Multiplicity issues arise prevalently in a variety of situations in clinical trials and statistical methods for multiple testing have gradually gained importance with the increasing number of complex clinical trial designs. In general, two types of multiple testing can be performed (Dmitrienko et al., 2009): union-intersection testing (UIT) and intersection-union testing (IUT). The UIT is of the interest in this dissertation. Thus, the familywise error rate (FWER) is required to be controlled in the strong sense. A number of methods have been developed for controlling the FWER, including single-step and stepwise procedures. In single-step approaches, such as the simple Bonferroni method, the rejection decision of a hypothesis does not depend on the decision of any other hypotheses. Single-step approaches can be improved in terms of power through stepwise approaches, while also controlling for the desired error rate. Besides, it is also possible to improve those procedures by a parametric approach. In the first project, we developed a new and powerful single-step progressive parametric multiple (SPPM) testing procedure for correlated normal test statistics. Through simulation studies, we demonstrate that SPPM improves power substantially when the correlation is moderate and/or the magnitude of eect sizes are similar. Group sequential designs (GSD) are clinical trials allowing interim looks with the possibility of early terminations due to ecacy, harm or futility, which can reduce the overall costs and timelines for the development of a new drug. However, repeated looks of data also have multiplicity issues and could inflate the type I error rate. The proper treatments to the error inflation have been discussed widely (Pocock, 1977), (O'Brien and Fleming, 1979), (Wang and Tsiatis, 1987), (Lan and DeMets, 1983). Most literature about GSD focuses on a single endpoint. GSD with multiple endpoints however, has also received considerable attention. The main focus of our second project is a GSD with multiple primary endpoints, in which the trial is to evaluate whether at least one of the endpoints is statistically signicant. In this study design, multiplicity issues arise from repeated interims and multiple endpoints. Therefore, the appropriate adjustments must be made to control the Type I error rate. Our second purpose here is to show that the combination of multiple endpoint and repeated interim analyses can lead to a more powerful design. Via the multivariate normal distribution, a method that allows for simultaneously consideration of interim analyses and all clinical endpoints was proposed. The new approach is derived from the closure principle, thus it can control type I error rate strongly. We evaluate the power under dierent scenarios and show that it compares favorably to other methods when correlation among endpoints is non-zero. In the group sequential design framework, another interesting topic is multiple arm multiple stage design (MAMS), where multiple arms are involved in the trial at the beginning with the flexibility about treatment selection or stopping decisions during the interim analyses. One of major hurdles of MAMS is the computational cost with the increasing number of arms and interim looks. Various designs were implemented to overcome this diculty (Thall et al., 1988; Schaid et al., 1990; Follmann et al., 1994; Stallard and Todd, 2003; Stallard and Friede, 2008; Magirr et al., 2012; Wason et al., 2017), but also control the FWER with the potential inflation from the multiple arm comparisons and multiple interim tests. Here, we consider a more flexible drop-the-loser design allowing the safety information in the treatment selection without a pre-specied dropping-arms mechanism and it still retains reasonable high power. The two dierent types of stopping boundaries are proposed for such a design. A sample size is also adjustable if the winner arm is dropped due to the safety considerations.

Title: Multidimensional statistics and applications to study genes Author: Mgr. Peter Bubelíny Department: Department of probability and mathematical statistics Supervisor: prof. Lev Klebanov, DrSc., KPMS MFF UK Abstract: Microarray data of gene expressions consist of thousands of genes and just some tens of observations. Moreover, genes are highly correlated between themselves and contain systematic errors. Hence the magnitude of these data does not afford us to estimate their correlation structure. In many statistical problems with microarray data, we have to test some thousands of hypotheses simultaneously. Due to dependence between genes, p-values of these hypotheses are dependent as well. In this work, we compared conve- nient multiple testing procedures reasonable for dependent hypotheses. The common manner to make microarray data more uncorrelated and partially eliminate systematic errors is normalizing them. We proposed some new normalizations and studied how different normalizations influence hypothe- ses testing. Moreover, we compared tests for finding differentially expressed genes or gene sets and identified some interesting properties of some tests such as bias of two-sample Kolmogorov-Smirnov test and interesting behav- ior of Hotelling's test for dependent components of observations. In the end of...

In this dissertation, we propose a computational approach using a resampling based permutation test as an alternative to MCP-Mod (a hybrid framework integrating the multiple comparison procedure and the modeling technique) and gMCP-Mod (generalized MCP-Mod) [11], [29] in the step of identifying significant dose-response signals via model selection. We name our proposed approach RMCP-Mod or gRMCP-Mod correspondingly. The RMCP-Mod/gRMCP-Mod transforms the drug dose comparisons into a dose-response model selection issue via multiple hypotheses testing, an area where not much extended researches have been done, and solve it using resampling based multiple testing procedures [38]. The proposed approach avoids the inclusion of the prior dose-response knowledge known as "guesstimates" used in the model selection step of the MCP-Mod/gMCP-Mod framework, and therefore reduces the uncertainty in the significant model identification. When a new drug is being developed to treat patients with a specified disease, one of the key steps is to discover an optimal drug dose or doses that would produce the desired clinical effect with an acceptable level of toxicity. In order to nd such a dose or doses (different doses may be able to produce the same or better clinical effect with similar acceptable toxicity), the underlying dose-response signals need to be identified and thoroughly examined through statistical analyses. A dose-response signal refers to the fact that a drug has different clinical effects at many quantitative dose levels. Statistically speaking, the dose-response signal is a numeric relationship curve (shape) between drug doses and the clinical effects in quantitative measures. It's often been a challenge to nd correct and accurate efficacy and/or safety dose-response signals that would best describe the dose-effect relationship in the drug development process via conventional statistical methods because the conventional methods tend to either focus on a fixed, small number of quantitative dosages or evaluate multiple pre-denied dose-response models without Type I error control. In searching for more efficient methods, a framework of combining both multiple comparisons procedure (MCP) and model-based (Mod) techniques acronymed MCP-Mod was developed by F. Bretz, J. C. Pinheiro, and M. Branson [11] to handle normally distributed, homoscedastic dose response observations. Subsequently, a generalized version of the MCP- Mod named gMCP-Mod which can additionally deal with binary, counts, or time-to-event dose-response data as well as repeated measurements over time was developed by J. C. Pinheiro, B. Bornkamp, E. Glimm and F. Bretz [29]. The MCP-Mod/gMCP-Mod uses the guesstimates" in the MCP step to pre-specify parameters of the candidate models; however, in situations where the prior knowledge of the dose-response information is difficult to obtain, the uncertainties could be introduced into the model selection process, impacting on the correctness of the model identification. Throughout the evaluation of its application to the hypothetical and real study examples as well as simulation comparisons to the MCP-Mod/gMCP-Mod, our proposed approach, RMCP-Mod/gRMCP-Mod seems a viable method that can be used in the practice with some further improvements and researches that are still needed in applications to broader dose-response data types. / Statistics