A Bayesian Approach to Ranking and Selection of Related Means with Alternatives to Analysis-of-Variance Methodology

A set of unknown normal means (treatment effects, say) {θ1, θ2, …, θk } is to be investigated. Two common questions in analysis of variance and ranking and selection are as follows: (a) What is the strength of evidence against the hypothesis H 0 of equality of means? (b) If H 0 is false, which mean is the largest (or smallest)? A Bayesian approach to the problem is taken, leading to calculation of the posterior probability of H 0 and the posterior probabilities that each mean is the largest, conditional on H 0 being false. A variety of exchangeable, nonexchangeable, informative, and noninformative prior assumptions are considered. Calculations involve, at worst, only low-dimensional numerical integration, in spite of the fact that the dimension k can be arbitrarily large. As an example, Table 1 presents, for each baseball team in the National League in 1984, the highest batting average obtained by any player on the team with at least 150 at bats. The observed batting averages are treated as sampl...