  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.
