Calculations of level probabilities for normal random variables with unequal variances with applications to Bartholomews test in unbalanced one-way models

An effective procedure is presented to compute the level probabilities under simple order of independent normal random variables with unequal variances. These level probabilities have an application in the unbalanced one-way layout for comparing k treatment effects where Bartholomew (1959. Biometrika 48, 325¯332, 1961a. J. Roy. Statist. Soc. Ser. B 23 239¯ 281, 1961b. Biometrika 48, 325¯332) proposed the likelihood ratio test for testing the homogeneity of the treatment effects against the simply ordered alternative hypothesis. Although there is some literature showing that Bartholomewʹs test has good properties, its null distribution for the unbalanced model has been difficult to calculate except for small k. The calculation of the level probabilities allows the computation of the p-values of Bartholomewʹs test for unbalanced models.