# Exploratory search for interactions in linear regression with normal distr. of residuals but non-normally distr. DV

I am conducting a very exploratory analysis (I.e., few to no specific predictions about which predictors/interactions are most relevant) assessing whether ten IVs (all binary) predict one DV (continuous). Preliminary analyses showed that the DV was non-normally distributed, but when running the full regression model, the residuals ARE normally distributed. I take this to mean it’s fine to use linear regression as the strategy for the overall model, but I am also interested in finding out whether any of my predictors interact with gender to predict the DV.

In other papers, I have seen people search for interaction terms of interest (again, this is very exploratory work) by running a series of ANOVAs. So, for example, I would test whether gender interacts with predictor 1 to predict the DV, then whether gender interacts with predictor 2 to predict the DV, and so on. Significant interaction terms are then added to the full regression model to see if they significantly improve model fit.

This leads to my question: given that my DV is non-normally distributed, can I use ANOVAs (which rely on normality) to look for possible interactions? I am thinking yes because I end up using linear regression anyway (and ANOVA is after all, a form of regression with similar assumptions). But, is there any reason I should actually look for interactions to include in my regression using some non-parametric technique?

When you say that the residuals are normally distributed, did you test that formally? If so, then it's probably reasonable to use a typical linear model. The key is that $$r = y - X\beta$$ is normal, since least squares is assuming that $$y$$ is a combination of a deterministic component from the regressors and an iid normal distribution error term that, crucially, is independent of $$X$$ and $$y$$. So the dependent variable not having a normal marginal distribution is fine.