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The Scheffe test is the most conservative, the Tukey test is best when many comparisons are made (when there are many groups), and the Bonferroni test is preferred when few comparisons are made. However, these post-hoc tests often support the same conclusions.3 To illustrate, let’s say the independent variable has three categories. Then, a post-hoc test will examine hypotheses for whether . In addition, these tests will also examine which categories have means that are not significantly different from each other, hence, providing homogeneous subsets. An example of this approach is given later in this chapter. Knowing such subsets can be useful when the independent variable has many categories (for example, classes of employees). Figure 13.1 ANOVA: Significant and Insignificant Differences Eta-squared (η2) is a measure of association for mixed nominal-interval variables and is appropriate for ANOVA. Its values range from zero to one, and it is interpreted as the percentage of variation explained. It is a directional measure, and computer programs produce two statistics, alternating specification of the dependent variable. Finally, ANOVA can be used for testing interval-ordinal relationships. We can ask whether the change in means follows a linear pattern that is either increasing or decreasing. For example, assume we want to know whether incomes increase according to the political orientation of respondents, when measured on a seven-point Likert scale that ranges from very liberal to very conservative. If a linear pattern of increase exists, then a linear relationship is said to exist between these variables. Most statistical software packages can test for a variety of progressive relationships. ANOVA Assumptions ANOVA assumptions are essentially the same as those of the t-test: (1) the dependent variable is continuous, and the independent variable is ordinal or nominal, (2) the groups have equal variances, (3) observations are independent, and (4) the variable is normally distributed in each of the groups. The assumptions are tested in a similar manner. Relative to the t-test, ANOVA requires a little more concern regarding the assumptions of normality and homogeneity. First, like the t-test, ANOVA is not robust for the presence of outliers, and analysts examine the presence of outliers for each group. Also, ANOVA appears to be less robust than the t-test for deviations from normality. Second, regarding groups having equal variances, our main concern with homogeneity is that there are no substantial differences in the amount of variance across the groups; the test of homogeneity is a strict test, testing for any departure from equal variances, and in practice, groups may have neither equal variances nor substantial differences in the amount of variances. In these instances, a visual finding of no substantial differences suffices. Other strategies for dealing with heterogeneity are variable transformations and the removal of outliers, which increase variance, especially in small groups. Such outliers are detected by examining boxplots for each group separately. Also, some statistical software packages (such as SPSS), now offer post-hoc tests when equal variances are not assumed.4 A Working Example The U.S. Environmental Protection Agency (EPA) measured the percentage of wetland loss in watersheds between 1982 and 1992, the most recent period for which data are available (government statistics are sometimes a little old).5 An analyst wants to know whether watersheds with large surrounding populations have
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Evan M. Berman (Essential Statistics for Public Managers and Policy Analysts)