Solved – Three-way ANOVA: How to interptret a significant three-way interaction with no significant post-hocs?

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?

My suggestion is to stop looking at asterisks and to start looking at plots.

For starters, you don't even know how well your model fits the data. Go back to your original 3-factor model and examine residual plots — residuals versus fitted values and residuals versus each factor's levels. If in that first plot you see a kind of horn shape, it would suggest that using a transformed response like square-root, log, or reciprocal might give you a better-fitting model. Sometimes, in such cases, transformations simplify the interaction structure too.

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 $C$.

Once you have that, then figure out how to do the corresponding post-hoc tests. Probably not a good idea to use hand calculations for this because there are too many sources of confusion with three factors. I'm not sure what software you're using but I or someone else might be able to show more specifically what to do.

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