Mixtures of g-Priors for Analysis of Variance Models with a Diverging Number of Parameters

Abstract

We consider Bayesian approaches for the hypothesis testing problem in the analysis-of-variance (ANOVA) models. With the aid of the singular value decomposition of the centered designed matrix, we reparameterize the ANOVA models with linear constraints for uniqueness into a standard linear regression model without any constraint. We derive the Bayes factors based on mixtures of $g$-priors and study their consistency properties with a growing number of parameters. It is shown that two commonly used hyper-priors on $g$ (the Zellner-Siow prior and the beta-prime prior) yield inconsistent Bayes factors due to the presence of an inconsistency region around the null model. We propose a new class of hyper-priors to avoid this inconsistency problem. Simulation studies on the two-way ANOVA models are conducted to compare the performance of the proposed procedures with that of some existing ones in the literature.