Selecting the normal population with the best regression value - a Bayesian approach

For the intra-class regression (multiple slopes) model, much attention has been given to the estimation and hypothesis testing problems concerning the parameters. However, few papers in the literature address the practical question of selecting the population with the largest regression value. The existing classical ranking and selection procedure employs a rule which always selects the population associated with the largest sample regression value computed by the least-squares method. Whereas the rule is reasonable if all variances of the sample estimates are equal, it may not be desirable when some of the variances are different because the rule ignores the unequal variances associated with the estimates. We propose a Bayesian approach to the selection problem by calculating, for a given value of the covariate, the posterior probabilities of each population mean being the largest. In the two examples considered in the paper, it is observed that there are occasions when the two largest sample regression values are close together and the larger one is associated with a larger variance, the population with the smaller sample regression value can be selected as the best population by our Bayesian procedure. Although calculation of the posterior probabilities may involve high dimensional numerical integration, the required computation can be handled efficiently by some Monte Carlo integration methods as described in the paper.