2.0 Situation - One-Way ANOVA

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In Chapter 1, tools for comparing the means of two groups were considered. More generally, these methods are used for a quantitative response and a categorical explanatory variable (group) which had two and only two levels. The MockJury data set actually contained three groups (Figure 2-1) with Beautiful, Average, and Unattractive rated pictures randomly assigned to the subjects for sentence ratings. In a situation with more than two groups, we have two choices. First, we could rely on our two group comparisons, performing tests for every possible pair (Beautiful vs Average, Beautiful vs Unattractive, and Average vs Unattractive). We spent Chapter 1 doing inferences for differences between Average and Unattractive. The other two comparisons would lead us to initially end up with three p-values and no direct answer about our initial question of interest - is there some overall difference in the average sentences provided across the groups? In this chapter, we will learn a new method, called Analysis of Variance, ANOVA, that directly assesses whether there is evidence of some overall difference in the means among the groups. This version of an ANOVA is called a One-Way ANOVA since there is just one 21 grouping variable. After we perform our One-Way ANOVA test for overall evidence of a difference, we will revisit the comparisons similar to those considered in Chapter 1 to get more details on specific differences among the pairs of groups - what we call pair-wise comparisons. An issue is created when you perform many tests simultaneously and we will augment our previous methods with an adjusted method for pairwise comparisons to make our results valid called Tukey's Honest Significant Difference.

To make this more concrete, we return to the original MockJury data, making side-by-side boxplots and beanplots (Figure 2-1) as well summarizing the sentences for the three groups using favstats.

> favstats(Years~Attr,data=MockJury)

> require(heplots)

> require(mosaic)

> data(MockJury)

> par(mfrow=c(1,2))

> boxplot(Years~Attr,data=MockJury)

> beanplot(Years~Attr,data=MockJury,log="",col="bisque",method="jitter")

> favstats(Years~Attr,data=MockJury)

 .group min Q1 median Q3 max mean sd n missing 1 Beautiful 1 2 3 6.5 15 4.333333 3.405362 39 0 2 Average 1 2 3 5.0 12 3.973684 2.823519 38 0 3 Unattractive 1 2 5 10.0 15 5.810811 4.364235 37 0

There are slight differences in the sample sizes in the three groups with 37 Unattractive, 38 Average and 39 Beautiful group responses, providing a data set has a total sample size of N=114. The Beautiful and Average groups do not appear to be very different with means of 4.33 and 3.97 years. In Chapter 1, we found moderate evidence regarding the difference in Average and Unattractive. It is less clear whether we might find evidence of a difference between Beautiful and Unattractive groups since we are comparing means of 5.81 and 4.33 years. All the distributions appear to be right skewed with relatively similar shapes. The variability in Average and Unattractive groups seems like it could be slightly different leading to an overall concern of whether the variability is the same in all the groups.

21In Chapter 3, we will discuss methods for when there are two categorical explanatory variables that is called the Two-Way ANOVA.