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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.

Note that the eight P values produced by two-way ANOVA are not corrected for the eight comparisons. It would seem logical to do so, but this is not traditionally (ever?) done in ANOVA.

If your data have repeated measures in any (or all) of the factors, Prism can either do repeated measures ANOVA or fit a mixed effects model. With no missing values, the P values and multiple comparisons tests are identical. If there are missing values, the results can only be interpreted if the reason for the value being missing is random. If a value is missing because it was too high to measure (or too low), then it is not missing randomly. If values are missing because a treatment is toxic, then the values are not randomly missing.

It might help to review our information on interpreting ANOVA table and mixed model results for two-way ANOVA results.

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.