Chapters 11&12

1. Chapters 11&12 Factorial and Mixed Factor ANOVA and ANCOVA

2. ANOVA Review • Compare 2+ mean scores • One way (1 factor or IV) • Repeated measures (multiple factors) • Main effects • Interactions • F-ratio • P-value • Post hoc tests and corrections • Within and between

3. Multiple Factor ANOVA • aka Factorial ANOVA; incorporates more than one IV (factor). • Only one DV • Factor = IV • Levels are the “groups” within each factor. • In the reaction time example, there was one factor (“drug”) with three levels (beta blocker, caffeine, and placebo). • Mixed factor is both within and between in the same analysis.

4. Factorial ANOVA Example • Studies are explained by their levels • 2 x 3 or 3 x 3 x 4 • The effect of three conditions of muscle glycogen at two different exercise intensities on blood lactate. There are 2 IV (factors: glycogen and exercise intensity) and 1 DV (blood lactate). • 3 levels of muscle glycogen: depleted, loaded, normal. • 2 levels of exercise intensity: 40% and 70% VO2max. • 2 x 3 ANOVA, two-way ANOVA. • 60 subjects randomized to the 6 cells (n = 10 per cell). Between subjects.

5. Factorial ANOVA Example • Each subject, after appropriate glycogen manipulation, performs 30 minute cycle ergometer ride at either low intensity (40%) or high intensity (70%). • Blood is sampled following ride for lactate level.

6. 3 F ratios in 2-way ANOVA • 2 “Main Effects” – a ‘main effect” looks at the effect of one IV while ignoring the other IV(s), i.e., “collapsed across” the other IV(s). Based on the “marginal means” (collapsed). • Main effect for Intensity – • based on “row” marginal means (collapsed across glycogen state). • If significant, look at mean values to see which one is larger (since there are only 2 means).

7. 3 F ratios in 2-way ANOVA • Main effect for glycogen state • Compare column marginal means. • If significant, perform follow-up procedures on the 3 means (collapsed across intensity). • Main effects are easily followed up if the “interaction” (see below) is not significant. • Each main effect is treated as a single factor ANOVA while ignoring the other factor. • If the interaction is significant, focus on the interaction even if the main effects are significant. Ignore the main effects

8. Exercise Intensity Marginal Means Glycogen Condition Exercise Intensity Glycogen Marginal Means

10. Main Effect for Intensity

12. Main Effect for Glycogen

13. 3 F ratios in 2-way ANOVA • Interaction – does the effect of one IV (factor) change across levels of the other factor(s). • Significant interaction indicates that the effects of muscle glycogen on blood [lactate] differs across levels of exercise intensity. • Or equivalently, a significant interaction indicates that the effects of exercise intensity on blood [lactate] differs across different levels of muscle glycogen. • Interactions tell you that the slopes of lines of the plotted data are not parallel. • In other words the groups did not react the same way.

14. Interactions • The first F ratio to consider is the highest order (most complicated) interaction. In this example, there is only one interaction. • If the interaction is significant, then ignore the main effects and analyze the interaction. • When a significant interaction occurs, the main effects can be misleading.

16. Interaction of Intensity and Glycogen

19. Interactions • Options for Follow Up Procedures • Perform multiple pairwise comparisons; need to control familywise Type I error rate. (Bonferroni) • Tests of Simple Main Effects • Compare cell means within the levels of each factor. • Examples: • 1. Perform two 1x3 ANOVAs; one for each level of exercise intensity. • 2. Perform three 1x2 ANOVAs (single df comparisons, t tests) for each level of glycogen state. • 3. Perform both 1 and 2 above.

20. Or and

21. Interactions • Options for Follow Up Procedures (continued) • Analysis of interaction comparisons – transform the factorial into a set of smaller factorials. • Plot interaction and describe. • The choice of follow-up procedure depends on the research question(s); one may be better in one situation vs. another.

22. Main Effect for Intensity Main Effect for Glycogen Interaction of Intensity and Glycogen

23. ANCOVA • Analysis of Covariance • Combined use of ANOVA and Regression • Adjust for covariate by regressing covariate on the DV, then doing an ANOVA on the adjusted DV. • Can remove pre-treatment variations (as measured by the covariate) from the post-treatment means prior to testing groups for differences in the DV. • Example – compare strength in subjects who did Swiss Ball exercise vs. controls. • Randomization may not equate groups on body weight. • Covary for body weight prior to comparing groups.

24. ANCOVA • Issues with ANCOVA • Covariate should be highly correlated with DV. • Covariate should not be correlated with IV. • Homogeneity of Regression • Slopes of regression lines between covariate and DV must be equal across levels of the IV. • Violation implies an interaction between the covariate and IV. • Groups may differ on other variables that are not adjusted. • Abuse – arguably inappropriate to correct for pre-existing group differences if those groups were not formed by randomization.

25. ANCOVA • Advantages • More Power – due to decreased variance that must be explained by the IV (smaller error term in the F ratio). • Covariate “accounts for” some of the variance in the DV   variance that must be explained by IV to reach significance. • Some suggest use of covariate solely to increase power. • Adjusts for pre-treatment differences between groups. • If pre-treatment differences exist because groups were not randomly formed, then ANCOVA will not magically eliminate the bias that may exist with non-random assignment.

26. Next Class • Tonight: factorial ANOVA and ANCOVA in lab and stat practice • Research paper due and stat practice • Final exam next week