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1. Introduction
Linear mixed models (LMM) have been used to find associations between continuous phenotypes and genetic variants, genes, and gene-environment (GE) interactions in unrelated and related subjects in genome-wide association (GWA) analysis. For unrelated subjects, the analysis can be performed within the generalized linear model framework, however, for related subjects as in the case of family data, one has to include the kinship matrix to take into account the correlation among the relatives for each family. In this paper, we are interested in testing GE interaction for discrete phenotypes. Generalized linear mixed models (GLMM) proposed by Breslow and Clayton [1] is an ideal statistical approach to detect such an interaction with non-continuous phenotypes, because it can treat the familiar effect on the phenotype as a random effect.
Gene-based GE interaction tests have previously been proposed for independent subjects [2,3,4]. While each GE interaction can be tested individually using one single nucleotide polymorphism (SNP) at a time, it is known that single SNP association is not as powerful as the gene-based analysis [2] due to the linkage disequilibrium (LD) present among the SNPs in a gene. Lin et al. [2] proposed a variance component test (VCT) of the interactions by treating the interactions as a random effect. This approach was extended to sequencing data with rare variants [3,5]. To overcome multicollinearity of the coefficients of the genetic markers, Lin et al. [2,3] applied ridge regression penalization of SNP coefficients and estimated the ridge penalty parameter with generalized cross-validation. However, this method is computationally demanding and their final test ignores the tuning of the ridge penalty parameter. Coombes [6] instead proposed treating the genetic coefficients as a random effect in a linear mixed model framework to perform the ridge penalization. This equivalence was initially proposed by Bishop and Tipping [7,8] for Bayesian ridge regression in linear models framework. While this approach was able to incorporate the ridge penalty into the test statistic, it was only developed for a quantitative phenotype [6].
Here, we propose a GLMM GE interaction framework for discrete and continuous phenotypes that treats the coefficients of genetic markers as random effects. Also, because the correlation among relatives cannot be ignored, this modeling framework incorporates the kinship matrix in the GLMM [9]. We test...