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

2. ANOVA Review Compare 2+ mean scores Main effects Interactions F-ratio
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. Glycogen Condition 40% D40 L40 N40 MM40% 70% D70 L70 N70 MM70% MMD MML
Exercise Intensity
Marginal Means
Glycogen Condition
Depleted
Loaded
Normal
40%
D40
L40
N40
MM40%
70%
D70
L70
N70
MM70%
MMD
MML
MMN
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. Depleted
Loaded
Normal
40%
D40
L40
N40
70%
D70
L70
N70 Or
D40
D70
L40
L70
N40
N70
D40
L40
N40
and
D70
L70
N70

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