Variance Component Testing in Generalized Linear Mixed Models

Generalized linear mixed models (GLMM) are used in situations where a number of characteristics (covariates) affect a nonnormal response variable and the responses are correlated. For example, in a number of biological applications, the responses are correlated due to common genetic or environmental factors. In many applications, the magnitude of the variance components corresponding to one or more of the random effects are of interest, especially the point null hypothesis that one or more of the variance components are zero. This work reviews a number of approaches for estimating the Bayes factor comparing the models with and without the random effects in question. The computations involved with finding Bayes factors for these models pose many challenges, and we discuss how one can overcome them. We perform a comparative study of the different approaches to compute Bayes factors for GLMMs by applying them to two different data sets. The first example employs a probit regression model with a single variance component to data from a natural selection study on turtles. The second example uses a disease mapping model from epidemiology, a Poisson regression model with two variance components being used to analyze the data. The importance sampling method is found to be the method of choice to compute the Bayes factor of interest for these problems. Chib’s method is also found to be efficient.