Bayes factors for variance components in the mixed linear model

Mithat Gonen, Texas Tech University

Abstract

The Bayes factor is a widely-used summary measure that can be used to test hypotheses in a Bayesian setting. It also performs well in problems of model selection. In this study, Bayes factors for variance components in the mixed linear model are derived. The formulation used avoids the assumption of a priori independence between the variance components by using a Dirichlet prior on the intraclass correlations. A reference prior, which results in a Bayes factor that is flexible and easy to use, is suggested. Hypothesis tests using the Bayes factor avoid difficulties of the classical tests, such as non-uniqueness and invalid asymptotics.

The priors on the nuisance parameters are chosen to be non-informative and the corresponding integrals are carried out analytically. However, for the parameters of interest, numerical methods have to be used. For this purpose, Monte Carlo methods have been investigated. Simple random sampling and Latin hypercube sampling are employed for simulating the prior and a Gibbs sampling scheme has been implemented for simulating the posterior. The resulting estimators are compared on a small data set.