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 and 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 proposal also
forms the basis for a rejection algorithm which allows samples to be
generated from the posterior distributions under the null and
alternative hypotheses. We show that our importance sampling estimator
has finite variance by exploiting a property of the rejection
algorithm. Both the importance sampling and rejection method are
straightforward to implement.
For large samples we develop a boundary Laplace approximation which is
accurate to O(1) in probability, and investigate its accuracy via
simulation. The relationship of this approximation to the Schwarz
criterion is then examined. 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 conclusions with those from classical
approaches.
Keywords: Boundary problem; Importance sampling; Laplace's method; Reference prior; Rejection method;
Schwarz criterion.