Adaptive Designs with Discrete Test Statistics and Consideration of Overrunning

Interim analyses are used in clinical trials in order to enable early decisions for medical, ethical, and economic reasons. However, it appears unfeasible to stop a trial during such an interim analysis. New patients will thus enter the trial while the interim analysis is ongoing. Moreover, depending on the event kinetics of the specific disease, the trial design, and the corresponding endpoints, some patients might still be unevaluable at the interim analysis due to not yet completed follow-up. Occurrence of these types of patients is characteristic for sequentially analyzed trials. Such patients are referred to as interim patients. In trials with multiple primary endpoints, another type of interim patients occurs. If some but not all null hypotheses can be rejected at the interim analysis, the trial might be continued to a second stage in order to answer the remaining questions. These second stage patients, however, provide new data to all trial questions including the already rejected ones and thus formally act as interim patients regarding the already rejected null hypotheses. Although all kinds of interim patients are not part of the interim analysis, the data collected on those patients have to be sent to the office of regulatory affairs and will be analyzed. If a smaller or contrasting treatment effect is observed in interim patients, this might lead to a withdrawal of an earlier superiority proof.

Presently, interim patients and their data are usually not considered in the confirmatory test. We offer a strategy to deal with interim patients in sequentially analyzed trials with discrete test statistics. The method covers sequentially analyzed single- and multi-arm trials with one or multiple primary endpoints.

When planning adaptive designs, it is common practice to assume that the stage-wise p-values are independent and standard uniformly distributed under the null hypothesis. In the context of discrete test statistics, this implies conservative tests. We provide an algorithm which iteratively optimizes an initially given design while adjusting for both discreteness of test statistics and interim patients. The algorithm is described verbally, graphically and formally to facilitate immediate implementation in computer software.

The optimized design exploits the aspired significance level better and is more powerful than the initial one. The algorithm applies to fixed sample and planned flexible adaptive designs for single- and multi-arm trials with one or multiple primary endpoints. The benefit increases with the number of interim patients.

When planning a trial with interim analyses, the rules for decisions must be adjusted to interim patients. Otherwise, the test procedure is conservative resulting in loss of power. This is essential in situations where the number of interim patients is important compared to the first stage, particularly in trials with multiple primary endpoints.