# Interpreting main effect when interaction is significant in regression

Before anyone mark this question as duplicate, I have read the other posts on this question on CV, the answers usually call for plotting the data and check what they look like. But this is not practical for me because I have continuous multi-dimensional data, and I need to conduct hundreds of these regressions.

Now onto the specifics. I am fitting a linear model:

$$y\sim space+orientation+velociy+space*orientation+space*velocity+orientation*velocity$$

where $y$ is an individual neuron's firing rate.

My question is whether $space$ has a meaningful significant main-effect and quantify it. This is easy to analyze when the interaction effects involving $space$ are not significant -- I can get a confidence-interval for the partial-$R^2$ between the full model:

$$y\sim space+orientation+velocity$$ and the partial model $$y\sim orientation+velocity$$

and use that as a metric. I used bootstrap because of heteroskedasticity and non-normal residual distribution.

However, when the interaction effects are significant, interpreting main effect is difficult. I can have the case where main-effect is significant as a result of the interaction, or the case where the interaction-effect explains variance in the DV in addition to the main-effect.

It is unpractical to plot out the data because all my predictors are continuous. Any suggestions on what statistical techniques I can use?

• Have you heard of scatterplot matrices? They might be useful to you here. – whuber Feb 21 '18 at 23:52
• I am not clear what it means for the main effect to be significant because of the interaction. – Dimitriy V. Masterov Feb 22 '18 at 0:09
• @DimitriyV.Masterov See the link for an example of interaction driving the main effects. – Asy Feb 23 '18 at 0:50
• @whuber Scatterplot matrices look useful. Is there a way to summarize them? – Asy Feb 23 '18 at 0:51
• The main point to looking at scatterplot matrices is to see aspects of the bivariate marginal distributions that would not be reflected in any summary that one might plan to make beforehand. Thus, any appropriate summary would depend on what you see. – whuber Feb 23 '18 at 15:01