# Interpreting lower order effects not contributing to the interaction terms, when the interaction is significant (C in a regression of A + B + C + A*B)

In a regression including 3 variables, and the interaction of 2 of those variables:

• Variable A
• Variable B
• Variable C
• Variable A * Variable B,

where the interaction of Variable A * Variable B is significant, should the regression coefficient for C be interpreted?

1. Under the tradition that if the interaction term A*B if significant, the main effects/conditional effects of A and B should not be
interpreted, does this logic of not interpreting the main effects
apply to any variables in the model not present in the interaction
term?

2. Or should the coefficient for Variable C be interpreted since it was not included in the interaction term?

• If so, how should C be interpreted, if significant? What is the meaning of it?
3. How does the presence of C effect the way A*B should be interpreted?

The answer to your question 2 is that it has the same interpretation as usual. That is to say that it implies that the levels of C are different. Assuming there is no AC and BC interaction that difference is maintained irrespective of the levels of A or B or their combination.

As for your question 3 the answer is no, subject to the same proviso about that being the only interaction.

I think that covers your question 1 as well in passing.

• Thank you very much for your explanation, Professor Dewey. I'm afraid I'm new to interactions and am unsure what it means that there are "levels" of C or of A or B? Would it be possible to please elaborate -- it would be greatly appreciated!!
– R G
Jul 16 '16 at 19:43
• In terms of interpreting C as usual, do I refrain from interpreting the results for A and B, and then interpret C's coefficient as for every 1-unit increase in C, a beta-increase in Y is expected, after controlling all associated covariates (would this include A, B, and A*B as held constant)? Then, in interpreting the interaction, do I just do so as with any 2-way interaction, or is there a special protocol for when including a third variable not present in the interaction?
– R G
Jul 16 '16 at 19:49
• Yes, and yes there is nothing special to do. Jul 16 '16 at 21:13