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.
Highlights
In the field of applied statistics, analysis-of-variance (ANOVA) is a collection of statistical models commonly used to test hypotheses about the presence of a group effect
We have studied Bayes factor consistency under the various mixtures of g-priors for the hypothesis testing problem in the multi-way ANOVA models with a diverging number of parameters
It has been shown that the Bayes factors based on the ZS prior and the BP prior are not always consistent due to the presence of an inconsistency region around the null model when k grows proportionally to n
Summary
In the field of applied statistics, analysis-of-variance (ANOVA) is a collection of statistical models commonly used to test hypotheses about the presence of a group (treatment) effect. We follow the suggestion of Ley and Steel (2012) and adopt a hyper-prior on g to reflect its uncertainty and randomness and to allow for the data to determine the inference on g This hyper-prior must be proper, because the null model does not involve g and the improper prior will yield the Bayes factor with an undefined normalizing constant. This paper fills the gap by studying Bayes factors consistency under various mixtures of g-priors in the ANOVA models. It deserves mentioning that the proposed results based on the second prior generalize some existing ones for the one-way/two-way ANOVA models studied by Maruyama (2012) and Wang and Sun (2013).
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