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Nonlinear interaction detection through model-based sufficient dimension reduction

In this paper we propose an efficient model-based sufficient dimension reduction method to detect interactions. We introduce a new class of multivariate adaptive varying index models (MAVIM) to investigate nonlinear interaction effects of the grouped covariates on multivariate response variables. Grouping the covariates through linear combinations in the MAVIM accommodates weak individual interaction effects as long as their joint interaction effects are strong enough to be detectable. We estimate the joint interaction effects by a weighted-profile least squares method that is numerically stable and computationally fast. The resultant profile least squares estimate is root-n consistent and asymptotically normal. We discuss how to choose an optimal weight to improve the estimation efficiency. We determine the structural dimension with a BIC-type criterion, and establish its consistency. The effectiveness of our proposal is illustrated through simulation studies and an analysis of Framingham heart study.