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

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value of 43.5 as the expected baseline grade. However, now that we’ve recoded our variables, the baseline corresponds to a student who has read the textbook and did attend class, and for that student we would expect a grade of 89.5. 16.6.4 How to encode non binary factors as contrasts At this point, I’ve shown you how we can view a 2 ˆ 2 ANOVA into a linear model. And it’s pretty easy to see how this generalises to a 2 ˆ 2 ˆ 2 ANOVA or a 2 ˆ 2 ˆ 2 ˆ 2 ANOVA... it’s the same thing, really: you just add a new binary variable for each of your factors. Where it begins to get trickier is when we consider factors that have more than two levels. Consider, for instance, the 3 ˆ 2 ANOVA that we ran earlier in this chapter using the clin.trial data. How can we convert the three-level drug factor into a numerical form that is appropriate for a regression? The answer to this question is pretty simple, actually. All we have to do is realise that a three-level factor can be redescribed as two binary variables. Suppose, for instance, I were to create a new binary variable called druganxifree. Whenever the drug variable is equal to "anxifree" we set druganxifree = 1. Otherwise, we set druganxifree = 0. This variable sets up a contrast, in this case between anxifree and the other two drugs. By itself, of course, the druganxifree contrast isn’t enough to fully capture all of the information in our drug variable. We need a second contrast, one that allows us to distinguish between joyzepam and the placebo. To do this, we can create a second binary contrast, called drugjoyzepam, which equals 1 if the drug is joyzepam, and 0 if it is not. Taken together, these two contrasts allows us to perfectly discriminate between all three possible drugs. The table below illustrates this: drug druganxifree drugjoyzepam "placebo" 0 0 "anxifree" 1 0 "joyzepam" 0 1 If the drug administered to a patient is a placebo, then both of the two contrast variables will equal 0. If the drug is Anxifree, then the druganxifree variable will equal 1, and drugjoyzepam will be 0. The reverse is true for Joyzepam: drugjoyzepam is 1, and druganxifree is 0. Creating contrast variables manually is not too difficult to do using basic R commands. For example, here’s how we would create the druganxifree variable: > druganxifree druganxifree [1] 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 The clin.trial$drug == "anxifree" part of the command returns a logical vector that has a value of TRUE if the drug is Anxifree, and a value of FALSE if the drug is Joyzepam or the placebo. The as.numeric() function just converts TRUE to 1 and FALSE to 0. Obviously, this command creates the druganxifree variable inside the workspace. If you wanted to add it to the clin.trial data frame, you’d use a commmand like this instead: > clin.trial$druganxifree
that contains the same data as clin.trial, but with the two factors represented in terms of the contrast variables: > clin.trial.2 clin.trial.2 druganxifree drugjoyzepam therapyCBT mood.gain 1 0 0 0 0.5 2 0 0 0 0.3 3 0 0 0 0.1 4 1 0 0 0.6 5 1 0 0 0.4 6 1 0 0 0.2 7 0 1 0 1.4 8 0 1 0 1.7 BLAH BLAH BLAH It’s not as pretty as the original clin.trial data, but it’s definitely saying the same thing. We have now recoded our three-level factor in terms of two binary variables, and we’ve already seen that ANOVA and regression behave the same way for binary variables. However, there are some additional complexities that arise in this case, which we’ll discuss in the next section. 16.6.5 The equivalence between ANOVA and regression for non-binary factors Now we have two different versions of the same data set: our original data frame clin.trial in which the drug variable is expressed as a single three-level factor, and the expanded data set clin.trial.2 in which it is expanded into two binary contrasts. Once again, the thing that we want to demonstrate is that our original 3 ˆ 2 factorial ANOVA is equivalent to a regression model applied to the contrast variables. Let’s start by re-running the ANOVA: > drug.anova summary( drug.anova ) Df Sum Sq Mean Sq F value Pr(>F) drug 2 3.45 1.727 26.15 1.9e-05 *** therapy 1 0.47 0.467 7.08 0.019 * Residuals 14 0.92 0.066 Obviously, there’s no surprises here. That’s the exact same ANOVA that we ran earlier, except for the fact that I’ve arbitrarily decided to rename the output variable as drug.anova for some stupid reason. 12 Next, let’s run a regression, using druganxifree, drugjoyzepam and therapyCBT as the predictors. Here’s what we get: > drug.regression summary( drug.regression ) BLAH BLAH BLAH Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.289 0.121 2.38 0.032 * 12 Future versions of this book will try to be a bit more consistent with the naming scheme for variables. One of the many problems with having to write a lengthy text very quickly to meet a teaching deadline is that you lose some internal consistency. - 529 -
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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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5. Descriptive statistics Any time
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mean of these observations is just:
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mean( afl.margins[1:5] ) [1] 36.6 A
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the median rises only to $62,500. I
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Next, let’s calculate means and m
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5.2 Measures of variability The sta
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English: which game value deviation
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and you get the same... no, wait...
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Mean − SD Mean Mean + SD Figure 5
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Negative Skew No Skew Positive Skew
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Platykurtic ("too flat") Mesokurtic
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e useful, let’s try this for a ne
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argument specifies the grouping var
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grumpiness score. 13 If the mean is
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Frequency 0 5 10 15 20 Frequency 0
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My grumpiness 40 50 60 70 80 90 4 6
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following extract illustrates the b
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Y1 4 5 6 7 8 9 10 Y2 3 4 5 6 7 8 9
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Okay, so let’s have a look at our
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pay 0.760 0.720 . 0.137 . 0.196 . d
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5 6.68 9.75 NA 5 6 5.99 5.04 72 6 B
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• Measures of variability. In con
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6. Drawing graphs Above all else sh
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delete the plot and then type a new
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Fibonacci 2 4 6 8 10 12 1 2 3 4 5 6
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high-level functions all rely on th
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type = ’p’ type = ’o’ type
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Fibonacci 2 4 6 8 10 12 1 2 3 4 5 6
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Figure 6.8: Altering the scale and
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y the height of the leftmost bar in
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6.3.1 Visual style of your histogra
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0 20 40 60 80 100 120 (a) (b) Figur
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0 20 40 60 80 100 120 Winning Margi
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0 50 100 150 200 250 300 Winning Ma
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6.5.3 Drawing multiple boxplots One
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Winning Margin 0 50 100 150 1987 19
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The second way do to it is to use a
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cor( x = parenthood ) # calculate c
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0 10 20 30 0 10 20 30 Adelaide Fitz
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value for mar is c(5.1, 4.1, 4.1, 2
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This takes the “active” figure
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7. Pragmatic matters The garden of
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in the command above I didn’t nam
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speaker ee onk oo pip makka-pakka 0
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2 7 3 3 1 3 3 -1 1 -1 BLAH BLAH BLA
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seen it before. In any case, those
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Table 7.2: Two more arithmetic oper
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fact R does provide a function for
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But suppose, on the other hand, tha
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is select several different variabl
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column number. So, if we want to pi
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case.2 upsy-daisy pip 2 case.4 makk
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Note that R will only allow you to
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7.6.2 Sorting a factor You can also
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Two other methods that I want to br
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At this point you should have two q
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(alcohol, caffeine or no drug). Bec
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3 3 female 4.6 643 7.4 226 7.3 412
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discuss melt() and cast() in a fair
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paste( hw, ng, sep = ".", collapse
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Table 7.3: The ordering of various
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Table 7.4: Standard escape characte
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sub( pattern = "a", replacement = "
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Figure 7.1: The booksales2.csv data
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note that if you don’t have the R
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etween the two. However, there are
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a third variable to the cross-tabs
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today print(today) # display the d
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encode numbers in binary, 29 0.1 is
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environment. And so when you type i
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8. Basic programming Machine dreams
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for ages, you figure out some reall
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Figure 8.2: A screenshot showing th
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8.1.5 Differences between scripts a
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to the second value in vector; and
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crunching, we tell R (on line 21) t
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What this does is create a function
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hundreds of other things besides. H
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Part IV. Statistical theory - 269 -
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me: I guess I don’t see that. Sur
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discussion of probability theory is
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In this case 11 of these 20 coin fl
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frequentists do this sometimes. And
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Probability of event 0.0 0.1 0.2 0.
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proceed to roll all 20 dice, what
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(a) Probability 0.00 0.05 0.10 0.15
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In other words, there is a 76.9% ch
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Probability Density 0.0 0.1 0.2 0.3
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Shaded Area = 68.3% Shaded Area = 9
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Probability Density 0.0 0.1 0.2 0.3
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• The χ 2 distribution is anothe
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work rather than rchisq() - the obs
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10.1.1 Defining a population A samp
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Figure 10.2: Biased sampling withou
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a stratified sampling technique you
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10.2 The law of large numbers In th
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Table 10.1: Ten replications of the
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Sample Size = 1 Sample Size = 2 Sam
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Sample Size = 1 Sample Size = 2 0.0
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efer to them. For instance, if true
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Average Sample Mean 96 98 100 102 1
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10.5 Estimating a confidence interv
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sense idea of what it means to say
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Average Attendance 0 10000 30000 19
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Here’s how to plot the means and
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Let’s suppose that this glorious
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Dan’s research hypothesis: “ESP
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the type I error rate. As Blackston
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Critical Regions for a Two−Sided
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Critical Region for a One−Sided T
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one based on Neyman’s approach to
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alternative hypothesis is a near ce
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Sampling Distribution for X if θ=.
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Table 11.2: A crude guide to unders
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ureaucratic process. It’s not par
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(binomial test was non significant)
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12. Categorical data analysis Now t
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observations that fall within the i
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it is when it predicts too many (wh
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df = 3 df = 4 df = 5 0 2 4 6 8 10 V
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That’s why I told R to use the up
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clubs 35 40 0.2 diamonds 51 60 0.3
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only one of many (albeit one of the
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That’s more or less what we’re
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Well, since these probabilities hav
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Chapek 9 has an unfortunate tendenc
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12.5 Assumptions of the test(s) All
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To get the test of independence, al
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This is a bit more output than we g
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subj.1 : 1 no :70 no :90 subj.10 :
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13. Comparing two means In the prev
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40 50 60 70 80 90 Grades Figure 13.
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Two Sided Test One Sided Test −1.
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lower.area print( lower.area ) [1]
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df = 2 df = 10 −4 −2 0 2 4 valu
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than I’ve done here. For instance
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Anastasia’s students Bernadette
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null hypothesis μ alternative hypo
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ut once again I’m going to start
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for the difference “(mean 1) minu
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null hypothesis μ alternative hypo
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13.5.1 The data The data set that w
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ciMean( x = chico$improvement ) 2.5
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Paired samples t-test Variables: gr
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Grouping variable: ID variable: tim
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In a one-sided confidence interval,
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are different tests. Secondly, I wa
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Table 13.1: A (very) rough guide to
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ut you no longer believe that the c
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Frequency 0 5 10 15 20 Normally Dis
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Sampling distribution of W (for nor
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We then count up the number of chec
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• A paired samples t-test is used
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14. Comparing several means (one-wa
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Mood Gain 0.5 1.0 1.5 placebo anxif
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This formula looks pretty much iden
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is true, then you’d expect all th
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mean square, MS w , can be viewed a
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3 placebo 0.1 0.45 -0.350 0.1225 4
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is choose an α level (i.e., accept
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names( my.anova ) [1] "coefficients
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etaSquared( x = my.anova ) eta.sq e
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One thing that bugs me slightly abo
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If we compare these three p-values
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• Homogeneity of variance. Notice
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Secondly, I did mention that it’s
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Frequency 0 1 2 3 4 5 6 7 Sample Qu
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Looking at this table, notice that
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• Post hoc analysis and correctio
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15. Linear regression The goal in t
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The Best Fitting Regression Line No
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• data. The data frame containing
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Figure 15.4: A 3D visualisation of
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Wonderful. A big number that doesn
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If our regression model has K predi
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You can see why this is handy, sinc
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what it is. The test for the signif
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Fortunately, confidence intervals f
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• Uncorrelated predictors. The id
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Page 493 and 494: Outcome High influence Predictor Fi
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Page 497 and 498: Standardized residuals −2 −1 0
Page 499 and 500: 78 Residuals vs Fitted Residuals
Page 501 and 502: In a lot of cases, the solution to
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Page 505 and 506: 15.10.1 Backward elimination Okay,
Page 507 and 508: + day 1 1.55760 1837.2 297.08 + bab
Page 509 and 510: etween those two SS values as a sum
Page 511 and 512: 16. Factorial ANOVA Over the course
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Page 515 and 516: drug 2 3.45 1.727 18.6 8.6e-05 ***
Page 517 and 518: means can be organised into the sam
Page 519 and 520: attributable to the “ANOVA model
Page 521 and 522: Only Factor A has an effect Only Fa
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Page 531 and 532: 16.4 Assumption checking As with on
Page 533 and 534: 16.5.1 The F test comparing two mod
Page 535 and 536: the degrees of freedom here is 15.
Page 537 and 538: tfm.2 grade attend reading 1 90 yes
Page 539 and 540: line, we obtain the following: Ŷ 7
Page 541: Estimate Std. Error t value Pr(>|t|
Page 545 and 546: What about the case when there does
Page 547 and 548: R comes with a variety of functions
Page 549 and 550: enlightening read. There are a lot
Page 551 and 552: 16.8 Post hoc tests Time to switch
Page 553 and 554: diff lwr upr p adj anxifree:no.ther
Page 555 and 556: 1 yes none 3.700 2 no none 5.550 3
Page 557 and 558: Analysis of Variance Table Response
Page 559 and 560: 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
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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
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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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