# 2.0 Situation - One-Way ANOVA

by

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

*since there is just one*

**One-Way ANOVA**^{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

*. 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*

**pair-wise comparisons***.*

**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.

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