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

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8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 IV Statistical theory 269 9 Introduction to probability 275 9.1 How are probability and statistics different? . . . . . . . . . . . . . . . . . . . . . . . . . 276 9.2 What does probability mean? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 9.3 Basic probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 9.4 The binomial distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 9.5 The normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 9.6 Other useful distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 10 Estimating unknown quantities from a sample 301 10.1 Samples, populations and sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 10.2 The law of large numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 10.3 Sampling distributions and the central limit theorem . . . . . . . . . . . . . . . . . . . . 309 10.4 Estimating population parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 10.5 Estimating a confidence interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 10.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 11 Hypothesis testing 327 11.1 A menagerie of hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 11.2 Two types of errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 11.3 Test statistics and sampling distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 332 11.4 Making decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 11.5 The p value of a test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 11.6 Reporting the results of a hypothesis test . . . . . . . . . . . . . . . . . . . . . . . . . . 338 11.7 Running the hypothesis test in practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 11.8 Effect size, sample size and power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 11.9 Some issues to consider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 11.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 V Statistical tools 349 12 Categorical data analysis 351 12.1 The χ 2 goodness-of-fit test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 12.2 The χ 2 test of independence (or association) . . . . . . . . . . . . . . . . . . . . . . . . . 363 12.3 The continuity correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 12.4 Effect size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 12.5 Assumptions of the test(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 12.6 The most typical way to do chi-square tests in R . . . . . . . . . . . . . . . . . . . . . . . 371 12.7 The Fisher exact test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 12.8 The McNemar test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 12.9 What’s the difference between McNemar and independence? . . . . . . . . . . . . . . . . 377 12.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 vii
13 Comparing two means 379 13.1 The one-sample z-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 13.2 The one-sample t-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 13.3 The independent samples t-test (Student test) . . . . . . . . . . . . . . . . . . . . . . . . 389 13.4 The independent samples t-test (Welch test) . . . . . . . . . . . . . . . . . . . . . . . . . 398 13.5 The paired-samples t-test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 13.6 One sided tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 13.7 Using the t.test() function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 13.8 Effect size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 13.9 Checking the normality of a sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 13.10 Testing non-normal data with Wilcoxon tests . . . . . . . . . . . . . . . . . . . . . . . . 420 13.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 14 Comparing several means (one-way ANOVA) 425 14.1 An illustrative data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 14.2 How ANOVA works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 14.3 Running an ANOVA in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437 14.4 Effect size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 14.5 Multiple comparisons and post hoc tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 14.6 Assumptions of one-way ANOVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 14.7 Checking the homogeneity of variance assumption . . . . . . . . . . . . . . . . . . . . . 447 14.8 Removing the homogeneity of variance assumption . . . . . . . . . . . . . . . . . . . . . 449 14.9 Checking the normality assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 14.10 Removing the normality assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 14.11 On the relationship between ANOVA and the Student t test . . . . . . . . . . . . . . . . 453 14.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 15 Linear regression 457 15.1 What is a linear regression model? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 15.2 Estimating a linear regression model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 15.3 Multiple linear regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 15.4 Quantifying the fit of the regression model . . . . . . . . . . . . . . . . . . . . . . . . . . 464 15.5 Hypothesis tests for regression models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466 15.6 Testing the significance of a correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 15.7 Regarding regression coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 15.8 Assumptions of regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474 15.9 Model checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 15.10 Model selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490 15.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 16 Factorial ANOVA 497 16.1 Factorial ANOVA 1: balanced designs, no interactions . . . . . . . . . . . . . . . . . . . 497 16.2 Factorial ANOVA 2: balanced designs, interactions allowed . . . . . . . . . . . . . . . . . 506 16.3 Effect size, estimated means, and confidence intervals . . . . . . . . . . . . . . . . . . . . 513 16.4 Assumption checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 16.5 The F test as a model comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 16.6 ANOVA as a linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 16.7 Different ways to specify contrasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532 16.8 Post hoc tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 16.9 The method of planned comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 16.10 Factorial ANOVA 3: unbalanced designs . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 16.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 viii
Page 2 and 3: Learning statistics with R: A tutor
Page 4 and 5: This book is published under a Crea
Page 6 and 7: Preface Table of Contents I Backgro
Page 10 and 11: VI Endings, alternatives and prospe
Page 12 and 13: February 16, 2015 Preface to Versio
Page 14 and 15: and density is discussed. A detaile
Page 17 and 18: 1. Why do we learn statistics? “T
Page 19 and 20: And finally, an invalid argument wi
Page 21 and 22: Admission rate (both genders) 0 20
Page 23 and 24: experiment, and they don’t get an
Page 25 and 26: 2. A brief introduction to research
Page 27 and 28: different ways: the length of time
Page 29 and 30: that when I ask 100 people how they
Page 31 and 32: Table 2.1: The relationship between
Page 33 and 34: 2.3 Assessing the reliability of a
Page 35 and 36: 2.5.1 Experimental research The key
Page 37 and 38: things “inside” the study. Let
Page 39 and 40: they lack face validity, you’ll f
Page 41 and 42: that age is growing at an incredibl
Page 43 and 44: then it can be a big problem. 2.7.7
Page 45 and 46: 2.7.10 Placebo effects The placebo
Page 47 and 48: it doesn’t, which one do you thin
Page 49: Part II. An introduction to R - 35
Page 52 and 53: go out into the real world. It’s
Page 54 and 55: download Rstudio. To understand why
Page 56 and 57: 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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- 72 -
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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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- 110 -
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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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- 274 -
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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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- 300 -
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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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Learning Statistics with R - A tutorial for psychology students and other beginners, 2018a

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