# Three-way ANOVA: How to interptret a significant three-way interaction with no significant post-hocs? [duplicate]

I have a three-way 2x2x3 ANOVA, which initially showed a significant AxBxC interaction

(but no significant two-way or main effects)

I've broken this down into two-way ANOVAs to try to interpret the result, however:

1) Splitting it by variable "A", doing two two-way ANOVAs (i.e. for each of two levels of "A"), gave no significant BxC interactions or main effects of B or C for either level of A

2) Splitting it by variable "B" gave no significant AxC interactions or main effects of A or C for either level of B.

My question is: How to I proceed with interpreting this?

Once you have a model that fits (and I think you should stick with one model and not split-out the data and fit several models), compute the cell means (predicted [transformed] response at each factor combination), and construct interaction plots -- for example, at each level of $A$, make a plot with levels of $C$ on the $x$ axis, means on the $y$ axis, and connect the dots at each level of $B$ (refer to an anova/design test for examples). These plots may suggest to you which comparisons of cell means tell the story. For example, maybe it's comparisons of $B$ for each combination of $A$ and $C$.