This dissertation addresses several different but related topics arising in the field of multiple testing, including weighted procedures and graphical approaches for controlling the familywise error rate (FWER), and stepwise procedures with control of the false discovery rate (FDR) for discrete data. It consists of three major parts. The first part investigates weighted procedures for controlling the FWER. In many statistical applications, hypotheses may be differentially weighted according to their different importance. Many weighted multiple testing procedures (wMTPs) have been developed for controlling the FWER. Among these procedures, two weighted Holm procedures are commonly used in practice: one is based on ordered weighted p-values and is called WHP; the alternative weighted Holm procedure that is based on ordered raw p-values is named WAR This part of dissertation studies statistical properties of these two weighted procedures and make recommendation for their applications. First, the corresponding closed testing procedures (CTPs) of both weighted procedures are obtained and the WHP is proved to be uniformly more powerful than the WAP. Following this, in order to provide an intuitive and clear way to communicate with non-statisticians, two procedures are visualized with graphical approaches through a common initial graph and their corresponding updating strategies. Next, the adjusted p-values are derived for these two procedures. Finally, the optimality of these two procedures is discussed and it is shown that the WHP is an optimal procedure in the sense that the procedure cannot be improved by increasing even one of its critical values without losing control over the FWER. Simulations were conducted to provide numerical evidence of superior performance of the WHP in terms of the FWER control and average power. In the second part of the dissertation, two graphical approaches are investigated. One is the original graphical approach which is introduced in Bretz et al. (2009) and widely used in clinical trials studies, and the other one is the default graphical approach, proposed in Burman et al. (2009). These two graphical approaches are commonly considered to be equivalent in the literature. However, this study shows that their equivalence can only be achieved under certain conditions or in the case of three hypotheses. When the conditions are satisfied, a general method is developed for deriving the equivalent graph. The nonuniqueness property of the original graphical approach is also discussed. Moreover, a simple and direct proof is offered for showing the FWER control of the original graphical approach. This is helpful for understanding the original graphical approach thoroughly and provides some guideline to develop new graphical approaches. In the third part of the dissertation, a new generalized step-up FDR controlling procedure is developed for discrete data. Most existing FDR controlling procedures are developed for continuous data, which are often conservative when analyzing discrete data. Lynch and Guo (2016) introduced a generalized stepwise procedure which generalizes the usual stepwise procedure to the case where each hypothesis is tested with a different set of critical constants. Under the framework of the generalized step-up approach, by taking the discreteness and heterogeneity properties of discrete data into account and fully utilizing known marginal distributions of true null p-values, a powerful generalized step-up procedure is proposed for discrete case. Theoretically, it is shown that the proposed procedure strongly controls the FDR under independence and is more powerful than the popular BH procedure. Nevertheless, some theoretical as well as simulation issues still remain to be fully addressed.
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