# Sensible to remove interaction term in this instance, to interpret a main effect?

Predictor A is a continuous predictor.

Predictor B is a dummy coded categorical predictor with three levels.

I run a regression including PA, PB and PA*PB.

The results indicate a significant effect of PB (level 3 vs. level 1) and an interaction between PA and PB (level 2 vs. level 1).

I understand that so long as the interaction is in the model, the effect of PB (level 3 vs. level 1) is only a simple effect. Does it make sense to remove the interaction between PA and PB in order to interpret the main effect of PB (level 3 vs. level 1)? Or am I stuck only being able to talk about the simple effect of PB (3 vs 1) because another level of PB is involved in an interaction?

I would not change the model in order to understand how to interpret it. And as a rule, it's best not to interpret main effects (what you call simple effects) separately when there exists an interaction in your model. This may be slightly less of a problem in your specific example, where the interaction is significant only for a subset of your levels. But it's still worth taking this precaution, because the model captures differences between levels even if they are not significant.

No problem arises if you consider the main effects and interactions jointly. This is difficult to do from a table of coefficients, however. Instead, plot out your fitted model in a manner that integrates the main effects and interactions - including your non-significant level differences. Visualising complex fitted models is a very important step in interpreting them. It allows you to compare levels in an intuitive manner while capturing the joint effect of all terms in your model.

The effects package is a good tool for this, if you use R:

Fox, J. (2003). Effect displays in R for generalised linear models. Journal of statistical software, 8(15), 1-27.

https://www.jstatsoft.org/article/view/v008i15/effect-displays-revised.pdf

And here's the package manual

No, I think you're interpreting this wrong.

First of all, you should have obtained five regression coefficients, for PA, PB2, PB3, PA*PB2 and PA*PB3. Of those, you say PB3 and PA*PB2 are statistically significant. For PB3, it is correct to say that it indicates an effect (a higher outcome y, whatever your y is) of level 3 compared to level 1.

The other term however (the interaction), what it really shows is that PA does have a statistically significant effect, but only among observations where PB==2. Whereas PA does not have an effect for participants at PB level 1 or 3.

In other words, it's not an effect of PB per se, it is that PB modifies the effect of PA on your outcome, so the effect of PA is different for each level of PB. That's a completely unrelated matter to the effect of PB on the outcome.

So, to answer your question, there's really no reason to remove the interaction term, you're free to interpret both effects as you see them in your model. One effect of level 3 PB compared to level 1, and one effect of PA but only for PB level 2 participants.

In fact IMHO you should definitely keep your interaction term in the model if (a) it is conceptually (based on your actual problem) more reasonable to include it than not, and (b) you get a better model fit (e.g. according to a likelihood ratio rest) with the interaction term than without.

• Thanks for the reply! "Whereas PA does not have an effect for participants at PB level 1 or 3." Is this because the PA x PB (3v1) interaction is not significant? Does that necessarily mean that it won't have an effect in either of those levels if I were to run simple effects analysis on those levels separately? – Dave Jun 6 '18 at 21:06
• @Dave Yes, and yes. – Theodore Lytras Jun 7 '18 at 21:58