Three-way ANOVA, also called three-factor ANOVA, determines how a response is affected by three factors. For example, you might measure a response to three different drugs, in men and women, with two different pretreatments. In this example, drug treatment is one factor, gender is the other, and pretreatment is the third. Read elsewhere to learn about interpreting the results.
ANOVA also assumes that all sets of replicates have the same SD overall, and that any differences between SDs are due to random sampling.
Three-way ANOVA works by comparing the differences among group means with the pooled standard deviations of the groups. If subjects were given more than one treatment sequentially, or the experimental design worked with sets of matched subjects, then you should use repeated measures ANOVA. Prism cannot calculate three-way ANOVA with repeated measures in any factor.
The term “error” refers to the difference between each value and the mean of all the replicates. The results of three-way ANOVA only make sense when the scatter is random – that whatever factor caused a value to be too high or too low affects only that one value. Prism cannot test this assumption. You must think about the experimental design. For example, the errors are not independent if you have six replicates, but these were obtained from two animals in triplicate. In this case, some factor may cause all values from one animal to be high or low.
Three-way ANOVA compares the means. It is possible to have a tiny P value – clear evidence that the population means are different – even if the distributions overlap considerably. In some situations – for example, assessing the usefulness of a diagnostic test – you may be more interested in the overlap of the distributions than in differences between means.
Don't mix up three way ANOVA with one way ANOVA with three groups. With three way ANOVA, there are three grouping variables, maybe gender, presence or absence of disease, and control vs. treated. With one-way ANOVA there is one grouping variable (perhaps treatment). If there are three alternative treatments, you need one-way ANOVA not three-way ANOVA.
Prism performs Type I ANOVA, also known as fixed-effect ANOVA. This tests for differences among the means of the particular groups you have collected data from. Different calculations are needed if you randomly selected groups from an infinite (or at least large) number of possible groups, and want to reach conclusions about differences among ALL the groups, even the ones you didn't include in this experiment.