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In clinical trials, several endpoints (EPs) are often evaluated to compare treatments in some therapeutic area. Suppose that there are two EPs in a clinical trial. We propose a new set of composite hypotheses for continuous variables, taking the relative clinical importance of the EPs into account. The main hypotheses were formulated to show that a treatment is so superior to the control treatment, which is not necessarily a placebo, in one EP, that the possible non-inferiority of the treatment by at most a certain value in the other EP can be compensated sufficiently, taking the clinical point of view into account. The maximum non-inferiority margin of one EP might not be a biologically unimportant difference in exchange for much superiority of the other EP. This formulation leads to a new composite EP and a very simple test statistic. The intersection-union principle was employed to derive the proposed test.

Often a treatment is assessed by co-primary endpoints so that a comprehensive picture of the treatment effect can be obtained. Co-primary endpoints can be different medical assessments angled at different aspects of a disease, therefore, are used collectively to strengthen evidence for the treatment effect. It is common sense that if a treatment is ineffective, the chance to show that the treatment is effective in all co-primary endpoints should be small. Therefore, it may not be necessary to require all the co-primary endpoints to be statistically significant at the 1-sided 0.025 level to control the error rate of wrongly approving an ineffective treatment. Rather it is reasonable to allow certain variation for the p -values within a range close to 0.025. In this paper, statistical methods are developed to derive decision rules to evaluate co-primary endpoints collectively. The decision rules control the error rate of wrongly accepting an ineffective treatment at the level of 0.025 for a study and the error rate at a slightly higher level for a treatment that works for all the co-primary endpoints except perhaps one. The decision rules also control the error rates for individual endpoints. Potential applications in clinical trials are presented., (2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

Ranking multivariate ordinal data and applying a non-parametric test is an analytical approach commonly employed to compare treatments. We study three types of ranking and demonstrate how to combine them. The ranking methods rest upon partial orders of the multidimensional measurements or upon the sum of ranks. Since their usage is simple as regards statistical assumptions and technical realization, they are also adapted for health professionals without deep statistical knowledge. Our goal is discussing differences between the approaches and disclosing possible statistical consequences of their usage., (2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.)

In a typical clinical trial, there are one or two primary endpoints, and a few secondary endpoints. When at least one primary endpoint achieves statistical significance, there is considerable interest in using results for the secondary endpoints to enhance characterization of the treatment effect. Because multiple endpoints are involved, regulators may require that the familywise type I error rate be controlled at a pre-set level. This requirement can be achieved by using "gatekeeping" methods. However, existing methods suffer from logical oddities such as allowing results for secondary endpoint(s) to impact the likelihood of success for the primary endpoint(s). We propose a novel and easy-to-implement gatekeeping procedure that is devoid of such deficiencies. A real data example and simulation results are used to illustrate efficiency gains of our method relative to existing methods.

This article discusses specific assumptions necessary for permutation multiple tests to control the Familywise Error Rate (FWER). At issue is that, in comparing parameters of the marginal distributions of two sets of multivariate observations, validity of permutation testing is affected by all the parameters in the joint distributions of the observations. We show the surprising fact that, in the case of a linear model with i.i.d. errors such as in the analysis of Quantitative Trait Loci (QTL), this issue has no impact on control of FWER, if the test statistic is of a particular form. On the other hand, in the analysis of gene expression levels or multiple safety endpoints, unless some assumption connecting the marginal distributions of the observations to their joint distributions is made, permutation multiple tests may not control FWER.

We consider the problem of drawing superiority inferences on individual endpoints following non-inferiority testing. Röhmel et al. (2006) pointed out this as an important problem which had not been addressed by the previous procedures that only tested for global superiority. Röhmel et al. objected to incorporating the non-inferiority tests in the assessment of the global superiority test by exploiting the relationship between the two, since the results of the latter test then depend on the non-inferiority margins specified for the former test. We argue that this is justified, besides the fact that it enhances the power of the global superiority test. We provide a closed testing formulation which generalizes the three-step procedure proposed by Röhmel et al. for two endpoints. For the global superiority test, Röhmel et al. suggest using the Läuter (1996) test which is modified to make it monotone. The resulting test not only is complicated to use, but the modification does not readily extend to more than two endpoints, and it is less powerful in general than several of its competitors. This is verified in a simulation study. Instead, we suggest applying the one-sided likelihood ratio test used by Perlman and Wu (2004) or the union-intersection t(max) test used by Tamhane and Logan (2004).

This is a discussion of the following two papers 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.