Regression modelling on stratified data with the lasso

We consider the estimation of regression models on strata defined using a categorical covariate, in order to identify interactions between this categorical covariate and the other predictors. A basic approach requires the choice of a reference stratum. We show that the performance of a penalized version of this approach depends on this arbitrary choice, and propose an approach that bypasses this at almost no additional computational cost. Regarding model selection consistency, our proposal mimics the strategy based on an optimal and covariate-specific choice for the reference stratum. An empirical study confirms that our proposal generally outperforms the basic approach in the identification and description of the interactions. An illustration on gene expression data is provided.