New Step Down Procedures for Control of the Familywise Error Rate

Abstract

The main research topic in this dissertation is the development of the closure method of multiple testing procedures. Considering a general procedure that allows the underlying test statistics as well as the associated parameters to be dependent, we first propose a step-down procedure controlling the FWER, which is defined as the probability of committing at least one false discovery. Holm (1979) first proposed a step-down procedure for multiple hypothesis testing with a control of the familywise error rate (FWER) under any kind of dependence. Under the normal distributional setup, Seneta and Chen (2005) sharpened the Holm procedure by taking into account the correlations between the test statistics. In this dissertation, the Seneta-Chen procedure is further modified yielding a more powerful FWER controlling procedure. We then advance our research and propose another step-down procedure to control the generalized FWER (k-FWER), which is defined as the probability of making at least k false discoveries. We compare our proposed k-FWER procedure with the Lehmann and Romano (2005) procedure. The proposed k-FWER procedure is more powerful, particularly when there is a strong dependence in the tests. When the proportion of true null hypotheses is expected to be small, the traditional tests are usually conservative by a factor associated with pi0, which is the proportion of true null hypotheses among all null hypotheses. Under independence, two procedures controlling the FWER and the k-FWER are proposed in this dissertation. Simulations are carried out to show that our procedures often provide much better FWER or k-FWER control and power than the traditional procedures.