Learning Statistics with R

543 Pages·2016·5.7 MB·English

by  Navarro D.

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ID:13570633 5 800090 850531
www.lulu.com
Learning statistics with R:
A tutorial for psychology students and other beginners
(Version 0.3)
Daniel Navarro
University of Adelaide
[email protected]
Online:
ua.edu.au/ccs/teaching/lsr
www.lulu.com/content/13570633
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This book was brought to you today by the letter ‘R’.
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iv
Table of Contents
Preface ix
I Background 1
1 Why do we learn statistics? 3
1.1 On the psychology of statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 The cautionary tale of Simpson’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Statistics in psychology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Statistics in everyday life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 There’s more to research methods than statistics . . . . . . . . . . . . . . . . . . . . . . 10
2 A brief introduction to research design 11
2.1 Introduction to psychological measurement . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Scales of measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Assessing the reliability of a measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 The “role” of variables: predictors and outcomes . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Experimental and non-experimental research . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Assessing the validity of a study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Confounds, artifacts and other threats to validity . . . . . . . . . . . . . . . . . . . . . . 24
2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
II An introduction to R 33
3 Getting started with R 35
3.1 Installing R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Typing commands at the R console . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Doing simple calculations with R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Storing a number as a variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 Using functions to do calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6 Storing many numbers as a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.7 Storing text data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.8 Storing “true or false” data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.9 Indexing vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.10 Quitting R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 Additional R concepts 67
4.1 Using comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Installing and loading packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Managing the workspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 Navigating the file system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.5 Loading and saving data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.6 Useful things to know about variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.7 Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.8 Data frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.9 Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
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4.10 Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.11 Generic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.12 Getting help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
III Working with data 103
5 Descriptive statistics 105
5.1 Measures of central tendency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2 Measures of variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.3 Skew and kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.4 Getting an overall summary of a variable . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.5 Descriptive statistics separately for each group . . . . . . . . . . . . . . . . . . . . . . . 128
5.6 Standard scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.7 Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.8 Handling missing values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6 Drawing graphs 147
6.1 An overview of R graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.2 An introduction to plotting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.3 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.4 Stem and leaf plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
6.5 Boxplots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.6 Scatterplots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.7 Bar graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.8 Saving image files using R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7 Pragmatic matters 185
7.1 Tabulating and cross-tabulating data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
7.2 Transforming and recoding a variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.3 A few more mathematical functions and operations . . . . . . . . . . . . . . . . . . . . . 193
7.4 Extracting a subset of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.5 Extracting a subset of a data frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
7.6 Sorting, flipping and merging data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.7 Reshaping a data frame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
7.8 Working with text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
7.9 Reading unusual data files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
7.10 Coercing data from one class to another . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
7.11 Other useful data structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
7.12 Miscellaneous topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
7.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
8 Basic programming 243
8.1 Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
8.2 Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.3 Conditional statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
8.4 Writing functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
8.5 Implicit loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
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IV Statistical theory 259
9 Introduction to probability 261
9.1 Probability theory v. statistical inference . . . . . . . . . . . . . . . . . . . . . . . . . . 262
9.2 Basic probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9.3 The binomial distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
9.4 The normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
9.5 Other useful distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
9.6 What does probability mean? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
10 Estimating population parameters from a sample 285
10.1 Samples, populations and sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
10.2 Estimating population means and standard deviations . . . . . . . . . . . . . . . . . . . 288
10.3 Sampling distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
10.4 The central limit theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
10.5 Estimating a confidence interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
11 Hypothesis testing 303
11.1 A menagerie of hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
11.2 Two types of errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
11.3 Test statistics and sampling distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 308
11.4 Making decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
11.5 The p value of a test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
11.6 Reporting the results of a hypothesis test . . . . . . . . . . . . . . . . . . . . . . . . . . 314
11.7 Running the hypothesis test in practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
11.8 Effect size, sample size and power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
11.9 Some issues to consider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
11.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
V Statistical tools 327
12 Categorical data analysis 329
12.1 The χ2 goodness-of-fit test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
12.2 The χ2 test of independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
12.3 The continuity correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
12.4 Effect size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
12.5 Assumptions of the test(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
12.6 The Fisher exact test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
12.7 The McNemar test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
12.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
13 Comparing two means 353
13.1 The one-sample z-test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
13.2 The one-sample t-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
13.3 The independent samples t-test (Student test) . . . . . . . . . . . . . . . . . . . . . . . . 365
13.4 The independent samples t-test (Welch test) . . . . . . . . . . . . . . . . . . . . . . . . . 375
13.5 The paired-samples t-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
13.6 Effect size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
13.7 Checking the normality of a sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
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13.8 Testing non-normal data with Wilcoxon tests . . . . . . . . . . . . . . . . . . . . . . . . 390
13.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
14 Comparing several means (one-way ANOVA) 395
14.1 An illustrative data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
14.2 How ANOVA works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
14.3 Running an ANOVA in R. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
14.4 Effect size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
14.5 Multiple comparisons and post hoc tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
14.6 Assumptions of one-way ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
14.7 Checking the homogeneity of variance assumption . . . . . . . . . . . . . . . . . . . . . 417
14.8 Removing the homogeneity of variance assumption . . . . . . . . . . . . . . . . . . . . . 419
14.9 Checking the normality assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
14.10 Removing the normality assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420
14.11 On the relationship between ANOVA and the Student t test . . . . . . . . . . . . . . . . 423
14.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
15 Linear regression 427
15.1 What is a linear regression model? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
15.2 Estimating a linear regression model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
15.3 Multiple linear regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
15.4 Quantifying the fit of the regression model . . . . . . . . . . . . . . . . . . . . . . . . . . 434
15.5 Hypothesis tests for regression models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
15.6 Regarding regression coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
15.7 Assumptions of regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
15.8 Model checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
15.9 Model selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
15.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
16 Factorial ANOVA 465
16.1 Factorial ANOVA 1: balanced designs, no interactions . . . . . . . . . . . . . . . . . . . 465
16.2 Factorial ANOVA 2: balanced designs, interactions allowed . . . . . . . . . . . . . . . . . 474
16.3 Effect size, estimated means, and confidence intervals . . . . . . . . . . . . . . . . . . . . 481
16.4 Assumption checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
16.5 The F test as a model comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
16.6 ANOVA as a linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
16.7 Different ways to specify contrasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
16.8 Post hoc tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
16.9 The method of planned comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
16.10 Factorial ANOVA 3: unbalanced designs . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
16.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
17 Epilogue 521
17.1 The undiscovered statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
17.2 Learning the basics, and learning them in R . . . . . . . . . . . . . . . . . . . . . . . . . 529
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531
viii
Preface
There’s a part of me that really doesn’t want to publish this book. It’s not finished.
AndwhenIsaythat,Imeanit. Thereferencingisspottyatbest,thechaptersummariesarejustlists
of section titles, there’s no index, there are no exercises for the reader, the organisation is suboptimal,
andthecoverageoftopicsisjustnotcomprehensiveenoughformyliking. Additionally,therearesections
withcontentthatI’mnothappywith,figuresthatreallyneedtoberedrawn,andI’vehadalmostnotime
to hunt down inconsistencies, typos, or errors. In other words, this book is not finished. If I didn’t have
a looming teaching deadline and a baby due in a few weeks, I really wouldn’t be making this available at
all.
Whatthismeansisthatifyouareanacademiclookingforteachingmaterials,aPh.D.studentlooking
tolearnR,orjustamemberofthegeneralpublicinterestedinstatistics,Iwouldadviseyoutobecautious.
What you’re looking at is a first draft, and it may not serve your purposes. If we were living in the days
when publishing was expensive and the internet wasn’t around, I would never consider releasing a book
inthisform. Thethoughtofsomeongshellingout$80forthis(whichiswhatacommercialpublishertold
me it would retail for when they offered to distribute it) makes me feel more than a little uncomfortable.
However, it’s the 21st century, so I can post the pdf on my website for free, and I can distribute hard
copies via a print-on-demand service for less than half what a textbook publisher would charge. And so
my guilt is assuaged, and I’m willing to share! With that in mind, you can obtain free soft copies and
cheap hard copies online, from the following webpages:
Soft copy: ua.edu.au/ccs/teaching/lsr
Hard copy: www.lulu.com/content/13570633
Even so, the warning still stands: what you are looking at is Version 0.3 of a work in progress. If and
whenithitsVersion1.0, Iwouldbewillingtostandbehindtheworkandsay, yes, thisisatextbookthat
I would encourage other people to use. At that point, I’ll probably start shamelessly flogging the thing
on the internet and generally acting like a tool. But until that day comes, I’d like it to be made clear
that I’m really ambivalent about the work as it stands.
All of the above being said, there is one group of people that I can enthusiastically endorse this book
to: the psychology students taking our undergraduate research methods classes (DRIP and DRIP:A) in
2013. For you, this book is ideal, because it was written to accompany your stats lectures. If a problem
arises due to a shortcoming of these notes, I can and will adapt content on the fly to fix that problem.
Effectively, you’ve got a textbook written specifically for your classes, distributed for free (electronic
copy) or at near-cost prices (hard copy). Better yet, the notes have been tested: Version 0.1 of these
notes was used in the 2011 class, Version 0.2 was used in the 2012 class, and now you’re looking at the
new and improved Version 0.3. I’m not saying these notes are titanium plated awesomeness on a stick –
thoughifyouwantedtosaysoonthestudentevaluationforms, thenyou’retotallywelcometo–because
they’re not. But I am saying that they’ve been tried out in previous years and they seem to work okay.
Besides, there’s a group of us around to troubleshoot if any problems come up, and you can guarantee
that at least one of your lecturers has read the whole thing cover to cover!
Okay, with all that out of the way, I should say something about what the book aims to be. At its
core,itisanintroductorystatisticstextbookpitchedprimarilyatpsychologystudents. Assuch,itcovers
the standard topics that you’d expect of such a book: study design, descriptive statistics, the theory of
hypothesis testing, t-tests, χ2 tests, ANOVA and regression. However, there are also several chapters
devotedtotheRstatisticalpackage,includingachapterondatamanipulationandanotheroneonscripts
and programming. Moreover, when you look at the content presented in the book, you’ll notice a lot of
topicsthataretraditionallysweptunderthecarpetwhenteachingstatisticstopsychologystudents. The
Bayesian/frequentist divide is openly disussed in the probability chapter, and the disagreement between
Neyman and Fisher about hypothesis testing makes an appearance. The difference between probability
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