President and ConsultantData Infoworks, Inc., Sunnyvale, CA
Bayesian Multiple Comparisons Using Dirichlet Process Priors
The problem of multiple comparisons is considered from a Bayesian viewpoint. The criterion of posterior probabilities of hypotheses is proposed as a decision making tool. The family of Dirichlet process priors is applied in the form of baseline prior/likelihood combinations, to obtain posterior probabilities for different hypotheses. The baseline prior/likelihood combinations considered here are beta/binomial, normal/inverted gamma with equal and unequal variances on treatment means. The analytical intractability of the solution is alleviated by using a sampling based algorithm, called Gibbs sampling. The posterior probabilities of the hypotheses are easily obtained as a by-product in evaluating the marginal posterior distributions of the parameters. The prior probabilities of the hypotheses depend directly on the concentration parameter of the Dirichlet process prior. A method for eliciting values for the concentration parameter, along with the necessary software, is introduced. The power of the proposed procedure is compared with that of Duncan's multiple range test. For certain choices of the concentration parameter and posterior probabilities of equality for a pair of means, the proposed procedure is shown to have higher power than Duncan's test. Examples are discussed for all three baseline prior/likelihood combinations, which also serve as demonstrations for typical analyses of data under the proposed procedure. The procedure is then extended to include covariables.