Bayes Factors and Approximations for Variance Component Models

Abstract

In this article we consider tests of variance components using Bayes factors. Such tests arise in many fields of application, including medicine, agriculture, and engineering. When using Bayes factors, the choice of prior distribution on the parameter of interest is of great importance; we propose a “unit-information” reference method for variance component models. The calculation of Bayes factors in this context is not straightforward; there are well-documented difficulties with Markov chain Monte Carlo approaches such as Gibbs sampling, and the usual Laplace approximation is not appropriate, due to the boundary null hypothesis. We describe both an importance sampling approach and an analytical approximation for calculating the numerator and denominator of the Bayes factor. The importance sampling approach is straightforward to implement and also forms the basis for a rejection algorithm that allows generation of samples from the posterior distributions under the null and alternative hypotheses. We suggest that the proposal for the rejection algorithm be based on the likelihood of a subset of the data. For large samples, we develop a boundary Laplace approximation that is accurate to order op1). We investigate the accuracy of the approximation via simulation, and examine its relationship to the Schwarz criterion. We illustrate the importance sampling/rejection method and boundary Laplace approximation on a number of examples, including a challenging two-way, highly unbalanced dataset and compare our methods with frequentist alternatives.

Key Words:

• Boundary problem
• Importance sampling
• Laplace's method
• Reference prior
• Rejection method
• Schwarz criterion.