The objective of this paper is to develop statistical methodology for non-inferiority hypotheses to censored, exponentially distributed time to event endpoints. Motivated by a recent clinical trial in depression, we consider a gold standard design where a test group is compared with an active reference and with a placebo group. The test problem is formulated in terms of a retention of effect hypothesis. Thus, the proposed Wald-type test procedure assures that the effect of the test group is better than a pre-specified proportion Delta of the treatment effect of the reference group compared with the placebo group. A sample size allocation rule to achieve optimal power is presented, which only depends on the pre-specified Delta and the probabilities for the occurrence of censoring. In addition, a pretest is presented for either the reference or the test group to ensure assay sensitivity in the complete test procedure. The actual type I error and the sample size formula of the proposed tests are explored asymptotically by means of a simulation study showing good small sample characteristics. To illustrate the procedure a randomized, double blind clinical trial in depression is evaluated. An R-package for implementation of the proposed tests and for sample size determination accompanies this paper on the author's web page.

Clinical noninferiority trials with three (or more) groups recently have received much attention, e.g. due to the fact that regulatory agencies often require that a placebo group has to be evaluated in addition to a new experimental drug and an active control. We discuss the likelihood ratio tests for binary endpoints and various noninferiority hypotheses. We find that, depending on the particular hypothesis, either the LR test reduces asymptotically to the intersection union test, or to a test which follows asymptotically a mixture of generalized χ 2 -distributions. The performance of this asymptotic is investigated and an exact modification is given. It is shown that this test considerably outperforms multiple testing methods with respect to power, such as the Bonferroni adjustment. The methods are illustrated with a cancer study where antiemetic agents were compared. Finally, we discuss the extension of the results to other settings, such as normal endpoints.

We consider a class of testing problems where the null space is the union of k − 1 subgraphs of the form h j (θ j ) ≤ θ k , with j = 1, . . . , k − 1, (θ 1 , . . . , θ k ) the unknown parameter, and h j given increasing functions. The data consist of k independent samples, assumed to be drawn from a distribution with parameter θ j , j = 1, . . . , k, respectively. An important class of examples covered by this setting is that of non-inferiority hypotheses, which have recently become important in the evaluation of drugs or therapies. When the true parameter approaches the boundary at a 1/ √ n rate, we give the explicit form of the asymptotic distribution of the log-likelihood ratio statistic. This extends previous work on the distribution of likelihood ratio statistics to local alternatives. We consider the prominent example of binomial data and illustrate the theory for k = 2 and 3 samples. We explain how this can be used for planning a non-inferiority trial. To this end we calculate the optimal sample ratios yielding the maximal power in a binomial non-inferiority trial.

This work would not have been possible without the help of many people. I wish to thank my thesis advisor Prof. Dr. Axel Munk for proposing this subject and for helpful discussions and Prof. Dr. Edgar Brunner for taking the Koreferat. To Dr. Alexander Schacht (Lilly Deutschland GmbH) I am indebted for providing me with the data set from a clinical trial in the treatment of depression. I would like to thank my colleagues at the Institute for Mathematical Stochastics, in particular Jörn Dannemann, for helpful discussions, inspiration and friendly warm atmosphere. During my time as a Ph.D. student I was a member of the Ph.D. Programm "Applied Statistics and Empirical Methods" in the Centre of Statistics and I gratefully acknowledge financial support from the "Georg-Lichtenberg-Programm" for the first eight month of my Ph.D. studies and the possibility of partaking in interesting scientific discourse. Last but certainly not least I express my deepest gratitude to my family for various forms of support, above all from my girlfriend Merle Schlichte as well as my parents Waltraud and Rainer.