Computational methods for adaptive and multiple inference

Adaptive designs provide an active statistical research topic and are increasingly applied in clinical trials. Multiple testing and simultaneous confidence intervals are related and similar important topics of statistical inference and are even more frequently applied in medical and biological sciences. As recent research indicates, modern statistical solutions for adaptive and multiple inference often require substantial computational efforts and rely on suitable and efficient numerical algorithms. For adaptive designs the conditional type I error principle is an example where complex numerical solutions are needed, in particular, for the computation of confidence intervals and for hypotheses with nuisance parameters. In multiple inference efficient numerical solutions are required e.g. when a large number of parameters are investigated, complex test procedures are used and/or simultaneous confidence intervals are computed. It is therefore important to develop new statistical approaches for adaptive designs and multiple inference that are computationally and statistically efficient. This research project consists of two parts in which problems from adaptive designs and multiple inference are considered. The first part deals with the extension and application of the conditional type I error principle in adaptive designs. We consider the construction of one- and two-sided confidence intervals for two- and multi-arm clinical trials that start with some classical group sequential or multiple test design and where sample sizes, the number of interim analyses, the decision boundaries and/or treatment arms are changed at interim analyses based on the information accumulated so far. We will also search for solutions of the conditional type I error principle in the presence of nuisance parameters, particularly, within the class of linear models, for two- and multi-armed designs. The second part deals with simultaneous parameter estimation. Here we will consider a completely new method for simultaneous confidence intervals. The method is based on a new class of intersection hypothesis tests and the closed testing principle. We will use penalized union intersection tests were the penalizations depend on the parameter values to be tested. Such simultaneous confidence intervals will resolve an open statistical problem, namely, the construction of simultaneous confidence intervals that provide multiple test procedures which are statistically as (or almost as) efficient as common step-wise multiple test procedures but always provide additional information to the multiple test for all parameters. The main difficulty with our new approach is again the implementation of the underlying closed test procedure. In this case we have to deal with an infinite number of intersection tests that are not nested. Hence the implementation is a demanding task that is of principle and theoretical nature and not only important from a computational point of view. We will also consider the extension of the method to adaptive closed test procedures.

The project dealt with statistical and numerical methods for adaptive study designs and multiple testing problems. We have, for instance, extended the multiple test methodology by a new class of simultaneous confidence intervals that provide similar power advantages as step-down tests, like e.g. the Holm test, and specific gate keeping procedures and, at the same time, give much more information than the known intervals of the Holm, other step-down and gate-keeping tests. In another research we derived numerical shortcuts for closed test procedures for logically related null hypotheses. The shortcut applies e.g. to the all pairwise comparison problem, and to adaptive closed testing procedures. We also extended the multiple test methodology for dose finding studies and derived new efficient multiples tests for cost-effectiveness studies. For adaptive two-stage designs with interim treatment selection we considered the important problem of selection bias and developed shrinkage estimates for two-stage designs. We showed that the new estimates reduce the selection bias and have a much higher precision than the MLE and the bias adjusted estimates from the literature. We furthermore extended an important adaptive testing methodology, namely for cases with nuisance parameters and used the new method to provide an exact test for seamless phase II/III designs where treatments are allocated by a response adaptive scheme in the first stage and the selected treatments are block randomized in the second stage. These types of designs were controversially discussed in the scientific and regulatory community, in particular, because of the lack for an exact testing method. In cooperation with statisticians from the FDA, we also clarified issues with treatment selection based on surrogates and the currently discussed promising zone approach for sample size recalculations.