Sliced Inverse Regression with Variable Selection and Interaction Detection

Variable selection methods play important roles in modeling high dimensional data and are keys to data-driven scientific discoveries. In this paper, we consider the problem of variable selection with interaction detection under the sliced inverse index modeling framework, in which the response is influenced by predictors through an unknown function of both linear combinations of predictors and interactions among them. Instead of building a predictive model of the response given combinations of predictors, we start by modeling the conditional distribution of predictors given responses. This inverse modeling perspective motivates us to propose a stepwise procedure based on likelihood-ratio tests that is effective and computationally efficient in detecting interaction with little assumptions on its parametric form. The proposed procedure is able to detect pairwise interactions among p predictors with a computational time of O(p) instead of O(p2) under moderate conditions. Consistency of the procedure in variable selection under a diverging number of predictors and sample size is established. Its excellent empirical performance in comparison with some existing methods is demonstrated through simulation studies as well as real data examples.