I ran a Generalized Linear Mixed Model in R and included an interaction effect between two predictors. The interaction was not significant, but the main effects (the two predictors) both were.
Now many textbook examples tell me that if there is a significant effect of the interaction, the main effects cannot be interpreted. But what if your interaction is not significant?
Can I conclude that the two predictors have an effect on the response? Or is it better to run a new model where I leave out the interaction? I prefer not to do so, because I would then have to control for multiple testing.
Best Answer
A little niggle
I hope that's not true. They should say that if there is an interaction term, say between X and Z called XZ, then the interpretation of the individual coefficients for X and for Z cannot be interpreted in the same way as if XZ were not present. You can definitely interpret it.
Question 2
If the interaction makes theoretical sense then there is no reason not to leave it in, unless concerns for statistical efficiency for some reason override concerns about misspecification and allowing your theory and your model to diverge.
Given that you have left it in, then interpret your model using marginal effects in the same way as if the interaction were significant. For reference, I include a link to Brambor, Clark and Golder (2006) who explain how to interpret interaction models and how to avoid the common pitfalls.
Think of it this way: you often have control variables in a model that turn out not to be significant, but you don't (or shouldn't) go chopping them out at the first sign of missing stars.
Question 1
You ask whether you can 'conclude that the two predictors have an effect on the response?' Apparently you can, but you can also do better. For the model with the interaction term you can report what effect the two predictors actually have on the dependent variable (marginal effects) in a way that is indifferent to whether the interaction is significant, or even present in the model.
The Bottom Line
If you remove the interaction you are re-specifying the model. This may be a reasonable thing to do for many reasons, some theoretical and some statistical, but making it easier to interpret the coefficients is not one of them.