Step-up multiple testing of parameters with unequally correlated estimates

We consider the problem of simultaneously testing k > or = to 2 hypotheses on parameters theta(1), ..., theta(k) using test statistics t(1), ..., t(k) such that a specified familywise error rate alpha is achieved. Dunnett and Tamhane (1992a) proposed a step-up multiple test procedure, in which testing starts with the hypothesis corresponding to the least significant test statistic and proceeds towards the most significant, stopping the first time a significant test result is obtained (and rejecting the hypotheses corresponding to that and any remaining test statistics). The parameter estimates used in the t statistics were assumed to be normally distributed with a common variance, which was a known multiple of an unknown sigma(2), and known correlations which were equal. In the present article, we show how the procedure can be extended to include unequally correlated parameter estimates. Unequal correlations occur, for example, in experiments involving comparisons among treatment groups with unequal sample sizes. We also compare the step-up and step-down multiple testing approaches and discuss applications to some biopharmaceutical testing problems.