Informative simultaneous confidence intervals for the fallback procedure

The fallback procedure is an extension of the hierarchical test procedure relaxing the predefined hierarchical order. It can be applied for example in dose-finding studies. If interest is in extending the fallback procedure to simultaneous confidence intervals, one finds proposals in the literature, which have, however, the drawback that noninformative rejections may arise. A noninformative rejection means that the confidence interval of a rejected null hypothesis contains all parameters of the alternative and thus gives no useful information about the true value of the effect parameter. We present a modification of the fallback procedure with corresponding simultaneous confidence intervals that is informative in every case where a hypothesis is rejected. The main idea consists of splitting the level between the null hypotheses and a nested family of informative hypotheses constituting the alternative. The splitting weights depend continuously on the parameter. The new method is represented by a simple graph and can be easily implemented by an explicit algorithm. We give an example and compare our approach with an existing extension of the fallback procedure to simultaneous confidence intervals by simulations in the context of a dose-finding clinical trial. As a result, we see that the problem of noninformative rejections can be completely removed by the informative fallback procedure, while the involved power loss can be controlled by careful planning.