Two-Stage Adaptive Designs with Correlated Test Statistics

Stepwise progressive parametric multiple testing procedure with correlated normal test statistics

Multiple testing problems are often seen in clinical trials. An appropriate testing procedure should be specified to deal with the potential inflation of type I error rate due to multiplicity. In this article, we propose a stepwise progressive parametric multiple (SPPM) testing procedure, which constructs the testing using the products of all the combinations of local [Formula: see text]-values and the critical values are determined by numerical integrations progressively using the closure principle. We have compared the performance of SPPM to several other procedures, and demonstrate the advantage of SPPM procedure, in terms of power, for the certain situations of multiple testing.

Adaptive Designs with Discrete Test Statistics and Consideration of Overrunning

Background: 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. Occur-rence 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.

Objectives: 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.

Methods: 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.

Results: 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.

Conclusions: 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.

[Modern study designs and analysis methods in clinical research]

BACKGROUND

Progress in the field of medical research requires further development of clinical trial methodology to overcome the challenges resulting from small patient populations and restricted resources.

METHODS

Classical single-stage designs with fixed sample sizes do not allow for interim analyses or design modifications. In contrast, adaptive designs adhere to established quality criteria while providing flexibility when conducting a clinical trial. In the face of new discoveries or information collected in the course of a trial, sample size adjustment, the selection of the target population and further design modifications can be performed. This enhances the chance of success of a clinical trial. Besides adaptive designs, classical approaches may be replaced or complemented by Bayesian methods. In a Bayesian approach prior knowledge can be efficiently included and hence the amount of information utilized in statistical analyses is increased. Furthermore, Bayes procedures allow the results of a statistical evaluation to be displayed very clearly.

CONCLUSION

Modern approaches, such as adaptive designs and Bayesian designs overcome the challenges in clinical research due to enhanced flexibility and efficiency. In addition, both approaches can be combined.

Adaptive designs with correlated test statistics

In clinical trials, the collected observations such as clustered data or repeated measurements are often correlated. As a consequence, test statistics in a multistage design are correlated. Adaptive designs were originally developed for independent test statistics. We present a general framework for two-stage adaptive designs with correlated test statistics. We show that the significance level for the Bauer-Köhne design is inflated for positively correlated test statistics from a bivariate normal distribution. The decision boundary for the second stage can be modified so that type one error is controlled. This general concept is expandable to other adaptive designs. In order to use these designs, the correlation between test statistics has to be estimated. For a known covariance matrix, we show how correlation can be determined within the framework of linear mixed models. A sample size reassessment rule is proposed and evaluated for an unknown covariance matrix by simulation. As Wald test statistics in linear mixed models have independent increments, we use this property to create valid test procedures. We compare these procedures with the proposed design in our simulations.

How practical are adaptive designs likely to be for confirmatory trials?

This is a discussion of the following papers appearing in this special issue on adaptive designs: 'Confirmatory Seamless Phase II/III Clinical trials with Hypotheses Selection at Interim: General Concepts' by Frank Bretz, Heinz Schmidli, Franz König, Amy Racine and Willi Maurer; and 'Confirmatory Seamless Phase II/III Clinical Trials with Hypotheses Selection at Interim: Applications and Practical Considerations' by Heinz Schmidli, Frank Bretz, Amy Racine and Willi Maurer.