Three-way ANOVA really is beyond "basic biostatistics". Multiple comparisons after three-way ANOVA stretch this definition even more. If you haven't taken the time to really understand three-way ANOVA, it is quite easy to be mislead by the results. Beware! It is not possible to understand three-way ANOVA only by reading these help screens.

Three-way ANOVA divides the total variability among values into eight components, the variability due to each of the factors (three components), due to each of the two-way interactions between two factors, due to three three-way interaction among all factors, and due to the variation among replicates (called residual or error variation). For each of those sources of variation, Prism reports the fraction of the variation attributed to that source, and (for all but the last) a P value testing the null hypothesis that the data are drawn from a population where that potential source of variation in fact contributes nothing to the overall variation among values.

Multiple comparisons testing is one of the most confusing topics in statistics. Since Prism offers nearly the same multiple comparisons tests for one-, two and three-way ANOVA, we have consolidated the information on multiple comparisons.