N$ observations in your data set that are constrained to satisfy 1 grand mean, your residual degrees of freedom now become $N-(R \\times C)-1+1$, or just $N-(R \\times C)$." ] }, { "attachments": {}, "cell_type": "markdown", "id": "6eee58bc", "metadata": {}, "source": [ "### Interpreting the results\n", "\n", "Now we are in a position to understand all the rows in the ANOVA table that we saw earlier. Even if you skimmed lightly over the math in the previous section (and you would be forgiven if you did), hopefully you will now have some intuition for the differences between main effects and interaction effects. But what does it all mean, in the end?\n", "\n", "Let's quickly remind ourselves of the results of our factorial ANOVA:" ] }, { "cell_type": "code", "execution_count": 17, "id": "c22ad370", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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SourceSSDFMSFp-uncnp2
0drug3.45321.72731.7140.0000.841
1therapy0.46710.4678.5820.0130.417
2drug * therapy0.27120.1362.4900.1250.293
3Residual0.653120.054NaNNaNNaN
\n", "
" ], "text/plain": [ " Source SS DF MS F p-unc np2\n", "0 drug 3.453 2 1.727 31.714 0.000 0.841\n", "1 therapy 0.467 1 0.467 8.582 0.013 0.417\n", "2 drug * therapy 0.271 2 0.136 2.490 0.125 0.293\n", "3 Residual 0.653 12 0.054 NaN NaN NaN" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "model2 = pg.anova(dv='mood_gain', between=['drug', 'therapy'], data=df, detailed=True)\n", "round(model2, 3)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "73ca4349", "metadata": {}, "source": [ "We can now see that while we do have a significant main effect of drug ($F_{2,12} = 31.7, p <.001$) and therapy type ($F_{1,12} = 8.6, p=.013$), there is no significant interaction between the two ($F_{2,12} = 2.5, p = 0.125$)." ] }, { "attachments": {}, "cell_type": "markdown", "id": "836263e3", "metadata": {}, "source": [ "There's a couple of very important things to consider when interpreting the results of factorial ANOVA. Firstly, there's the same issue that we had with one-way ANOVA, which is that if you obtain a significant main effect of (say) `drug`, it doesn't tell you anything about which drugs are different to one another. To find that out, you need to run additional analyses. We'll talk about some analyses that you can run in the sections on [contrasts](contrasts) and [post-hoc tests](posthoc2). The same is true for interaction effects: knowing that there's a significant interaction doesn't tell you anything about what kind of interaction exists. Again, you'll need to run additional analyses. \n", "\n", "Secondly, there's a very peculiar interpretation issue that arises when you obtain a significant interaction effect but no corresponding main effect. This happens sometimes. For instance, in the crossover interaction shown in {numref}`fig-anovas-avec-interaction`), this is exactly what you'd find: in this case, neither of the main effects would be significant, but the interaction effect would be. This is a difficult situation to interpret, and people often get a bit confused about it. The general advice that statisticians like to give in this situation is that you shouldn't pay much attention to the main effects when an interaction is present. The reason they say this is that, although the tests of the main effects are perfectly valid from a mathematical point of view, when there is a significant interaction effect, the main effects rarely test interesting hypotheses. [Recall](factanovahyp) that the null hypothesis for a main effect is that the *marginal means* are equal to each other, and that a marginal mean is formed by averaging across several different groups. But if you have a significant interaction effect, then you *know* that the groups that comprise the marginal mean aren't homogeneous, so it's not really obvious why you would even care about those marginal means. \n", "\n", "Here's what I mean. Again, let's stick with a clinical example. Suppose that we had a $2 \\times 2$ design comparing two different treatments for phobias (e.g., systematic desensitisation vs flooding), and two different anxiety reducing drugs (e.g., Anxifree vs Joyzepam). Now suppose what we found was that Anxifree had no effect when desensitisation was the treatment, and Joyzepam had no effect when flooding was the treatment. But both were pretty effective for the other treatment. This is a classic crossover interaction, and what we'd find when running the ANOVA is that there is no main effect of drug, but a significant interaction. Now, what does it actually *mean* to say that there's no main effect? Well, it means that, if we average over the two different psychological treatments, then the *average* effect of Anxifree and Joyzepam is the same. But why would anyone care about that? When treating someone for phobias, it is never the case that a person can be treated using an \"average\" of flooding and desensitisation: that doesn't make a lot of sense. You either get one or the other. For one treatment, one drug is effective; and for the other treatment, the other drug is effective. The interaction is the important thing; the main effect is kind of irrelevant. \n", "\n", "This sort of thing happens a lot: the main effect are tests of marginal means, and when an interaction is present we often find ourselves not being terribly interested in marginal means, because they imply averaging over things that the interaction tells us shouldn't be averaged! Of course, it's not always the case that a main effect is meaningless when an interaction is present. Often you can get a big main effect and a very small interaction, in which case you can still say things like \"drug A is generally more effective than drug B\" (because there was a big effect of drug), but you'd need to modify it a bit by adding that \"the difference in effectiveness was different for different psychological treatments\". In any case, the main point here is that whenever you get a significant interaction you should stop and *think* about what the main effect actually means in this context. Don't automatically assume that the main effect is interesting. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "07bd5740", "metadata": {}, "source": [ "### Effect sizes\n", "\n", "The effect size calculations for a factorial ANOVA is pretty similar to [those used in one way ANOVA](etasquared). Specifically, we can use $\\eta^2$ (eta-squared) as simple way to measure how big the overall effect is for any particular term. As before, $\\eta^2$ is defined by dividing the sum of squares associated with that term by the total sum of squares. For instance, to determine the size of the main effect of Factor A, we would use the following formula\n", "\n", "$$\n", "\\eta_A^2 = \\frac{\\mbox{SS}_{A}}{\\mbox{SS}_{T}}\n", "$$\n", "\n", "As before, this can be interpreted in much the same way as $R^2$ in regression.[^R] It tells you the proportion of variance in the outcome variable that can be accounted for by the main effect of Factor A. It is therefore a number that ranges from 0 (no effect at all) to 1 (accounts for *all* of the variability in the outcome). Moreover, the sum of all the $\\eta^2$ values, taken across all the terms in the model, will sum to the the total $R^2$ for the ANOVA model. If, for instance, the ANOVA model fits perfectly (i.e., there is no within-groups variability at all!), the $\\eta^2$ values will sum to 1. Of course, that rarely if ever happens in real life.\n", "\n", "However, when doing a factorial ANOVA, there is a second measure of effect size that people like to report, known as partial $\\eta^2$. This is the effect size measure that `pingouin` gives you by default. The idea behind partial $\\eta^2$ (which is sometimes denoted $_p\\eta^2$ or $\\eta^2_p$) is that, when measuring the effect size for a particular term (say, the main effect of Factor A), you want to deliberately ignore the other effects in the model (e.g., the main effect of Factor B). That is, you would pretend that the effect of all these other terms is zero, and then calculate what the $\\eta^2$ value would have been. This is actually pretty easy to calculate. All you have to do is remove the sum of squares associated with the other terms from the denominator. In other words, if you want the partial $\\eta^2$ for the main effect of Factor A, the denominator is just the sum of the SS values for Factor A and the residuals:\n", "\n", "$$\n", "\\mbox{partial } \\eta^2_A = \\frac{\\mbox{SS}_{A}}{\\mbox{SS}_{A} + \\mbox{SS}_{R}}\n", "$$\n", "\n", "This will always give you a larger number than $\\eta^2$, which the cynic in me suspects accounts for the popularity of partial $\\eta^2$. And once again you get a number between 0 and 1, where 0 represents no effect. However, it's slightly trickier to interpret what a large partial $\\eta^2$ value means. In particular, you can't actually compare the partial $\\eta^2$ values across terms! Suppose, for instance, there is no within-groups variability at all: if so, SS$_R = 0$. What that means is that *every* term has a partial $\\eta^2$ value of 1. But that doesn't mean that all terms in your model are equally important, or indeed that they are equally large. All it mean is that all terms in your model have effect sizes that are large *relative to the residual variation*. It is not comparable across terms.\n", "\n", "[^R]: This chapter seems to be setting a new record for the number of different things that the letter R can stand for: so far we have R referring to the number of rows in our table of means, the residuals in the model, and now the correlation coefficient in a regression. Sorry: we clearly don't have enough letters in the alphabet. However, I've tried pretty hard to be clear on which thing R is referring to in each case.\n", "\n", "To see what I mean by this, it's useful to see a concrete example. We can get `pingouin` to report $\\eta^2$ instead of partial $\\eta^2$ by specifying this in the `effsize` argument when we run our ANOVA:" ] }, { "cell_type": "code", "execution_count": 18, "id": "0e24f51c", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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SourceSSDFMSFp-uncn2
0drug3.45321.72731.7140.0000.713
1therapy0.46710.4678.5820.0130.096
2drug * therapy0.27120.1362.4900.1250.056
3Residual0.653120.054NaNNaNNaN
\n", "
" ], "text/plain": [ " Source SS DF MS F p-unc n2\n", "0 drug 3.453 2 1.727 31.714 0.000 0.713\n", "1 therapy 0.467 1 0.467 8.582 0.013 0.096\n", "2 drug * therapy 0.271 2 0.136 2.490 0.125 0.056\n", "3 Residual 0.653 12 0.054 NaN NaN NaN" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "model2 = pg.anova(dv='mood_gain', \n", " between=['drug', 'therapy'], \n", " data=df, \n", " detailed=True, \n", " effsize='n2')\n", "round(model2, 3)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "e2641e95", "metadata": {}, "source": [ "Looking at the $\\eta^2$ values first, we see that `drug` accounts for 71.3\\% of the variance (i.e. $\\eta^2 = 0.713$) in `mood_gain`, whereas `therapy` only accounts for 9.6\\%. This leaves a total of 19.1\\% of the variation unaccounted for (i.e., the residuals constitute 19.1\\% of the variation in the outcome). Overall, this implies that we have a very large effect[^bigeffect] of `drug` and a modest effect of `therapy`. \n", "\n", "[^bigeffect]: Implausibly large, I would think: the artificiality of this data set is really starting to show!\n", "\n", "Now let's look at the partial $\\eta^2$ values. \n" ] }, { "cell_type": "code", "execution_count": 19, "id": "0de77fdd", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
SourceSSDFMSFp-uncnp2
0drug3.45321.72731.7140.0000.841
1therapy0.46710.4678.5820.0130.417
2drug * therapy0.27120.1362.4900.1250.293
3Residual0.653120.054NaNNaNNaN
\n", "
" ], "text/plain": [ " Source SS DF MS F p-unc np2\n", "0 drug 3.453 2 1.727 31.714 0.000 0.841\n", "1 therapy 0.467 1 0.467 8.582 0.013 0.417\n", "2 drug * therapy 0.271 2 0.136 2.490 0.125 0.293\n", "3 Residual 0.653 12 0.054 NaN NaN NaN" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "model2 = pg.anova(dv='mood_gain', \n", " between=['drug', 'therapy'], \n", " data=df, \n", " detailed=True, \n", " effsize='np2')\n", "round(model2, 3)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "ca624119", "metadata": {}, "source": [ "Because the effect of `therapy` isn't all that large, controlling for it doesn't make much of a difference, so the partial $\\eta^2$ for `drug` doesn't increase very much, and we obtain a value of $_p\\eta^2 = 0.789$). In contrast, because the effect of `drug` was very large, controlling for it makes a big difference, and so when we calculate the partial $\\eta^2$ for `therapy` you can see that it rises to $_p\\eta^2 = 0.336$. The question that we have to ask ourselves is, what do these partial $\\eta^2$ values actually *mean*? The way I generally interpret the partial $\\eta^2$ for the main effect of Factor A is to interpret it as a statement about a hypothetical experiment in which *only* Factor A was being varied. So, even though in *this* experiment we varied both A and B, we can easily imagine an experiment in which only Factor A was varied: the partial $\\eta^2$ statistic tells you how much of the variance in the outcome variable you would expect to see accounted for in that experiment. However, it should be noted that this interpretation -- like many things associated with main effects -- doesn't make a lot of sense when there is a large and significant interaction effect. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "45fbeceb", "metadata": {}, "source": [ "(factorialanovaassumptions)=\n", "## Assumption checking\n", "\n", "As with one-way ANOVA, the key assumptions of factorial ANOVA are homogeneity of variance (all groups have the same standard deviation), normality of the residuals, and independence of the observations. The first two are things we can test for. The third is something that you need to assess yourself by asking if there are any special relationships between different observations. What about homogeneity of variance and normality of the residuals? As it turns out, these are pretty easy to check: it's no different to the checks we did for a one-way ANOVA.\n", "\n", "### Levene test for homogeneity of variance\n", "\n", "To test whether the groups have the same variance, we can use the Levene test. The theory behind the Levene test has [already been discussed](levene), so I won't discuss it again. The funny thing is, though, that while pacakges exist to perform a Levene test on factorial ANOVA models in e.g. R, I have not been able to find a package that does the same in Python. As of the date of writing this, I can only find examples of Levene's tests being performed on one-way models in Python. However, as far as I can tell, all that one needs to do is to create a new column in the dataframe that represents the interaction of the two factors, and run the Levene's test on that. This seems to give the same result as e.g. the R function `levene()` run on a two-way model with interactions, so I offer it here as my solution to this problem:\n", "\n" ] }, { "cell_type": "code", "execution_count": 20, "id": "352bfe32", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Wpvalequal_var
levene0.095450.99123True
\n", "
" ], "text/plain": [ " W pval equal_var\n", "levene 0.09545 0.99123 True" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\n", "df['interaction'] = df['drug']+df['therapy']\n", "\n", "pg.homoscedasticity(data=df, \n", " dv='mood_gain', \n", " group='interaction').round(5)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "1249612f", "metadata": {}, "source": [ "The fact that the Levene test is non-significant means that we can safely assume that the homogeneity of variance assumption is not violated." ] }, { "attachments": {}, "cell_type": "markdown", "id": "2ac77ad3", "metadata": {}, "source": [ "### Normality of residuals\n", "\n", "As with one-way ANOVA, we can test for the normality of residuals in a [straightforward fashion](anovanormality). At the time of writing, `pingouin` doesn't have a way to extract the residuals, so we will need to use `statsmodels` to run our ANOVA first, and then use the `.resid` method to extract the residuals from the model. Once we have done that, we can examine those residuals in a few different ways. It's generally a good idea to examine them graphically, by drawing histograms (i.e., `sns.histplot()` function) and QQ plots (i.e., `pg.qqplot()` function. If you want a formal test for the normality of the residuals, then we can run the Shapiro-Wilk test (i.e., `pg.normality()`). If we wanted to check the residuals with respect to our two-way ANOVA (including interactions), we could first get the residuals using the following code:" ] }, { "cell_type": "code", "execution_count": 21, "id": "e9badcb6", "metadata": {}, "outputs": [], "source": [ "import statsmodels.api as sm\n", "from statsmodels.formula.api import ols\n", "\n", "formula = 'mood_gain ~ drug + therapy + drug:therapy'\n", "\n", "model = ols(formula, data=df).fit()\n", "res = model.resid" ] }, { "attachments": {}, "cell_type": "markdown", "id": "b84077fb", "metadata": {}, "source": [ "As you can see from the `formula` variable above, in `statsmodels` we need to explicitly ask it to model the interactions for us, while `pingouin` does this automatically. We specify the interaction using `:`, so the interaction of `drug` and `therapy`is written as `drug:therapy`.\n", "\n", "Now we can do our normality test:" ] }, { "cell_type": "code", "execution_count": 22, "id": "6ee2f46f", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
Wpvalnormal
00.9288160.185093True
\n", "
" ], "text/plain": [ " W pval normal\n", "0 0.928816 0.185093 True" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pg.normality(res)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "b86b61a9", "metadata": {}, "source": [ "I haven’t included the plots (you can draw them yourself if you want to see them), but you can see from the non-significance of the Shapiro-Wilk test that normality isn’t violated here." ] }, { "attachments": {}, "cell_type": "markdown", "id": "33423cb2", "metadata": {}, "source": [ "(omnibusF)=\n", "## The $F$ test as a model comparison\n", "\n", "At this point, I want to talk in a little more detail about what the $F$-tests in an ANOVA are actually doing. In the context of ANOVA, I've been referring to the $F$-test as a way of testing whether a particular term in the model (e.g., main effect of Factor A) is significant. This interpretation is perfectly valid, but it's not necessarily the most useful way to think about the test. In fact, it's actually a fairly limiting way of thinking about what the $F$-test does. Consider the clinical trial data we've been working with in this chapter. Suppose I want to see if there are *any* effects of any kind that involve `therapy`. I'm not fussy: I don't care if it's a main effect or an interaction effect.[^imagine] One thing I could do is look at the output for `model2` earlier: in this model we did see a main effect of therapy ($p=.013$) but we did not see an interaction effect ($p=.125$). That's kind of telling us what we want to know, but it's not quite the same thing. What we really want is a single test that *jointly* checks the main effect of therapy and the interaction effect. \n", "\n", "Given the way that I've been describing the ANOVA $F$-test up to this point, you'd be tempted to think that this isn't possible. On the other hand, if you recall the [chapter on regression](modelselreg), we were able to use $F$-tests to make comparisons between a wide variety of regression models. Perhaps something of that sort is possible with ANOVA? And of course, the answer here is yes. The thing that you really need to understand is that the $F$-test, as it is used in both ANOVA and regression, is really a comparison of *two* statistical models. One of these models is the full model (alternative hypothesis), and the other model is a simpler model that is missing one or more of the terms that the full model includes (null hypothesis). The null model cannot contain any terms that are not in the full model. In the example I gave above, the full model is `model2`, and it contains a main effect for therapy, a main effect for drug, and the drug by therapy interaction term. The null model would be `model1` since it contains only the main effect of drug. \n", "\n", "### The $F$ test comparing two models\n", "\n", "Let's frame this in a slightly more abstract way. We'll say that our full model can be written as a `statsmodels` formula that contains several different terms, say `Y ~ A + B + C + D` (remember that `~` here can be read as \"predicted by\"). Our null model only contains some subset of these terms, say `Y ~ A + B`. Some of these terms might be main effect terms, others might be interaction terms. It really doesn't matter. The only thing that matters here is that we want to treat some of these terms as the \"starting point\" (i.e. the terms in the null model, `A` and `B`), and we want to see if including the other terms (i.e., `C` and `D`) leads to a significant improvement in model performance, over and above what could be achieved by a model that includes only `A` and `B`.\n", "\n", "\n", "[^imagine]: There could be all sorts of reasons for doing this, I would imagine." ] }, { "attachments": {}, "cell_type": "markdown", "id": "7964f149", "metadata": {}, "source": [ "Is there a way of making this comparison directly?\n", "\n", "To answer this, let's go back to fundamentals. As we saw back when we learned about [one-way ANOVA](ANOVA), the $F$-test is constructed from two kinds of quantity: sums of squares (SS) and degrees of freedom (df). These two things define a mean square value (MS = SS/df), and we obtain our $F$ statistic by contrasting the MS value associated with \"the thing we're interested in\" (the model) with the MS value associated with \"everything else\" (the residuals). What we want to do is figure out how to talk about the SS value that is associated with the *difference* between two models. It's actually not all that hard to do. \n", "\n", "Let's start with the fundamental rule that we used throughout the chapter on regression:\n", "\n", "$$\n", "\\mbox{SS}_{T} = \\mbox{SS}_{M} + \\mbox{SS}_{R} \n", "$$\n", "\n", "That is, the total sums of squares (i.e., the overall variability of the outcome variable) can be decomposed into two parts: the variability associated with the model $\\mbox{SS}_{M}$, and the residual or leftover variability, $\\mbox{SS}_{R}$. However, it's kind of useful to rearrange this equation slightly, and say that the SS value associated with a model is defined like this...\n", "\n", "$$\n", "\\mbox{SS}_{M} = \\mbox{SS}_{T} - \\mbox{SS}_{R} \n", "$$\n", "\n", "Now, in our scenario, we have two models: the null model (M0) and the full model (M1):\n", "\n", "$$\n", "\\mbox{SS}_{M0} = \\mbox{SS}_{T} - \\mbox{SS}_{R0}\n", "$$\n", "\n", "$$\n", "\\mbox{SS}_{M1} = \\mbox{SS}_{T} - \\mbox{SS}_{R1} \n", "$$\n", "\n", "Next, let's think about what it is we *actually* care about here. What we're interested in is the *difference* between the full model and the null model. So, if we want to preserve the idea that what we're doing is an \"analysis of the variance\" (ANOVA) in the outcome variable, what we should do is define the SS associated with the difference to be equal to the difference in the SS:\n", "\n", "$$\n", "\\begin{align}\n", "\\mbox{SS}_{\\Delta} &= \\mbox{SS}_{M1} - \\mbox{SS}_{M0}\\\\\n", "&= (\\mbox{SS}_{T} - \\mbox{SS}_{R1}) - (\\mbox{SS}_{T} - \\mbox{SS}_{R0} ) \\\\\n", "&= \\mbox{SS}_{R0} - \\mbox{SS}_{R1}\n", "\\end{align}\n", "$$\n", "\n", "Now that we have our degrees of freedom, we can calculate mean squares and $F$ values in the usual way. Specifically, we're interested in the mean square for the difference between models, and the mean square for the residuals associated with the *full* model (M1), which are given by\n", "\n", "$$\n", "\\begin{align}\n", "\\mbox{MS}_{\\Delta} &= \\frac{\\mbox{SS}_{\\Delta} }{ \\mbox{df}_{\\Delta} } \\\\\n", "\\mbox{MS}_{R1} &= \\frac{ \\mbox{SS}_{R1} }{ \\mbox{df}_{R1} }\\\\\n", "\\end{align}\n", "$$\n", "\n", "Finally, taking the ratio of these two gives us our $F$ statistic:\n", "\n", "$$\n", "F = \\frac{\\mbox{MS}_\\Delta} {\\mbox{MS}_{R1}}\n", "$$\n", "\n", "\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "b43a3623", "metadata": {}, "source": [ "### Running the test in Python\n", "\n", "At this point, it may help to go back to our concrete example. The null model here is `model1`, which stipulates that there is a main effect of drug, but no other effects exist. We expressed this via the model formula `mood_gain ~ drug`. The alternative model here is `model3`, which stipulates that there is a main effect of drug, a main effect of therapy, and an interaction. This model corresponds to the formula `mood_gain ~ drug + therapy + drug:therapy`. The key thing here is that if we compare `model1` to `model3`, we’re lumping the main effect of therapy and the interaction term together. Running this test in Python is straightforward: we just input both models to the `anova_lm` function from `statsmodels`, and it will run the exact F -test that I outlined above." ] }, { "cell_type": "code", "execution_count": 23, "id": "f770232c", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
df_residssrdf_diffss_diffFPr(>F)
015.01.3916670.0NaNNaNNaN
112.00.6533333.00.7383334.5204080.024242
\n", "
" ], "text/plain": [ " df_resid ssr df_diff ss_diff F Pr(>F)\n", "0 15.0 1.391667 0.0 NaN NaN NaN\n", "1 12.0 0.653333 3.0 0.738333 4.520408 0.024242" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "from statsmodels.formula.api import ols\n", "from statsmodels.stats.anova import anova_lm\n", "\n", "df = pd.read_csv(\"https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/clintrial.csv\")\n", "\n", "formula1 = 'mood_gain ~ drug'\n", "formula2 = 'mood_gain ~ drug + therapy + drug:therapy'\n", "\n", "model1 = ols(formula1, data=df).fit()\n", "model2 = ols(formula2, data=df).fit()\n", "\n", "anovaResults = anova_lm(model1, model2)\n", "anovaResults" ] }, { "cell_type": "markdown", "id": "654a0c87", "metadata": {}, "source": [ "Let's see if we can reproduce this $F$-test ourselves. Firstly, if you go back and look at the ANOVA tables that we printed out for `model1` and `model3` you can reassure yourself that the RSS values printed in this table really do correspond to the residual sum of squares associated with these two models. So let's type them in as variables:" ] }, { "cell_type": "code", "execution_count": 24, "id": "b8ec6850", "metadata": {}, "outputs": [], "source": [ " ss_res_null = 1.391667\n", " ss_res_full = 0.653333" ] }, { "cell_type": "markdown", "id": "91df920f", "metadata": {}, "source": [ "Now, following the procedure that I described above, we will say that the \"between model\" sum of squares, is the difference between these two residual sum of squares values. So, if we do the subtraction, we discover that the sum of squares associated with those terms that appear in the full model but not the null model is:" ] }, { "cell_type": "code", "execution_count": 25, "id": "9dcc4fa5", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.7383339999999999" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ " ss_diff = ss_res_null - ss_res_full \n", " ss_diff" ] }, { "cell_type": "markdown", "id": "c89b4380", "metadata": {}, "source": [ "Right. Next, as always we need to convert these SS values into MS (mean square) values, which we do by dividing by the degrees of freedom. The degrees of freedom associated with the full-model residuals hasn't changed from our original ANOVA for `model2`: it's the total sample size $N$, minus the total number of groups $G$ that are relevant to the model. We have 18 people in the trial and 6 possible groups (i.e., 2 therapies $\\times$ 3 drugs), so the degrees of freedom here is 12. The degrees of freedom for the null model are calculated similarly. The only difference here is that there are only 3 relevant groups (i.e., 3 drugs), so the degrees of freedom here is 15. And, because the degrees of freedom associated with the difference is equal to the difference in the two degrees of freedom, we arrive at the conclusion that we have $15-12 = 3$ degrees of freedom. Now that we know the degrees of freedom, we can calculate our MS values:" ] }, { "cell_type": "code", "execution_count": 26, "id": "0a107207", "metadata": {}, "outputs": [], "source": [ " ms_res = ss_res_full / 12\n", " ms_diff = ss_diff / 3 " ] }, { "cell_type": "markdown", "id": "e951e0f8", "metadata": {}, "source": [ "Okay, now that we have our two MS values, we can divide one by the other, and obtain an $F$-statistic ..." ] }, { "cell_type": "code", "execution_count": 27, "id": "644ccf98", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "4.520414551231913" ] }, "execution_count": 27, "metadata": {}, "output_type": "execute_result" } ], "source": [ " F_stat = ms_diff / ms_res \n", " F_stat" ] }, { "cell_type": "markdown", "id": "e5226a3e", "metadata": {}, "source": [ "... and, just as we had hoped, this turns out to be nearly identical to the $F$-statistic that the `anova_lm()` function produced earlier. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "a1c074c0", "metadata": {}, "source": [ "(anovalm)=\n", "## ANOVA as a linear model\n", "\n", "One of the most important things to understand about ANOVA and regression is that they're basically the same thing. On the surface of it, you wouldn't think that this is true: after all, the way that I've described them so far suggests that ANOVA is primarily concerned with testing for group differences, and regression is primarily concerned with understanding the correlations between variables. And as far as it goes, that's perfectly true. But when you look under the hood, so to speak, the underlying mechanics of ANOVA and regression are awfully similar. In fact, if you think about it, you've already seen evidence of this. ANOVA and regression both rely heavily on sums of squares (SS), both make use of $F$ tests, and so on. Looking back, it's hard to escape the feeling that the chapters on ANOVA and regression were a bit repetitive. \n", "\n", "The reason for this is that ANOVA and regression are both kinds of **_linear models_**. In the case of regression, this is kind of obvious. The regression equation that we use to define the relationship between predictors and outcomes *is* the equation for a straight line, so it's quite obviously a linear model. When we use e.g. a `pingouin` command like `pg.linear_regression([predictor1, predictor2], outcome)` what we're really working with is the somewhat uglier linear model:\n", "\n", "$$\n", "Y_{p} = b_1 X_{1p} + b_2 X_{2p} + b_0 + \\epsilon_{p}\n", "$$\n", "\n", "where $Y_p$ is the outcome value for the $p$-th observation (e.g., $p$-th person), $X_{1p}$ is the value of the first predictor for the $p$-th observation, $X_{2p}$ is the value of the second predictor for the $p$-th observation, the $b_1$, $b_2$ and $b_0$ terms are our regression coefficients, and $\\epsilon_{p}$ is the $p$-th residual. If we ignore the residuals $\\epsilon_{p}$ and just focus on the regression line itself, we get the following formula:\n", "\n", "$$\n", "\\hat{Y}_{p} = b_1 X_{1p} + b_2 X_{2p} + b_0\n", "$$\n", "\n", "where $\\hat{Y}_p$ is the value of $Y$ that the regression line predicts for person $p$, as opposed to the actually-observed value $Y_p$. The thing that isn't immediately obvious is that we can write ANOVA as a linear model as well. However, it's actually pretty straightforward to do this. Let's start with a really simple example: rewriting a $2 \\times 2$ factorial ANOVA as a linear model. " ] }, { "cell_type": "markdown", "id": "73ffb9de-fb23-4667-9c0a-42ef4d32505e", "metadata": { "scrolled": true }, "source": [ "### Some data\n", "\n", "To make things concrete, let's suppose that our outcome variable is the `grade` that a student receives in my class, a ratio-scale variable corresponding to a mark from 0\\% to 100\\%. There are two predictor variables of interest: whether or not the student turned up to lectures (the `attend` variable), and whether or not the student actually read the textbook (the `reading` variable). We'll say that `attend = 1` if the student attended class, and `attend = 0` if they did not. Similarly, we'll say that `reading = 1` if the student read the textbook, and `reading = 0` if they did not. \n", "\n", "Okay, so far that's simple enough. The next thing we need to do is to wrap some maths around this (sorry!). For the purposes of this example, let $Y_p$ denote the `grade` of the $p$-th student in the class. This is not quite the same notation that we used earlier in this chapter: previously, we've used the notation $Y_{rci}$ to refer to the $i$-th person in the $r$-th group for predictor 1 (the row factor) and the $c$-th group for predictor 2 (the column factor). This extended notation was really handy for describing how the SS values are calculated, but it's a pain in the current context, so I'll switch notation here. Now, the $Y_p$ notation is visually simpler than $Y_{rci}$, but it has the shortcoming that it doesn't actually keep track of the group memberships! That is, if I told you that $Y_{0,0,3} = 35$, you'd immediately know that we're talking about a student (the 3rd such student, in fact) who didn't attend the lectures (i.e., `attend = 0`) and didn't read the textbook (i.e. `reading = 0`), and who ended up failing the class (`grade = 35`). But if I tell you that $Y_p = 35$ all you know is that the $p$-th student didn't get a good grade. We've lost some key information here. Of course, it doesn't take a lot of thought to figure out how to fix this: what we'll do instead is introduce two new variables $X_{1p}$ and $X_{2p}$ that keep track of this information. In the case of our hypothetical student, we know that $X_{1p} = 0$ (i.e., `attend = 0`) and $X_{2p} = 0$ (i.e., `reading = 0`). So the data might look like this:" ] }, { "cell_type": "markdown", "id": "caef9b8f-ecda-49fc-9468-36697ccfb639", "metadata": {}, "source": [ "| person $p$ | grade $Y_p$ | attendance $X_{1p}$ | reading $X_{2p}$ | |\n", "|------------|-------------|---------------------|------------------|---|\n", "| 1 | 90 | 1 | 1 | |\n", "| 2 | 87 | 1 | 1 | |\n", "| 3 | 75 | 0 | 1 | |\n", "| 4 | 60 | 1 | 0 | |\n", "| 5 | 35 | 0 | 0 | |\n", "| 6 | 50 | 0 | 0 | |\n", "| 7 | 65 | 1 | 0 | |\n", "| 8 | 70 | 0 | 1 | |" ] }, { "cell_type": "markdown", "id": "40be45ec-07d7-4deb-857b-e3a963e3dca7", "metadata": {}, "source": [ "This isn't anything particularly special, of course: it's exactly the format in which we expect to see our data! In other words, if your data have been stored as a ``pandas`` data frame, then you're probably expecting to see something that looks like the ``rtfm1`` data frame:" ] }, { "cell_type": "code", "execution_count": 137, "id": "e031a15a-c31a-456a-8e58-922823ae39cd", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
gradeattendreading
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" ], "text/plain": [ " grade attend reading\n", "0 90 1 1\n", "1 87 1 1\n", "2 75 0 1\n", "3 60 1 0\n", "4 35 0 0\n", "5 50 0 0\n", "6 65 1 0\n", "7 70 0 1" ] }, "execution_count": 137, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rtfm1 = pd.read_csv('https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/rtfm1.csv')\n", "rtfm1" ] }, { "cell_type": "markdown", "id": "54ded97e-e298-407b-8795-e6f4603b2d9a", "metadata": {}, "source": [ "Well, sort of. I suspect that a few readers are probably frowning a little at this point. Earlier on in the book (for example back when we talked about the [McNemar test](mcnemar)),we have typically represented nominal scale variables such as `attend` and `reading` as text, rather than encoding them as numeric variables. The ``rtfm1`` data frame doesn't do this, but the `rtfm2` data frame does, and so you might instead be expecting to see data like this:" ] }, { "cell_type": "code", "execution_count": 138, "id": "bb20d924-0c93-4344-97e1-7668f356feae", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
gradeattendreading
090yesyes
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" ], "text/plain": [ " grade attend reading\n", "0 90 yes yes\n", "1 87 yes yes\n", "2 75 no yes\n", "3 60 yes no\n", "4 35 no no\n", "5 50 no no\n", "6 65 yes no\n", "7 70 no yes" ] }, "execution_count": 138, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rtfm2 = pd.read_csv('https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/rtfm2.csv')\n", "rtfm2" ] }, { "cell_type": "markdown", "id": "12caf1dc-eae8-4bcc-af18-c00d8f85394e", "metadata": {}, "source": [ "However, for the purposes of this section it's important that we be able to switch back and forth between these two different ways of thinking about the data. After all, our goal in this section is to look at some of the mathematics that underpins ANOVA, and if we want to do that we need to be able to see the numerical representation of the data (in `rtfm1`) as well as the more meaningful factor representation (in `rtfm2`). We will [return to this problem](changingbaseline) however, because it is not irrelevant which numbers we use to represent our categories. For now, however, we are on safe ground with our 0's and 1's, so let's proceed and confirm that this data set corresponds to a balanced design:" ] }, { "cell_type": "code", "execution_count": 144, "id": "a79d642f-b1c3-407b-a895-03a20ff88c09", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
readingnoyes
attend
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" ], "text/plain": [ "reading no yes\n", "attend \n", "no 2 2\n", "yes 2 2" ] }, "execution_count": 144, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pd.crosstab(rtfm2['attend'], rtfm2['reading'])" ] }, { "cell_type": "markdown", "id": "ebae46e1-fb5a-43cb-ad27-6e386778561b", "metadata": {}, "source": [ "For each possible combination of the `attend` and `reading` variables, we have exactly two students. If we're interested in calculating the mean `grade` for each of these cells," ] }, { "cell_type": "code", "execution_count": 145, "id": "f18e2df4-07e0-4434-8f53-703c59c08f9b", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
grade
attendreading
nono42.5
yes72.5
yesno62.5
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" ], "text/plain": [ " grade\n", "attend reading \n", "no no 42.5\n", " yes 72.5\n", "yes no 62.5\n", " yes 88.5" ] }, "execution_count": 145, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rtfm_agg = rtfm2.groupby(['attend', 'reading']).mean()\n", "rtfm_agg" ] }, { "cell_type": "markdown", "id": "629e1d54-f922-4ab4-9bb2-465d83b9ac25", "metadata": {}, "source": [ "Looking at this table, one gets the strong impression that reading the text and attending the class both matter a lot. " ] }, { "cell_type": "markdown", "id": "ebaf7665-a259-4560-bf38-b16136b6da1c", "metadata": {}, "source": [ "### ANOVA with binary factors as a regression model\n", "\n", "Okay, let's get back to talking about the mathematics. We now have our data expressed in terms of three numeric variables: the continuous variable $Y$, and the two binary variables $X_1$ and $X_2$. What I want to you to recognise is that our 2$\\times$2 factorial ANOVA is *exactly* equivalent to the regression model \n", "\n", "$$\n", "Y_{p} = b_1 X_{1p} + b_2 X_{2p} + b_0 + \\epsilon_p\n", "$$\n", "\n", "This is, of course, the exact same equation that I used earlier to describe a two-predictor regression model! The only difference is that $X_1$ and $X_2$ are now *binary* variables (i.e., values can only be 0 or 1), whereas in a regression analysis we expect that $X_1$ and $X_2$ will be continuous. There's a couple of ways I could try to convince you of this. One possibility would be to do a lengthy mathematical exercise, proving that the two are identical. However, I'm going to go out on a limb and guess that most of the readership of this book will find that to be annoying rather than helpful. Instead, I'll explain the basic ideas, and then rely on Python to show that that ANOVA analyses and regression analyses aren't just similar, they're identical for all intents and purposes. Let's start by running this as an ANOVA." ] }, { "cell_type": "markdown", "id": "159d3e91-67ba-45f7-9ec3-78f53a13902b", "metadata": {}, "source": [ "Now, you may remember that [back when we ran one-way ANOVAs](introduceaov), I said I thought the `statsmodels` package was great, but I also gushed about how we could use good old `pingouin` too. For this example, I am going to use `statsmodels`. I'll explain why in a bit, but first let's run our ANOVA:" ] }, { "cell_type": "code", "execution_count": 147, "id": "d8afc8e5-3a97-4a88-9090-75c814abd4b3", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
sum_sqdfFPR(>F)
attend648.01.021.6000.006
reading1568.01.052.2670.001
Residual150.05.0NaNNaN
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" ], "text/plain": [ " sum_sq df F PR(>F)\n", "attend 648.0 1.0 21.600 0.006\n", "reading 1568.0 1.0 52.267 0.001\n", "Residual 150.0 5.0 NaN NaN" ] }, "execution_count": 147, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import statsmodels.api as sm\n", "from statsmodels.formula.api import ols\n", "\n", "# Here we define our model.\n", "\n", "model = ols('grade ~ attend + reading', data=rtfm1).fit()\n", "sm.stats.anova_lm(model, typ=2).round(3)" ] }, { "cell_type": "markdown", "id": "b0422951-6311-43c2-9bc2-cf8a5a36957e", "metadata": {}, "source": [ "So, by reading the key numbers off the ANOVA table and the table of means that we presented earlier, we can see that the students obtained a higher grade if they attended class $(F_{1,5} = 26.1, p = .0056)$ and if they read the textbook $(F_{1,5} = 52.3, p = .0008)$." ] }, { "cell_type": "markdown", "id": "45d62e8e-426f-428f-818b-9588f2aa0654", "metadata": {}, "source": [ "Now let's think about the same analysis from a linear regression perspective. In the `rtfm1` data set, we have encoded `attend` and `reading` as if they were numeric predictors. In this case, this is perfectly acceptable. There really is a sense in which a student who turns up to class (i.e. `attend = 1`) has in fact done \"more attendance\" than a student who does not (i.e. `attend = 0`). So it's not at all unreasonable to include it as a predictor in a regression model. It's a little unusual, because the predictor only takes on two possible values, but it doesn't violate any of the assumptions of linear regression. And it's easy to interpret. If the regression coefficient for `attend` is greater than 0, it means that students that attend lectures get higher grades; if it's less than zero, then students attending lectures get lower grades. The same is true for our `reading` variable.\n", "\n", "Wait a second... *why* is this true? It's something that is intuitively obvious to everyone who has taken a few stats classes and is comfortable with the maths, but it *isn't* clear to everyone else at first pass. To see why this is true, it helps to look closely at a few specific students. Let's start by considering the 6th and 7th students in our data set (i.e. $p=6$ and $p=7$). Neither one has read the textbook, so in both cases we can set `reading = 0`. Or, to say the same thing in our mathematical notation, we observe $X_{2,6} = 0$ and $X_{2,7} = 0$. However, student number 7 did turn up to lectures (i.e., `attend = 1`, $X_{1,7} = 1$) whereas student number 6 did not (i.e., `attend = 0`, $X_{1,6} = 0$). Now let's look at what happens when we insert these numbers into the general formula for our regression line. For student number 6, the regression predicts that" ] }, { "cell_type": "markdown", "id": "9751fd9b-7dbc-45dd-8b7a-89f32f73ff62", "metadata": {}, "source": [ "$$\n", "\\begin{array}\n", "\\hat{Y}_{6} &=& b_1 X_{1,6} + b_2 X_{2,6} + b_0 \\\\\n", "&=& (b_1 \\times 0) + ( b_2 \\times 0) + b_0 \\\\\n", "&=& b_0\n", "\\end{array}\n", "$$" ] }, { "cell_type": "markdown", "id": "b891a7cf-06b1-42b1-bad7-d2dc5ac5a40a", "metadata": {}, "source": [ "So we're expecting that this student will obtain a grade corresponding to the value of the intercept term $b_0$. What about student 7? This time, when we insert the numbers into the formula for the regression line, we obtain the following:" ] }, { "cell_type": "markdown", "id": "e88d71f7-68eb-4c4e-bee8-a65c80296bf3", "metadata": {}, "source": [ "$$\n", "\\begin{array}\n", "\\hat{Y}_{7} &=& b_1 X_{1,7} + b_2 X_{2,7} + b_0 \\\\\n", "&=& (b_1 \\times 1) + ( b_2 \\times 0) + b_0 \\\\\n", "&=& b_1 + b_0\n", "\\end{array}\n", "$$" ] }, { "cell_type": "markdown", "id": "d6cc3c74-09d4-42fa-9da3-54bcd915ee9b", "metadata": {}, "source": [ "Because this student attended class, the predicted grade is equal to the intercept term $b_0$ *plus* the coefficient associated with the `attend` variable, $b_1$. So, if $b_1$ is greater than zero, we're expecting that the students who turn up to lectures will get higher grades than those students who don't. If this coefficient is negative, we're expecting the opposite: students who turn up at class end up performing much worse. In fact, we can push this a little bit further. What about student number 1, who turned up to class ($X_{1,1} = 1$) *and* read the textbook ($X_{2,1} = 1$)? If we plug these numbers into the regression, we get\n", "\n", "$$\n", "\\begin{array}\n", "\\hat{Y}_{1} &=& b_1 X_{1,1} + b_2 X_{2,1} + b_0 \\\\\n", "&=& (b_1 \\times 1) + ( b_2 \\times 1) + b_0 \\\\\n", "&=& b_1 + b_2 + b_0\n", "\\end{array}\n", "$$" ] }, { "cell_type": "markdown", "id": "f09395ba-041b-4315-ae22-c616ccc436a8", "metadata": {}, "source": [ "So if we assume that attending class helps you get a good grade (i.e., $b_1 > 0$) and if we assume that reading the textbook also helps you get a good grade (i.e., $b_2 >0$), then our expectation is that student 1 will get a grade that that is higher than student 6 and student 7. \n", "\n", "And at this point, you won't be at all suprised to learn that the regression model predicts that student 3, who read the book but didn't attend lectures, will obtain a grade of $b_2 + b_0$. I won't bore you with yet another regression formula. Instead, what I'll do is show you the following table of *expected grades*:" ] }, { "cell_type": "markdown", "id": "6c11cbb7-f589-489a-a94f-31aa735fa0df", "metadata": {}, "source": [ "| | | read textbook? ||\n", "|---------------|---------|----------------|-------------------|\n", "| | | **no** | **yes** |\n", "| **attended?** | **no** | $b_0$ | $b_0 + b_2$ |\n", "| | **yes** | $b_0 + b_1$\" | $b_0 + b_1 + b_2$ |" ] }, { "cell_type": "markdown", "id": "104edecf-15b1-4e38-a348-660df6fb80b4", "metadata": {}, "source": [ "As you can see, the intercept term $b_0$ acts like a kind of \"baseline\" grade that you would expect from those students who don't take the time to attend class or read the textbook. Similarly, $b_1$ represents the boost that you're expected to get if you come to class, and $b_2$ represents the boost that comes from reading the textbook. In fact, if this were an ANOVA you might very well want to characterise $b_1$ as the main effect of attendance, and $b_2$ as the main effect of reading! In fact, for a simple $2 \\times 2$ ANOVA that's *exactly* how it plays out. " ] }, { "cell_type": "markdown", "id": "5fa1588c-c18e-4cbf-9213-c275a47f0ff7", "metadata": {}, "source": [ "Okay, now that we're really starting to see why ANOVA and regression are basically the same thing, let's actually run our regression using a regression function to convince ourselves that this is really true. And this brings us to the first reason for using `statsmodels` for this example rather than `pingouin`. If you look back at the code we used to run our ANOVA, you will see that it was this:\n", "\n", "`model = ols('grade ~ attend + reading', data=rtfm2).fit()`\n", "`sm.stats.anova_lm(model, typ=2).round(4)```\n", "\n", "That is to say, we already ran a regression! Our variable `model` literally is a linear regression, and all we did with `sm.stats.anova_lm` was dress up the results in an ANOVA costume[^halloween]\n", "\n", "Let's take a closer look at the output of that regression we did. `statsmodels` spits out a lot of information, but I just want to zoom in on the part that is relevant for us now, which can be found like this:\n", "\n", "[^halloween]: Wait a minute... I just got an idea for my next Halloween costume 🎃" ] }, { "cell_type": "code", "execution_count": 148, "id": "1cc7a2ed-ee68-40f9-b16f-b25478a50964", "metadata": { "scrolled": true }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/Users/ethan/opt/miniconda3/envs/pythonbook3/lib/python3.11/site-packages/scipy/stats/_stats_py.py:1736: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=8\n", " warnings.warn(\"kurtosistest only valid for n>=20 ... continuing \"\n" ] }, { "data": { "text/html": [ "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
coef std err t P>|t| [0.025 0.975]
Intercept 25.5000 6.423 3.970 0.011 8.990 42.010
attend 18.0000 3.873 4.648 0.006 8.044 27.956
reading 28.0000 3.873 7.230 0.001 18.044 37.956
" ], "text/latex": [ "\\begin{center}\n", "\\begin{tabular}{lcccccc}\n", "\\toprule\n", " & \\textbf{coef} & \\textbf{std err} & \\textbf{t} & \\textbf{P$> |$t$|$} & \\textbf{[0.025} & \\textbf{0.975]} \\\\\n", "\\midrule\n", "\\textbf{Intercept} & 25.5000 & 6.423 & 3.970 & 0.011 & 8.990 & 42.010 \\\\\n", "\\textbf{attend} & 18.0000 & 3.873 & 4.648 & 0.006 & 8.044 & 27.956 \\\\\n", "\\textbf{reading} & 28.0000 & 3.873 & 7.230 & 0.001 & 18.044 & 37.956 \\\\\n", "\\bottomrule\n", "\\end{tabular}\n", "\\end{center}" ], "text/plain": [ "" ] }, "execution_count": 148, "metadata": {}, "output_type": "execute_result" } ], "source": [ "model.summary().tables[1]" ] }, { "cell_type": "markdown", "id": "998f52e6-6138-4b7a-a5d4-57410708225d", "metadata": {}, "source": [ "There's a few interesting things to note here. First, notice that the intercept term is 43.5, which is close to the \"group\" mean of 42.5 observed for those two students who didn't read the text or attend class. Second, notice that we have the regression coefficient of $b_1 = 18.0$ for the attendance variable, suggesting that those students that attended class scored 18\\% higher than those who didn't. So our expectation would be that those students who turned up to class but didn't read the textbook would obtain a grade of $b_0 + b_1$, which is equal to $43.5 + 18.0 = 61.5$. Again, this is similar to the observed group mean of 62.5. You can verify for yourself that the same thing happens when we look at the students that read the textbook.\n", "\n", "Actually, we can push a little further in establishing the equivalence of our ANOVA and our regression. Look at the $p$-values associated with the `attend` variable and the `reading` variable in the regression output. They're identical to the ones we encountered earlier when running the ANOVA. This might seem a little surprising, since the test used when running our regression model calculates a $t$-statistic and the ANOVA calculates an $F$-statistic. However, if you can remember all the way back to the chapter on [probability](probability), I mentioned that there's a relationship between the $t$-distribution and the $F$-distribution: if you have some quantity that is distributed according to a $t$-distribution with $k$ degrees of freedom and you square it, then this new squared quantity follows an $F$-distribution whose degrees of freedom are 1 and $k$. We can check this with respect to the $t$ statistics in our regression model. For the `attend` variable we get a $t$ value of 4.648. If we square this number we end up with 21.604, which is identical to the corresponding $F$ statistic in our ANOVA. [I love it when a plan comes together.](https://www.youtube.com/watch?v=NsUFBm1uENs)\n" ] }, { "cell_type": "markdown", "id": "76fcaa5a-c7b3-497d-9fe3-5a01701e199f", "metadata": {}, "source": [ "I mentioned there was a second reason I didn't use `pingouin` for this example. This is because as far as I can tell, when performing an ANOVA `pingouin` always calculates not only the main effects, but also the interaction, thus giving slightly different results. In order to keep things simple (and maintain parity with [LSR](http://learningstatisticswithr.com/), I decided to go with `statsmodels` and not specify any interactions. Just to be sure though, let's run the ANOVA with `pingouin`, and then run the regression in `statsmodels` with a little ANOVA dressing on top, and confirm that we get the same result:" ] }, { "cell_type": "code", "execution_count": 149, "id": "ec480505-abcd-43fa-abdd-a5622915820a", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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SourceSSDFMSFp-uncnp2
0attend648.01648.018.2540.0130.820
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" ], "text/plain": [ " Source SS DF MS F p-unc np2\n", "0 attend 648.0 1 648.0 18.254 0.013 0.820\n", "1 reading 1568.0 1 1568.0 44.169 0.003 0.917\n", "2 attend * reading 8.0 1 8.0 0.225 0.660 0.053\n", "3 Residual 142.0 4 35.5 NaN NaN NaN" ] }, "execution_count": 149, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pg.anova(dv='grade', between=['attend', 'reading'], data=rtfm1).round(3)" ] }, { "cell_type": "code", "execution_count": 150, "id": "b8cbf2e1-be49-4206-a2ea-35dc01501053", "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/html": [ "
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sum_sqdfFPR(>F)
attend648.01.018.2540.013
reading1568.01.044.1690.003
attend:reading8.01.00.2250.660
Residual142.04.0NaNNaN
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" ], "text/plain": [ " sum_sq df F PR(>F)\n", "attend 648.0 1.0 18.254 0.013\n", "reading 1568.0 1.0 44.169 0.003\n", "attend:reading 8.0 1.0 0.225 0.660\n", "Residual 142.0 4.0 NaN NaN" ] }, "execution_count": 150, "metadata": {}, "output_type": "execute_result" } ], "source": [ "model = ols('grade ~ attend + reading + attend:reading', data=rtfm1).fit()\n", "sm.stats.anova_lm(model, typ=2).round(3)" ] }, { "cell_type": "markdown", "id": "d8e11264-5512-4524-9ae5-f91f6fd8497d", "metadata": {}, "source": [ "Yup. Same thing." ] }, { "cell_type": "markdown", "id": "cfe6ea9b-adfe-47cc-8b79-6c128ad6dda9", "metadata": {}, "source": [ "(changingbaseline) =\n", "### Changing the baseline category\n", "\n", "At this point, you're probably convinced that the ANOVA and the regression are actually identical to each other. So there's one last thing I should show you. What happens if I use the data from `rtfm2` to run the regression? In `rtfm2`, we coded the `attend` and `reading` variables as text rather than as numeric variables. Does this matter? In this case, it turns out that it doesn't. The only differences are superficial:" ] }, { "cell_type": "code", "execution_count": 151, "id": "968f5b6d-66aa-43fb-812a-7a8a212e528f", "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/Users/ethan/opt/miniconda3/envs/pythonbook3/lib/python3.11/site-packages/scipy/stats/_stats_py.py:1736: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=8\n", " warnings.warn(\"kurtosistest only valid for n>=20 ... continuing \"\n" ] }, { "data": { "text/html": [ "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
coef std err t P>|t| [0.025 0.975]
Intercept 25.5000 6.423 3.970 0.011 8.990 42.010
attend 18.0000 3.873 4.648 0.006 8.044 27.956
reading 28.0000 3.873 7.230 0.001 18.044 37.956
" ], "text/latex": [ "\\begin{center}\n", "\\begin{tabular}{lcccccc}\n", "\\toprule\n", " & \\textbf{coef} & \\textbf{std err} & \\textbf{t} & \\textbf{P$> |$t$|$} & \\textbf{[0.025} & \\textbf{0.975]} \\\\\n", "\\midrule\n", "\\textbf{Intercept} & 25.5000 & 6.423 & 3.970 & 0.011 & 8.990 & 42.010 \\\\\n", "\\textbf{attend} & 18.0000 & 3.873 & 4.648 & 0.006 & 8.044 & 27.956 \\\\\n", "\\textbf{reading} & 28.0000 & 3.873 & 7.230 & 0.001 & 18.044 & 37.956 \\\\\n", "\\bottomrule\n", "\\end{tabular}\n", "\\end{center}" ], "text/plain": [ "" ] }, "execution_count": 151, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# With factors encoded numerically\n", "model = ols('grade ~ attend + reading', data=rtfm1).fit()\n", "model.summary().tables[1]" ] }, { "cell_type": "code", "execution_count": 157, "id": "aa85b1f1-ff2b-4a68-9617-1b755c1aa023", "metadata": { "scrolled": true }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/Users/ethan/opt/miniconda3/envs/pythonbook3/lib/python3.11/site-packages/scipy/stats/_stats_py.py:1736: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=8\n", " warnings.warn(\"kurtosistest only valid for n>=20 ... continuing \"\n" ] }, { "data": { "text/html": [ "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
coef std err t P>|t| [0.025 0.975]
Intercept 43.5000 3.354 12.969 0.000 34.878 52.122
attend[T.yes] 18.0000 3.873 4.648 0.006 8.044 27.956
reading[T.yes] 28.0000 3.873 7.230 0.001 18.044 37.956
" ], "text/latex": [ "\\begin{center}\n", "\\begin{tabular}{lcccccc}\n", "\\toprule\n", " & \\textbf{coef} & \\textbf{std err} & \\textbf{t} & \\textbf{P$> |$t$|$} & \\textbf{[0.025} & \\textbf{0.975]} \\\\\n", "\\midrule\n", "\\textbf{Intercept} & 43.5000 & 3.354 & 12.969 & 0.000 & 34.878 & 52.122 \\\\\n", "\\textbf{attend[T.yes]} & 18.0000 & 3.873 & 4.648 & 0.006 & 8.044 & 27.956 \\\\\n", "\\textbf{reading[T.yes]} & 28.0000 & 3.873 & 7.230 & 0.001 & 18.044 & 37.956 \\\\\n", "\\bottomrule\n", "\\end{tabular}\n", "\\end{center}" ], "text/plain": [ "" ] }, "execution_count": 157, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# With factors encoded as text\n", "model = ols('grade ~ attend + reading', data=rtfm2).fit()\n", "model.summary().tables[1]" ] }, { "cell_type": "markdown", "id": "f90d2b4b-5d16-4b6e-8746-46d813cacf91", "metadata": {}, "source": [ "The only thing that is different is that `statsmodels` labels the two variables differently: the output now refers to `attend[T.yes]` and `reading[T.yes]`. You can probably guess what this means. When `statsmodels` refers to `reading[T.yes]` it's trying to indicate that it is assuming that \"yes = 1\" and \"no = 0\". This is important for two reasons, one having to do with textually-encoded categories, and one having to do with numerically-encoded categories. \n", "\n", "Let's take the textually-encoded categories first. Suppose we wanted to say that \"yes = 0\" and \"no = 1\". We could still run this as a regression model, but now all of our coefficients will go in the opposite direction, because the effect of `reading[T.no]` would be referring to the consequences of *not* reading the textbook." ] }, { "cell_type": "code", "execution_count": 161, "id": "872f7533-0885-40e4-aa0e-f5598e351eec", "metadata": {}, "outputs": [ { "ename": "PatsyError", "evalue": "Error evaluating factor: TypeError: 'Series' object is not callable\n grade ~ C(attend) + C(reading(reference=\"yes\"))\n ^^^^^^^^^^^^^^^^^^^^^^^^^^^", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)", "File \u001b[0;32m~/opt/miniconda3/envs/pythonbook3/lib/python3.11/site-packages/patsy/compat.py:36\u001b[0m, in \u001b[0;36mcall_and_wrap_exc\u001b[0;34m(msg, origin, f, *args, **kwargs)\u001b[0m\n\u001b[1;32m 35\u001b[0m \u001b[38;5;28;01mtry\u001b[39;00m:\n\u001b[0;32m---> 36\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mf\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43margs\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mkwargs\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 37\u001b[0m \u001b[38;5;28;01mexcept\u001b[39;00m \u001b[38;5;167;01mException\u001b[39;00m \u001b[38;5;28;01mas\u001b[39;00m e:\n", "File \u001b[0;32m~/opt/miniconda3/envs/pythonbook3/lib/python3.11/site-packages/patsy/eval.py:169\u001b[0m, in \u001b[0;36mEvalEnvironment.eval\u001b[0;34m(self, expr, source_name, inner_namespace)\u001b[0m\n\u001b[1;32m 168\u001b[0m code \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mcompile\u001b[39m(expr, source_name, \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124meval\u001b[39m\u001b[38;5;124m\"\u001b[39m, \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mflags, \u001b[38;5;28;01mFalse\u001b[39;00m)\n\u001b[0;32m--> 169\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28meval\u001b[39m(code, {}, VarLookupDict([inner_namespace]\n\u001b[1;32m 170\u001b[0m \u001b[38;5;241m+\u001b[39m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39m_namespaces))\n", "File \u001b[0;32m:1\u001b[0m\n", "\u001b[0;31mTypeError\u001b[0m: 'Series' object is not callable", "\nThe above exception was the direct cause of the following exception:\n", "\u001b[0;31mPatsyError\u001b[0m Traceback (most recent call last)", "Cell \u001b[0;32mIn[161], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m model \u001b[38;5;241m=\u001b[39m \u001b[43mols\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[38;5;124;43mgrade ~ C(attend) + C(reading(reference=\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43myes\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43m))\u001b[39;49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mdata\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mrtfm2\u001b[49m\u001b[43m)\u001b[49m\u001b[38;5;241m.\u001b[39mfit()\n\u001b[1;32m 2\u001b[0m model\u001b[38;5;241m.\u001b[39msummary()\u001b[38;5;241m.\u001b[39mtables[\u001b[38;5;241m1\u001b[39m]\n", "File \u001b[0;32m~/opt/miniconda3/envs/pythonbook3/lib/python3.11/site-packages/statsmodels/base/model.py:203\u001b[0m, in \u001b[0;36mModel.from_formula\u001b[0;34m(cls, formula, data, subset, drop_cols, *args, **kwargs)\u001b[0m\n\u001b[1;32m 200\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m missing \u001b[38;5;241m==\u001b[39m \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mnone\u001b[39m\u001b[38;5;124m'\u001b[39m: \u001b[38;5;66;03m# with patsy it's drop or raise. let's raise.\u001b[39;00m\n\u001b[1;32m 201\u001b[0m missing \u001b[38;5;241m=\u001b[39m \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mraise\u001b[39m\u001b[38;5;124m'\u001b[39m\n\u001b[0;32m--> 203\u001b[0m tmp \u001b[38;5;241m=\u001b[39m \u001b[43mhandle_formula_data\u001b[49m\u001b[43m(\u001b[49m\u001b[43mdata\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;28;43;01mNone\u001b[39;49;00m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mformula\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mdepth\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43meval_env\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 204\u001b[0m \u001b[43m \u001b[49m\u001b[43mmissing\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mmissing\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 205\u001b[0m ((endog, exog), missing_idx, design_info) \u001b[38;5;241m=\u001b[39m tmp\n\u001b[1;32m 206\u001b[0m max_endog \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mcls\u001b[39m\u001b[38;5;241m.\u001b[39m_formula_max_endog\n", "File \u001b[0;32m~/opt/miniconda3/envs/pythonbook3/lib/python3.11/site-packages/statsmodels/formula/formulatools.py:63\u001b[0m, in \u001b[0;36mhandle_formula_data\u001b[0;34m(Y, X, formula, depth, missing)\u001b[0m\n\u001b[1;32m 61\u001b[0m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[1;32m 62\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m data_util\u001b[38;5;241m.\u001b[39m_is_using_pandas(Y, \u001b[38;5;28;01mNone\u001b[39;00m):\n\u001b[0;32m---> 63\u001b[0m result \u001b[38;5;241m=\u001b[39m \u001b[43mdmatrices\u001b[49m\u001b[43m(\u001b[49m\u001b[43mformula\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mY\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mdepth\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mreturn_type\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[38;5;124;43mdataframe\u001b[39;49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[43m,\u001b[49m\n\u001b[1;32m 64\u001b[0m \u001b[43m \u001b[49m\u001b[43mNA_action\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43mna_action\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 65\u001b[0m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[1;32m 66\u001b[0m result \u001b[38;5;241m=\u001b[39m dmatrices(formula, Y, depth, return_type\u001b[38;5;241m=\u001b[39m\u001b[38;5;124m'\u001b[39m\u001b[38;5;124mdataframe\u001b[39m\u001b[38;5;124m'\u001b[39m,\n\u001b[1;32m 67\u001b[0m NA_action\u001b[38;5;241m=\u001b[39mna_action)\n", "File \u001b[0;32m~/opt/miniconda3/envs/pythonbook3/lib/python3.11/site-packages/patsy/highlevel.py:309\u001b[0m, in \u001b[0;36mdmatrices\u001b[0;34m(formula_like, data, eval_env, NA_action, return_type)\u001b[0m\n\u001b[1;32m 299\u001b[0m \u001b[38;5;250m\u001b[39m\u001b[38;5;124;03m\"\"\"Construct two design matrices given a formula_like and data.\u001b[39;00m\n\u001b[1;32m 300\u001b[0m \n\u001b[1;32m 301\u001b[0m \u001b[38;5;124;03mThis function is identical to :func:`dmatrix`, except that it requires\u001b[39;00m\n\u001b[0;32m (...)\u001b[0m\n\u001b[1;32m 306\u001b[0m \u001b[38;5;124;03mSee :func:`dmatrix` for details.\u001b[39;00m\n\u001b[1;32m 307\u001b[0m \u001b[38;5;124;03m\"\"\"\u001b[39;00m\n\u001b[1;32m 308\u001b[0m eval_env \u001b[38;5;241m=\u001b[39m EvalEnvironment\u001b[38;5;241m.\u001b[39mcapture(eval_env, reference\u001b[38;5;241m=\u001b[39m\u001b[38;5;241m1\u001b[39m)\n\u001b[0;32m--> 309\u001b[0m (lhs, rhs) \u001b[38;5;241m=\u001b[39m \u001b[43m_do_highlevel_design\u001b[49m\u001b[43m(\u001b[49m\u001b[43mformula_like\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mdata\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43meval_env\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 310\u001b[0m \u001b[43m \u001b[49m\u001b[43mNA_action\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mreturn_type\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 311\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m lhs\u001b[38;5;241m.\u001b[39mshape[\u001b[38;5;241m1\u001b[39m] \u001b[38;5;241m==\u001b[39m \u001b[38;5;241m0\u001b[39m:\n\u001b[1;32m 312\u001b[0m \u001b[38;5;28;01mraise\u001b[39;00m PatsyError(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mmodel is missing required outcome variables\u001b[39m\u001b[38;5;124m\"\u001b[39m)\n", "File \u001b[0;32m~/opt/miniconda3/envs/pythonbook3/lib/python3.11/site-packages/patsy/highlevel.py:164\u001b[0m, in \u001b[0;36m_do_highlevel_design\u001b[0;34m(formula_like, data, eval_env, NA_action, return_type)\u001b[0m\n\u001b[1;32m 162\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mdata_iter_maker\u001b[39m():\n\u001b[1;32m 163\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28miter\u001b[39m([data])\n\u001b[0;32m--> 164\u001b[0m design_infos \u001b[38;5;241m=\u001b[39m \u001b[43m_try_incr_builders\u001b[49m\u001b[43m(\u001b[49m\u001b[43mformula_like\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mdata_iter_maker\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43meval_env\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 165\u001b[0m \u001b[43m \u001b[49m\u001b[43mNA_action\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 166\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m design_infos \u001b[38;5;129;01mis\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m \u001b[38;5;28;01mNone\u001b[39;00m:\n\u001b[1;32m 167\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m build_design_matrices(design_infos, data,\n\u001b[1;32m 168\u001b[0m NA_action\u001b[38;5;241m=\u001b[39mNA_action,\n\u001b[1;32m 169\u001b[0m return_type\u001b[38;5;241m=\u001b[39mreturn_type)\n", "File \u001b[0;32m~/opt/miniconda3/envs/pythonbook3/lib/python3.11/site-packages/patsy/highlevel.py:66\u001b[0m, in \u001b[0;36m_try_incr_builders\u001b[0;34m(formula_like, data_iter_maker, eval_env, NA_action)\u001b[0m\n\u001b[1;32m 64\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;28misinstance\u001b[39m(formula_like, ModelDesc):\n\u001b[1;32m 65\u001b[0m \u001b[38;5;28;01massert\u001b[39;00m \u001b[38;5;28misinstance\u001b[39m(eval_env, EvalEnvironment)\n\u001b[0;32m---> 66\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mdesign_matrix_builders\u001b[49m\u001b[43m(\u001b[49m\u001b[43m[\u001b[49m\u001b[43mformula_like\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mlhs_termlist\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 67\u001b[0m \u001b[43m \u001b[49m\u001b[43mformula_like\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mrhs_termlist\u001b[49m\u001b[43m]\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 68\u001b[0m \u001b[43m \u001b[49m\u001b[43mdata_iter_maker\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 69\u001b[0m \u001b[43m \u001b[49m\u001b[43meval_env\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 70\u001b[0m \u001b[43m \u001b[49m\u001b[43mNA_action\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 71\u001b[0m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[1;32m 72\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;01mNone\u001b[39;00m\n", "File \u001b[0;32m~/opt/miniconda3/envs/pythonbook3/lib/python3.11/site-packages/patsy/build.py:693\u001b[0m, in \u001b[0;36mdesign_matrix_builders\u001b[0;34m(termlists, data_iter_maker, eval_env, NA_action)\u001b[0m\n\u001b[1;32m 689\u001b[0m factor_states \u001b[38;5;241m=\u001b[39m _factors_memorize(all_factors, data_iter_maker, eval_env)\n\u001b[1;32m 690\u001b[0m \u001b[38;5;66;03m# Now all the factors have working eval methods, so we can evaluate them\u001b[39;00m\n\u001b[1;32m 691\u001b[0m \u001b[38;5;66;03m# on some data to find out what type of data they return.\u001b[39;00m\n\u001b[1;32m 692\u001b[0m (num_column_counts,\n\u001b[0;32m--> 693\u001b[0m cat_levels_contrasts) \u001b[38;5;241m=\u001b[39m \u001b[43m_examine_factor_types\u001b[49m\u001b[43m(\u001b[49m\u001b[43mall_factors\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 694\u001b[0m \u001b[43m \u001b[49m\u001b[43mfactor_states\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 695\u001b[0m \u001b[43m \u001b[49m\u001b[43mdata_iter_maker\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 696\u001b[0m \u001b[43m \u001b[49m\u001b[43mNA_action\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 697\u001b[0m \u001b[38;5;66;03m# Now we need the factor infos, which encapsulate the knowledge of\u001b[39;00m\n\u001b[1;32m 698\u001b[0m \u001b[38;5;66;03m# how to turn any given factor into a chunk of data:\u001b[39;00m\n\u001b[1;32m 699\u001b[0m factor_infos \u001b[38;5;241m=\u001b[39m {}\n", "File \u001b[0;32m~/opt/miniconda3/envs/pythonbook3/lib/python3.11/site-packages/patsy/build.py:443\u001b[0m, in \u001b[0;36m_examine_factor_types\u001b[0;34m(factors, factor_states, data_iter_maker, NA_action)\u001b[0m\n\u001b[1;32m 441\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m data \u001b[38;5;129;01min\u001b[39;00m data_iter_maker():\n\u001b[1;32m 442\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m factor \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mlist\u001b[39m(examine_needed):\n\u001b[0;32m--> 443\u001b[0m value \u001b[38;5;241m=\u001b[39m \u001b[43mfactor\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43meval\u001b[49m\u001b[43m(\u001b[49m\u001b[43mfactor_states\u001b[49m\u001b[43m[\u001b[49m\u001b[43mfactor\u001b[49m\u001b[43m]\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mdata\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 444\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m factor \u001b[38;5;129;01min\u001b[39;00m cat_sniffers \u001b[38;5;129;01mor\u001b[39;00m guess_categorical(value):\n\u001b[1;32m 445\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m factor \u001b[38;5;129;01mnot\u001b[39;00m \u001b[38;5;129;01min\u001b[39;00m cat_sniffers:\n", "File \u001b[0;32m~/opt/miniconda3/envs/pythonbook3/lib/python3.11/site-packages/patsy/eval.py:568\u001b[0m, in \u001b[0;36mEvalFactor.eval\u001b[0;34m(self, memorize_state, data)\u001b[0m\n\u001b[1;32m 567\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21meval\u001b[39m(\u001b[38;5;28mself\u001b[39m, memorize_state, data):\n\u001b[0;32m--> 568\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[38;5;28;43mself\u001b[39;49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43m_eval\u001b[49m\u001b[43m(\u001b[49m\u001b[43mmemorize_state\u001b[49m\u001b[43m[\u001b[49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43meval_code\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m]\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 569\u001b[0m \u001b[43m \u001b[49m\u001b[43mmemorize_state\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 570\u001b[0m \u001b[43m \u001b[49m\u001b[43mdata\u001b[49m\u001b[43m)\u001b[49m\n", "File \u001b[0;32m~/opt/miniconda3/envs/pythonbook3/lib/python3.11/site-packages/patsy/eval.py:551\u001b[0m, in \u001b[0;36mEvalFactor._eval\u001b[0;34m(self, code, memorize_state, data)\u001b[0m\n\u001b[1;32m 549\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21m_eval\u001b[39m(\u001b[38;5;28mself\u001b[39m, code, memorize_state, data):\n\u001b[1;32m 550\u001b[0m inner_namespace \u001b[38;5;241m=\u001b[39m VarLookupDict([data, memorize_state[\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mtransforms\u001b[39m\u001b[38;5;124m\"\u001b[39m]])\n\u001b[0;32m--> 551\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mcall_and_wrap_exc\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43mError evaluating factor\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m,\u001b[49m\n\u001b[1;32m 552\u001b[0m \u001b[43m \u001b[49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[43m,\u001b[49m\n\u001b[1;32m 553\u001b[0m \u001b[43m \u001b[49m\u001b[43mmemorize_state\u001b[49m\u001b[43m[\u001b[49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43meval_env\u001b[39;49m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m]\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43meval\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 554\u001b[0m \u001b[43m \u001b[49m\u001b[43mcode\u001b[49m\u001b[43m,\u001b[49m\n\u001b[1;32m 555\u001b[0m \u001b[43m \u001b[49m\u001b[43minner_namespace\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43minner_namespace\u001b[49m\u001b[43m)\u001b[49m\n", "File \u001b[0;32m~/opt/miniconda3/envs/pythonbook3/lib/python3.11/site-packages/patsy/compat.py:43\u001b[0m, in \u001b[0;36mcall_and_wrap_exc\u001b[0;34m(msg, origin, f, *args, **kwargs)\u001b[0m\n\u001b[1;32m 39\u001b[0m new_exc \u001b[38;5;241m=\u001b[39m PatsyError(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;132;01m%s\u001b[39;00m\u001b[38;5;124m: \u001b[39m\u001b[38;5;132;01m%s\u001b[39;00m\u001b[38;5;124m: \u001b[39m\u001b[38;5;132;01m%s\u001b[39;00m\u001b[38;5;124m\"\u001b[39m\n\u001b[1;32m 40\u001b[0m \u001b[38;5;241m%\u001b[39m (msg, e\u001b[38;5;241m.\u001b[39m\u001b[38;5;18m__class__\u001b[39m\u001b[38;5;241m.\u001b[39m\u001b[38;5;18m__name__\u001b[39m, e),\n\u001b[1;32m 41\u001b[0m origin)\n\u001b[1;32m 42\u001b[0m \u001b[38;5;66;03m# Use 'exec' to hide this syntax from the Python 2 parser:\u001b[39;00m\n\u001b[0;32m---> 43\u001b[0m exec(\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mraise new_exc from e\u001b[39m\u001b[38;5;124m\"\u001b[39m)\n\u001b[1;32m 44\u001b[0m \u001b[38;5;28;01melse\u001b[39;00m:\n\u001b[1;32m 45\u001b[0m \u001b[38;5;66;03m# In python 2, we just let the original exception escape -- better\u001b[39;00m\n\u001b[1;32m 46\u001b[0m \u001b[38;5;66;03m# than destroying the traceback. But if it's a PatsyError, we can\u001b[39;00m\n\u001b[1;32m 47\u001b[0m \u001b[38;5;66;03m# at least set the origin properly.\u001b[39;00m\n\u001b[1;32m 48\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m \u001b[38;5;28misinstance\u001b[39m(e, PatsyError):\n", "File \u001b[0;32m:1\u001b[0m\n", "\u001b[0;31mPatsyError\u001b[0m: Error evaluating factor: TypeError: 'Series' object is not callable\n grade ~ C(attend) + C(reading(reference=\"yes\"))\n ^^^^^^^^^^^^^^^^^^^^^^^^^^^" ] } ], "source": [ "model = ols('grade ~ C(attend) + C(reading(reference=\"yes\"))', data=rtfm2).fit()\n", "model.summary().tables[1]" ] }, { "cell_type": "code", "execution_count": 139, "id": "e9ea7c4b-024b-4b20-abb8-9a0f3584208a", "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " grade attend reading\n", "0 90 2 1\n", "1 87 2 1\n", "2 75 1 1\n", "3 60 2 0\n", "4 35 1 0\n", "5 50 1 0\n", "6 65 2 0\n", "7 70 1 1" ] }, "execution_count": 141, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rtfm3 = rtfm1\n", "rtfm3['attend'] =[2,2,1,2,1,1,2,1]\n", "rtfm3" ] }, { "cell_type": "code", "execution_count": 1, "id": "05d52668-4d20-4b4f-af25-5c196037236c", "metadata": {}, "outputs": [ { "ename": "NameError", "evalue": "name 'ols' is not defined", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)", "Cell \u001b[0;32mIn[1], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m model \u001b[38;5;241m=\u001b[39m ols(\u001b[38;5;124m'\u001b[39m\u001b[38;5;124mgrade ~ attend + reading\u001b[39m\u001b[38;5;124m'\u001b[39m, data\u001b[38;5;241m=\u001b[39mrtfm3)\u001b[38;5;241m.\u001b[39mfit()\n\u001b[1;32m 2\u001b[0m model\u001b[38;5;241m.\u001b[39msummary()\u001b[38;5;241m.\u001b[39mtables[\u001b[38;5;241m1\u001b[39m]\n", "\u001b[0;31mNameError\u001b[0m: name 'ols' is not defined" ] } ], "source": [ "model = ols('grade ~ attend + reading', data=rtfm3).fit()\n", "model.summary().tables[1]" ] }, { "attachments": {}, "cell_type": "markdown", "id": "5129dee9", "metadata": {}, "source": [ "(encodingnonbinary)=\n", "### How to encode non binary factors as contrasts\n", "\n", "At this point, I've shown you how we can view a $2\\times 2$ ANOVA into a linear model. And it's pretty easy to see how this generalises to a $2 \\times 2 \\times 2$ ANOVA or a $2 \\times 2 \\times 2 \\times 2$ ANOVA... it's the same thing, really: you just add a new binary variable for each of your factors. Where it begins to get trickier is when we consider factors that have more than two levels. Consider, for instance, the $3 \\times 2$ ANOVA that we ran earlier in this chapter using the `clin.trial` data. How can we convert the three-level `drug` factor into a numerical form that is appropriate for a regression?\n", "\n", "The answer to this question is pretty simple, actually. All we have to do is realise that a three-level factor can be redescribed as *two* binary variables. Suppose, for instance, I were to create a new binary variable called `druganxifree`. Whenever the `drug` variable is equal to `\"anxifree\"` we set `druganxifree = 1`. Otherwise, we set `druganxifree = 0`. This variable sets up a **_contrast_**, in this case between anxifree and the other two drugs. By itself, of course, the `druganxifree` contrast isn't enough to fully capture all of the information in our `drug` variable. We need a second contrast, one that allows us to distinguish between joyzepam and the placebo. To do this, we can create a second binary contrast, called `drugjoyzepam`, which equals 1 if the drug is joyzepam, and 0 if it is not. Taken together, these two contrasts allows us to perfectly discriminate between all three possible drugs. The table below illustrates this:" ] }, { "attachments": {}, "cell_type": "markdown", "id": "c8cb1b8a", "metadata": {}, "source": [ "### The equivalence between ANOVA and regression for non-binary factors" ] }, { "attachments": {}, "cell_type": "markdown", "id": "023eb81c", "metadata": {}, "source": [ "### Degrees of freedom as parameter counting!" ] }, { "attachments": {}, "cell_type": "markdown", "id": "9822b55a", "metadata": {}, "source": [ "### A postscript" ] }, { "attachments": {}, "cell_type": "markdown", "id": "7312a906", "metadata": {}, "source": [ "(contrasts)=\n", "## Different ways to specify contrasts" ] }, { "attachments": {}, "cell_type": "markdown", "id": "73844ec3", "metadata": {}, "source": [ "### Treatment contrasts" ] }, { "attachments": {}, "cell_type": "markdown", "id": "8a371e42", "metadata": {}, "source": [ "### Helmert contrasts" ] }, { "attachments": {}, "cell_type": "markdown", "id": "e3143af1", "metadata": {}, "source": [ "### Sum to zero contrasts" ] }, { "attachments": {}, "cell_type": "markdown", "id": "099029dc", "metadata": {}, "source": [ "### Setting contrasts" ] }, { "attachments": {}, "cell_type": "markdown", "id": "601681f7", "metadata": {}, "source": [ "(posthoc2)=\n", "## Post hoc tests" ] }, { "attachments": {}, "cell_type": "markdown", "id": "750fbb0a", "metadata": {}, "source": [ "(unbalancedanova)=\n", "## Unbalanced designs" ] }, { "attachments": {}, "cell_type": "markdown", "id": "7b40d4f6", "metadata": {}, "source": [ "### The coffee data" ] }, { "attachments": {}, "cell_type": "markdown", "id": "f764979f", "metadata": {}, "source": [ "### \"Standard ANOVA\" does not exist for unbalanced designs" ] }, { "attachments": {}, "cell_type": "markdown", "id": "79bdb60a", "metadata": {}, "source": [ "### Type I sum of squares" ] }, { "attachments": {}, "cell_type": "markdown", "id": "c5c57864", "metadata": {}, "source": [ "### Type III sum of squares" ] }, { "attachments": {}, "cell_type": "markdown", "id": "ab95a1b2", "metadata": {}, "source": [ "### Type II sum of squares" ] }, { "attachments": {}, "cell_type": "markdown", "id": "4e7be2d7", "metadata": {}, "source": [ "### Effect sizes (and non-additive sums of squares)" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a14fdc6d", "metadata": {}, "source": [ "## Summary" ] }, { "attachments": {}, "cell_type": "markdown", "id": "02399b77", "metadata": {}, "source": [] } ], "metadata": { "celltoolbar": "Tags", "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.3" } }, "nbformat": 4, "nbformat_minor": 5 } { "cells": [ { "attachments": {}, "cell_type": "markdown", "id": "imperial-facing", "metadata": {}, "source": [ "(bayes)=\n", "# Bayesian Statistics" ] }, { "attachments": {}, "cell_type": "markdown", "id": "4f7dec85", "metadata": {}, "source": [ "> *In our reasonings concerning matter of fact, there are all imaginable degrees of assurance, from the highest certainty to the lowest species of moral evidence. A wise man, therefore, proportions his belief to the evidence.* \n", "-- [David Hume](http://en.wikiquote.org/wiki/David_Hume)." ] }, { "attachments": {}, "cell_type": "markdown", "id": "e7d85f99", "metadata": {}, "source": [ "The ideas I've presented to you in this book describe inferential statistics from the frequentist perspective. I'm not alone in doing this. In fact, almost every textbook given to undergraduate psychology students presents the opinions of the frequentist statistician as *the* theory of inferential statistics, the one true way to do things. I have taught this way for practical reasons. The frequentist view of statistics dominated the academic field of statistics for most of the 20th century, and this dominance is even more extreme among applied scientists. It was and is current practice among psychologists to use frequentist methods, although this is changing. Because frequentist methods are ubiquitous in scientific papers, every student of statistics needs to understand those methods, otherwise they will be unable to make sense of what those papers are saying! Unfortunately -- in my opinion at least -- the current practice in psychology is often misguided, and the reliance on frequentist methods is partly to blame. In this chapter I explain why I think this, and provide an introduction to Bayesian statistics, an approach that I think is generally superior to the orthodox approach.\n", "\n", "This chapter comes in two parts. First I talk about [what Bayesian statistics are all about](basicbayes), covering the basic mathematical rules for how it works as well as an explanation for [why I think the Bayesian approach is so useful](whybayes). Afterwards, I provide a brief overview of how you can do Bayesian versions of [chi-square tests](bayescontingency), [$t$-tests](ttestbf), [regression](bayesregression)) and [ANOVA](bayesanova)." ] }, { "attachments": {}, "cell_type": "markdown", "id": "cd46bb5a", "metadata": {}, "source": [ "(basicbayes)=\n", "## Probabilistic reasoning by rational agents\n", "\n", "From a Bayesian perspective, statistical inference is all about *belief revision*. I start out with a set of candidate hypotheses $h$ about the world. I don't know which of these hypotheses is true, but I _do_ have some beliefs about which hypotheses are plausible and which are not. When I observe the data $d$, I have to revise those beliefs. If the data are consistent with a hypothesis, my belief in that hypothesis is strengthened. If the data inconsistent with the hypothesis, my belief in that hypothesis is weakened. That's it! At the end of this section I'll give a precise description of how Bayesian reasoning works, but first I want to work through a simple example in order to introduce the key ideas. Consider the following reasoning problem:\n", "\n", "> *I'm carrying an umbrella. Do you think it will rain?*\n", "\n", "In this problem, I have presented you with a single piece of data ($d =$ I'm carrying the umbrella), and I'm asking you to tell me your beliefs about whether it's raining. You have two possible **_hypotheses_**, $h$: either it rains today or it does not. How should you solve this problem? " ] }, { "attachments": {}, "cell_type": "markdown", "id": "c40bf9e0", "metadata": {}, "source": [ "(priors)=\n", "### Priors: what you believed before\n", "\n", "The first thing you need to do ignore what I told you about the umbrella, and write down your pre-existing beliefs about rain. This is important: if you want to be honest about how your beliefs have been revised in the light of new evidence, then you *must* say something about what you believed before those data appeared! So, what might you believe about whether it will rain today? You probably know that I live in Australia, and that much of Australia is hot and dry. And in fact you're right: the city of Adelaide where I live has a Mediterranean climate, very similar to southern California, southern Europe or northern Africa. I'm writing this in January, and so you can assume it's the middle of summer. In fact, you might have decided to take a quick look on Wikipedia[^adelaide] and discovered that Adelaide gets an average of 4.4 days of rain across the 31 days of January. Without knowing anything else, you might conclude that the probability of January rain in Adelaide is about 15\\%, and the probability of a dry day is 85\\%. If this is really what you believe about Adelaide rainfall (and now that I've told it to you, I'm betting that this really *is* what you believe) then what I have written here is your **_prior distribution_**, written $P(h)$:\n", "\n", "[^adelaide]: http://en.wikipedia.org/wiki/Climate_of_Adelaide" ] }, { "attachments": {}, "cell_type": "markdown", "id": "ec6dbb75", "metadata": {}, "source": [ "| Hypothesis | Degree of Belief |\n", "|------------|------------------|\n", "| Rainy day | 0.15 |\n", "| Dry day | 0.85 |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "ecbfb8f9", "metadata": {}, "source": [ "### Likelihoods: theories about the data\n", "\n", "To solve the reasoning problem, you need a theory about my behaviour. When does Dan carry an umbrella? You might guess that I'm not a complete idiot[^idiot], and I try to carry umbrellas only on rainy days. On the other hand, you also know that I have young kids, and you wouldn't be all that surprised to know that I'm pretty forgetful about this sort of thing. Let's suppose that on rainy days I remember my umbrella about 30\\% of the time (I really am awful at this). But let's say that on dry days I'm only about 5\\% likely to be carrying an umbrella. So you might write out a little table like this:\n", "\n", "[^idiot]: It's a leap of faith, I know, but let's run with it okay?" ] }, { "attachments": {}, "cell_type": "markdown", "id": "9e3ee8ce", "metadata": {}, "source": [ "| Hypothesis | Umbrella | No umbrella |\n", "|------------|----------|-------------|\n", "| Rainy day | 0.30 | 0.70 |\n", "| Dry day | 0.05 | 0.95 |" ] }, { "attachments": {}, "cell_type": "markdown", "id": "d1f477f7", "metadata": {}, "source": [ "It's important to remember that each cell in this table describes your beliefs about what data $d$ will be observed, *given* the truth of a particular hypothesis $h$. This \"conditional probability\" is written $P(d|h)$, which you can read as \"the probability of $d$ given $h$\". In Bayesian statistics, this is referred to as **_likelihood_** of data $d$ given hypothesis $h$.[^likelihood]\n", "\n", "[^likelihood]: Um. I hate to bring this up, but some statisticians would object to me using the word \"likelihood\" here. The problem is that the word \"likelihood\" has a very specific meaning in frequentist statistics, and it's not quite the same as what it means in Bayesian statistics. As far as I can tell, Bayesians didn't originally have any agreed upon name for the likelihood, and so it became common practice for people to use the frequentist terminology. This wouldn't have been a problem, except for the fact that the way that Bayesians use the word turns out to be quite different to the way frequentists do. This isn't the place for yet another lengthy history lesson, but to put it crudely: when a Bayesian says \"*a* likelihood function\" they're usually referring one of the *rows* of the table. When a frequentist says the same thing, they're referring to the same table, but to them \"*a* likelihood function\" almost always refers to one of the *columns*. This distinction matters in some contexts, but it's not important for our purposes." ] }, { "attachments": {}, "cell_type": "markdown", "id": "3f350d58", "metadata": {}, "source": [ "### The joint probability of data and hypothesis\n", "\n", "At this point, all the elements are in place. Having written down the priors and the likelihood, you have all the information you need to do Bayesian reasoning. The question now becomes, *how* do we use this information? As it turns out, there's a very simple equation that we can use here, but it's important that you understand why we use it, so I'm going to try to build it up from more basic ideas.\n", "\n", "Let's start out with one of the rules of probability theory. I listed it in a table way back in the chapter on [probability](basicprobability), but I didn't make a big deal out of it at the time and you probably ignored it. The rule in question is the one that talks about the probability that *two* things are true. In our example, you might want to calculate the probability that today is rainy (i.e., hypothesis $h$ is true) *and* I'm carrying an umbrella (i.e., data $d$ is observed). The **_joint probability_** of the hypothesis and the data is written $P(d,h)$, and you can calculate it by multiplying the prior $P(h)$ by the likelihood $P(d|h)$. Mathematically, we say that:\n", "\n", "$$\n", "P(d,h) = P(d|h) P(h)\n", "$$\n", "\n", "So, what is the probability that today is a rainy day *and* I remember to carry an umbrella? As we discussed earlier, the prior tells us that the probability of a rainy day is 15\\%, and the likelihood tells us that the probability of me remembering my umbrella on a rainy day is 30\\%. So the probability that both of these things are true is calculated by multiplying the two:\n", "\n", "$$\n", "\\begin{align}\n", "P(\\mbox{rainy}, \\mbox{umbrella}) & = & P(\\mbox{umbrella} | \\mbox{rainy}) \\times P(\\mbox{rainy}) \\\\\n", "& = & 0.30 \\times 0.15 \\\\\n", "& = & 0.045\n", "\\end{align}\n", "$$\n", "\n", "In other words, _before being told anything about what actually happened_, you think that there is a 4.5\\% probability that today will be a rainy day and that I will remember an umbrella. However, there are of course *four* possible things that could happen, right? So let's repeat the exercise for all four. If we do that, we end up with the following table:\n", "\n", "| | Umbrella | No-umbrella |\n", "|-------|----------|-------------|\n", "| Rainy | 0.045 | 0.105 |\n", "| Dry | 0.0425 | 0.8075 |\n", "\n", "This table captures all the information about which of the four possibilities are likely. To really get the full picture, though, it helps to add the row totals and column totals. That gives us this table:\n", "\n", "| | Umbrella | No-umbrella | Total |\n", "|-------|----------|-------------|-------|\n", "| Rainy | 0.0450 | 0.1050 | 0.15 |\n", "| Dry | 0.0425 | 0.8075 | 0.85 |\n", "| Total | 0.0875 | 0.9125 | 1 |\n", "\n", "\n", "This is a very useful table, so it's worth taking a moment to think about what all these numbers are telling us. First, notice that the row sums aren't telling us anything new at all. For example, the first row tells us that if we ignore all this umbrella business, the chance that today will be a rainy day is 15\\%. That's not surprising, of course: that's our prior. The important thing isn't the number itself: rather, the important thing is that it gives us some confidence that our calculations are sensible! Now take a look at the column sums, and notice that they tell us something that we haven't explicitly stated yet. In the same way that the row sums tell us the probability of rain, the column sums tell us the probability of me carrying an umbrella. Specifically, the first column tells us that on average (i.e., ignoring whether it's a rainy day or not), the probability of me carrying an umbrella is 8.75\\%. Finally, notice that when we sum across all four logically-possible events, everything adds up to 1. In other words, what we have written down is a proper probability distribution defined over all possible combinations of data and hypothesis.\n", "\n", "\n", "Now, because this table is so useful, I want to make sure you understand what all the elements correspond to, and how they written:\n", "\n", "| | Umbrella | No-umbrella | |\n", "|-------|--------------------|-----------------------|----------|\n", "| Rainy | $P$(Umbrella, Rainy) | $P$(No-umbrella, Rainy) | $P$(Rainy) |\n", "| Dry | $P$(Umbrella, Dry) | $P$(No-umbrella, Dry) | $P$(Dry) |\n", "| | $P$(Umbrella) | $P$(No-umbrella) | |\n", "\n", "Finally, let's use \"proper\" statistical notation. In the rainy day problem, the data corresponds to the observation that I do or do not have an umbrella. So we'll let $d_1$ refer to the possibility that you observe me carrying an umbrella, and $d_2$ refers to you observing me not carrying one. Similarly, $h_1$ is your hypothesis that today is rainy, and $h_2$ is the hypothesis that it is not. Using this notation, the table looks like this:\n", "\n", "| | d1 | d2 | |\n", "|-------|----------|----------|-------|\n", "| $h_1$ | $P(h_1,d_1)$ | $P(h_1,d_2)$ | $P(h_1)$ |\n", "| $h_2$ | $P(h_2,d_1)$ | $P(h_2,d_2)$ | $P(h_2)$ |\n", "| | $P(d_1)$ | $P(d_2)$ | |\n", "\n", "\n", "\n" ] }, { "attachments": {}, "cell_type": "markdown", "id": "22fb7c61", "metadata": {}, "source": [ "### Updating beliefs using Bayes' rule\n", "\n", "The table we laid out in the last section is a very powerful tool for solving the rainy day problem, because it considers all four logical possibilities and states exactly how confident you are in each of them _before being given any data_. It's now time to consider what happens to our beliefs when we are actually given the data. In the rainy day problem, you are told that I really *am* carrying an umbrella. This is something of a surprising event: according to our table, the probability of me carrying an umbrella is only 8.75\\%. But that makes sense, right? A person carrying an umbrella on a summer day in a hot dry city is pretty unusual, and so you really weren't expecting that. Nevertheless, the problem tells you that it is true. No matter how unlikely you thought it was, you must now adjust your beliefs to accommodate the fact that you now *know* that I have an umbrella.[^umbrella] To reflect this new knowledge, our *revised* table must have the following numbers:\n", "\n", "[^umbrella]: If we were being a bit more sophisticated, we could extend the example to accommodate the possibility that I'm lying about the umbrella. But let's keep things simple, shall we?\n", "\n", "| | Umbrella | No-umbrella |\n", "|-------|----------|-------------|\n", "| Rainy | | 0 |\n", "| Dry | | 0 |\n", "| Total | 1 | 0 |\n", "\n", "In other words, the facts have eliminated any possibility of \"no umbrella\", so we have to put zeros into any cell in the table that implies that I'm not carrying an umbrella. Also, because you know for a fact that I am carrying an umbrella, so the column sum on the left must be 1 to correctly describe the fact that $P(\\mbox{umbrella})=1$.\n", "\n", "What two numbers should we put in the empty cells? Again, let's not worry about the maths, and instead think about our intuitions. When we wrote out our table the first time, it turned out that those two cells had almost identical numbers, right? We worked out that the joint probability of \"rain and umbrella\" was 4.5\\%, and the joint probability of \"dry and umbrella\" was 4.25\\%. In other words, before I told you that I am in fact carrying an umbrella, you'd have said that these two events were almost identical in probability, yes? But notice that *both* of these possibilities are consistent with the fact that I actually am carrying an umbrella. From the perspective of these two possibilities, very little has changed. I hope you'd agree that it's *still* true that these two possibilities are equally plausible. So what we expect to see in our final table is some numbers that preserve the fact that \"rain and umbrella\" is *slightly* more plausible than \"dry and umbrella\", while still ensuring that numbers in the table add up. Something like this, perhaps?\n", "\n", "| | Umbrella | No-umbrella |\n", "|----------|-------------|----------------|\n", "| Rainy | 0.514 | 0 |\n", "| Dry | 0.486 | 0 |\n", "| Total | 1 | 0 |\n", "\n", "What this table is telling you is that, after being told that I'm carrying an umbrella, you believe that there's a 51.4\\% chance that today will be a rainy day, and a 48.6\\% chance that it won't. That's the answer to our problem! The **_posterior probability_** of rain $P(h|d)$ given that I am carrying an umbrella is 51.4\\%\n", "\n", "How did I calculate these numbers? You can probably guess. To work out that there was a 0.514 probability of \"rain\", all I did was take the 0.045 probability of \"rain and umbrella\" and divide it by the 0.0875 chance of \"umbrella\". This produces a table that satisfies our need to have everything sum to 1, and our need not to interfere with the relative plausibility of the two events that are actually consistent with the data. To say the same thing using fancy statistical jargon, what I've done here is divide the joint probability of the hypothesis and the data $P(d,h)$ by the **_marginal probability_** of the data $P(d)$, and this is what gives us the posterior probability of the hypothesis *given* that we know the data have been observed. To write this as an equation:[^nothingnew]\n", " \n", "[^nothingnew]: You might notice that this equation is actually a restatement of the same basic rule I listed at the start of the last section. If you multiply both sides of the equation by $P(d)$, then you get $P(d) P(h| d) = P(d,h)$, which is the rule for how joint probabilities are calculated. So I'm not actually introducing any \"new\" rules here, I'm just using the same rule in a different way.\n", "\n", "$$\n", "P(h | d) = \\frac{P(d,h)}{P(d)}\n", "$$\n", "\n", "However, remember what I said at the start of the last section, namely that the joint probability $P(d,h)$ is calculated by multiplying the prior $P(h)$ by the likelihood $P(d|h)$. In real life, the things we actually know how to write down are the priors and the likelihood, so let's substitute those back into the equation. This gives us the following formula for the posterior probability:\n", "\n", "$$\n", "P(h | d) = \\frac{P(d|h) P(h)}{P(d)}\n", "$$\n", "\n", "And this formula, folks, is known as **_Bayes' rule_**. It describes how a learner starts out with prior beliefs about the plausibility of different hypotheses, and tells you how those beliefs should be revised in the face of data. In the Bayesian paradigm, all statistical inference flows from this one simple rule." ] }, { "attachments": {}, "cell_type": "markdown", "id": "020e5bfa", "metadata": {}, "source": [] }, { "attachments": {}, "cell_type": "markdown", "id": "4e55f10a", "metadata": {}, "source": [ "(bayesianhypothesistests)=\n", "## Bayesian hypothesis tests\n", "\n", "When we first discussed [hypothesis testing](hypothesistesting), I described the orthodox approach. It took an entire chapter to describe, because null hypothesis testing is a very elaborate contraption that people find very hard to make sense of. In contrast, the Bayesian approach to hypothesis testing is incredibly simple. Let's pick a setting that is closely analogous to the orthodox scenario. There are two hypotheses that we want to compare, a null hypothesis $h_0$ and an alternative hypothesis $h_1$. Prior to running the experiment, we have some beliefs $P(h)$ about which hypotheses are true. We run an experiment and obtain data $d$. Unlike frequentist statistics, Bayesian statistics does allow us to talk about the probability that the null hypothesis is true. Better yet, it allows us to calculate the **_posterior probability of the null hypothesis_**, using Bayes' rule:\n", "\n", "$$\n", "P(h_0 | d) = \\frac{P(d|h_0) P(h_0)}{P(d)}\n", "$$\n", "\n", "This formula tells us exactly how much belief we should have in the null hypothesis after having observed the data $d$. Similarly, we can work out how much belief to place in the alternative hypothesis using essentially the same equation. All we do is change the subscript:\n", "\n", "$$\n", "P(h_1 | d) = \\frac{P(d|h_1) P(h_1)}{P(d)}\n", "$$\n", "\n", "It's all so simple that I feel like an idiot even bothering to write these equations down, since all I'm doing is copying Bayes rule from the previous section.[^itscomplicated]\n", "\n", "[^itscomplicated]: Obviously, this is a highly simplified story. All the complexity of real life Bayesian hypothesis testing comes down to how you calculate the likelihood $P(d|h)$ when the hypothesis $h$ is a complex and vague thing. I'm not going to talk about those complexities in this book, but I do want to highlight that although this simple story is true as far as it goes, real life is messier than I'm able to cover in an introductory stats textbook." ] }, { "attachments": {}, "cell_type": "markdown", "id": "d997c010", "metadata": {}, "source": [ "(bayesfactors)=\n", "### The Bayes factor\n", "\n", "In practice, most Bayesian data analysts tend not to talk in terms of the raw posterior probabilities $P(h_0|d)$ and $P(h_1|d)$. Instead, we tend to talk in terms of the **_posterior odds_** ratio. Think of it like betting. Suppose, for instance, the posterior probability of the null hypothesis is 25\\%, and the posterior probability of the alternative is 75\\%. The alternative hypothesis is three times as probable as the null, so we say that the *odds* are 3:1 in favour of the alternative. Mathematically, all we have to do to calculate the posterior odds is divide one posterior probability by the other:\n", "\n", "$$\n", "\\frac{P(h_1 | d)}{P(h_0 | d)} = \\frac{0.75}{0.25} = 3\n", "$$\n", "\n", "Or, to write the same thing in terms of the equations above:\n", "\n", "$$\n", "\\frac{P(h_1 | d)}{P(h_0 | d)} = \\frac{P(d|h_1)}{P(d|h_0)} \\times \\frac{P(h_1)}{P(h_0)}\n", "$$\n", "\n", "Actually, this equation is worth expanding on. There are three different terms here that you should know. On the left hand side, we have the posterior odds, which tells you what you believe about the relative plausibilty of the null hypothesis and the alternative hypothesis *after* seeing the data. On the right hand side, we have the **_prior odds_**, which indicates what you thought *before* seeing the data. In the middle, we have the **_Bayes factor_**, which describes the amount of evidence provided by the data:\n", "\n", "$$\n", "\\begin{array}{ccccc}\\displaystyle\n", "\\frac{P(h_1 | d)}{P(h_0 | d)} &=& \\displaystyle\\frac{P(d|h_1)}{P(d|h_0)} &\\times& \\displaystyle\\frac{P(h_1)}{P(h_0)} \\\\[6pt] \\\\[-2pt]\n", "\\uparrow && \\uparrow && \\uparrow \\\\[6pt]\n", "\\mbox{Posterior odds} && \\mbox{Bayes factor} && \\mbox{Prior odds}\n", "\\end{array}\n", "$$\n", "\n", "The Bayes factor (sometimes abbreviated as **_BF_**) has a special place in the Bayesian hypothesis testing, because it serves a similar role to the $p$-value in orthodox hypothesis testing: it quantifies the strength of evidence provided by the data, and as such it is the Bayes factor that people tend to report when running a Bayesian hypothesis test. The reason for reporting Bayes factors rather than posterior odds is that different researchers will have different priors. Some people might have a strong bias to believe the null hypothesis is true, others might have a strong bias to believe it is false. Because of this, the polite thing for an applied researcher to do is report the Bayes factor. That way, anyone reading the paper can multiply the Bayes factor by their own *personal* prior odds, and they can work out for themselves what the posterior odds would be. In any case, by convention we like to pretend that we give equal consideration to both the null hypothesis and the alternative, in which case the prior odds equals 1, and the posterior odds becomes the same as the Bayes factor." ] }, { "attachments": {}, "cell_type": "markdown", "id": "7d934836", "metadata": {}, "source": [ "### Interpreting Bayes factors\n", "\n", "One of the really nice things about the Bayes factor is the numbers are inherently meaningful. If you run an experiment and you compute a Bayes factor of 4, it means that the evidence provided by your data corresponds to betting odds of 4:1 in favour of the alternative. However, there have been some attempts to quantify the standards of evidence that would be considered meaningful in a scientific context. The two most widely used are from {cite:ts}`Jeffreys1961` and {cite:ts}`Kass1995`. Of the two, I tend to prefer the {cite:ts}`Kass1995` table because it's a bit more conservative. So here it is:" ] }, { "attachments": {}, "cell_type": "markdown", "id": "7d9fe758", "metadata": {}, "source": [ "| Bayes factor | Interpretation |\n", "|--------------|----------------------|\n", "| 1 - 3 | Negligible evidence |\n", "| 3 - 20 | Positive evidence |\n", "| 20 - 150 | Strong evidence |\n", "| $>$ 150 | Very strong evidence |\n", "\n", "\n", "And to be perfectly honest, I think that even the Kass and Raftery standards are being a bit charitable. If it were up to me, I'd have called the \"positive evidence\" category \"weak evidence\". To me, anything in the range 3:1 to 20:1 is \"weak\" or \"modest\" evidence at best. But there are no hard and fast rules here: what counts as strong or weak evidence depends entirely on how conservative you are, and upon the standards that your community insists upon before it is willing to label a finding as \"true\". \n", "\n", "In any case, note that all the numbers listed above make sense if the Bayes factor is greater than 1 (i.e., the evidence favours the alternative hypothesis). However, one big practical advantage of the Bayesian approach relative to the orthodox approach is that it also allows you to quantify evidence *for* the null. When that happens, the Bayes factor will be less than 1. You can choose to report a Bayes factor less than 1, but to be honest I find it confusing. For example, suppose that the likelihood of the data under the null hypothesis $P(d|h_0)$ is equal to 0.2, and the corresponding likelihood $P(d|h_1)$ under the alternative hypothesis is 0.1. Using the equations given above, Bayes factor here would be:\n", "\n", "$$ \n", "\\mbox{BF} = \\frac{P(d|h_1)}{P(d|h_0)} = \\frac{0.1}{0.2} = 0.5\n", "$$\n", "\n", "Read literally, this result tells is that the evidence in favour of the alternative is 0.5 to 1. I find this hard to understand. To me, it makes a lot more sense to turn the equation \"upside down\", and report the amount op evidence in favour of the *null*. In other words, what we calculate is this:\n", "\n", "$$ \n", "\\mbox{BF}^\\prime = \\frac{P(d|h_0)}{P(d|h_1)} = \\frac{0.2}{0.1} = 2\n", "$$\n", "\n", "And what we would report is a Bayes factor of 2:1 in favour of the null. Much easier to understand, and you can interpret this using the table above." ] }, { "attachments": {}, "cell_type": "markdown", "id": "06b02e71", "metadata": {}, "source": [ "(whybayes)=\n", "## Why be a Bayesian?\n", "\n", "Up to this point I've focused exclusively on the logic underpinning Bayesian statistics. We've talked about the idea of \"probability as a degree of belief\", and what it implies about how a rational agent should reason about the world. The question that you have to answer for yourself is this: how do *you* want to do your statistics? Do you want to be an orthodox statistician, relying on sampling distributions and $p$-values to guide your decisions? Or do you want to be a Bayesian, relying on Bayes factors and the rules for rational belief revision? And to be perfectly honest, I can't answer this question for you. Ultimately it depends on what you think is right. It's your call, and your call alone. That being said, I can talk a little about why *I* prefer the Bayesian approach. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "6170c12d", "metadata": {}, "source": [ "### Statistics that mean what you think they mean\n", "\n", "> *You keep using that word. I do not think it means what you think it means* \n", "-- Inigo Montoya, The Princess Bride [^inigo]\n", "\n", "\n", "To me, one of the biggest advantages to the Bayesian approach is that it answers the right questions. Within the Bayesian framework, it is perfectly sensible and allowable to refer to \"the probability that a hypothesis is true\". You can even try to calculate this probability. Ultimately, isn't that what you *want* your statistical tests to tell you? To an actual human being, this would seem to be the whole *point* of doing statistics: to determine what is true and what isn't. Any time that you aren't exactly sure about what the truth is, you should use the language of probability theory to say things like \"there is an 80\\% chance that Theory A is true, but a 20\\% chance that Theory B is true instead\". \n", "\n", "This seems so obvious to a human, yet it is explicitly forbidden within the orthodox framework. To a frequentist, such statements are a nonsense because \"the theory is true\" is not a repeatable event. A theory is true or it is not, and no probabilistic statements are allowed, no matter how much you might want to make them. There's a reason why, back when we talked about [$p$-values](pvalue), I repeatedly warned you *not* to interpret the $p$-value as the probability that the null hypothesis is true. There's a reason why almost every textbook on statstics is forced to repeat that warning. It's because people desperately *want* that to be the correct interpretation. Frequentist dogma notwithstanding, a lifetime of experience of teaching undergraduates and of doing data analysis on a daily basis suggests to me that most actual humans think that \"the probability that the hypothesis is true\" is not only meaningful, it's the thing we care *most* about. It's such an appealing idea that even trained statisticians fall prey to the mistake of trying to interpret a $p$-value this way. For example, here is a quote from an official Newspoll report in 2013, explaining how to interpret their (frequentist) data analysis:[^newspoll]\n", "\n", "[^newspoll]: http://about.abc.net.au/reports-publications/appreciation-survey-summary-report-2013/\n", "\n", "> Throughout the report, where relevant, statistically significant changes have been noted. All significance tests have been based on the 95 percent level of confidence. **This means that if a change is noted as being statistically significant, there is a 95 percent probability that a real change has occurred**, and is not simply due to chance variation. (emphasis added)\n", "\n", "Nope! That's *not* what $p<.05$ means. That's *not* what 95\\% confidence means to a frequentist statistician. The bolded section is just plain wrong. Orthodox methods cannot tell you that \"there is a 95\\% chance that a real change has occurred\", because this is not the kind of event to which frequentist probabilities may be assigned. To an ideological frequentist, this sentence should be meaningless. Even if you're a more pragmatic frequentist, it's still the wrong definition of a $p$-value. It is simply not an allowed or correct thing to say if you want to rely on orthodox statistical tools. \n", "\n", "On the other hand, let's suppose you are a Bayesian. Although the bolded passage is the wrong definition of a $p$-value, it's pretty much exactly what a Bayesian means when they say that the posterior probability of the alternative hypothesis is greater than 95\\%. And here's the thing. If the Bayesian posterior is actually thing you *want* to report, why are you even trying to use orthodox methods? If you want to make Bayesian claims, all you have to do is be a Bayesian and use Bayesian tools. \n", "\n", "Speaking for myself, I found this to be a the most liberating thing about switching to the Bayesian view. Once you've made the jump, you no longer have to wrap your head around counterinuitive definitions of $p$-values. You don't have to bother remembering why you can't say that you're 95\\% confident that the true mean lies within some interval. All you have to do is be honest about what you believed before you ran the study, and then report what you learned from doing it. Sounds nice, doesn't it? To me, this is the big promise of the Bayesian approach: you do the analysis you really want to do, and express what you really believe the data are telling you.\n", "\n", "[^inigo]: http://www.imdb.com/title/tt0093779/quotes. I should note in passing that I'm not the first person to use this quote to complain about frequentist methods. Rich Morey and colleagues had the idea first. I'm shamelessly stealing it because it's such an awesome pull quote to use in this context and I refuse to miss any opportunity to quote *The Princess Bride*." ] }, { "attachments": {}, "cell_type": "markdown", "id": "86aea727", "metadata": {}, "source": [ "> *If [$p$] is below .02 it is strongly indicated that the [null] hypothesis fails to account for the whole of the facts. We shall not often be astray if we draw a conventional line at .05 and consider that [smaller values of $p$] indicate a real discrepancy.* \n", "-- Sir Ronald {cite:ts}`Fisher1925`\n", "\n", "\n", "Consider the quote above by Sir Ronald Fisher, one of the founders of what has become the orthodox approach to statistics. If anyone has ever been entitled to express an opinion about the intended function of $p$-values, it's Fisher. In this passage, taken from his classic guide *Statistical Methods for Research Workers*, he's pretty clear about what it means to reject a null hypothesis at $p<.05$. In his opinion, if we take $p<.05$ to mean there is \"a real effect\", then \"we shall not often be astray\". This view is hardly unusual: in my experience, most practitioners express views very similar to Fisher's. In essence, the $p<.05$ convention is assumed to represent a fairly stringent evidentiary standard.\n", "\n", "Well, how true is that? One way to approach this question is to try to convert $p$-values to Bayes factors, and see how the two compare. It's not an easy thing to do because a $p$-value is a fundamentally different kind of calculation to a Bayes factor, and they don't measure the same thing. However, there have been some attempts to work out the relationship between the two, and it's somewhat surprising. For example, {cite:ts}`Johnson2013` presents a pretty compelling case that (for $t$-tests at least) the $p<.05$ threshold corresponds roughly to a Bayes factor of somewhere between 3:1 and 5:1 in favour of the alternative. If that's right, then Fisher's claim is a bit of a stretch. Let's suppose that the null hypothesis is true about half the time (i.e., the prior probability of $H_0$ is 0.5), and we use those numbers to work out the posterior probability of the null hypothesis given that it has been rejected at $p<.05$. Using the data from {cite:ts}`Johnson2013`, we see that if you reject the null at $p<.05$, you'll be correct about 80\\% of the time. I don't know about you, but in my opinion an evidentiary standard that ensures you'll be wrong on 20\\% of your decisions isn't good enough. The fact remains that, quite contrary to Fisher's claim, if you reject at $p<.05$ you shall quite often go astray. It's not a very stringent evidentiary threshold at all. \n", "\n", "\n", "## The $p$-value is a lie.\n", "\n", "\n", "> *The cake is a lie.* \n", "*The cake is a lie.* \n", "*The cake is a lie.* \n", "*The cake is a lie.* \n", "-- Portal[^portal]\n", "\n", "[^portal]: http://knowyourmeme.com/memes/the-cake-is-a-lie\n", "\n", "\n", "Okay, at this point you might be thinking that the real problem is not with orthodox statistics, just the $p<.05$ standard. In one sense, that's true. The recommendation that {cite:ts}`Johnson2013` gives is not that \"everyone must be a Bayesian now\". Instead, the suggestion is that it would be wiser to shift the conventional standard to something like a $p<.01$ level. That's not an unreasonable view to take, but in my view the problem is a little more severe than that. In my opinion, there's a fairly big problem built into the way most (but not all) orthodox hypothesis tests are constructed. They are grossly naive about how humans actually do research, and because of this most $p$-values are wrong. \n", "\n", "Sounds like an absurd claim, right? Well, consider the following scenario. You've come up with a really exciting research hypothesis and you design a study to test it. You're very diligent, so you run a power analysis to work out what your sample size should be, and you run the study. You run your hypothesis test and out pops a $p$-value of 0.072. Really bloody annoying, right? \n", "\n", "What should you do? Here are some possibilities:\n", "\n", "\n", "1. You conclude that there is no effect, and try to publish it as a null result\n", "1. You guess that there might be an effect, and try to publish it as a \"borderline significant\" result\n", "1. You give up and try a new study\n", "1. You collect some more data to see if the $p$ value goes up or (preferably!) drops below the \"magic\" criterion of $p<.05$\n", "\n", "\n", "Which would *you* choose? Before reading any further, I urge you to take some time to think about it. Be honest with yourself. But don't stress about it too much, because you're screwed no matter what you choose. Based on my own experiences as an author, reviewer and editor, as well as stories I've heard from others, here's what will happen in each case:\n", "\n", "\n", "- Let's start with option 1. If you try to publish it as a null result, the paper will struggle to be published. Some reviewers will think that $p=.072$ is not really a null result. They'll argue it's borderline significant. Other reviewers will agree it's a null result, but will claim that even though some null results *are* publishable, yours isn't. One or two reviewers might even be on your side, but you'll be fighting an uphill battle to get it through.\n", "\n", "- Okay, let's think about option number 2. Suppose you try to publish it as a borderline significant result. Some reviewers will claim that it's a null result and should not be published. Others will claim that the evidence is ambiguous, and that you should collect more data until you get a clear significant result. Again, the publication process does not favour you.\n", "\n", "- Given the difficulties in publishing an \"ambiguous\" result like $p=.072$, option number 3 might seem tempting: give up and do something else. But that's a recipe for career suicide. If you give up and try a new project else every time you find yourself faced with ambiguity, your work will never be published. And if you're in academia without a publication record you can lose your job. So that option is out.\n", "\n", "- It looks like you're stuck with option 4. You don't have conclusive results, so you decide to collect some more data and re-run the analysis. Seems sensible, but unfortunately for you, if you do this all of your $p$-values are now incorrect. *All* of them. Not just the $p$-values that you calculated for *this* study. All of them. All the $p$-values you calculated in the past and all the $p$-values you will calculate in the future. Fortunately, no-one will notice. You'll get published, and you'll have lied.\n", "\n", "Wait, what? How can that last part be true? I mean, it sounds like a perfectly reasonable strategy doesn't it? You collected some data, the results weren't conclusive, so now what you want to do is collect more data until the the results *are* conclusive. What's wrong with that?\n", "\n", "Honestly, there's nothing wrong with it. It's a reasonable, sensible and rational thing to do. In real life, this is exactly what every researcher does. Unfortunately, the [theory of null hypothesis testing](hypothesistesting) as I described it *forbids* you from doing this.[^nhst] The reason is that the theory assumes that the experiment is finished and all the data are in. And because it assumes the experiment is over, it only considers *two* possible decisions. If you're using the conventional $p<.05$ threshold, those decisions are:\n", "\n", "[^nhst]: In the interests of being completely honest, I should acknowledge that not all orthodox statistical tests that rely on this silly assumption. There are a number of *sequential analysis* tools that are sometimes used in clinical trials and the like. These methods are built on the assumption that data are analysed as they arrive, and these tests aren't horribly broken in the way I'm complaining about here. However, sequential analysis methods are constructed in a very different fashion to the \"standard\" version of null hypothesis testing. They don't make it into any introductory textbooks, and they're not very widely used in the psychological literature. The concern I'm raising here is valid for every single orthodox test I've presented so far, and for almost every test I've seen reported in the papers I read.\n", "\n", "| Outcome | Action |\n", "|----------------------|-----------------|\n", "| $p$ less than .05 | Reject the null |\n", "| $p$ greater than .05 | Retain the null |\n", "\n", "What *you're* doing is adding a third possible action to the decision making problem. Specifically, what you're doing is using the $p$-value itself as a reason to justify continuing the experiment. And as a consequence you've transformed the decision-making procedure into one that looks more like this:\n", "\n", "| Outcome | Action |\n", "|------------------------|-----------------------------------------|\n", "| $p$ less than .05 | Stop the experiment and reject the null |\n", "| $p$ between .05 and .1 | Continue the experiment |\n", "| $p$ greater than .1 | Stop the experiment and retain the null |\n", "\n", "The \"basic\" theory of null hypothesis testing isn't built to handle this sort of thing, not in the form I described in the chapter on [null hypothesis significance testing](hypothesistesting). If you're the kind of person who would choose to \"collect more data\" in real life, it implies that you are *not* making decisions in accordance with the rules of null hypothesis testing. Even if you happen to arrive at the same decision as the hypothesis test, you aren't following the decision *process* it implies, and it's this failure to follow the process that is causing the problem.[^lier] Your $p$-values are a lie.\n", "\n", "[^lier]: A [related problem](http://xkcd.com/1478/)\n", "\n", "Worse yet, they're a lie in a dangerous way, because they're all *too small*. To give you a sense of just how bad it can be, consider the following (worst case) scenario. Imagine you're a really super-enthusiastic researcher on a tight budget who didn't pay any attention to my warnings above. You design a study comparing two groups. You desperately want to see a significant result at the $p<.05$ level, but you really don't want to collect any more data than you have to (because it's expensive). In order to cut costs, you start collecting data, but every time a new observation arrives you run a $t$-test on your data. If the $t$-tests says $p<.05$ then you stop the experiment and report a significant result. If not, you keep collecting data. You keep doing this until you reach your pre-defined spending limit for this experiment. Let's say that limit kicks in at $N=1000$ observations. As it turns out, the truth of the matter is that there is no real effect to be found: the null hypothesis is true. So, what's the chance that you'll make it to the end of the experiment and (correctly) conclude that there is no effect? In an ideal world, the answer here should be 95\\%. After all, the whole *point* of the $p<.05$ criterion is to control the Type I error rate at 5\\%, so what we'd hope is that there's only a 5\\% chance of falsely rejecting the null hypothesis in this situation. However, there's no guarantee that will be true. You're breaking the rules: you're running tests repeatedly, \"peeking\" at your data to see if you've gotten a significant result, and all bets are off. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "535c0ac2", "metadata": {}, "source": [ "```{figure} ../img/bayes/adapt.png\n", ":name: fig-adapt\n", ":width: 600px\n", ":align: center\n", "\n", "An illustration of the difference between how the mean and the median should be interpreted. The mean is basically the “centre of gravity” of the data set: if you imagine that the histogram of the data is a solid object, then the point on which you could balance it (as if on a see-saw) is the mean. In contrast, the median is the middle observation. Half of the observations are smaller, and half of the observations are larger.\n", "```" ] }, { "attachments": {}, "cell_type": "markdown", "id": "0790c463", "metadata": {}, "source": [ "So how bad is it? The answer is shown as the solid black line in {numref}`fig-adapt`, and it's *astoundingly* bad. If you peek at your data after every single observation, there is a 49\\% chance that you will make a Type I error. That's, um, quite a bit bigger than the 5\\% that it's supposed to be. By way of comparison, imagine that you had used the following strategy. Start collecting data. Every single time an observation arrives, run a [*Bayesian* $t$-test](ttestbf) and look at the Bayes factor. I'll assume that {cite:ts}`Johnson2013` is right, and I'll treat a Bayes factor of 3:1 as roughly equivalent to a $p$-value of .05.[^whythreeone] This time around, our trigger happy researcher uses the following procedure: if the Bayes factor is 3:1 or more in favour of the null, stop the experiment and retain the null. If it is 3:1 or more in favour of the alternative, stop the experiment and reject the null. Otherwise continue testing. Now, just like last time, let's assume that the null hypothesis is true. What happens? As it happens, I ran the simulations for this scenario too, and the results are shown as the dashed line in {numref}`fig-adapt`. It turns out that the Type I error rate is much much lower than the 49\\% rate that we were getting by using the orthodox $t$-test.\n", "\n", "[^whythreeone]: Some readers might wonder why I picked 3:1 rather than 5:1, given that {cite:ts}`Johnson2013` suggests that $p=.05$ lies somewhere in that range. I did so in order to be charitable to the $p$-value. If I'd chosen a 5:1 Bayes factor instead, the results would look even better for the Bayesian approach.\n", "\n", "In some ways, this is remarkable. The entire *point* of orthodox null hypothesis testing is to control the Type I error rate. Bayesian methods aren't actually designed to do this at all. Yet, as it turns out, when faced with a \"trigger happy\" researcher who keeps running hypothesis tests as the data come in, the Bayesian approach is much more effective. Even the 3:1 standard, which most Bayesians would consider unacceptably lax, is much safer than the $p<.05$ rule. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "9ce4a648", "metadata": {}, "source": [ "## Evidentiary standards you can believe" ] }, { "attachments": {}, "cell_type": "markdown", "id": "fe43f6aa", "metadata": {}, "source": [ "### Is it really this bad?\n", "\n", "The example I gave in the previous section is a pretty extreme situation. In real life, people don't run hypothesis tests every time a new observation arrives. So it's not fair to say that the $p<.05$ threshold \"really\" corresponds to a 49\\% Type I error rate (i.e., $p=.49$). But the fact remains that if you want your $p$-values to be honest, then you either have to switch to a completely different way of doing hypothesis tests, or you must enforce a strict rule: *no peeking*. You are *not* allowed to use the data to decide when to terminate the experiment. You are *not* allowed to look at a \"borderline\" $p$-value and decide to collect more data. You aren't even allowed to change your data analyis strategy after looking at data. You are strictly required to follow these rules, otherwise the $p$-values you calculate will be nonsense.\n", "\n", "And yes, these rules are surprisingly strict. As a class exercise a couple of years back, I asked students to think about this scenario. Suppose you started running your study with the intention of collecting $N=80$ people. When the study starts out you follow the rules, refusing to look at the data or run any tests. But when you reach $N=50$ your willpower gives in... and you take a peek. Guess what? You've got a significant result! Now, sure, you know you *said* that you'd keep running the study out to a sample size of $N=80$, but it seems sort of pointless now, right? The result is significant with a sample size of $N=50$, so wouldn't it be wasteful and inefficient to keep collecting data? Aren't you tempted to stop? Just a little? Well, keep in mind that if you do, your Type I error rate at $p<.05$ just ballooned out to 8\\%. When you report $p<.05$ in your paper, what you're *really* saying is $p<.08$. That's how bad the consequences of \"just one peek\" can be.\n", "\n", "Now consider this ... the scientific literature is filled with $t$-tests, ANOVAs, regressions and chi-square tests. When I wrote this book I didn't pick these tests arbitrarily. The reason why these four tools appear in most introductory statistics texts is that these are the bread and butter tools of science. None of these tools include a correction to deal with \"data peeking\": they all assume that you're not doing it. But how realistic is that assumption? In real life, how many people do you think have \"peeked\" at their data before the experiment was finished and adapted their subsequent behaviour after seeing what the data looked like? Except when the sampling procedure is fixed by an external constraint, I'm guessing the answer is \"most people have done it\". If that has happened, you can infer that the reported $p$-values are wrong. Worse yet, because we don't know what decision process they actually followed, we have no way to know what the $p$-values *should* have been. You can't compute a $p$-value when you don't know the decision making procedure that the researcher used. And so the reported $p$-value remains a lie. \n", "\n", "Given all of the above, what is the take home message? It's not that Bayesian methods are foolproof. If a researcher is determined to cheat, they can always do so. Bayes' rule cannot stop people from lying, nor can it stop them from rigging an experiment. That's not my point here. My point is the same one I made at the very [beginning of the book](whywhywhy): the reason why we run statistical tests is to protect us from ourselves. And the reason why \"data peeking\" is such a concern is that it's so tempting, *even for honest researchers*. A theory for statistical inference has to acknowledge this. Yes, you might try to defend $p$-values by saying that it's the fault of the researcher for not using them properly. But to my mind that misses the point. A theory of statistical inference that is so completely naive about humans that it doesn't even consider the possibility that the researcher might *look at their own data* isn't a theory worth having. In essence, my point is this:\n", "\n", "\n", "> *Good laws have their origins in bad morals.* \n", "-- Ambrosius Macrobius[^Macrobius]\n", "\n", "[^Macrobius]: http://www.quotationspage.com/quotes/Ambrosius_Macrobius/\n", "\n", "\n", "Good rules for statistical testing have to acknowledge human frailty. None of us are without sin. None of us are beyond temptation. A good system for statistical inference should still work even when it is used by actual humans. Orthodox null hypothesis testing does not.[^orthodoxysux]\n", "\n", "[^orthodoxysux]: Okay, I just *know* that some knowledgeable frequentists will read this and start complaining about this section. Look, I'm not dumb. I absolutely know that if you adopt a sequential analysis perspective you can avoid these errors within the orthodox framework. I also know that you can explictly design studies with interim analyses in mind. So yes, in one sense I'm attacking a \"straw man\" version of orthodox methods. However, the straw man that I'm attacking is the one that *is used by almost every single practitioner*. If it ever reaches the point where sequential methods become the norm among experimental psychologists and I'm no longer forced to read 20 extremely dubious ANOVAs a day, I promise I'll rewrite this section and dial down the vitriol. But until that day arrives, I stand by my claim that *default* Bayes factor methods are much more robust in the face of data analysis practices as they exist in the real world. *Default* orthodox methods suck, and we all know it." ] }, { "attachments": {}, "cell_type": "markdown", "id": "93c3616d", "metadata": {}, "source": [ "(bayescontingency)=\n", "## Bayesian analysis of contingency tables\n", "\n", "Time to change gears. Up to this point I've been talking about what Bayesian inference is and why you might consider using it. I now want to briefly describe how to do Bayesian versions of various statistical tests. The discussions in the next few sections are not as detailed as I'd like, but I hope they're enough to help you get started. So let's begin.\n", "\n", "The first kind of statistical inference problem I discussed in this were [categorical data analysis problems](chisquare). In that chapter, I talked about several different statistical problems that you might be interested in, but the one that appears most often in real life is the analysis of *contingency tables*. In this kind of data analysis situation, we have a cross-tabulation of one variable against another one, and the goal is to find out if there is some *association* between these variables. The data set I used to illustrate this problem is found in the `chapek9` data. Let's take a look:" ] }, { "cell_type": "code", "execution_count": 1, "id": "f37a719b", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" ], "text/plain": [ " species choice\n", "0 robot flower\n", "1 human data\n", "2 human data\n", "3 human data\n", "4 robot data" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import pandas as pd\n", "\n", "df = pd.read_csv('https://raw.githubusercontent.com/ethanweed/pythonbook/main/Data/chapek9.csv')\n", "df.head()" ] }, { "attachments": {}, "cell_type": "markdown", "id": "88783caf", "metadata": {}, "source": [ "In this data set, we supposedly sampled 180 beings and measured two things. First, we checked whether they were humans or robots, as captured by the `species` variable. Second, we asked them to nominate whether they most preferred flowers, puppies, or data. When we produce the cross-tabulation, we get this as the results:" ] }, { "cell_type": "code", "execution_count": 5, "id": "b1e41d33", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
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" ], "text/plain": [ "species human robot\n", "choice \n", "data 65 44\n", "flower 13 30\n", "puppy 15 13" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "observations = pd.crosstab(index=df[\"choice\"], columns=df[\"species\"],margins=False)\n", "observations" ] }, { "attachments": {}, "cell_type": "markdown", "id": "158fc10c", "metadata": {}, "source": [ "Surprisingly, the humans seemed to show a much stronger preference for data than the robots did. At the time we speculated that this might have been because the questioner was a large robot carrying a gun, and the humans might have been scared. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "98b14380", "metadata": {}, "source": [ "### The orthodox text\n", "\n", "Just to refresh your memory, when we analysed these data before, we ran a $chi$-square test:" ] }, { "cell_type": "code", "execution_count": 10, "id": "0a0a8c89", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "chi2 = 10.722\n", "p = 0.005\n", "degrees of freedom = 2\n" ] } ], "source": [ "from scipy.stats import chi2_contingency\n", "\n", "chi2, p, dof, ex = chi2_contingency(observations)\n", "\n", "print(\"chi2 = \", round(chi2,3))\n", "print(\"p = \", round(p,3))\n", "print(\"degrees of freedom = \", dof)\n" ] }, { "attachments": {}, "cell_type": "markdown", "metadata": {}, "source": [ "Because we found a small $p$ value (in this case $p<.01$), we concluded that the data are inconsistent with the null hypothesis of no association, and we rejected it. " ] }, { "attachments": {}, "cell_type": "markdown", "id": "176b3d85", "metadata": {}, "source": [ "### The Bayesian test\n", "\n", "How do we run an equivalent test as a Bayesian? Well, like every other bloody thing in statistics, there's a lot of different ways you *could* do it." ] }, { "attachments": {}, "cell_type": "markdown", "id": "1988e362", "metadata": {}, "source": [] }, { "attachments": {}, "cell_type": "markdown", "id": "36fc4cc7", "metadata": {}, "source": [ "### Writing up the results" ] }, { "attachments": {}, "cell_type": "markdown", "id": "36db1a47", "metadata": {}, "source": [ "### Other sampling plans" ] }, { "attachments": {}, "cell_type": "markdown", "id": "713ef507", "metadata": {}, "source": [ "(ttestbf)=\n", "## Bayesian $t$-tests" ] }, { "attachments": {}, "cell_type": "markdown", "id": "06e30ebb", "metadata": {}, "source": [ "### Independent samples $t$-test" ] }, { "attachments": {}, "cell_type": "markdown", "id": "b3b2f313", "metadata": {}, "source": [ "### Paired samples $t$-test" ] }, { "attachments": {}, "cell_type": "markdown", "id": "5025fdeb", "metadata": {}, "source": [ "(bayesregression)=\n", "## Bayesian regression" ] }, { "attachments": {}, "cell_type": "markdown", "id": "ec1dca42", "metadata": {}, "source": [ "### A quick refresher" ] }, { "attachments": {}, "cell_type": "markdown", "id": "8dc9bbe2", "metadata": {}, "source": [ "### The Bayesian version" ] }, { "attachments": {}, "cell_type": "markdown", "id": "9a8b91de", "metadata": {}, "source": [ "### Finding the best model" ] }, { "attachments": {}, "cell_type": "markdown", "id": "5fe83094", "metadata": {}, "source": [ "### Extracting Bayes factors for all included terms" ] }, { "attachments": {}, "cell_type": "markdown", "id": "099e79e8", "metadata": {}, "source": [ "(bayesanova)=\n", "## Bayesian ANOVA" ] }, { "attachments": {}, "cell_type": "markdown", "id": "a0a93bb9", "metadata": {}, "source": [ "### A quick refresher" ] }, { "attachments": {}, "cell_type": "markdown", "id": "43ca2015", "metadata": {}, "source": [ "### The Bayesian version" ] }, { "attachments": {}, "cell_type": "markdown", "id": "5dffe7b0", "metadata": {}, "source": [ "### Constructing Bayesian Type II tests" ] }, { "attachments": {}, "cell_type": "markdown", "id": "c826eaac", "metadata": {}, "source": [ "## Summary" ] }, { "attachments": {}, "cell_type": "markdown", "id": "78e73888", "metadata": {}, "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.3" } }, "nbformat": 4, "nbformat_minor": 5 } { "cells": [ { "cell_type": "markdown", "id": "elegant-syndicate", "metadata": {}, "source": [ "# Epilogue" ] }, { "cell_type": "code", "execution_count": null, "id": "super-petroleum", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.2" } }, "nbformat": 4, "nbformat_minor": 5 }  Xapian GlassnLNrB ; !WFkyl]MA5!paP?+ zkZI, seVG9* o\L>-lQC4&scSE6' xkZL>+ wYL;,xk\NA0!qdV</whZL:)yiXG9, qcPC6% q ` O < -  { l ^ C 2 !  x j ] O A 3 $    r d V C 5 $   y k Z ? .  yhZK:}l^M<+rWI:* |aF+raTC2%seR7&`0.9034421`0.90338403746572691`0.903384 1  ` 0.9031` 0.9 1  `0.8995151` 0.8981` 0.8971`0.8953941`0.890668 1`0.8833331` 0.88 1`0.8679401`0.866249 1` 0.86043481 1` 0.861`0.856988 1`0.8500001` 0.851`0.849966 1`0.844244 1`0.842961 1` 0.8411` 0.84091` 0.841` 0.8251` 0.8201` 0.821` 0.8191`0.81652143688850311`0.816420516344841`0.81628673948959821`0.81623650600024271`0.81610271914787861`0.8161018215248351` 0.81611` 0.8161  ` 0.8141` 0.81231` 0.8121` 0.8111` 0.811` 0.80751` 0.80042036 1` 0.80 1` 0.81  ` 0.79371` 0.7891`0.78322005568903541`0.78040752894019821` 0.7741` 0.75 1` 0.7431` 0.7411` 0.741`0.7395611`0.73833399999999991`0.7383331` 0.7321` 0.73 1`0.7216961`0.721 `0.7166671` 0.7131`0.7127621`0.71275454045129331`0.71 1`0.7014071` 0.70 1` 0.71` 0.69 1` 0.681` 0.67 1`0.666667 1` 0.6601`0.65515154600143431`0.6533331` 0.65331` 0.6531`0.6453211`0.6394981`0.6356201`0.632005 1` 0.63 1` 0.6281`0.6279491  `0.6261771`0.6253971`0.6218571` 0.6161`0.6147231`0.6053091`0.60530883073806761`0.6033331` 0.6031` 0.601` 0.6 1  ` 0.5991` 0.5971` 0.5931` 0.59 1`0.5865611`0.574276 1`0.5714461` 0.57 1` 0.56924183 1`0.5678031` 0.5661`0.5659641  ` 0.5631` 0.5601` 0.56 1` 0.5581` 0.55351` 0.5531`0.55 1 ` 0.5491` 0.54851` 0.5411` 0.5401` 0.535771`0.5338541` 0.53 1` 0.52751` 0.5271` 0.5261` 0.5221` 0.521` 0.5171` 0.5141`0.50416912403709371`0.5041691`0.500918 1`0.500000 1` 0.50 1` 0.5.41` 0.5.31`0.5 E1  `0.4948211`0.490539 1`0.48868316145543881` 0.4861` 0.481`0.4753461` 0.471`0.468179 1` 0.46721` 0.4671`0.465301 1`0.4617291`0.455649 1`0.454794 1`0.451788 1`0.4500001` 0.451` 0.44885781 1`0.445448 1`0.444097 1`0.439356 1`0.4300001` 0.431`0.4250001` 0.421` 0.41701` 0.4171` 0.411` 0.401` 0.41 `0.3920031` 0.381`0.3640111` 0.361` 0.351` 0.34600569 1` 0.3361`0.33333333333333331` 0.3331` 0.321`0.300000000000000041` 0.301` 0.31  ` 0.29331` 0.2931` 0.2881` 0.2821`0.2810691` 0.281` 0.2731`0.2716761`0.27111` 0.2711` 0.27041` 0.271` 0.2661` 0.26141`0.261198 1` 0.261`0.251901 1`0.251 `0.248681 1`0.246331 1`0.244469 1`0.244058 1` 0.2415587 1` 0.2341` 0.2331` 0.23041` 0.2251` 0.221` 0.2181`0.2157651` 0.2151`0.2136981` 0.211`0.2054861` 0.20251` 0.2022036 1`0.20220358121717247 1` 0.20 1`0.21 ` 0.192 1` 0.191`0.1850931`0.1849001` 0.181` 0.167 1`0.1667211`0.1633331`0.1600671` 0.161`0.1504461`0.15 1`0.1455581` 0.14441`0.1426251` 0.141` 0.1361` 0.13561` 0.131` 0.1251` 0.12461` 0.12251` 0.12 1` 0.11321` 0.10891` 0.10501` 0.1051`0.1039521` 0.10241`0.101097188056387571` 0.10031`0.1000001` 0.101`0.1 #1`0.0984081  ` 0.0981` 0.0961` 0.095451`0.0933931`0.0927781`0.091` 0.08751` 0.08641` 0.081`0.0767971`0.0764791  ` 0.0761` 0.0721`0.070738 1` 0.071`0.068885 1`0.068365 1`0.0681661`0.064349557865161611`0.0636881`0.0613291`0.061107 1`0.0583091` 0.0581` 0.0561` 0.05591` 0.05441` 0.0541` 0.053611` 0.0531`0.051 `0.049744 1` 0.04841` 0.04501` 0.0451` 0.04351` 0.0431` 0.042529491`0.0425291` 0.04251`0.04 1` 0.0391` 0.03881` 0.038 1`0.0377418520240214 1` 0.0371093751` 0.0371`0.036145218781445441`0.0361451` 0.0361`0.03571428571428571 1` 0.0351` 0.0331` 0.03241` 0.03 1` 0.0271`0.0266781` 0.0261` 0.0251` 0.02471366 1`0.0242421`0.0238457437649397531` 0.02251`0.021613 1` 0.021264741`0.02097873567785172 1` 0.0201`0.021`0.0180731` 0.0161` 0.01571` 0.0151` 0.01441` 0.01361` 0.0131` 0.01261`0.0119228718824698771` 0.0111` 0.01051`0.0104341  ` 0.0101` 0.01 1 ` 0.00641` 0.0061`0.0058331`0.005 1`0.004697213134214071 1`0.004697 1`0.004613 1` 0.00441`0.004249 1`0.003827 1`0.0036211`0.0033281`0.003310 1` 0.0031` 0.00251` 0.00221`0.002154 1` 0.0021`0.0018111`0.001 1`0.000523 1`0.000325 1` 0.00011`0.0000911` 0.00008671`0.0000861`0.0000301` 0.00000321`0.0000031  `0.0000001`0.00001` 0.000 1`0.00 1 `0.01  ` 0,71` 0,31` 0,250001` 0,21,1 1` 0,21 1` 0,20,40,601` 0,17 1` 0,120 1` 0,101 1` 0,100 1`0,11 ` 0,0,31` 0,01 `(012X-GXM`J1&}-U@R>`xlvK `445452314666553810411187247841988577172215113421338813148139721855314519197741699889981431110bJlearning statistics with python1. why do we learn statistics?*2. a brief introduction to research design3. getting started with python4. more python concepts5. descriptive statistics6. drawing graphs7. data wrangling8. basic programming9. statistical theory10. introduction to probability/11. estimating unknown quantities from a sample12. hypothesis testing13. categorical data analysis14. comparing two means+15. comparing several means (one-way anova)16. linear regression17. factorial anova18. bayesian statistics 19. epilogue20. references`104119885`A1. why do we learn statistics?learning statistics with python`- valuesmaptitle:0;wordcount:1;geo.position:2` languageeng`kindfulltext`datafullPath`{n[@2!{jOB2yfYC7* ~rdM?3% qcS<( th\D5 veRE* wfVE6(udUF7&sfK=/!fYK<.TH7'}n_RC5&rdR+wiOA4 vA3% tgXI7* }ocUE8* v A 4 &  t b U F 7 (  } i [ N ? 0 !   z k Z K ? .    q b S D 5 &  weQD7( sf1$yj[N?0!xiZK>1"vj\N5` 15.10.31` 15.10.21` 15.10.11` 15.101` 15.11` 15.0000001`15.0 1`15,2,3,4,5,6,7,8,9,121`2151. ` 142.21` 142.01` 140 1` 14.9871`14.95182846479696 1` 14.9.21` 14.9.11` 14.91` 14.8.21` 14.8.11` 14.81` 14.7.41` 14.7.31` 14.7.21` 14.7.11` 14.71` 14.61` 14.5.31` 14.5.21` 14.5.11` 14.51` 14.466667 1` 14.46666667 1` 14.4.21` 14.4.11` 14.41` 14.3.81` 14.3.71` 14.3.61` 14.3.51` 14.3.41` 14.3.31` 14.3.21` 14.3.11` 14.31` 14.2.31` 14.2.21` 14.2.11` 14.21` 14.141` 14.131` 14.121` 14.111` 14.101` 14.1.41` 14.1.31` 14.1.21` 14.1.11` 14.11` 14.0 1`2141 ` 13th 1` 1361` 1339756836 1` 132.7151` 132.301` 132.1381`132.00101` 132.0011` 132.001` 1321` 131.941` 13.8 1` 13.7731` 13.7.1 1` 13.7 1` 13.6 1` 13.53333333 1` 13.533333 1` 13.5 1` 13.4 1` 13.341` 13.3 1` 13.2.3 1`13.2.2.2 1`13.2.2.1 1` 13.2.2 1` 13.2.1 1` 13.2 1` 13.1.9 1` 13.1.8 1` 13.1.7 1` 13.1.6 1` 13.1.5 1` 13.1.4 1` 13.1.3 1` 13.1.2 1` 13.1.1 1` 13.1 1` 13.01` 13,0001`2131` 12941` 12871` 1281` 127 1` 126.361`126.27871` 1261` 125.971`125.96561` 125.961` 125 1` 12311` 1231` 122 1` 121 1` 120.411` 1201` 12.9691` 12.9.3 1` 12.9.2 1` 12.9.1 1` 12.9 1` 12.8.3 1` 12.8.2 1` 12.8.1 1` 12.8 1` 12.781` 12.7500001` 12.751` 12.741` 12.7 1` 12.6.2 1` 12.6.1 1` 12.6 1` 12.501` 12.5.3 1` 12.5.2 1` 12.5.1 1` 12.5 1` 12.4.3 1` 12.4.2 1` 12.4.1 1` 12.4 1` 12.3 1` 12.280435 1` 12.221` 12.2 1` 12.10 1` 12.1.2 1` 12.1.1 1` 12.1 1` 12.0761` 12.0700001` 12.071` 12.033333 1` 12.03 1` 12.01  ` 12,161` 12,11`2121  ` 11th1` 119.971`119.93011` 119.9301` 119.931` 119.8421` 119.7931` 119 1` 118 1` 11721` 1171` 116.0000001` 1161` 11511` 115.9581` 115 1 ` 114 1` 1131` 112.8781` 111.0 1` 1111` 1101` 11.91` 11.71`11.661  ` 11.6 1` 11.5.4 1` 11.5.3 1` 11.5.2 1` 11.5.1 1` 11.5 1` 11.421716 1` 11.4.2 1` 11.4.1 1` 11.4 1` 11.3.3 1` 11.3.2 1` 11.3.1 1` 11.3 1` 11.2 1` 11.131562 1` 11.1.5 1` 11.1.4 1` 11.1.3 1` 11.1.2 1` 11.1.1 1` 11.1 1` 11.051` 11,971`2111` 10th1` 10s1` 109 1`108,25,593,375,393,3411` 1081 ` 107 1 ` 106 1` 105.2 1` 1051` 1041` 103.5 1`103 1` 1024 1` 102.8 1`102.5083916244877 1` 1021` 101.2 1` 1011` 100th1` 10011` 10000 1 `10001` 100.0000001` 100.000001`100,2,3,4,5,6,7,8,9,101` 100,000,0001`$1001 %A` 10.922195 1` 10.841` 10.761` 10.757745 1` 10.7221` 10.721572 1`10.721571792595633 1`10.7 1` 10.6 1` 10.5.1 1` 10.5 1` 10.4.2 1` 10.4.1 1` 10.4 1` 10.34 1` 10.3.1 1` 10.3 1` 10.2.3 1` 10.2.2 1` 10.2.1 1` 10.2 1`10.181  ` 10.1211` 10.1 1` 10.0811` 10.051  `10.049671` 10.0491` 10.04871`10.0 1 ` 10,000s1`10,000 1` 10+1`2101 - ` 1.9991` 1.971`1.960201263621358 1`1.9602012636213575 1` 1.96 1  `1.9599641`1.9599639845400545 1`1.959963984540054 1` 1.9421` 1.941` 1.91` 1.861`1.85913139073970051`1.8000001` 1.81` 1.7951`1.79127727372778421` 1.7551` 1.731` 1.7271` 1.72671`1.7266671` 1.711` 1.7081` 1.7.0 1` 1.7 1 `1.660335028063474e1` 1.66 1`1.6583891`1.6510381` 1.651`1.6448541` 1.641` 1.6241` 1.61` 1.5561` 1.5401` 1.501`1.51 ` 1.491`1.4833331` 1.48261` 1.48 1` 1.471`1.451 `1.4479521`1.41  `1.39180000000000011`1.3916671` 1.391` 1.36144597 1` 1.36 1`1.3541831` 1.35375252 1` 1.351` 1.3071` 1.30681`1.3000001` 1.3 1 `1.2815521` 1.281` 1.21 ` 1.15 1` 1.141` 1.12.0 1` 1.11.0 1` 1.111`1.1094001`1.1055701` 1.1.11`1.11 ` 1.04 1`1.0364333894937898 1` 1.031` 1.02290263 1`1.0204091` 1.021`1.0158841` 1.01581`1.0151191` 1.01 1` 1.00241`1.0000000000000011`1.00000000000000091`1.00000000000000011`1.0000001  ` 1.00001`1.091  ` 1,71` 1,61` 1,51` 1,25 1` 1,200,0001` 1,2,3,4,5,6,71` 1,2,3,41` 1,2,31` 1,2,2,11` 1,161` 1,121` 1,11 1` 1,10000 1` 1,100 1` 1,1.2,2,2.21`1,1,2,3,5,8,131` 1,1,0.1 1` 1,11  ` 1,000,0001` 1,01  `21 1D 2zsTFKvN"` 0x111b6a4d01` 0th1` 091` 081` 0721` 062 1` 06 1` 057 1` 0541` 051 1`05;1  ` 049 1` 0441`041` 038 1` 036 1` 034 1` 03 1` 025 1` 0221` 021 1` 0201` 02 1` 0191` 0131`01 1` 00561` 0051`001 1` 00081` 0000000001 1`!000000000000000000000000136 1` 000 1` 001`0.99999999999999991`0.99999833966497191`0.9999841`0.9978661`0.9970241` 0.994079471` 0.991231` 0.991`0.98988318443298341`0.9898641` 0.9791` 0.978735261`0.975 1` 0.9701` 0.971` 0.96911` 0.9691` 0.9661`0.96019023656845091` 0.960191`0.9541161` 0.9541`0.95086860926029911`0.95 1` 0.931`0.9288161` 0.92441` 0.921` 0.9171` 0.91251`0.90940196586125251`0.9078171` 0.904715~sgUKA5*weektrackstrnshaltregplu number_of_hmean_jarq highe foll eleg deal clmbelagee69.053.1415.110.904pcH;,zl`QE9,o`SD5( |m^QD5&xl7* |m`QB5& vgXK>/ ~rdXC7*{l]OB3$ucVG8mY>1% {l`N;( whYM?0uiZK<- QF:- tgVE8'tgVJ<. y_E5$  q c U G 9 +    t e X J < / !  r ` Q > 1 ~ q d /   } o b S D 5 ' |pbVA/!l_QE8eXJ<0$~o`K9-!sfTF9,` 3.101` 3.11`3.0902321`3.06434955786516161` 3.0411` 3.04091` 3.041` 3.0311` 3.021` 3.001` 3.01` 3,3,0.0011` 3,20 1`031J+ T'+ :` 2pt1`2p1` 2nd 1 `2k1`2d1` 2_p1` 2_a1`2_1` 2;111` 2991` 2951` 2921` 29.0114921`291 ` 287.481` 287.431`28.910743260859241` 28.61` 28.00001`281` 2781` 2731` 2721` 27.9561` 27.21` 27.01` 271 ` 2631` 2621` 26.321` 26.151` 26.11`26.0736363591082741` 26.0736361` 26.071`26 1 ` 25th1` 2591` 2531` 2521` 2511` 25.7500001` 25,043,7501`25 B1` 24501` 2401` 24.59 1` 24.01`24 1` 2371` 2361` 2351` 2341` 2331` 230 1` 23.9 1` 23.68 1` 23.61` 23.5 1` 23.1 1` 23.09 1` 23.031` 23.0248061`231 ` 2261` 225.0 1` 2221`22.99998 1` 22.9998 1` 22.5 1` 22.216667 1` 22.21666667 1`221 ` 21st1` 2171` 215.241`215.23828653684431` 215.21` 21.6041` 21.6001` 21.333 1 ` 21.31` 21.161` 21.01 1`&21M1+`20th 1` 2061  ` 2051` 2048 1` 2041` 2031` 20241` 20231` 20221`220211` 20181` 20161` 20141` 20131`20111`20101 ` 201.71` 2011` 20081` 20071`2006 1`20051`20041 ` 20031`20021` 20001`2001  `20.854399960170911` 20.851` 20.78333333 1` 20.783333 1` 20.0000001` 20.01`1201 ` 2.9251` 2.8671` 2.801` 2.81 ` 2.78e1` 2.751` 2.71831` 2.7.91` 2.7.81` 2.7.71` 2.7.61` 2.7.51` 2.7.41` 2.7.31` 2.7.21` 2.7.121` 2.7.111` 2.7.101` 2.7.11` 2.71` 2.6.51` 2.6.41` 2.6.31` 2.6.21` 2.6.11` 2.61 ` 2.5th 1` 2.591e1` 2.581`2.5758291` 2.541` 2.5.21` 2.5.11`2.51 ` 2.4901` 2.48981` 2.41 `2.3263481` 2.31 `2.262157162740992 1`2.262157162740991 1` 2.261`2.2596055351576811` 2.2591`2.2547131`2.254712867006931` 2.251`2.2416591` 2.2.61` 2.2.51` 2.2.41` 2.2.31` 2.2.21` 2.2.11` 2.21`2.1840671` 2.181` 2.1621` 2.15e1`2.1457300163209325e1`2.12426457862480021` 2.1201` 2.115432391`2.1154321` 2.1.21` 2.1.11`2.11  `2.0742321` 2.071`2.0468211`2.0341871` 2.031`2.0000000000000011`2.000000 1` 2.0 1 ` 2,971` 2,9.491` 2,71` 2,61` 2,31`2,2,1,2,1,1,2,11` 2,21` 2,151` 2,121` 2,11 1` 2,11`2+1`221 )5_$,yebm` 1st1`1p1`1996 1`19951`1994 1` 19901` 199 1`1988 1` 1987 1`19861 ` 19851` 19841` 19831` 19811` 19801` 198 1`19791` 1978 1` 1976 1`19751 `19741`19731` 197 1` 19681` 1967 1  ` 19651` 19631`19611` 19601` 196.0 1` 196 1` 1954 1` 19521` 19511` 195 1`1947 1`19461 ` 19451` 19381` 19371` 1934 1` 193171` 193131` 19251` 1923 1 ` 1922679674 1`1922 1 `1920s1  ` 192 1` 19111` 1911` 19081` 19071` 1900 1` 19.51` 19.4 1 ` 19.11`019m19` 18991` 1871` 1861` 18541`1838.72248832000041` 1837.1559161` 1837.0922281` 1831` 180 1` 18.9.31` 18.9.21` 18.9.11` 18.91` 18.8.41` 18.8.31` 18.8.21` 18.8.11` 18.81`18.72448042232567 1` 18.7.21` 18.7.11` 18.71` 18.6107781` 18.611` 18.6.41` 18.6.31` 18.6.21` 18.6.11` 18.6 1` 18.5.11` 18.51` 18.41` 18.3.11` 18.31` 18.2541` 18.2.21` 18.2.11` 18.21` 18.101` 18.1.41` 18.1.31` 18.1.21` 18.1.11` 18.11` 18.0441` 18.011` 18.0000001 ` 18.00001` 18.01  `1181& o` 17th1` 1771` 176.0000001` 1761` 1751` 1741` 17361` 1731` 1721` 1713 1` 1711` 1701` 17.9471` 17.91` 17.8.61` 17.8.51` 17.8.41` 17.8.31` 17.8.21` 17.8.11` 17.81` 17.71` 17.6.41` 17.6.31` 17.6.21` 17.6.11` 17.61` 17.5.71` 17.5.61` 17.5.51` 17.5.41` 17.5.31` 17.5.21` 17.5.11` 17.51` 17.4.21` 17.4.11` 17.41` 17.3.21` 17.3.11` 17.31` 17.2.51` 17.2.41` 17.2.31` 17.2.21` 17.2.11` 17.21`17.12238658574764 1` 17.1.51` 17.1.41` 17.1.31` 17.1.21` 17.1.11` 17.11`2171 ` 1691` 1681` 167 1` 1661` 1651  ` 1641` 1631` 1621` 160 1` 16.99 1` 16.9.61` 16.9.51` 16.9.41` 16.9.31` 16.9.21` 16.9.11` 16.91` 16.81` 16.7.31` 16.7.21` 16.7.11` 16.71` 16.6.31` 16.6.21` 16.6.11` 16.61` 16.5.31` 16.5.21` 16.5.11` 16.51` 16.4.11` 16.41` 16.31` 16.2.21` 16.2.11` 16.21` 16.1721` 16.17151` 16.171` 16.131` 16.111` 16.10.31` 16.10.21` 16.10.11` 16.101` 16.11` 16.0161`&161` 15th 1` 1591` 1571` 1568.01` 1531` 150.01`1501`15.9 1` 15.81` 15.7.11` 15.71` 15.6.11` 15.61` 15.521` 15.5.51` 15.5.41` 15.5.31` 15.5.21` 15.5.11` 15.51`15.485850070591503 1` 15.41` 15.3.11` 15.31` 15.2.51` 15.2.41` 15.2.31` 15.2.21` 15.2.11` 15.21` 15.121` 15.111 udSG9+ ~paP6& qbQ@2$zj^PD8*}j\M>2${l`TG;-!|obTH<0sg\J ~qdWH9,yh[N@2"ug[N4( r`RF:.sfTA4' sfTF:.! paTB6*ugYK=/viYK=,|obQC5  r ` S F 7 (  y l Z L @ 4 (   ~ r f Z K : +   u g Y H 6 *   o _ Q C 5 '  rdSB(}lM> veSF:-!xcTG:) thO@3% `3.09023` 69.0000001`69 1 ` 6891` 6861` 6841` 6801` 68.621` 68.3 1` 681 `679.8345129870131` 6771`675.97181689049591`675.97181689049581`67.844215137914151` 67.841` 67.81` 67.51`67 1 ` 66th1` 66.2500001` 66.01`661 ` 65.31` 65.01` 65,0001`651` 648.01` 6451` 6431`64 1` 63.81`63.710001` 63.711` 63.51` 63.1521741` 63.0500001`631` 6251` 62131` 6211` 62.51` 62.0500001`62.000001` 62,5001`62 1 ` 6111` 61.51` 61.0000001`611` 6071` 600px 1` 600 1` 60.0 1` 60,80 1`$60,58,24,26,34,42,31,30,33,2,91` 60,0001`60 *1`6.9769231` 6.971`6.9652001` 6.951` 6.891` 6.771` 6.71`6.681  `6.6161371` 6.6 1` 6.581` 6.51` 6.481`6.4754360883393791`6.4754361`6.4600001` 6.4311`6.4250001` 6.421`6.4056121` 6.41` 6.3.4.11` 6.3.41` 6.3.31` 6.3.21` 6.3.11` 6.31`6.2925001`6.2850001` 6.2.41` 6.2.31` 6.2.21` 6.2.11` 6.21`6.1485281` 6.131` 6.11` 6.081` 6.01` 6,61`061e3 ` 5th1`5991.5771` 593.31` 5931` 5911` 59.7000001`591` 588.81` 588.71` 5841` 5831` 582.91` 581.01` 580.91` 580 1` 58.3850001` 58,3331` 581` 579.01` 5751` 5701` 57.7000001` 57.6421`57.000001` 571 ` 5691` 5681` 5671` 5651` 5611` 5601` 56.9800001` 56.316667 1` 56.31666667 1` 56.0000001 ` 56.01`56/1&` 5551` 5541` 5531` 5521` 5511` 5501 ` 55.0000001` 55.01`551 ` 5491` 54.01` 541 `537797731` 53.41` 53.1000001` 53 1` 52.68333333 1` 52.683333 1` 52.31` 52.2671` 52.1221` 521` 51.81` 51.7500001` 51.61` 51.41`51 1 ` 50th1` 50000 1` 5001` 50.5000001` 50.51` 50.0 1` 50,1001` 50,0001` 50 :1 ` 5.9 1` 5.81 `5.7749181` 5.761` 5.731` 5.7.21` 5.7.11` 5.71` 5.681` 5.6.51` 5.6.41` 5.6.31` 5.6.21` 5.6.11` 5.6 1 ` 5.561` 5.51` 5.4941`5.4426171` 5.4.21` 5.4.11` 5.4 1 ` 5.391` 5.31`5.276692 1` 5.251` 5.2341` 5.2.71` 5.2.61` 5.2.51` 5.2.41` 5.2.31` 5.2.21` 5.2.11` 5.21`5.141  ` 5.1.71` 5.1.61` 5.1.51` 5.1.41` 5.1.31` 5.1.21` 5.1.11` 5.11` 5.01` 5,6,7,81` 5,41`151  E  ` 4th1` 4981` 4921` 491` 4881` 4861` 48.7845691` 48.61` 481` 475.2311`471` 4641` 4610795 1` 4610025 1`461 ` 456451` 4531` 45.271`451 ` 4451` 4441` 44351` 4431` 4421` 4411` 44.6000001` 44.61` 44.1691`44 1 ` 43921` 4391` 43451` 43211` 4321` 43.50001` 43.51`43 1 ` 42951` 42941` 42931` 42921` 42911` 4220361125 1` 42.9000001` 42.91` 42.51` 42.191`421 ` 4171` 4121` 41.761` 41.42311` 41.4231` 41.421` 41.0000001`41.000001`411 ` 4051` 4041` 4011` 400001`4001` 40.71` 40.667 1 ` 40,801`40 1 ` 4.9 1 ` 4.8451`4.8400001` 4.841  ` 4.821` 4.8.31` 4.8.21` 4.8.11` 4.81`4.741666666666667 1` 4.741` 4.71` 4.6621` 4.6481` 4.6.11` 4.61 `4.5204145512319131`4.5204081` 4.511` 4.50 1`4.51 ` 4.4.31` 4.4.21` 4.4.11` 4.41 ` 4.3.11` 4.31` 4.261` 4.251`4.2062221` 4.2.31` 4.2.21` 4.2.11` 4.21` 4.191` 4.18.41` 4.18.31` 4.18.21` 4.18.11` 4.181` 4.171` 4.161` 4.151` 4.141` 4.13.21` 4.13.11` 4.131` 4.121` 4.111` 4.10.21` 4.10.11` 4.101` 4.1 1` 4.01` 4,610,795 1` 4,609,795 1` 4,5,61` 4,4,100 1` 4,101`141" k`3rd1 `3d1` 3981` 3931` 39.9961`39 1 ` 3861` 38.9451`381` 376.361` 3761` 3751` 3711` 37.9561` 37.751`371` 3681` 3671` 3641` 36.601` 36.61` 361` 3591514384 1` 3531` 350 1 ` 35.51`35.301136363636371` 35.3011361` 35.01`351 ` 3491` 34771` 3461` 3451` 344 1` 34211` 3421` 3411` 34.8781` 34.1 1`34 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Why do we learn statistics? C-ethanweed.github.io/pythonbook/01.02-studydesign.html2. A brief introduction to research design Cethanweed.github.io/pythonbook/02.01-getting_started_with_python.html3. Getting Started with Python Cethanweed.github.io/pythonbook/02.02-more_python_concepts.html4. More Python Concepts Cethanweed.github.io/pythonbook/03.01-descriptives.html5. Descriptive statistics Cethanweed.github.io/pythonbook/03.02-drawing_graphs.html6. Drawing Graphs Cethanweed.github.io/pythonbook/03.03-pragmatic_matters.html7. Data Wrangling Cethanweed.github.io/pythonbook/03.04-basic_programming.html8. Basic Programming Cethanweed.github.io/pythonbook/04.01-intro-to-probability.html9. Statistical theory Cethanweed.github.io/pythonbook/04.02-probability.html10. Introduction to Probability Cethanweed.github.io/pythonbook/04.03-estimation.html11. Estimating unknown quantities from a sample Cethanweed.github.io/pythonbook/04.04-hypothesis-testing.html12. 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