Date of Award

12-2020

Degree Name

Doctor of Philosophy

Department

Statistics

First Advisor

Dr. Joshua Naranjo

Second Advisor

Dr. Joseph McKean

Third Advisor

Dr. Georgiana Fisher

Fourth Advisor

Dr. Dror Rom

Keywords

Multiple comparisons, simes, Hochberg procedure, Hudson procedure

Abstract

One of the major concerns with multiple tests of significance is controlling the family wise error rate. Various methods have been developed to ensure that the false positive rate be maintained at some prespecified level. One of the most well know being the Bonferroni procedure. Simes presented an improved Bonferroni procedure for testing the global hypothesis that is more powerful and less conservative, especially with positively correlated tests. While Simes’s procedure is more powerful, it does not allow for making inferences on the individual hypotheses. However, the Simes procedure has since become the foundation of many p-value based multiple testing procedures. Hochberg and Hommel are two examples of procedures that have extended the Simes procedure to make inferences on individual hypotheses. Though the procedures by Hochberg and Hommel are based on the Simes procedure, they are conservative as they may not be able to reject any of the individual hypotheses when Simes’s test rejects the global null hypothesis. It is this disconnect between the global test and tests for the individual hypotheses where improvements may be made on the power to reject at least one individual hypothesis. Simes second conjecture was to reject the individual hypotheses H(1), ..., H(j), where j = max{j : P(j) j /m}. It can easily be shown that this method would not control the family wise error rate even for independent tests. An extended single-step Simes testing procedure is presented that rejects a subset of the hypotheses rejected in Simes’s second conjecture. This new procedure rejects at least one hypothesis when the Simes global test rejects, making it the most powerful Simes based procedure for the rejection of at least one hypothesis. This procedure is shown to have strong control of the family wise error for three non-negatively correlated normals with the MTP2 property. Further, simulation studies conducted show that the procedure may control the family wise error rate for as many as thirty or more hypotheses for non-negatively correlated normals with the MTP2 property as well as non-negatively correlated chi-square and T test statistics.

Access Setting

Dissertation-Open Access

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