How Prism computes multiple comparison tests after one-, two-, and three-way ANOVA.

Prism  6 and 7 

This eight-page document (.DOCX) also in PDF format reviews exactly how Prism does the calculations. These are all standard methods, but this document concisely explains the algorithms we use. All of the methods come from Maxwell and Delaney. 

Prism 8 

Prism 8 adds more tests and fits a mixed model to do (essentially) repeated measures ANOVA with missing values. This 20 page document (pdf) explains exactly how Prism does these tests (here is the .docx version).