Variable importance assessments and backward variable selection for multi-sample problems

Abstract Variable selection for multi-sample problems is of great interest in statistics. Existing methods for addressing this problem have some limits or disadvantages. In this paper, we propose distance-based variable importance measures to deal with these problems, which are inspired by the Multi-response permutation procedure (MRPP), Energy distance (ED) and Distance components (DISCO) analysis. The proposed variable importance assessments can effectively measure the importance of an individual dimension by quantifying its influence on the differences between multivariate distributions across treatment groups. An importance-measure-based backward selection (IM-BWS) algorithm is developed that can be used in variable selection for multi-sample problems to discover important variables. We propose a modified MRPP based on the IM-BWS procedure for improving the power performance of the original MRPP. Our proposed methods are model-free, work for high-dimensional data, and can capture important variables under different models. Both simulations and real data applications demonstrate that our proposed method enjoys good properties and has advantages over other existing methods.