Bayesian Model Selection Approach to Analysis of Variance Under Heteroscedasticity

The classical Bayesian approach to analysis of variance assumes the homoscedastic condition and uses conventional uniform priors on the location parameters and on the logarithm of the common scale. The problem has been developed as one of estimation of location parameters. We argue that this does not lead to an appropriate Bayesian solution. A solution based on a Bayesian model selection procedure is proposed. Our development is in the general heteroscedastic setting in which a frequentist exact test does not exist. The Bayes factor involved uses intrinsic and fractional priors which are used instead of the usual default prior distributions for which the Bayes factor is not well defined. The behaviour of these Bayes factors is compared with the Bayesian information criterion of Schwarz and the frequentist asymptotic approximations of Welch and Brown and Forsythe.