Dokument: Asymptotic and Exact Results on FWER and FDR in Multiple Hypotheses Testing

Titel:Asymptotic and Exact Results on FWER and FDR in Multiple Hypotheses Testing
URL für Lesezeichen:https://docserv.uni-duesseldorf.de/servlets/DocumentServlet?id=16990
URN (NBN):urn:nbn:de:hbz:061-20110113-092759-1
Kollektion:Dissertationen
Sprache:Englisch
Dokumententyp:Wissenschaftliche Abschlussarbeiten » Dissertation
Medientyp:Text
Autor: Gontscharuk, Veronika [Autor]
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Dateien vom 07.01.2011 / geändert 07.01.2011
Beitragende:Apl. Prof. Dr. Finner, Helmut [Gutachter]
Prof. Dr. Janssen, Arnold [Gutachter]
Prof. Dr. Blanchard, Gilles [Gutachter]
Dewey Dezimal-Klassifikation:500 Naturwissenschaften und Mathematik » 510 Mathematik
Beschreibung:Nowadays, multiple hypotheses testing has become a Promising area of statistics. In medicine, biology, pharmacology, epidemiology and even marketing, many hypotheses often have to be tested simultaneously. In some applications like genome-wide association studies, there may be several hundreds of thousands hypotheses to be tested.

An important concept in multiple testing is controlling a suitable Type I error rate. The Family-Wise Error Rate (FWER) is a classical error rate criterion and denotes the probability of one or more false rejections. Unfortunately, the FWER is often too restrictive if the number of hypotheses is very large. In 1995, Benjamini and Hochberg introduced an alternative error rate called the False Discovery Rate (FDR). The FDR denotes the expected proportion of falsely rejected hypotheses among all rejections. Typically, multiple test procedures controlling the FDR are more powerful than multiple tests controlling the FWER. However, if the number of true hypotheses is large and almost all hypotheses are true, procedures controlling the FWER may be a good alternative to tests controlling the FDR.

In this work we deal with multiple test procedures that control one of the aforementioned multiple error rates for independent test statistics and dependent ones as well. In the case of dependent test statistics, asymptotic considerations play a decisive role. Chapter 1 is an introduction into basic concepts and problems concerning multiple hypotheses testing.

In Chapter 2 we discuss a possibility to improve the power of some classical multiple tests controlling the FWER by applying a plug-in estimate for the number of true null hypotheses. We investigate several plug-in estimates and prove FWER control of Bonferroni, Šidàk and so-called step-down plug-in multiple test procedures. Moreover, we obtain some asymptotic results and compare the power of plug-in tests with the power of the Corresponding classical procedures.

In Chapter 3 we restrict our attention to exact control of the FDR for step-up-down (SUD) test procedures. We give a recursive scheme which allows to calculate critical values such that the corresponding FDR equals the pre-specified FDR bounding curve. This scheme is numerically extremely sensitive so that computation of feasible solutions remains a challenging problem. We introduce alternative FDR bounding curves and study their connection to rejection curves as well as the existence of valid sets of critical values leading to these FDR bounding curves. In order to compute feasible critical values two further approaches are presented.

In Chapter 4 we focus on situations where some kind of weak dependence occurs. We consider models where the empirical cumulative distribution function of p-values corresponding to true null hypotheses is asymptotically bounded by the distribution function of a uniform variate. Important examples of weak dependence like block-dependence of test statistics and pairwise
comparisons are investigated in more detail. We prove that large classes of plug-in tests and SUD procedures control the corresponding error rate under weak dependence at least asymptotically. Various numerical examples illustrate our theoretical results.
Lizenz:In Copyright
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Fachbereich / Einrichtung:Mathematisch- Naturwissenschaftliche Fakultät » WE Mathematik » Mathematische Statistik und Wahrscheinlichkeitstheorie
Dokument erstellt am:13.01.2011
Dateien geändert am:13.01.2011
Promotionsantrag am:08.06.2010
Datum der Promotion:27.10.2010
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