More on ANOVA

Overview ANOVA as Regression Comparison Methods

ANOVA AS REGRESSION Predict scores on Y (the DV) Predictors are dummy variables indicating group membership

Dummy Variables Group membership is categorical Need one less dummy variable than the number of groups If you are in the group, your score on that dummy variable = 1 If you are not in that group, your score on that dummy variable = 0

Example of Dummy Variables for Three Groups X1X1 X2X2 Group 110 Group 201 Group 300

Regression Equation for ANOVA b o is mean of base group b 1 and b 2 indicate differences between base group and each of the other two groups

COMPARISON METHODS A significant F-test tells you that the groups differ, but not which groups. Multiple comparison methods provide specific comparisons of group means.

Planned Contrasts Decide which groups (or combinations) you wish to compare before doing the ANOVA. The comparisons must be orthogonal to each other (statistically independent).

Choosing Weights Assign a weight to each group. The weights have to add up to zero. Weights for the two sides must balance. Check for orthogonality of each pair of comparisons.

Example of a Planned Comparison GroupWeight Placebo+2 Treatment A-1 Treatment B-1 This compares the average of Treatments A and B to the Placebo mean.

Another Planned Comparison GroupWeight Placebo 0 Treatment A-1 Treatment B +1 This one leaves out the Placebo group and compares the two treatments.

Check for Orthogonality GroupC 1C 2 Placebo+20 Treatment A-1-1 Treatment B-1 +1 Multiply the weights and then add up the products. The two comparisons are orthogonal if the sum is zero. 0 +1

Non-Orthogonal Comparisons GroupC 1C 2 Placebo+2+1 Treatment A-10 Treatment B-1 -1 These two comparisons do not ask independent questions

Selecting Comparisons Maximum number of comparisons is number of groups minus 1. Start with the most important comparison. Then find a second comparison that is orthogonal to the first one. Each comparison must be orthogonal to every other comparison.

How Planned Contrasts Work A Sum of Squares is computed for each contrast, depending on the weights. An F-test for the contrast is then computed.

SPSS Contrasts Deviation: compare each group to the overall mean Simple: compare a reference group to each of the other groups Difference: compare the mean of each group to the mean of all previous group means

More SPSS Contrasts Helmert: compare the mean of each group to the mean of all subsequent group means Repeated: compare the mean of each group to the mean of the subsequent group Polynomial: compare the pattern of means across groups to a function (e.g., linear, quadratic, cubic)

POST HOC COMPARISONS Done after an ANOVA has been done Need not be orthogonal Less powerful than planned contrasts

Fisher’s LSD Least Significant Difference Pairwise comparisons only Liberal

Bonferroni Pairwise comparisons only Divide alpha by number of tests More conservative than LSD

Tukey’s HSD Similar to Bonferroni, but more powerful with large number of means Pairwise comparisons only Critical value increases with number of groups

Take-Home Points ANOVA is a special case of linear regression. There are lots of ways to compare specific means.