Variable Selection for Generalized Single-Index Varying-Coefficient Models with Applications to Synergistic G × E Interactions

Complex diseases such as type 2 diabetes are influenced by both environmental and genetic risk factors, leading to a growing interest in identifying gene–environment (G × E) interactions. A three-step variable selection method for single-index varying-coefficients models was proposed in recent research. This method selects varying and constant-effect genetic predictors, as well as non-zero loading parameters, to identify genetic factors that interact linearly or nonlinearly with a mixture of environmental factors to influence disease risk. In this paper, we extend this approach to a binary response setting given that many complex human diseases are binary traits. We also establish the oracle property for our variable selection method, demonstrating that it performs as well as if the correct sub-model were known in advance. Additionally, we assess the performance of our method through finite-sample simulations with both continuous and discrete gene variables. Finally, we apply our approach to a type 2 diabetes dataset, identifying potential genetic factors that interact with a combination of environmental variables, both linearly and nonlinearly, to influence the risk of developing type 2 diabetes.