The Factorial ANOVA Is Done, Now What Do I Do?
After conducting a factorial ANOVA, one typically inspects the results of that ANOVA and then decides what additional analyses are needed. It is often recommended that this take place in a top-down fashion, inspecting the highest-order interaction term first and then moving down to interactions of the next lower order, an so on until reaching the main effects.
Two basic principles are:
· If an interaction is significant, conduct tests of simple (conditional) effects to help explain the interaction, and
· Effects which do not participate in higher-order interactions are easier to interpret than are those that do.
Consider a three-way analysis. If the triple interaction, AxBxC is significant, one might decide to test the simple (conditional) interaction of AxB at each level of C. If the AxB interaction at C=1 is significant, one might then decide to test the simple, simple (doubly conditional) main effects of A at each level of B for those cells where C=1. If the AxB interaction at C=2 is not significant, then one is likely to want to look the (simple main) effects of A and of B for those cells where C=1.
If the triple interaction is not significant, one next looks at the two-way interactions. Suppose that AxB is significant but the other two interactions are not. The significant AxB interaction might then be followed by tests of the simple main effects of A at each level of B. For each significant simple main effect of A, when there are more than two levels of A, one might want to conduct pairwise comparisons or more complex contrasts among A’s marginal means for the data at the specified level of B. Since the main effect of C does not participate in any significant interactions, it can now be more simply interpreted -- if there are more than two levels of C, one might want to conduct pairwise comparisons or more complex contrasts involving the marginal means of C.
In some situations one might be justified in interpreting main effects even when they do participate in significant interactions, especially when those interactions are monotonic. For example, even though AxB is significant, if the direction of the effect of A is the same at all levels of B, there may be some merit in talking about the main effect of A, ignoring B.
The most important thing to keep in mind is that the contrasts that are made (interactions, simple interactions, main effects, simple main effects, contrasts involving marginal means, and so on) should be contrasts help you answer questions of interest about the data. My presentation here has been rather abstract, treating A, B, and C as generic factors. When A, B, and C are particular variables, the recommendations given here may or may not make good sense. When they do not make good sense, do not follow them -- make the comparisons that do make good sense!
B. J. White, graduate student in PSYC 6431 in the Spring of 2002, prepared the following ANOVA Flow Chart based on the generic recommendations made above. Thanks, B.J.