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

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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 ~ drug model, let’s instead do it using therapy as the predictor. If we run this ANOVA, here’s what we get: > summary( aov( mood.gain ~ therapy, data = clin.trial )) Df Sum Sq Mean Sq F value Pr(>F) therapy 1 0.47 0.467 1.71 0.21 Residuals 16 4.38 0.274 Overall, it looks like there’s no significant effect here at all but, as we’ll see in Chapter 16 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 p1, 16q “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: > t.test( mood.gain ~ therapy, data = clin.trial, var.equal = TRUE ) Two Sample t-test data: mood.gain by therapy t = -1.3068, df = 16, p-value = 0.2098 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -0.84495 0.20051 sample estimates: mean in group none mean in group CBT 0.72222 1.04444 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 tp16q “´1.3068. However, there is a fairly straightforward relationship here. If we square the t-statistic > 1.3068 ^ 2 [1] 1.7077 we get the F -statistic from before. 14.12 Summary 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 will appear in a later version of this book. 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 Chapter 16. In terms of what we have discussed, the key topics were: • The basic logic behind how ANOVA works (Section 14.2) and how to run one in R(Section 14.3). • How to compute an effect size for an ANOVA (Section 14.4) - 454 -
• Post hoc analysis and corrections for multiple testing (Section 14.5). • The assumptions made by ANOVA (Section 14.6). • How to check the homogeneity of variance assumption (Section 14.7) and what to do if it is violated (Section 14.8). • How to check the normality assumption (Section 14.9) and what to do if it is violated (Section 14.10). 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 Sahai and Ageel (2000). 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. - 455 -
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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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Page 419 and 420: Paired samples t-test Variables: gr
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Page 423 and 424: In a one-sided confidence interval,
Page 425 and 426: are different tests. Secondly, I wa
Page 427 and 428: Table 13.1: A (very) rough guide to
Page 429 and 430: ut you no longer believe that the c
Page 431 and 432: Frequency 0 5 10 15 20 Normally Dis
Page 433 and 434: Sampling distribution of W (for nor
Page 435 and 436: We then count up the number of chec
Page 437 and 438: • A paired samples t-test is used
Page 439 and 440: 14. Comparing several means (one-wa
Page 441 and 442: Mood Gain 0.5 1.0 1.5 placebo anxif
Page 443 and 444: This formula looks pretty much iden
Page 445 and 446: is true, then you’d expect all th
Page 447 and 448: mean square, MS w , can be viewed a
Page 449 and 450: 3 placebo 0.1 0.45 -0.350 0.1225 4
Page 451 and 452: is choose an α level (i.e., accept
Page 453 and 454: names( my.anova ) [1] "coefficients
Page 455 and 456: etaSquared( x = my.anova ) eta.sq e
Page 457 and 458: One thing that bugs me slightly abo
Page 459 and 460: If we compare these three p-values
Page 461 and 462: • Homogeneity of variance. Notice
Page 463 and 464: Secondly, I did mention that it’s
Page 465 and 466: Frequency 0 1 2 3 4 5 6 7 Sample Qu
Page 467: Looking at this table, notice that
Page 471 and 472: 15. Linear regression The goal in t
Page 473 and 474: The Best Fitting Regression Line No
Page 475 and 476: • data. The data frame containing
Page 477 and 478: Figure 15.4: A 3D visualisation of
Page 479 and 480: Wonderful. A big number that doesn
Page 481 and 482: If our regression model has K predi
Page 483 and 484: You can see why this is handy, sinc
Page 485 and 486: what it is. The test for the signif
Page 487 and 488: Fortunately, confidence intervals f
Page 489 and 490: • Uncorrelated predictors. The id
Page 491 and 492: Outlier Outcome Predictor Figure 15
Page 493 and 494: Outcome High influence Predictor Fi
Page 495 and 496: Cook’s distance 0.00 0.02 0.04 0.
Page 497 and 498: Standardized residuals −2 −1 0
Page 499 and 500: 78 Residuals vs Fitted Residuals
Page 501 and 502: In a lot of cases, the solution to
Page 503 and 504: Signif. codes: 0 *** 0.001 ** 0.01
Page 505 and 506: 15.10.1 Backward elimination Okay,
Page 507 and 508: + day 1 1.55760 1837.2 297.08 + bab
Page 509 and 510: etween those two SS values as a sum
Page 511 and 512: 16. Factorial ANOVA Over the course
Page 513 and 514: At this point we have 12 different
Page 515 and 516: drug 2 3.45 1.727 18.6 8.6e-05 ***
Page 517 and 518: 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
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Analysis of Variance Table Response
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III tests (which are simple) before
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sugar 2.13 2 4.0446 0.045426 * milk
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even simpler. The main effect of su
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However, when we do the same thing
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Part VI. Endings, alternatives and
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first I want to work through a simp
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This is a very useful table, so it
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P pd|hqP phq P ph|dq “ P pdq And
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Bayesian approach relative to the o
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hardly unusual: in my experience, m
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Cumulative Probability of Type I Er
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are wrong. Worse yet, because we do
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library( BayesFactor ) # ...because
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make sense to people who already un
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-------------- [1] Non-indep. (a=1)
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Before moving on, it’s worth high
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17.6.2 The Bayesian version Okay, s
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[2] day : 0.2724027 ˘0% [3] baby.s
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therapy 0.4672 1 8.5816 0.01262 * d
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command that I use when I want to g
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• Other types of correlations. In
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measurements mean that independence
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the list that I’ve given above is
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So if you were forced to guess my w
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to be able to do more than ANOVA, m
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Fisher, R. A. (1922a). On the inter
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