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.

翻译：现有文献通过考虑环境混合物对疾病风险的共同影响,显示了综合基因-环境相互作用分析的力量。在这项工作中,我们提议了一个功能不同的指数系数模型,用于对雌性和多种环境变量进行纵向测量,并评估对长视疾病特征的遗传影响如何不因环境影响的混合而线性地改变。我们从一个对疼痛敏感度研究中得出一个估计程序,根据这种研究,SNP的影响不因剂量水平和其他环境变量的结合而线性地改变,从而影响病人的血液压力和心率。我们通过蒙特卡洛模拟研究评估我们估算和测试程序的执行情况。我们通过应用一个实际的对疼痛敏感度研究,对SNP的影响不因剂量水平和其他环境变量的结合而改变。