On selecting the best treatment in a generalized linear model

The problem of selecting the best treatment is studied under generalized linear models. For certain balanced designs, it is shown that simple rules are Bayes with respect to any non-informative prior on the treatment effects under any monotone invariant loss. When the nuisance parameters such as block effects are assumed to follow a uniform (improper) prior or a normal prior, Bayes rules are obtained for the normal linear model under more suitable balanced designs, keeping the generality of the loss and the generality of the non-informativeness on the prior of the treatment effects. These results are extended to certain types of informative priors on the treatment effects. When the designs are unbalanced, algorithms based on the Gibbs sampler and the Laplace method are provided to compute the Bayes rules.