Abstract
It has been a long history of using interactions in regression analysis to investigate alterations in covariate-effects on response variables. In this article, we aim to address two kinds of new challenges arising from the inclusion of such high-order effects in the regression model for complex data. The first kind concerns a situation where interaction effects of individual covariates are weak but those of combined covariates are strong, and the other kind pertains to the presence of nonlinear interactive effects directed by low-effect covariates. We propose a new class of semiparametric models with varying index coefficients, which enables us to model and assess nonlinear interaction effects between grouped covariates on the response variable. As a result, most of the existing semiparametric regression models are special cases of our proposed models. We develop a numerically stable and computationally fast estimation procedure using both profile least squares method and local fitting. We establish both estimation consistency and asymptotic normality for the proposed estimators of index coefficients as well as the oracle property for the nonparametric function estimator. In addition, a generalized likelihood ratio test is provided to test for the existence of interaction effects or the existence of nonlinear interaction effects. Our models and estimation methods are illustrated by simulation studies, and by an analysis of child growth data to evaluate alterations in growth rates incurred by mother’s exposures to endocrine disrupting compounds during pregnancy. Supplementary materials for this article are available online.
Highlights
Regression analysis has played a central role in studying relationships between variables in the statistical literature
Generalizing the single index coefficient regression model (Xia and Li, 1999), we propose a new class of semiparametric models with varying index coefficients, which enables us to model and assess nonlinear interaction effects between grouped covariates on the response variable
We propose a new class of semiparametric models with varying index coefficients, which allows us to study nonlinear interactive effects that are of scientific importance in the understanding of the response-covariate relationship
Summary
Regression analysis has played a central role in studying relationships between variables in the statistical literature. Note that when parameters β are fixed by known values or by their root-n consistent estimates, the SiCM is simplified as to be a VCM in which each coefficient function is univariate, and some of the existing nonparametric smoothing methods proposed for the VCM may be directly applied; for example, the kernel-based method (Cai et al (2000); Fan and Zhang (2008)) and the spline-based method (Huang et al (2004)). Several methods have been proposed for estimation in multivariate additive models, summarized as follows It is shown in Stone (1985) that the one-step LS B-spline estimators of the additive nonparametric functions have the univariate convergence rate, but no asymptotic distribution is available. All technical details including detailed proofs are provided in the Appendix
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