Bayesian model selection approach to the one way analysis of variance under homoscedasticity

Abstract

An objective Bayesian model selection procedure is proposed for the one way analysis of variance under homoscedasticity. Bayes factors for the usual default prior distributions are not well defined and thus Bayes factors for intrinsic priors are used instead. The intrinsic priors depend on a training sample which is typically a unique random vector. However, for the homoscedastic ANOVA it is not the case. Nevertheless, we are able to illustrate that the Bayes factors for the intrinsic priors are not sensitive to the minimal training sample chosen; furthermore, we propose an alternative pooled prior that yields similar Bayes factors. To compute these Bayes factors Bayesian computing methods are required when the sample sizes of the involved populations are large. Finally, a one to one relationship—which we call the calibration curve—between the posterior probability of the null hypothesis and the classical $p$ value is found, thus allowing comparisons between these two measures of evidence. The behavior of the calibration curve as a function of the sample size is studied and conclusions relating both procedures are stated.