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 disease risk. Motivated by that, we propose a functional varying index coefficient model for longitudinal measurements of a phenotypic trait and 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 functions and penalized splines 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 blood pressure and heart rate of patients. In order to deal with discrete measurements for risk of disease, we further extend our proposed FVICM to a generalized varying index coefficient model (gFVICM) to binary longitudinal traits. We apply penalized splines to approximate the nonparametric varying index coefficients and develop an estimation procedure based on the quadratic inference functions. The asymptotic normality established in the theoretical results enables us to develop a model selection criteria and construct a test statistic based on the quadratic inference function. In hypothesis test, we investigate the linearity of GE interactions using the proposed testing procedure. The utility of the method is further demonstrated through a pain sensitivity case study in which SNP effects are nonlinearly modulated by the combination of environmental mixtures to affect high blood pressure.Genetic pleiotropy refers to the situation in which a single gene influences multiple traits and so it is considered as a major factor that underlies genetic correlation among traits. For some complex diseases, there are multiple phenotypes that can used to diagnose or to quantify the risk of diseases and usually they have shared genetic determinations. In multivariate longitudinal data, multiple response variables are jointly measured over time from the same individual. It is appropriate to take into account the correlation between multivariate longitudinal responses. Therefore, we propose the joint partially linear varying coefficient models and the testing framework to jointly test the association of genetic factors with bivariate phenotypic values adjusting for environmental factors. We extended the quadratic inference functions to deal with the longitudinal correlations and used penalized splines for the approximation of nonparametric coefficients. The proposed method is illustrated by applying to a real data set from a pain sensitivity study, in which systolic blood pressure (SBP) and diastolic blood pressure (DBP) were correlated longitudinal quantified phenotypes of SNP effects.
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