How do you investigate lower-order interactions which include one factor that is part of a significant higher-order interaction?
Let's assume a research design with 4 factors (A, B, C, D), each with two levels, and I want to predict an outcome variable. Using linear mixed models (lmer in R), we first define a full model with the four-way interaction. Using drop1 reveals no significant four-way-interaction, therefore we drop it from the model and include all possible three-way-interactions. Using drop1 reveals now an interaction of AxBxC. To follow up on that, we run separate models for the two levels of A, e.g., to investigate how the interaction of BxC depends on the levels of A.
But having run these analyses, what can I say about a possible interaction of CxD? Does it make sense to have a separate look at that, even though C interacts with AxB?