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I have a simple design with two factors (each with two levels) and their interaction term. I am analyzing it with mixed models.

The interaction term is not significant. Do I need to fit another model without the interaction term before performing post-hoc tests, or is it more appropriate to perform post-hoc tests on the full model with the non-significant interaction term?

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It depends on what type of post-hoc test you are trying to conduct. The Tukey HSD test compares all possible pairwise combinations. If you have 4 levels and have fit the model with an interaction term, then you don't need to do post-hoc tests. With 4 possible levels and 3 contrasts, each stratum-specific effect is modeled individually. For the sake of interpretation, you might fit a post-hoc test just to summarize these results in a different or more detailed fashion. For consistency's sake, the same results will only be obtained if you keep the interaction term in the model.

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  • $\begingroup$ Thanks! I am not using Tukey HS; I am running specific post-hoc tests using linear combinations of the model parameters. The linear combinations that I need to answer my specific questions depend on whether or not I include the interaction term. Any thought? $\endgroup$ – Cristiano Jun 6 '18 at 17:37
  • $\begingroup$ @Cristiano Thanks for explaining. Still, the model with the interaction term seems preferable because it renders empirical predictions of the stratum specific effect, and thus is more general than the model which omits that term. This is an example where the dichotomy "statistically significant [vs not]" can lead to decisions that impose rather restrictive assumptions. $\endgroup$ – AdamO Jun 6 '18 at 17:39

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