# ANOVA: do I interpret the significant main effects not involved in interactions?

My statistical model is Dependent ~ A * Sex + B * Sex. (i.e., Dependent ~ A + B + A:Sex + B:Sex). I have prior reason to expect that Sex will interact significantly with A and B, but no reason to expect that A and B will interact. When I run my data, I get these results.

• A (significant main effect)
• B (not significant)
• A:Sex (not significant)
• B:Sex (significant interaction)

That is, there is a significant main effect of A, and an interaction between B and Sex. My understanding of ANOVA is that since B:Sex is significant, I need to segregate the data by the levels of one of the factors (I will choose Sex) and test the effect of B on each subset separately. But, should I interpret the main effect of A before doing that? Or should I segregate the data, and retain the independent variables A and B in the test for each subset?

• instead of segregating the data, you should fit a model with A, B, Sex and BSex as covariates. (why there is no Sex in you model? ASex means sex in the model.) Nov 21, 2018 at 23:25
• Oh, I was using R notation, so A * Sex means A, Sex, and the interaction of A and Sex. Nov 21, 2018 at 23:32
• Did you have any a priori hypotheses about which of the two interactions you would expect to be significant? Did you power your study to be able to detect significant interactions? Nov 22, 2018 at 2:32
• Yes, I believe both interactions could be significant, and there plenty of power. :-) Nov 22, 2018 at 12:57
• Are A and B continuous variables in your model? Nov 23, 2018 at 2:31

It is unclear from your question whether you are still engaged in model selection, or whether you have settled on your model and are now only interested in interpretation of the outcome. In view of the description, I am going to assume that you are still undertaking model selection, and you are interested in comparing the present model with one that removes the interaction of A and Sex.

In these kinds of models it is generally a bad idea to monkey with the data to try to segregate it. It is much simpler just to fit the two nested models of interest and then use a standard goodness-of-fit test comparing the nested models to see if there is any evidence to include the extra parameters in the larger model. In your case, the comparison of interest is between these nested models:

Model 1: Dependent ~ A + B + Sex + A:Sex + B:Sex

Model 2: Dependent ~ A + B + Sex + B:Sex

It should not be difficult to calculate the goodness-of-fit statistics for these two models and then perform a comparison, noting that Model 2 is nested within Model 1. This would usually be done with a chi-squared test comparing the log-deviance of the models.

• Thank you for your response. Unfortunately I intended the former option: I have my model, an am just trying to interpret the results correctly. I've edited the question to make it clearer. Nov 22, 2018 at 12:58
• If you have already fit your model, and you're happy with that model, then no issue of data segregation or additional testing arises. You just take the outputs of that model and interpret them in the normal manner for regression coefficients. Presence of conditional statistical associations can be done by using the T-tests in the output.
– Ben
Nov 22, 2018 at 22:58
• When I learned to do ANOVA (not in a statistics department), it was presented as standard practice that, when an interaction arises, one divides the data by the levels of one of the factors and does another ANOVA on the subsets (also here, which I found as a sanity check: r-bloggers.com/…) Is that not the case? Nov 23, 2018 at 15:35