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More on ANOVA

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

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More on ANOVA

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  1. More on ANOVA

  2. Overview • ANOVA as Regression • Comparison Methods

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

  4. 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

  5. Example of Dummy Variables for Three Groups

  6. Regression Equation for ANOVA • bo is mean of base group • b1 and b2 indicate differences between base group and each of the other two groups

  7. 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

  8. 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).

  9. 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.

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

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

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

  13. Non-Orthogonal Comparisons Group C 1 C 2 Placebo +2 +1 Treatment A -1 0 Treatment B -1 -1 +2 0 +1 These two comparisons do not ask independent questions

  14. 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.

  15. 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.

  16. 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

  17. 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)

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

  19. Fisher’s LSD • Least Significant Difference • Pairwise comparisons only • Liberal

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

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

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

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