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:
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?