On the level probabilities for useful partially ordered alternatives in the analysis of variance

In the analysis of variance, we often encounter situations in which we want to test the null hypothesis of homogeneity of the normal means against various partially ordered alternative hypotheses. We study likelihood ratio tests for three useful types of alternatives: d-star, bipartite and broom tree. Especially, we give computational formulas for the level probabilities of the alternative types. The results permit us to obtain critical values for practical use.