Learning Statistics with R - A tutorial for psychology students and other beginners, 2018a

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So if you were forced to guess my weight, you’d guess that I’m closer to her than to them. Maybe you’d make some correction for the fact that Adam and I are males and Briony and Carol are females. The statistical name for this kind of guessing is “imputation”. Doing imputation safely is hard, but important, especially when the missing data are missing in a systematic way. Because of the fact that people who are overweight are often pressured to feel poorly about their weight (often thanks to public health campaigns), we actually have reason to suspect that the people who are not responding are more likely to be overweight than the people who do respond. Imputing a weight to Dan means that the number of overweight people in the sample will probably rise from 1 out of 3 (if we ignore Dan), to 2 out of 4 (if we impute Dan’s weight). Clearly this matters. But doing it sensibly is more complicated than it sounds. Earlier, I suggested you should treat me like Carol, since we gave the same answer to the exercise question. But that’s not quite right: there is a systematic difference between us. She answered the question, and I didn’t. Given the social pressures faced by overweight people, isn’t it likely that I’m more overweight than Carol? And of course this is still ignoring the fact that it’s not sensible to impute a single weight to me, as if you actually knew my weight. Instead, what you need to do it is impute a range of plausible guesses (referred to as multiple imputation), in order to capture the fact that you’re more uncertain about my weight than you are about Carol’s. And let’s not get started on the problem posed by the fact that Elaine didn’t send in the survey. As you can probably guess, dealing with missing data is an increasingly important topic. In fact, I’ve been told that a lot of journals in some fields will not accept studies that have missing data unless some kind of sensible multiple imputation scheme is followed. • Power analysis. In Chapter 11 I discussed the concept of power (i.e., how likely are you to be able to detect an effect if it actually exists), and referred to power analysis, a collection of tools that are useful for assessing how much power your study has. Power analysis can be useful for planning a study (e.g., figuring out how large a sample you’re likely to need), but it also serves a useful role in analysing data that you already collected. For instance, suppose you get a significant result, and you have an estimate of your effect size. You can use this information to estimate how much power your study actually had. This is kind of useful, especially if your effect size is not large. For instance, suppose you reject the null hypothesis p ă .05, but you use power analysis to figure out that your estimated power was only .08. The significant result means that, if the null hypothesis was in fact true, there was a 5% chance of getting data like this. But the low power means that, even if the null hypothesis is false, the effect size was really as small as it looks, there was only an 8% chance of getting data like the one you did. This suggests that you need to be pretty cautious, because luck seems to have played a big part in your results, one way or the other! • Data analysis using theory-inspired models. In a few places in this book I’ve mentioned response time (RT) data, where you record how long it takes someone to do something (e.g., make a simple decision). I’ve mentioned that RT data are almost invariably non-normal, and positively skewed. Additionally, there’s a thing known as the speed-accuracy tradeoff: if you try to make decisions too quickly (low RT), you’re likely to make poorer decisions (lower accuracy). So if you measure both the accuracy of a participant’s decisions and their RT, you’ll probably find that speed and accuracy are related. There’s more to the story than this, of course, because some people make better decisions than others regardless of how fast they’re going. Moreover, speed depends on both cognitive processes (i.e., time spend thinking) but also physiological ones (e.g., how fast can you move your muscles). It’s starting to sound like analysing this data will be a complicated process. And indeed it is, but one of the things that you find when you dig into the psychological literature is that there already exist mathematical models (called “sequential sampling models”) that describe how people make simple decisions, and these models take into account a lot of the factors I mentioned above. You won’t find any of these theoretically-inspired models in a standard statistics textbook. Standard stats textbooks describe standard tools, tools that could meaningfully be applied in lots of different disciplines, not just psychology. ANOVA is an example of a standard - 594 -
tool: it is just as applicable to psychology as to pharmacology. Sequential sampling models are not: they are psychology-specific, more or less. This doesn’t make them less powerful tools: in fact, if you’re analysing data where people have to make choices quickly, you should really be using sequential sampling models to analyse the data. Using ANOVA or regression or whatever won’t work as well, because the theoretical assumptions that underpin them are not well-matched to your data. In contrast, sequential sampling models were explicitly designed to analyse this specific type of data, and their theoretical assumptions are extremely well-matched to the data. Obviously, it’s impossible to cover this sort of thing properly, because there are thousands of context-specific models in every field of science. Even so, one thing that I’d like to do in later versions of the book is to give some case studies that are of particular relevance to psychologists, just to give a sense for how psychological theory can be used to do better statistical analysis of psychological data. So, in later versions of the book I’ll probably talk about how to analyse response time data, among other things. 18.2 Learning the basics, and learning them in R Okay, that was... long. And even that listing is massively incomplete. There really are a lot of big ideas in statistics that I haven’t covered in this book. It can seem pretty depressing to finish a 600-page textbook only to be told that this only the beginning, especially when you start to suspect that half of the stuff you’ve been taught is wrong. For instance, there are a lot of people in the field who would strongly argue against the use of the classical ANOVA model, yet I’ve devote two whole chapters to it! Standard ANOVA can be attacked from a Bayesian perspective, or from a robust statistics perspective, or even from a “it’s just plain wrong” perspective (people very frequently use ANOVA when they should actually be using mixed models). So why learn it at all? As I see it, there are two key arguments. Firstly, there’s the pure pragmatism argument. Rightly or wrongly, ANOVA is widely used. If you want to understand the scientific literature, you need to understand ANOVA. And secondly, there’s the “incremental knowledge” argument. In the same way that it was handy to have seen one-way ANOVA before trying to learn factorial ANOVA, understanding ANOVA is helpful for understanding more advanced tools, because a lot of those tools extend on or modify the basic ANOVA setup in some way. For instance, although mixed models are way more useful than ANOVA and regression, I’ve never heard of anyone learning how mixed models work without first having worked through ANOVA and regression. You have to learn to crawl before you can climb a mountain. Actually, I want to push this point a bit further. One thing that I’ve done a lot of in this book is talk about fundamentals. I spent a lot of time on probability theory. I talked about the theory of estimation and hypothesis tests in more detail than I needed to. When talking about R, I spent a lot of time talking about how the language works, and talking about things like writing your own scripts, functions and programs. I didn’t just teach you how to draw a histogram using hist(), I tried to give a basic overview of how the graphics system works. Why did I do all this? Looking back, you might ask whether I really needed to spend all that time talking about what a probability distribution is, or why there was even a section on probability density. If the goal of the book was to teach you how to run a t-test or an ANOVA, was all that really necessary? Or, come to think of it, why bother with R at all? There are lots of free alternatives out there: PSPP, for instance, is an SPSS-like clone that is totally free, has simple “point and click” menus, and can (I think) do every single analysis that I’ve talked about in this book. And you can learn PSPP in about 5 minutes. Was this all just a huge waste of everyone’s time??? The answer, I hope you’ll agree, is no. The goal of an introductory stats is not to teach ANOVA. It’s not to teach t-tests, or regressions, or histograms, or p-values. The goal is to start you on the path towards becoming a skilled data analyst. And in order for you to become a skilled data analyst, you need - 595 -
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Learning statistics with R: A tutor
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This book is published under a Crea
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Preface Table of Contents I Backgro
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8.6 Summary . . . . . . . . . . . .
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VI Endings, alternatives and prospe
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February 16, 2015 Preface to Versio
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and density is discussed. A detaile
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1. Why do we learn statistics? “T
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And finally, an invalid argument wi
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Admission rate (both genders) 0 20
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experiment, and they don’t get an
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2. A brief introduction to research
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different ways: the length of time
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that when I ask 100 people how they
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Table 2.1: The relationship between
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2.3 Assessing the reliability of a
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2.5.1 Experimental research The key
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things “inside” the study. Let
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they lack face validity, you’ll f
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that age is growing at an incredibl
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then it can be a big problem. 2.7.7
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2.7.10 Placebo effects The placebo
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it doesn’t, which one do you thin
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Part II. An introduction to R - 35
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go out into the real world. It’s
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download Rstudio. To understand why
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Most of this text is pretty uninter
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citation() To cite R in publication
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Table 3.1: Basic arithmetic operati
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Division / and Multiplication *, th
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sales * royalty [1] 2450 As far as
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x to the power of 0.5”. Personall
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As you can see, specifying the argu
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Figure 3.3: If you’ve typed the n
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sales.by.month sales.by.month [1]
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profit / days.per.month [1] 0.00000
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not true of R. R is not infinitely
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Table 3.3: Some more logical operat
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In other words, any.sales.this.mont
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and gotten exactly the same result.
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Figure 3.6: The options window in R
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print( keeper ) [1] 8.539539 So, fr
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to import an SPSS data file into R,
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The output here is telling you that
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Figure 4.3: The Rstudio dialog box
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Figure 4.5: The Rstudio “Environm
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this should be obvious already. But
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Figure 4.6: The “file panel” is
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from this file into my workspace. T
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2 February 28 100 high 3 March 31 2
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Figure 4.9: The Rstudio window for
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4.6.1 Special values The first thin
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x y x * y Error in x * y : non-nu
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4.7.1 Introducing factors Suppose,
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factors in 7.11.2, but for now you
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An alternative method is to use the
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4.11 Generic functions There’s on
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load (base) R Documentation The R D
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Warning Saved R objects are binary
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5. Descriptive statistics Any time
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mean of these observations is just:
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mean( afl.margins[1:5] ) [1] 36.6 A
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the median rises only to $62,500. I
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Next, let’s calculate means and m
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5.2 Measures of variability The sta
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English: which game value deviation
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and you get the same... no, wait...
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Mean − SD Mean Mean + SD Figure 5
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Negative Skew No Skew Positive Skew
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Platykurtic ("too flat") Mesokurtic
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e useful, let’s try this for a ne
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argument specifies the grouping var
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grumpiness score. 13 If the mean is
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Frequency 0 5 10 15 20 Frequency 0
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My grumpiness 40 50 60 70 80 90 4 6
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following extract illustrates the b
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Y1 4 5 6 7 8 9 10 Y2 3 4 5 6 7 8 9
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Okay, so let’s have a look at our
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pay 0.760 0.720 . 0.137 . 0.196 . d
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5 6.68 9.75 NA 5 6 5.99 5.04 72 6 B
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• Measures of variability. In con
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6. Drawing graphs Above all else sh
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delete the plot and then type a new
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Fibonacci 2 4 6 8 10 12 1 2 3 4 5 6
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high-level functions all rely on th
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type = ’p’ type = ’o’ type
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Fibonacci 2 4 6 8 10 12 1 2 3 4 5 6
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Figure 6.8: Altering the scale and
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y the height of the leftmost bar in
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6.3.1 Visual style of your histogra
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0 20 40 60 80 100 120 (a) (b) Figur
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0 20 40 60 80 100 120 Winning Margi
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0 50 100 150 200 250 300 Winning Ma
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6.5.3 Drawing multiple boxplots One
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Winning Margin 0 50 100 150 1987 19
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The second way do to it is to use a
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cor( x = parenthood ) # calculate c
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0 10 20 30 0 10 20 30 Adelaide Fitz
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value for mar is c(5.1, 4.1, 4.1, 2
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This takes the “active” figure
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7. Pragmatic matters The garden of
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in the command above I didn’t nam
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speaker ee onk oo pip makka-pakka 0
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2 7 3 3 1 3 3 -1 1 -1 BLAH BLAH BLA
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seen it before. In any case, those
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Table 7.2: Two more arithmetic oper
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fact R does provide a function for
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But suppose, on the other hand, tha
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is select several different variabl
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column number. So, if we want to pi
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case.2 upsy-daisy pip 2 case.4 makk
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Note that R will only allow you to
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7.6.2 Sorting a factor You can also
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Two other methods that I want to br
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At this point you should have two q
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(alcohol, caffeine or no drug). Bec
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3 3 female 4.6 643 7.4 226 7.3 412
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discuss melt() and cast() in a fair
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paste( hw, ng, sep = ".", collapse
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Table 7.3: The ordering of various
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Table 7.4: Standard escape characte
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sub( pattern = "a", replacement = "
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Figure 7.1: The booksales2.csv data
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note that if you don’t have the R
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etween the two. However, there are
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a third variable to the cross-tabs
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today print(today) # display the d
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encode numbers in binary, 29 0.1 is
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environment. And so when you type i
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8. Basic programming Machine dreams
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for ages, you figure out some reall
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Figure 8.2: A screenshot showing th
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8.1.5 Differences between scripts a
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to the second value in vector; and
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crunching, we tell R (on line 21) t
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What this does is create a function
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hundreds of other things besides. H
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Part IV. Statistical theory - 269 -
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me: I guess I don’t see that. Sur
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discussion of probability theory is
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In this case 11 of these 20 coin fl
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frequentists do this sometimes. And
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Probability of event 0.0 0.1 0.2 0.
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proceed to roll all 20 dice, what
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(a) Probability 0.00 0.05 0.10 0.15
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In other words, there is a 76.9% ch
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Probability Density 0.0 0.1 0.2 0.3
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Shaded Area = 68.3% Shaded Area = 9
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Probability Density 0.0 0.1 0.2 0.3
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• The χ 2 distribution is anothe
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work rather than rchisq() - the obs
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10.1.1 Defining a population A samp
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Figure 10.2: Biased sampling withou
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a stratified sampling technique you
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10.2 The law of large numbers In th
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Table 10.1: Ten replications of the
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Sample Size = 1 Sample Size = 2 Sam
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Sample Size = 1 Sample Size = 2 0.0
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efer to them. For instance, if true
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Average Sample Mean 96 98 100 102 1
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10.5 Estimating a confidence interv
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sense idea of what it means to say
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Average Attendance 0 10000 30000 19
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Here’s how to plot the means and
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Let’s suppose that this glorious
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Dan’s research hypothesis: “ESP
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the type I error rate. As Blackston
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Critical Regions for a Two−Sided
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Critical Region for a One−Sided T
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one based on Neyman’s approach to
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alternative hypothesis is a near ce
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Sampling Distribution for X if θ=.
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Table 11.2: A crude guide to unders
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ureaucratic process. It’s not par
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(binomial test was non significant)
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12. Categorical data analysis Now t
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observations that fall within the i
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it is when it predicts too many (wh
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df = 3 df = 4 df = 5 0 2 4 6 8 10 V
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That’s why I told R to use the up
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clubs 35 40 0.2 diamonds 51 60 0.3
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only one of many (albeit one of the
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That’s more or less what we’re
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Well, since these probabilities hav
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Chapek 9 has an unfortunate tendenc
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12.5 Assumptions of the test(s) All
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To get the test of independence, al
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This is a bit more output than we g
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subj.1 : 1 no :70 no :90 subj.10 :
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13. Comparing two means In the prev
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40 50 60 70 80 90 Grades Figure 13.
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Two Sided Test One Sided Test −1.
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lower.area print( lower.area ) [1]
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df = 2 df = 10 −4 −2 0 2 4 valu
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than I’ve done here. For instance
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Anastasia’s students Bernadette
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null hypothesis μ alternative hypo
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ut once again I’m going to start
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for the difference “(mean 1) minu
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null hypothesis μ alternative hypo
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13.5.1 The data The data set that w
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ciMean( x = chico$improvement ) 2.5
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Paired samples t-test Variables: gr
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Grouping variable: ID variable: tim
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In a one-sided confidence interval,
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are different tests. Secondly, I wa
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Table 13.1: A (very) rough guide to
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ut you no longer believe that the c
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Frequency 0 5 10 15 20 Normally Dis
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Sampling distribution of W (for nor
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We then count up the number of chec
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• A paired samples t-test is used
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14. Comparing several means (one-wa
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Mood Gain 0.5 1.0 1.5 placebo anxif
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This formula looks pretty much iden
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is true, then you’d expect all th
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mean square, MS w , can be viewed a
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3 placebo 0.1 0.45 -0.350 0.1225 4
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is choose an α level (i.e., accept
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names( my.anova ) [1] "coefficients
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etaSquared( x = my.anova ) eta.sq e
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One thing that bugs me slightly abo
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If we compare these three p-values
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• Homogeneity of variance. Notice
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Secondly, I did mention that it’s
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Frequency 0 1 2 3 4 5 6 7 Sample Qu
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Looking at this table, notice that
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• Post hoc analysis and correctio
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15. Linear regression The goal in t
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The Best Fitting Regression Line No
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• data. The data frame containing
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Figure 15.4: A 3D visualisation of
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Wonderful. A big number that doesn
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If our regression model has K predi
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You can see why this is handy, sinc
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what it is. The test for the signif
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Fortunately, confidence intervals f
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• Uncorrelated predictors. The id
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Outlier Outcome Predictor Figure 15
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Outcome High influence Predictor Fi
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Cook’s distance 0.00 0.02 0.04 0.
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Standardized residuals −2 −1 0
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78 Residuals vs Fitted Residuals
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In a lot of cases, the solution to
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Signif. codes: 0 *** 0.001 ** 0.01
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15.10.1 Backward elimination Okay,
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+ day 1 1.55760 1837.2 297.08 + bab
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etween those two SS values as a sum
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16. Factorial ANOVA Over the course
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At this point we have 12 different
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drug 2 3.45 1.727 18.6 8.6e-05 ***
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means can be organised into the sam
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attributable to the “ANOVA model
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Only Factor A has an effect Only Fa
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The effect of CBT (difference betwe
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Now, if you go back to the formulas
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Here’s what I mean. Again, let’
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Speaking of interaction effects, he
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16.4 Assumption checking As with on
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16.5.1 The F test comparing two mod
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the degrees of freedom here is 15.
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tfm.2 grade attend reading 1 90 yes
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line, we obtain the following: Ŷ 7
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Estimate Std. Error t value Pr(>|t|
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that contains the same data as clin
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What about the case when there does
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R comes with a variety of functions
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enlightening read. There are a lot
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16.8 Post hoc tests Time to switch
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diff lwr upr p adj anxifree:no.ther
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1 yes none 3.700 2 no none 5.550 3
Page 557 and 558: Analysis of Variance Table Response
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Page 561 and 562: sugar 2.13 2 4.0446 0.045426 * milk
Page 563 and 564: even simpler. The main effect of su
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Page 588 and 589: -------------- [1] Non-indep. (a=1)
Page 590 and 591: Before moving on, it’s worth high
Page 592 and 593: 17.6.2 The Bayesian version Okay, s
Page 594 and 595: [2] day : 0.2724027 ˘0% [3] baby.s
Page 596 and 597: therapy 0.4672 1 8.5816 0.01262 * d
Page 598 and 599: command that I use when I want to g
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Page 602 and 603: • Other types of correlations. In
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Page 612 and 613: Fisher, R. A. (1922a). On the inter
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