Linear mixed model interaction not significant but post-hoc tests significant Suppose I have a linear mixed model with three binary factors A, B, C.
The main effects are all significant but only the interaction A*B is significant, none of the interactions including C is significant.
I then do post-hoc pairwise t-tests with Bonferroni correction.
These pairwise tests reveal that the influence of C is only significant for one out of the four possible values of (A,B). For all other values of (A,B) the post-hoc test is clearly insignificant (even without Bonferroni correction).
What does that mean?
 A: Having a significant AxB interaction term implies that the effect of A changes with a corresponding change in the value of B, and vice versa.  This is to say nothing of C, other than that by including it in the model, you are partitioning variance unique to C out of your error term.  It is quite possible to have a significant two-way interaction between A and B without C influencing that relationship.  One way to look at significant interaction terms is to view them as "permission" to probe further to look for conditional effects; finding a significant AxB interaction suggests that follow-up comparisons ought to be done for A at different levels of B, and vice versa.  The finding that the effect of C is significant in one of your comparisons would usually be ignored, given the interaction term is not significant (although, as another user pointed out, significance is an arbitrary threshold, so p-values of .06 and .05 don't really represent a categorical difference in interpretation).  Furthermore, it also possible to find a significant AxB interaction, and then, upon probing for conditional effects, you find no significant comparisons.  The significant interaction is telling you "the effect of A changes significantly as values of B change", but says nothing about the effect of A at any one particular value of B.
