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

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age.group data.frame( age, age.group ) age age.group 1 60 older 2 58 older 3 24 adult BLAH BLAH BLAH 10 2 young 11 9 young It can also be useful to tabulate the output, just to see if you’ve got a nice even division of the sample: > table( age.group ) age.group young adult older 2 6 3 In the example above, I made all the decisions myself. Much like the hist() function that we saw in Chapter 6, if you want to you can delegate a lot of the choices to R. 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 R 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: > age.group2 table( age.group2 ) age.group2 (1.94,21.3] (21.3,40.7] (40.7,60.1] 2 6 3 This output takes a little bit of interpretation, but it’s not complicated. What R hasdoneisdetermined 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. The formatting on those labels might look a bit funny to those of you who haven’t studied a lot of maths, but it’s pretty simple. When R describes the first category as corresponding to the range p1.94, 21.3s what it’s saying is that the range consists of those numbers that are larger than 1.94 but less than or equal to 21.3. In other words, the weird asymmetric brackets is Rs way of telling you that if there happens to be a value that is exactly equal to 21.3, then it belongs to the first category, not the second one. Obviously, this isn’t actually possible since I’ve only specified the ages to the nearest whole number, but R doesn’t know this and so it’s trying to be precise just in case. This notation is actually pretty standard, but I suspect not everyone reading the book will have - 202 -
seen it before. In any case, those labels are pretty ugly, so it’s usually a good idea to specify your own, meaningful labels to the categories. Before moving on, I should take a moment to talk a little about the mechanics of the cut() function. Notice that R 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 quantiles() 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 lsr package has a function called quantileCut() that does exactly this: > age.group3 table( age.group3 ) age.group3 (1.94,27.3] (27.3,33.7] (33.7,60.1] 4 3 4 Notice the difference in the boundaries that the quantileCut() function 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. More generally, regardless of whether you’re using the original cut() function or the quantileCut() 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. 4 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. 7.3 A few more mathematical functions and operations In Section 7.2 I discussed the ideas behind variable transformations, and showed that a lot of the transformations that you might want to apply to your data are based on fairly simple mathematical functions and operations, of the kind that we discussed in Chapter 3. In this section I want to return to that discussion, and mention several other mathematical functions and arithmetic operations that I didn’t bother to mention when introducing you to R, but are actually quite useful for a lot of real world data analysis. Table 7.1 gives a brief overview of the various mathematical functions I want to talk about (and 4 If you’ve read further into the book, and are re-reading this section, then a good example of this would be someone choosingtodoanANOVAusingage.group3 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. - 203 -
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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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Page 169 and 170: • Measures of variability. In con
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Page 177 and 178: high-level functions all rely on th
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Page 183 and 184: Figure 6.8: Altering the scale and
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Page 199 and 200: The second way do to it is to use a
Page 201 and 202: cor( x = parenthood ) # calculate c
Page 203 and 204: 0 10 20 30 0 10 20 30 Adelaide Fitz
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Page 207 and 208: This takes the “active” figure
Page 209 and 210: 7. Pragmatic matters The garden of
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Page 219 and 220: Table 7.2: Two more arithmetic oper
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Page 237 and 238: At this point you should have two q
Page 239 and 240: (alcohol, caffeine or no drug). Bec
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Page 243 and 244: discuss melt() and cast() in a fair
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Page 253 and 254: Figure 7.1: The booksales2.csv data
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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
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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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- 586 -
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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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