Mathematics Ph.D. Dissertations

Title

A Partitioning Approach for the Selection of the Best Treatment

Date of Award

2013

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Statistics

First Advisor

John T. Chen (Advisor)

Second Advisor

Arjun K. Gupta (Committee Member)

Third Advisor

Wei Ning (Committee Member)

Fourth Advisor

Haowen Xi (Committee Member)

Abstract

To select the best treatment among several treatments is essentially a multiple comparisons problem. Traditionally, when dealing with multiple comparisons, there is one main argument: with multiplicity adjustment or without adjustment. If multiplicity adjustment is made such as the Bonferroni method, the simultaneous inference becomes too conservative. Moreover, in the conventional methods of multiple comparisons, such as the Tukey's all pairwise multiple comparisons, although the simultaneous confidence intervals could be obtained, the best treatment cannot be distinguished efficiently. Therefore, in this dissertation, we propose several novel procedures using partitioning principle to develop more efficient simultaneous confidence sets to select the best treatment.

The method of partitioning principle for efficacy and toxicity for ordered treatments can be found in Hsu and Berger (1999). In this dissertation, by integrating the Bonferroni inequality, the partition approach is applied to unordered treatments for the inference of the best one.

With the introduction of multiple comparison methodologies, we mainly focus on the all pairwise multiple comparisons. This is because all the treatments should be compared when we select the best treatment. These procedures could be used in different data forms. Chapter 2 talks about how to utilize the procedure in dichotomous outcomes and the analysis of contingency tables, especially with the Fisher's Exact Test. Chapter 3 discusses the procedures in nonparametric field. With Mann-Whitney test, these procedures become more robust. Chapter 4 addresses the procedures with continuous data under normality. In Chapter 5 we apply the procedures to analyze a prostate cancer study.

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