A Hybrid Approximation Bayesian Test of Variance Components for Longitudinal Data

Abstract

The test of variance components of possibly correlated random effects in generalized linear mixed models (GLMMs) can be used to examine if there exists heterogeneous effects. The Bayesian test with Bayes factors offers a flexible method. In this article, we focus on the performance of Bayesian tests under three reference priors and a conjugate prior: an approximate uniform shrinkage prior, modified approximate Jeffreys' prior, half-normal unit information prior and Wishart prior. To compute Bayes factors, we propose a hybrid approximation approach combining a simulated version of Laplace's method and importance sampling techniques to test the variance components in GLMMs.

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“…Besides, we specify our prior beliefs with assuring conjugacy for each of the unknown parameters β and D proposed by Chib and Jeliazkov (2001) and Tsai (2010). Note that we use the number of Monte Carlo samples R = 10,000 for all analyses.…”

“…For simplicity in the following Bayesian analyses, we assume that the random effect b i follows a multivariate normal distribution with a q × 1 zero mean vector and a q × q covariance matrix D, that is b i ∼ N q (0, D). Besides, we specify our prior beliefs with assuring conjugacy for each of the unknown parameters β and D proposed by Chib and Jeliazkov (2001) and Tsai (2010). The vague prior distribution on β is assumed to be a noninformative multivariate normal distribution, N p (β 0 , B) with β 0 = 0 and B = 100 × I p , and on D is an inverse Wishart prior, that is D −1 ∼ W (V, υ) with the specification of a scale matrix V = I q and a degree of freedom υ = q.…”

“…To study the accuracy of the approximation approach proposed in Section 2.1, we considered three applications using normal, Poisson, and binary datasets, and the results of the proposed approach were compared with Chib's method (Chib and Jeliazkov, 2001) and the hybrid approximation approach (Tsai, 2010). The first application concerns a set of growth data from 11 girls and 16 boys (Potthoff and Roy, 1964) and the distance (mm) from the center of pituitary to pterygomaxillary fissure was recorded for each subject.…”