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

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The last part of the ANOVA table is the calculation of the p values. Once again, there is nothing new here: for each of our two factors, what we’re trying to do is test the null hypothesis that there is no relationship between the factor and the outcome variable (I’ll be a bit more precise about this later on). To that end, we’ve (apparently) followed a similar strategy that we did in the one way ANOVA, and have calculated an F -statistic for each of these hypotheses. To convert these to p values, all we need to do is note that the that the sampling distribution for the F statistic under the null hypothesis (that the factor in question is irrelevant) is an F distribution: and that two degrees of freedom values are those corresponding to the factor, and those corresponding to the residuals. For the drug factor we’re talking about an F distribution with 2 and 14 degrees of freedom (I’ll discuss degrees of freedom in more detail later). In contrast, for the therapy factor sampling distribution is F with 1 and 14 degrees of freedom. If we really wanted to, we could calculate the p value ourselves using the pf() function (see Section 9.6). Just to prove that there’s nothing funny going on, here’s what that would look like for the drug variable: > pf( q=26.15, df1=2, df2=14, lower.tail=FALSE ) [1] 1.871981e-05 And as you can see, this is indeed the p value reported in the ANOVA table above. At this point, I hope you can see that the ANOVA table for this more complicated analysis corresponding to model.2 should be read in much the same way as the ANOVA table for the simpler analysis for model.1. In short, it’s telling us that the factorial ANOVA for our 3 ˆ 2 design found a significant effect of drug (F 2,14 “ 26.15,pă .001) as well as a significant effect of therapy (F 1,14 “ 7.08,p“ .02). Or, to use the more technically correct terminology, we would say that there are two main effects of drug and therapy. At the moment, it probably seems a bit redundant to refer to these as “main” effects: but it actually does make sense. Later on, we’re going to want to talk about the possibility of “interactions” between the two factors, and so we generally make a distinction between main effects and interaction effects. 16.1.3 How are the sum of squares calculated? In the previous section I had two goals: firstly, to show you that the R commands needed to do factorial ANOVA are pretty much the same ones that we used for a one way ANOVA. The only difference is the formula argument to the aov() function. Secondly, I wanted to show you what the ANOVA table looks like in this case, so that you can see from the outset that the basic logic and structure behind factorial ANOVA is the same as that which underpins one way ANOVA. Try to hold onto that feeling. It’s genuinely true, insofar as factorial ANOVA is built in more or less the same way as the simpler oneway ANOVA model. It’s just that this feeling of familiarity starts to evaporate once you start digging into the details. Traditionally, this comforting sensation is replaced by an urge to murder the the authors of statistics textbooks. Okay, let’s start looking at some of those details. The explanation that I gave in the last section illustrates the fact that the hypothesis tests for the main effects (of drug and therapy in this case) are F -tests, but what it doesn’t do is show you how the sum of squares (SS) values are calculated. Nor does it tell you explicitly how to calculate degrees of freedom (df values) though that’s a simple thing by comparison. Let’s assume for now that we have only two predictor variables, Factor A and Factor B. If we use Y to refer to the outcome variable, then we would use Y rci to refer to the outcome associated with the i-thmemberofgrouprc (i.e., level/row r for Factor A and level/column c forFactorB).Thus, if we use Ȳ to refer to a sample mean, we can use the same notation as before to refer to group means, marginal means and grand means: that is, Ȳrc is the sample mean associated with the rth level of Factor Aandthecth level of Factor B, Ȳr.would be the marginal mean for the rth level of Factor A, Ȳ.c would be the marginal mean for the cth level of Factor B, and Ȳ.. is the grand mean. In other words, our sample - 502 -
means can be organised into the same table as the population means. For our clinical trial data, that table looks like this: no therapy CBT total placebo Ȳ 11 Ȳ 12 Ȳ 1. anxifree Ȳ 21 Ȳ 22 Ȳ 2. joyzepam Ȳ 31 Ȳ 32 Ȳ 3. total Ȳ .1 Ȳ .2 Ȳ .. And if we look at the sample means that I showed earlier, we have Ȳ11 “ 0.30, Ȳ12 “ 0.60 etc. In our clinical trial example, the drugs factor has 3 levels and the therapy factor has 2 levels, and so what we’re trying to run is a 3 ˆ 2 factorial ANOVA. However, we’ll be a little more general and say that Factor A (the row factor) has R levels and Factor B (the column factor) has C levels, and so what we’re runnning here is an R ˆ C factorial ANOVA. Now that we’ve got our notation straight, we can compute the sum of squares values for each of the two factors in a relatively familiar way. For Factor A, our between group sum of squares is calculated by assessing the extent to which the (row) marginal means Ȳ1., Ȳ2. etc, are different from the grand mean Ȳ .. . We do this in the same way that we did for one-way ANOVA: calculate the sum of squared difference between the Ȳi. values and the Ȳ.. values. Specifically, if there are N people in each group, then we calculate this: Rÿ SS A “pN ˆ Cq `Ȳr. ´ Ȳ..˘2 As with one-way ANOVA, the most interesting 3 part of this formula is the `Ȳ r. ´ Ȳ..˘2 bit, which corresponds to the squared deviation associated with level r. All that this formula does is calculate this squared deviation for all R levels of the factor, add them up, and then multiply the result by N ˆ C. The reason for this last part is that there are multiple cells in our design that have level r on Factor A: in fact, there are C of them, one corresponding to each possible level of Factor B! For instance, in our toy example, there are two different cells in the design corresponding to the anxifree drug: one for people with no.therapy, and one for the CBT group. Not only that, within each of these cells there are N observations. So, if we want to convert our SS value into a quantity that calculates the between-groups sum of squares on a “per observation” basis, we have to multiply by by N ˆ C. The formula for factor B is of course the same thing, just with some subscripts shuffled around: SS B “pN ˆ Rq r“1 Cÿ c“1 `Ȳ.c ´ Ȳ..˘2 Now that we have these formulas, we can check them against the R output from the earlier section. First, notice that we calculated all the marginal means (i.e., row marginal means Ȳr. and column marginal means Ȳ.c) earlier using aggregate(), and we also calculated the grand mean. Let’s repeat those calculations, but this time we’ll save the results to varibles so that we can use them in subsequent calculations: > drug.means therapy.means grand.mean
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Learning statistics with R: A tutor
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This book is published under a Crea
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Preface Table of Contents I Backgro
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8.6 Summary . . . . . . . . . . . .
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VI Endings, alternatives and prospe
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February 16, 2015 Preface to Versio
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and density is discussed. A detaile
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1. Why do we learn statistics? “T
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And finally, an invalid argument wi
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Admission rate (both genders) 0 20
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experiment, and they don’t get an
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2. A brief introduction to research
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different ways: the length of time
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that when I ask 100 people how they
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Table 2.1: The relationship between
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2.3 Assessing the reliability of a
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2.5.1 Experimental research The key
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things “inside” the study. Let
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they lack face validity, you’ll f
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that age is growing at an incredibl
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then it can be a big problem. 2.7.7
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2.7.10 Placebo effects The placebo
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it doesn’t, which one do you thin
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Part II. An introduction to R - 35
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go out into the real world. It’s
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download Rstudio. To understand why
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Most of this text is pretty uninter
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citation() To cite R in publication
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Table 3.1: Basic arithmetic operati
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Division / and Multiplication *, th
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sales * royalty [1] 2450 As far as
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x to the power of 0.5”. Personall
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As you can see, specifying the argu
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Figure 3.3: If you’ve typed the n
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sales.by.month sales.by.month [1]
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profit / days.per.month [1] 0.00000
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not true of R. R is not infinitely
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Table 3.3: Some more logical operat
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In other words, any.sales.this.mont
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and gotten exactly the same result.
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Figure 3.6: The options window in R
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print( keeper ) [1] 8.539539 So, fr
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to import an SPSS data file into R,
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The output here is telling you that
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Figure 4.3: The Rstudio dialog box
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Figure 4.5: The Rstudio “Environm
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this should be obvious already. But
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Figure 4.6: The “file panel” is
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from this file into my workspace. T
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2 February 28 100 high 3 March 31 2
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Figure 4.9: The Rstudio window for
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4.6.1 Special values The first thin
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x y x * y Error in x * y : non-nu
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4.7.1 Introducing factors Suppose,
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factors in 7.11.2, but for now you
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An alternative method is to use the
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4.11 Generic functions There’s on
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load (base) R Documentation The R D
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Warning Saved R objects are binary
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5. Descriptive statistics Any time
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mean of these observations is just:
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mean( afl.margins[1:5] ) [1] 36.6 A
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the median rises only to $62,500. I
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Next, let’s calculate means and m
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5.2 Measures of variability The sta
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English: which game value deviation
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and you get the same... no, wait...
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Mean − SD Mean Mean + SD Figure 5
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Negative Skew No Skew Positive Skew
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Platykurtic ("too flat") Mesokurtic
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e useful, let’s try this for a ne
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argument specifies the grouping var
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grumpiness score. 13 If the mean is
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Frequency 0 5 10 15 20 Frequency 0
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My grumpiness 40 50 60 70 80 90 4 6
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following extract illustrates the b
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Y1 4 5 6 7 8 9 10 Y2 3 4 5 6 7 8 9
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Okay, so let’s have a look at our
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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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Page 473 and 474: The Best Fitting Regression Line No
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Page 479 and 480: Wonderful. A big number that doesn
Page 481 and 482: If our regression model has K predi
Page 483 and 484: You can see why this is handy, sinc
Page 485 and 486: what it is. The test for the signif
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Page 489 and 490: • Uncorrelated predictors. The id
Page 491 and 492: Outlier Outcome Predictor Figure 15
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
Page 513 and 514: At this point we have 12 different
Page 515: drug 2 3.45 1.727 18.6 8.6e-05 ***
Page 519 and 520: attributable to the “ANOVA model
Page 521 and 522: Only Factor A has an effect Only Fa
Page 523 and 524: The effect of CBT (difference betwe
Page 525 and 526: Now, if you go back to the formulas
Page 527 and 528: Here’s what I mean. Again, let’
Page 529 and 530: Speaking of interaction effects, he
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 and 542: Estimate Std. Error t value Pr(>|t|
Page 543 and 544: that contains the same data as clin
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
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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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