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.

Keywords:

• Bayes factor
• Generalized linear
• Jeffreys' prior
• Laplace approximation
• Uniform shrinkage prior
• Unit-information prior

Mathematics Subject Classification:

• Primary 62F15
• Secondary 62F03

Acknowledgments

The author would like to thank Professor Chuhsing Kate Hsiao from Department of Public Health and Institute of Epidemiology, National Taiwan University for stimulating discussions about this work, and for making many valuable suggestions. The author is also grateful to two referees for their helpful and constructive comments, and for inspiring the author to adopt a more appropriate importance sampling density.