Abstract: Gene-environment (G×E) interactions play key roles in many complex diseases. An increasing number of epidemiological studies have shown the combined effect of multiple environmental exposures on disease risk. However, no appropriate statistical models have been developed to conduct a rigorous assessment of such combined effects when G×E interactions are considered. In this paper, we propose a partial linear varying multi-index coefficient model (PLVMICM) to assess how multiple environmental factors act jointly to modify individual genetic risk on complex disease. Our model includes the varying-index coefficient model as a special case, where discrete variables are admitted as the linear part. Thus PLVMICM  allows one to study nonlinear interaction effects between genes and continuous environments as well as linear interactions between genes and discrete environments, simultaneously. We derive a profile method to estimate parametric parameters and a B-spline backfitted kernel method to estimate nonlinear interaction functions. Consistency and asymptotic normality of the parametric and nonparametric estimates are established under some regularity conditions. Hypothesis testing for the parametric coefficients and nonparametric functions are conducted. Results show that the statistics for testing the parametric coefficients and the non-parametric functions asymptotically follow a χ²-distribution with different degrees of freedom. The utility of the method is demonstrated through extensive simulations and a case study.

Key words and phrases: Association study, backfitting, B-spline, single index model, varying coefficient model.