The use of multiple comparisons tests after repeated measures ANOVA is a tricky topic that many statistics texts avoid. We follow methods suggested by Maxwell and Delaney(1).
With repeated measures one-way, two-way, and three-way ANOVA, Prism computes the multiple comparisons tests in two different ways, depending on whether you ask Prism (on the Repeated Measures tab of the ANOVA dialog) to assume sphericity.
If you assume sphericity
The multiple comparisons tests performed by Prism use the mean square residual for all comparisons. This is a pooled value that assess variability in all the groups. If you assume that variability really is the same in all groups (with any differences due to chance) this gives you more power. This makes sense, as you get to use data from all time points to assess variability, even when comparing only two times.
With two-way and three-way ANOVA, the pooled error term Prism uses depends on the kind of comparison. With unbalanced data, Prism compares least-squares means calculated from the model.
If you do not assume sphericity
If you check the option to not assume sphericity, Prism does two things differently.
•It applies the Geisser-Greenhouse correction when computing the P values in the ANOVA table.
•It computes the multiple comparisons differently. For each comparison of two groups, it uses only the data in those two groups (essentially performing a paired t test). This makes sense when scatter increases with time, so later treatments give a more variable response than earlier treatments. It uses the method described on pages 552-555 of Maxwell(1).
•With two-way and three-way ANOVA, a comparison between two groups that contain the same subjects (groups that differ only in a repeated measures factor) is a paired t test. A comparison between groups that contain different subjects (groups that differ in a factor without repeated measures) is an unpaired t test that does not assume equal variances (Welch's t test).
When you choose not to assume sphericity, some multiple comparisons will have more power (and narrower confidence intervals) than they would if you did not assume sphericity. But others will have less power (and wider confidence intervals).
The Geisser-Greenhouse correction adjusts only the tests in the ANOVA table. It is not applied to the degrees of freedom of the multiple comparisons, which come from the paired or unpaired t tests.
When Prism fits a mixed effects model, sphericity is always assumed, so the multiple comparisons use the fitted model. With the Kenward-Roger or Satterthwaite method, their degrees of freedom come from that method.
Reference
1. Scott E. Maxwell, Harold D. Delaney, Designing Experiments and Analyzing Data: A Model Comparison Perspective, Second Edition. IBSN:0805837183
