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

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therapy 0.4672 1 8.5816 0.01262 * drug:therapy 0.2711 2 2.4898 0.12460 That’s pretty clearly showing us evidence for a main effect of drug at p ă .001, an effect of therapy at p ă .05 and no interaction. 17.7.2 The Bayesian version How do we do the same thing using Bayesian methods? The BayesFactor package contains a function called anovaBF() that does this for you. It uses a pretty standard formula and data structure, so the command should look really familiar. Just like we did with regression, it will be useful to save the output to a variable: > models models Bayes factor analysis -------------- [1] drug : 245.9026 ˘0% [2] therapy : 0.7316007 ˘0% [3] drug + therapy : 698.3343 ˘0.96% [4] drug + therapy + drug:therapy : 688.3077 ˘1.3% Against denominator: Intercept only --- Bayes factor type: BFlinearModel, JZS This looks very similar to the output we obtained from the regressionBF() function, and with good reason. Remember what I said back in Section 16.6: under the hood, ANOVA is no different to regression, and both are just different examples of a linear model. Becasue of this, the anovaBF() reports the output in much the same way. For instance, if we want to identify the best model we could use the same commands that we used in the last section. One variant that I find quite useful is this: > models/max(models) Bayes factor analysis -------------- [1] drug : 0.3521273 ˘0.96% [2] therapy : 0.001047637 ˘0.96% [3] drug + therapy : 1 ˘0% [4] drug + therapy + drug:therapy : 0.9856421 ˘1.62% Against denominator: mood.gain ~ drug + therapy --- Bayes factor type: BFlinearModel, JZS - 582 -
By “dividing” the models output by the best model (i.e., max(models)), what R is doing is using the best model (which in this case is drugs + therapy) as the denominator, which gives you a pretty good sense of how close the competitors are. For instance, the model that contains the interaction term is almost as good as the model without the interaction, since the Bayes factor is 0.98. In other words, the data do not clearly indicate whether there is or is not an interaction. 17.7.3 Constructing Bayesian Type II tests Okay, that’s all well and good, you might be thinking, but what do I report as the alternative to the p-value? In the classical ANOVA table, you get a single p-value for every predictor in the model, so you can talk about the significance of each effect. What’s the Bayesian analog of this? It’s a good question, but the answer is tricky. Remember what I said in Section 16.10 about ANOVA being complicated. Even in the classical version of ANOVA there are several different “things” that ANOVA might correspond to. Specifically, I discussed how you get different p-values depending on whether you use Type I tests, Type II tests or Type III tests. To work out which Bayes factor is analogous to “the” p-value in a classical ANOVA, you need to work out which version of ANOVA you want an analog for. For the purposes of this section, I’ll assume you want Type II tests, because those are the ones I think are most sensible in general. As I discussed back in Section 16.10, TypeIItestsfor a two-way ANOVA are reasonably straightforward, but if you have forgotten that section it wouldn’t be a bad idea to read it again before continuing. Assuming you’ve had a refresher on Type II tests, let’s have a look at how to pull them from the Bayes factor table. Suppose we want to test the main effect of drug. The null hypothesis for this test corresponds to a model that includes an effect of therapy, but no effect of drug. The alternative hypothesis is the model that includes both. In other words, what we want is the Bayes factor corresponding to this comparison: Null model: mood.gain ~ therapy Alternative model: mood.gain ~ therapy + drug As it happens, we can read the answer to this straight off the table because it corresponds to a comparison between the model in line 2 of the table and the model in line 3: the Bayes factor in this case represents evidence for the null of 0.001 to 1. Or, more helpfully, the odds are about 1000 to 1 against the null. The main effect of therapy can be calculated in much the same way. In this case, the null model is the one that contains only an effect of drug, and the alternative is the model that contains both. So the relevant comparison is between lines 2 and 1 in the table. The odds in favour of the null here are only 0.35 to 1. Again, I find it useful to frame things the other way around, so I’d refer to this as evidence of about 3 to 1 in favour of an effect of therapy. Finally, in order to test an interaction effect, the null model here is one that contains both main effects but no interaction. The alternative model adds the interaction. That is: Null model: mood.gain ~ drug + therapy Alternative model: mood.gain ~ drug + therapy + drug:therapy If we look those two models up in the table, we see that this comparison is between the models on lines 3 and 4 of the table. The odds of 0.98 to 1 imply that these two models are fairly evenly matched. You might be thinking that this is all pretty laborious, and I’ll concede that’s true. At some stage I might consider adding a function to the lsr package that would automate this process and construct something like a “Bayesian Type II ANOVA table” from the output of the anovaBF() function. However, I haven’t had time to do this yet, nor have I made up my mind about whether it’s really a good idea to do this. In the meantime, I thought I should show you the trick for how I do this in practice. The - 583 -
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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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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
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Page 582 and 583: are wrong. Worse yet, because we do
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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 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
Page 604 and 605: measurements mean that independence
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Page 608 and 609: So if you were forced to guess my w
Page 610 and 611: to be able to do more than ANOVA, m
Page 612 and 613: Fisher, R. A. (1922a). On the inter
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