Drug Information Journal - March 2009 - (Page 177) BIOSTATISTICS 177 John Lawrence US Food and Drug Administration, Silver Spring, Maryland Xiaobai Li Center for Biostatistics, The Ohio State University, Columbus, Ohio A Multiple Comparisons Procedure for Comparing All Arms in Three-Arm Clinical Trials In three-arm studies, there are often multiple hypotheses to be tested among the arms. In this article, we describe a multiple comparisons procedure that controls the familywise error rate in the strong sense. The procedure is appropriate when one of the arms is a control, and both treatment arms will be compared to the control as well as to each other. The procedure is shown to be consistent and the asymptotic relative efficiency (compared to the uniformly most powerful test for the individual comparisons) is close to 1 in many scenarios. Furthermore, the special case where the arms are placebo, active control, and test drug with one goal of showing noninferiority is examined. Key Words Multiple comparisons; Strong familywise error rate; Combination drugs; Active control; Noninferiority Correspondence Address John Lawrence, US Food and Drug Administration, 10903 New Hampshire Ave., Silver Spring, MD 20993. The views expressed are those of the authors and not necessarily those of the US Food and Drug Administration. INTRODUCTION In this article, we discuss the analysis of threearm clinical trials where the arms will be compared with respect to a single endpoint. Furthermore, there is an expected ordering of the three arms under a postulated alternative hypothesis. An example is a trial that measures blood pressure of hypertensive patients as the endpoint. There is a placebo control arm, a single drug (drug A) arm, and a third arm for the combination of two drugs (drug A + drug B). It is expected that drug A is better than placebo and the combination is better than drug A alone. In this example, there are three possible hypotheses to be tested: namely, blood pressure reduction in the monotherapy group is better than in the placebo group; the reduction in the combination group is better than in the placebo group; and the reduction in the combination group is better than in the monotherapy group. Since there are three hypotheses to be tested within a single trial, it may be desirable to control the strong familywise error rate for the three hypotheses. There are several well-known procedures for multiple comparisons that control the strong familywise error rate, but some are not appropriate for this situation. Dunnett’s method (1) is used to compare multiple treatment arms with a control, but would not allow for the comparison of the two active drug groups. The Tukey-Kramer method (2,3) can be used to compare all groups. The Bonferroni test or its closure (4) are general procedures for multiple comparisons. Although these latter two procedures allow for all the comparisons of interest and control the familywise error rate, they are not designed specifically for the scenario under consideration and are not as powerful as the procedure that is described in this article under many scenarios. The closure dominates the single-step Bonferroni test. Hommel’s procedure (5) is the closure of Simes’s test (6). This is also a general multiple comparisons procedure but does not control the familywise error rate in general. Hochberg’s step-up procedure (7) is also valid under conditions where Simes’s test is valid, but is dominated by Hommel’s procedure. We compare the proposed procedure with some of these procedures in the example section. M AT H E M AT I C A L D E S C R I P T I O N OF HYPOTHESES There are three groups labeled groups 1, 2, and 3. We will describe the hypotheses for a time-toevent endpoint and a continuous endpoint. If the endpoint is a time-to-event, assume a proportional hazards model. The hazard function for group 1 is h(t); the hazard function for group 2 is e−θ1h(t); and for group 3 it is e−θ1− θ2h(t). The three alternative hypotheses are H 1 : A 2 3 θ1 > 0, H A : θ2 > 0, and H A : θ1 + θ2 > 0. Submitted for publication: May 5, 2008 Accepted for publication: September 9, 2008 Drug Information Journal, Vol. 43, pp. 177–1 2009 • 0092-8615/2009 84, Printed in the USA. All rights reserved. Copyright © 2009 Drug Information Association, Inc.
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