(Joint work with Sylvia Schmidt University Bremen, FB 03)
Step-down tests (e.g. the Holm procedure) uniformly improve single-step tests (e.g. the Bonferroni test) with regard to power. However, when extending step-down tests to simultaneous confidence intervals (SCIs) the resulting confidence intervals are becoming complex and have the rather unfavorable property that they often provide no additional information to the sheer hypothesis test. We speak in this case of a non-informative rejection, because the confidence interval provides no information on the parameter value in the alternative. Non-informative rejections are particularly problematic in clinical trials with multiple treatments where an informative rejection is required to obtain valid estimates of the treatment effects. The extension of single-step tests to confidence intervals do not have this deficiency and almost always provide additional information to the hypothesis test. As a consequence step-downs tests when extended to SCIs do not uniformly improve single-step tests with regard to informative rejections. After a brief introduction to step-down tests and related simultaneous confidence intervals, we present a construction of a new class of simultaneous confidence intervals which uniformly improve the Bonferroni test with regard to informative rejections. This is done by using a dual family of weighted Bonferroni tests with weights depending continuously on the parameter values. We prove that the resulting vector of lower confidence bounds lies on a specific one-dimensional manifold in the multi-dimensional parameter space and provide simple algorithms for the numerical determination. The method is extended to union-intersection tests like the Dunnett procedure, is illustrated with a numerical example and is investigated in a comparative simulation study.