Abstract
We consider the inference problem of estimating covariate and genetic effects in a family-based case-control study where families are ascertained on the basis of the number of cases within the family. However, our interest lies not only in estimating the fixed covariate effects but also in estimating the random effects parameters that account for varying correlations among family members. These random effects parameters, though weakly identifiable in a strict theoretical sense, are often hard to estimate due to the small number of observations per family. A hierarchical Bayesian paradigm is a very natural route in this context with multiple advantages compared with a classical mixed effects estimation strategy based on the integrated likelihood. We propose a fully flexible Bayesian approach allowing nonparametric modeling of the random effects distribution using a Dirichlet process prior and provide estimation of both fixed effect and random effects parameters using a Markov chain Monte Carlo numerical integration scheme. The nonparametric Bayesian approach not only provides inference that is less sensitive to parametric specification of the random effects distribution but also allows possible uncertainty around a specific genetic correlation structure. The Bayesian approach has certain computational advantages over its mixed-model counterparts. Data from the Prostate Cancer Genetics Project, a family-based study at the University of Michigan Comprehensive Cancer Center including families having one or more members with prostate cancer, are used to illustrate the proposed methods. A small-scale simulation study is carried out to compare the proposed nonparametric Bayes methodology with a parametric Bayesian alternative.