A new class of powerful and informative simultaneous confidence intervals

Step-down tests uniformly improve single-step tests with regard to power and the average number of rejected hypotheses. However, when extended to simultaneous confidence intervals (SCIs), the resulting SCIs often provide no additional information to the sheer hypothesis test. We speak, in this case, of a non-informative rejection. Non-informative rejections are particularly problematic in clinical trials with multiple treatments, where an informative rejection is required to obtain useful estimates of the treatment effects. The extension of single-step tests to confidence intervals does not have this deficiency. As a consequence, step-down tests, when extended to SCIs, do not uniformly improve single-step tests with regard to informative rejections. To overcome this deficiency, we suggest the construction of a new class of simultaneous confidence intervals that uniformly improve the Bonferroni and Holm SCIs with regard to informative rejections. This can be achieved using a dual family of weighted Bonferroni tests, with the weights depending continuously on the parameter values. We provide a simple algorithm for these computations and show that the resulting lower confidence bounds have an attractive shrinkage property. The method is extended to union-intersection tests, such as the Dunnett procedure, and is investigated in a comparative simulation study. We further illustrate the utility of the method with an example from a real clinical trial in which two experimental treatments are compared with an active comparator with respect to non-inferiority and superiority.