Bayes Factor Consistency for Unbalanced ANOVA Models

TL;DR

This paper investigates the consistency of Bayes factors in unbalanced one-way fixed-effects ANOVA models, providing explicit formulas and analyzing asymptotic scenarios relevant to practical experimental designs.

Contribution

It extends existing Bayes factor consistency results to unbalanced ANOVA models using Zellner's g-prior with a closed-form expression.

Findings

Bayes factor is consistent when the number of units increases.

Bayes factor is consistent when the observations per unit increase.

Results extend to unbalanced designs beyond balanced models.

Abstract

In practical situations, most experimental designs often yield unbalanced data which have different numbers of observations per unit because of cost constraints, or missing data, etc. In this paper, we consider the Bayesian approach to hypothesis testing or model selection under the one-way unbalanced fixed-effects ANOVA model. We adopt Zellner's g-prior with the beta-prime distribution for g, which results in an explicit closed-form expression of the Bayes factor without integral representation. Furthermore, we investigate the model selection consistency of the Bayes factor under three different asymptotic scenarios: either the number of units goes to infinity, the number of observations per unit goes to infinity, or both go to infinity. The results presented extend some existing ones of the Bayes factor for the balanced ANOVA models in the literature.