Hello!

I'm working on a set of analyses for my dissertation, where linear regression / general linear model assumptions were violated per the residuals, and my outcome variables had multi-modal distributions, which means an OLS wasn't the best approach. Quantile regression was a perfect fit for me, and I found significant effects -- but I have two issues:

1) I need to figure out a way to probe for interaction effects within quantiles. From my understanding -- the mean and median are not great representations of the data, given the abnormal, multi-modal distributions. This means I can't stratify by (+/-) 1 standard deviation of the mean... OR do a median split on the moderator variable... right? If not, what method could I use to clarify the interaction effect that would make sense within relevant quantiles?

2) I am aware of the issue of multiple comparisions/ hypothesis tests. However, I can't seem to find a correction that doesn't implement a very heavy / strict penalty. I ruled out the need to do a Bonferonni correction, and landed on using a false discovery rate (the Benjamin-Hochberg adjustment). but... there's no overall model summary/F-test per model, with a corresponding model p-value. I only have the MAE and pseudo R-squared for each quantile. I detected different effects for multiple variables at various quantiles, so using the p-value of each predictor within each quantile makes the "number of tests" increase exponentially. (e.g., one model has 4 quantiles and runs tests within each quantile to capture what's happening at that area of the distrubition... which is conceptually equivalent to running 4 regressions within each section of the outcome's distribution).

Does anyone have guidance on one or both of these points?

TLDR:

1 -- How do you probe for significant interaction effects for quantile regression?

2 -- How do you correct for multiple comparisons and is it even necessary?

I'd appreciate your help!!