Mathematics Ph.D. Dissertations

Novel Step-Down Multiple Testing Procedures Under Dependence

Date of Award

2014

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Statistics

First Advisor

John Chen (Advisor)

Second Advisor

Steve Jex (Committee Member)

Third Advisor

Chen Hanfeng (Committee Member)

Fourth Advisor

Shang Junfeng (Committee Member)

Abstract

Many scientific experiments subject to rigorous statistical analysis involve the simultaneous evaluation of more than one hypothesis. One typical example is clinical trials comparing different drug regimens in terms of their therapeutic values and side effects. In such simultaneous comparison cases, the classical separate statistical inference does not work for these multiple related inference problems, because it can cause the error rate to be undesirably large due to ignoring the multiplicity of the problems. Multiple testing procedures adjust statistical inferences from an experiment for multiplicity and thus enable better decision making.

Consider the problem of simultaneously testing null hypotheses H1,..., Hn. The usual approach to dealing with the multiplicity problem is to restrict attention to procedures that control the familywise error rate (FWER), the probability of at least one false rejection. The Bonferroni procedure and its stepwise modification, Holm's procedure are among the most widely used procedures strongly controlling FWER. Lehmann and Romano (2005) generalized the concept of FWER and proposed procedures controlling the k-FWER, the probability of committing at k false rejections. Lehmann and Romano's step-down k-FWER controlling procedure (LR procedure) is the generalization of Holm's procedures. Seneta and Chen (2005) firstly improved Holm's procedure by incorporating the bivariate distribution information of the test statistics. This dissertation is mainly influenced by their work, in which they derived a step-down procedure (CS procedure) that is more powerful than Holm's and still strongly controls the FWER under arbitrary dependence.

In Chapter 3, we first propose three novel step-down FWER controlling procedures that improve both the CS and Holm procedures, by utilizing the bivariate distribution information of the test statistics. We also propose a fast algorithm to calculate the critical values of the new procedures. The theoretical proof and simulation study are both presented to demonstrate that the new procedures can strongly control the FWER and are more powerful than Holm's and the CS procedures under arbitrary dependence structure.

In addition, in Chapter 4, we also propose one new step-down k-FWER controlling procedure that improves the LR procedure under arbitrary dependence by using bivariate distribution of the test statistics. Moreover, a model-based, rather than probability-inequality-based, procedure is found for controlling k-FWER when all test statistics are mutually independent. Numerical analysis shows that, for k > 1, the critical values of k-FWER controlling procedures can exceed beyond the predetermined level α , without violating the k-FWER controlling at α. This is a main difference between FWER and k-FWER controlling procedures. The simulation study investigates the k-FWER control and the power of the new procedure by comparing it to the LR procedure.

Lastly, in Chapter 5, we make an attempt to answer two questions: First, as the uniformly most powerful (UMP) test can be defined for one individual hypothesis, if the similar concepts for multiple testing problems could be defined? Second, how to find such sets of tests that make the power of a certain multiple testing procedure as large as possible. We propose an answer for the Bonferroni procedure in the case that all test statistics are mutually independent. Specifically, we show that in the non-parametric setting the sign test is a UMP test for testing the population median when the probability distribution of the random sample is continuous. Moreover, it also makes the Bonferroni procedure achieve as much power as possible under the continuity assumption. A simulation study follows to support our results.

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