Functional varying index coefficient model for dynamic gene-environment interactions

Rooted in genetics, human complex diseases are largely influenced by environmental factors. Existing literature has shown the power of integrative gene-environment interaction analysis by considering the joint effect of environmental mixtures on a disease risk. In this work, we propose a functional varying index coefficient model for longitudinal measurements of a phenotypic trait together with multiple environmental variables, 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. Theoretical results such as estimation consistency and asymptotic normality of the estimates are established. In addition, we propose a hypothesis testing procedure to assess the significance of the nonparametric index coefficient function. We evaluate the performance of our estimation and testing procedure through Monte Carlo simulation studies. The proposed method is illustrated by applying to a real data set from a pain sensitivity study in which SNP effects are nonlinearly modulated by the combination of dosage levels and other environmental variables to affect patients' blood pressure and heart rate.