On Inferring Order Relations in Analysis of Variance

In the interpretation of an analysis of variance, the experimenter often wishes to express his results in something like the following terms: 'Treatment A is 'worse' (or 'better') than at least one of the treatments B, C, and D, say'. Such statemeents are particularly meaningful whenever one wants to screen out inferior treatments or select a subset containing the best treatment. To be more specific, in a one-way analysis of variance the following multiple comparison problem appears to be of practical importance: Let the variables Xii be independently normal with expectations tj and variance o-2 (i = t1 * I and j = 1, ni). We wish to infer from our analysis statements of the following type: