I know there are similar questions on this topic but I have checked all of them and haven't found the specific answer to my (simplistic) questions. There are 3 explanatory variables: A(3 levels, between), B(2 levels, within), C (4 levels, within). The 3-way interaction AxBxC is not significant. Of the 2-way interactions, only AxB is significant. My first question is whether it is even legitimate to interpret 2-way interactions if the 3-way is not significant? My second question is whether it is legitimate to decompose the significant 2-way term by looking at it at each level of the third variable (C)? My final question is whether this discussion is modified if the main effect of C is found to be significant. Thanks!
It is quite legitimate to interpret 2-way interactios in the absence of 3-way interaction. The converse (interpreting a 3-way interaction in the absence of lower order effects) is what would be debatable.
Not quite sure what you mean by "decompose the significant 2-way term by looking at it at each level of the third variable". You mean performing different ANOVA analysis, splitting the sample by levels of C? I guess if C matters for the AxB effect, this would show up in a significant 3-way interaction, see no advantage in doing what you propose.