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The genetic basis of complex traits involves the function of many genes with small effects as well as complex gene-gene and gene-environment interactions. As one of the major players in complex diseases, the role of gene-environment interactions has been increasingly recognized. Motivated by epidemiology studies to evaluate the joint effect of environmental mixtures, we developed a functional varying-index coefficient model (FVICM) to assess the combined effect of environmental mixtures and their interactions with genes, under a longitudinal design with quantitative traits. Built upon the previous work, we extend the FVICM model to accommodate binary longitudinal traits through the development of a generalized functional varying-index coefficient model (gFVICM). This model examines how the genetic effects on a disease trait are nonlinearly influenced by a combination of environmental factors. We derive an estimation procedure for the varying-index coefficient functions using quadratic inference functions combined with penalized splines. A hypothesis testing procedure is proposed to evaluate the significance of the nonparametric index functions. Extensive Monte Carlo simulations are conducted to evaluate the performance of the method under finite samples. The utility of the method is further demonstrated through a case study with a pain sensitivity dataset. SNPs were found to have their effects on blood pressure nonlinearly influenced by a combination of environmental factors. 

Human complex diseases are affected by both genetic and environmental factors. When multiple environmental risk factors are present, the interaction effect between a gene and the environmental mixture can be larger than the addition of individual interactions, resulting in the so-called synergistic gene–environment (GxE) interactions. Existing literature has shown the power of synergistic gene-environment interaction analysis with cross-sectional traits. In this work, we propose a functional varying index coefficient model for longitudinal traits together with multiple longitudinal environmental risk factors and assess how the genetic effects on a longitudinal disease trait are nonlinearly modified by a mixture of environmental influences. We derive an estimation procedure for the nonparametric functional varying index coefficients under the quadratic inference function and penalized spline framework. We evaluate some theoretical properties such as estimation consistency and asymptotic normality of the estimates. We further propose a hypothesis testing procedure to assess the significance of the synergistic GxE effect. The performance of the estimation and testing procedure is evaluated through Monte Carlo simulation studies. Finally, the utility of the method is illustrated by a real dataset from a pain sensitivity study in which SNP effects are nonlinearly modulated by a mixture of drug dosages and other environmental variables to affect patients’ blood pressure and heart rate.