How does Prism compute post tests following repeated measures two-way ANOVA?
Prism only offers post tests comparing two columns at a certain row, but the repeated measures factor can be by row or by column. The calculations are different.
Each column represents a different time point ("Matching by rows")
You are looking at one treatment (for each post test) and comparing two time points. This is sort of like a paired t test. In this case, the ANOVA table partitions the overall variation into that due to the treatment factor, that due to the repeated measurement factor, that due to variation between subjects, and remaining experimental (residual) error. The appropriate MS value to use in the post test is MSresidual, and the corresponding DF value is DFresidual. Prism has always handled this situation correctly.
Each row represents a different time point ("Matching by columns").
For each post test, you are comparing two groups at one time point. So for that comparison, the fact that the study was repeated measures really doesn't matter. At each time point you are looking at, you have one group of subjects treated one way and another group of subjects treated the other way. While the ANOVA table separated variation between subjects from experimental (residual) error, this distinction is irrelevant to the post test. So the correct MS value to use in the post test combines both the MSresidual and the MSsubject. This MS value (which is called MSwithincells and is not shown on the ANOVA table), can be computed from the sum-of-squares (SS) terms in the ANOVA table.
MSwithincells = [(SSsubject + SSresidual) / (DFsubject + DFresidual)]
The appropriate degrees of freedom for the post test is the sum of DFsubject and DFresidual.
MSwithincells can also be defined in this equivalent form:
(MSsubject*DFsubject + MSresidual*DFresidual) / (DFsubject + DFresidual)
You can see that MSwithincells is a weighted average of MSsubject and MSresidual, so its value lies between MSsubject and MSresidual.
When you perform regular (not repeated measures) ANOVA, the MSresidual term accounts for all the variability not explained by either the row or column factor. So the MSresidual term in ordinary two-way ANOVA is identical to MSwithincells defined above for repeated measures ANOVA.
For each post test, the t ratio is computed as the difference between means divided by: SQRT(MS*( (1/Na) + (1/NB))), where MS is the appropriate means square value as computed above, and Na and Nb are the number of subjects in each of the two groups you are comparing.
Note that there is a bug in Prism 3 and early releases of Prism 4 (up to 4.02 and 4.0b, but fixed in 4.03 and 4.0c). These versions of Prism incorrectly uses the MSsubject and DFsubject terms when performing post tests. The final versions of Prism 4 (4.03 and 4.0c) and Prism 5 do the calculations correctlty.
References:
G Keppel and TD Wickens, page 452.
SE Maxwell and HD Delaney, page 604.