Hierarchical Testing of a Primary and a Secondary Endpoint in a Group Sequential Design With Different Information Times

Reverse graphical approaches for multiple test procedures

The graphical approach has been proposed as a general framework for clinical trial designs involving multiple hypotheses, where decisions are made only based on the observed marginal p -values. The graphical approach starts from a graph that includes all hypotheses as vertices and gradually removes some vertices when their corresponding hypotheses are rejected. In this paper, we propose a reverse graphical approach, which starts from a set of singleton graphs and gradually adds vertices into graphs until rejection of a set of hypotheses is made. Proofs of familywise error rate control are provided. A simulation study is conducted for statistical power analysis, and a case study is included to illustrate how the proposed approach can be applied to clinical studies.

On dependence assumption in p-value based multiple test procedures

There are various multiple comparison procedures used in confirmatory clinical studies and exploratory research for multiplicity adjustment. Among them are the Hochberg and Benjamini-Hochberg procedures. A common misconception is that these procedures control the type I error rate properly if the test statistics are independent or positively correlated. In fact, a much stronger positive dependence assumption needs to be satisfied to guarantee the type I error rate control. We give a comprehensive review of the dependence conditions used in multiple testing procedures. We show that a weaker positive dependence assumption may result an inflation of type I error rate by a factor of 2 and discuss the type I error rate control under certain negative dependence conditions.

Group sequential Holm and Hochberg procedures

The problem of testing multiple hypotheses using a group sequential procedure often arises in clinical trials. We review several group sequential Holm (GSHM) type procedures proposed in the literature and clarify the relationships between them. In particular, we show which procedures are equivalent or, if different, which are more powerful and what are their pros and cons. We propose a step-up group sequential Hochberg (GSHC) procedure as a reverse application of a particular step-down GSHM procedure. We conducted an extensive simulation study to evaluate the familywise error rate (FWER) and power properties of that GSHM procedure and the GSHC procedure and found that the GSHC procedure controls FWER more closely and is more powerful. All procedures are illustrated with a common numerical example, the data for which are chosen to bring out the differences between them. A real case study is also presented to illustrate application of these procedures. R programs for applying the proposed procedures, additional simulation results, and the proof of the FWER control of the GSHC procedure in a special case are provided in Supplementary Material.

Improved group sequential Holm procedures for testing multiple correlated hypotheses over time

Clinical trials can typically feature two different types of multiple inference: testing of more than one null hypothesis and testing at multiple time points. These modes of multiplicity are closely related mathematically but distinct statistically and philosophically. Regulatory agencies require strong control of the family-wise error rate (FWER), the risk of falsely rejecting any null hypothesis at any analysis. The correlations between test statistics at interim analyses and the final analysis are therefore routinely used in group sequential designs to achieve less conservative critical values. However, the same type of correlations between different comparisons, endpoints or sub-populations are less commonly used. As a result, FWER is in practice often controlled conservatively for commonly applied procedures.Repeated testing of the same null hypothesis may give changing results, when the hypothesis is rejected at an interim but accepted at the final analysis. The mathematically correct overall rejection is at odds with an inference theoretic approach and with common sense. We discuss these two issues, of incorporating correlations and how to interpret time-changing conclusions, and provide case studies where power can be increased while adhering to sound statistical principles.

Sample size optimization and initial allocation of the significance levels in group sequential trials with multiple endpoints

We consider multistage tests of multiple hypotheses under a flexible setting of calendar time and information fraction, focusing on the case where there are two hypotheses under testing. Explicit expressions of statistical powers are derived. With a proof of existence and uniqueness of solution, we develop a numerical method to search the optimal sample size. The proposed method allows us to find the suitable allocation of initial significance level along with the minimum sample size for group sequential designs, with and without hierarchical structures among different endpoints.

On selecting the critical boundary functions in group-sequential trials with two time-to-event outcomes

We investigate selection of critical boundary functions for testing the hypotheses of two time-to-event outcomes as both primary endpoints or a primary and a secondary endpoint in group-sequential clinical trials, where (1) the effect sizes of endpoints are unequal, or (2) one endpoint is for short-term evaluation and the other for long-term evaluation. Bonferroni-Holm and fixed-sequence procedures are considered. We assess the effects of the magnitudes of the hazard ratios and the correlation between the endpoints on statistical powers and provide guidance for consideration.