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I'm working with a response variable with a poisson distribution and also with a mixed factor, therefore I'm using the glmer library for generalized linear mixed models. There are 3 explanatory variables: A(two levels), B (3 levels) and C(2 levels). After running the model, it showed that the 3-way interaction (ABC) was non significant. Yet, two 2-way interactions are indeed significant (AB and also BC, p<0.05). It's an experimental design, there's no correlation between variables and also there is no overdispersion nor outliers.

How can I proceed with this model? Should I keep the 3-way interaction and analyse the combination of all factors in post hoc tests? Or should I drop it? Also, is it appropiate to run 2 separate Tukey tests for each 2-way interaction that was significant?

Any help or recommendation is welcome. Also books or papers explaining what to do in such scenario. Thanks!

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If your 3-way interaction is not significant, I would remove that from the model and run the 2-way interaction. This will help with your model fit.

I would then see which interactions are significant and run the pairwise comparisons and plot the data.

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