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

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Fisher, R. A. (1922a). On the interpretation of χ 2 from contingency tables, and the calculation of p. Journal of the Royal Statistical Society, 84 , 87–94. 351, 374 Fisher, R. A. (1922b). On the mathematical foundation of theoretical statistics. Philosophical Transactions of the Royal Society A, 222 , 309–368. 281 Fisher, R. A. (1925). Statistical methods for research workers. Edinburgh, UK: Oliver and Boyd. 563 Fox, J., & Weisberg, S. (2011). An R companion to applied regression (2nd ed.). Los Angeles: Sage. 79, 475, 488 Friendly, M. (2011). Histdata: Data sets from the history of statistics and data visualization [Computer software manual]. Retrieved from http://CRAN.R-project.org/package=HistData (R package version 0.6-12) 158 Gelman, A., & Stern, H. (2006). The difference between “significant” and “not significant” is not itself statistically significant. The American Statistician, 60 , 328-331. 347 Gunel, E., & Dickey, J. (1974). Bayes factors for independence in contingency tables. Biometrika, 545-557. 572 Hays, W. L. (1994). Statistics (5th ed.). Fort Worth, TX: Harcourt Brace. 433 Hedges, L. V. (1981). Distribution theory for glass’s estimator of effect size and related estimators. Journal of Educational Statistics, 6 , 107–128. 414 Hedges, L. V., & Olkin, I. (1985). Statistical methods for meta-analysis. New York: Academic Press. 414 Hogg, R. V., McKean, J. V., & Craig, A. T. (2005). Introduction to mathematical statistics (6th ed.). Upper Saddle River, NJ: Pearson. 365 Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6 , 65–70. 445 Hothersall, D. (2004). History of psychology. McGraw-Hill. 29 Hsu, J. C. (1996). Multiple comparisons: Theory and methods. London, UK: Chapman and Hall. 444, 537 Ioannidis, J. P. A. (2005). Why most published research findings are false. PLoS Med, 2 (8), 697-701. 33 Jeffreys, H. (1961). The theory of probability (3rd ed.). Oxford. 561 Johnson, V. E. (2013). Revised standards for statistical evidence. Proceedings of the National Academy of Sciences(48), 19313–19317. 564, 567 Kahneman, D., & Tversky, A. (1973). On the psychology of prediction. Psychological Review, 80 , 237-251. 29 Kass, R. E., & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90 , 773-795. 561 Keynes, J. M. (1923). A tract on monetary reform. London: Macmillan and Company. 309 Kruschke, J. K. (2011). Doing Bayesian data analysis: A tutorial with R and BUGS. Burlington, MA: Academic Press. 584 Kruskal, W. H., & Wallis, W. A. (1952). Use of ranks in one-criterion variance analysis. Journal of the American Statistical Association, 47 , 583–621. 451 Kühberger, A., Fritz, A., & Scherndl, T. (2014). Publication bias in psychology: A diagnosis based on the correlation between effect size and sample size. Public Library of Science One, 9 ,1-8.33 Larntz, K. (1978). Small-sample comparisons of exact levels for chi-squared goodness-of-fit statistics. Journal of the American Statistical Association, 73 , 253–263. 371 Lee, M. D., & Wagenmakers, E.-J. (2014). Bayesian cognitive modeling: A practical course. Cambridge University Press. 585 Lehmann, E. L. (2011). Fisher, Neyman, and the creation of classical statistics. Springer. 346 Levene, H. (1960). Robust tests for equality of variances. In I. O. et al (Ed.), Contributions to probability and statistics: Essays in honor of harold hotelling (pp. 278–292). Palo Alto, CA: Stanford University Press. 447 Long, J., & Ervin, L. (2000). Using heteroscedasticity consistent standard errors in thee linear regression model. The American Statistician, 54 , 217-224. 488 Matloff, N., & Matloff, N. S. (2011). The art of R programming: A tour of statistical software design. - 598 -
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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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- 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
Page 561 and 562: sugar 2.13 2 4.0446 0.045426 * milk
Page 563 and 564: even simpler. The main effect of su
Page 565 and 566: However, when we do the same thing
Page 567: Part VI. Endings, alternatives and
Page 570 and 571: first I want to work through a simp
Page 572 and 573: This is a very useful table, so it
Page 574 and 575: P pd|hqP phq P ph|dq “ P pdq And
Page 576 and 577: Bayesian approach relative to the o
Page 578 and 579: hardly unusual: in my experience, m
Page 580 and 581: Cumulative Probability of Type I Er
Page 582 and 583: are wrong. Worse yet, because we do
Page 584 and 585: library( BayesFactor ) # ...because
Page 586 and 587: make sense to people who already un
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 596 and 597: therapy 0.4672 1 8.5816 0.01262 * d
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
Page 606 and 607: the list that I’ve given above is
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
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