Semiparametric Kernel-Based Regression for Evaluating Interaction Between Pathway Effect and Covariate

Pathway-based analysis has the ability to detect subtle changes in response variables that could be missed when using gene-based analysis. Since genes interact with other covariates such as environmental or clinical variables, so do pathways, which are sets of genes that serve particular cellular or physiological functions. However, since pathways are sets of genes and since environmental or clinical variables do not have parametric relationships with response variables, it is difficult to model unknown interaction terms between high-dimensional variables and low-dimensional variables as environmental or clinical variables. In this paper, we propose a semiparametric interaction model for two unknown functions to evaluate the interaction between a pathway and environmental or clinical variable: for the pathway, we use an unknown high-dimensional function, and for environmental or clinical variable, we use an unknown low-dimensional function. We model the environmental or clinical variable nonparametrically via a natural cubic spline. We model both the pathway effect and the interaction between the pathway and environmental or clinical effect nonparametrically via a kernel machine. Since both interactions among genes within the same pathway and the interaction between the pathway and the environmental or clinical variables are complex, we allow for the possibility that a pathway is interacting with environmental or clinical variables and the genes within the same pathway are interacting with each other. We illustrate our approach using simulated data and genetic pathway data for type II diabetes. Supplementary materials accompanying this paper appear online.