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Confidence Intervals for Functions of Variance Components Kok-Leong Chiang Iowa State University

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Confidence Intervals for Functions of Variance Components Kok-Leong Chiang Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 2000 Confidence intervals for functions of variance components Kok-Leong Chiang Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Statistics and Probability Commons Recommended Citation Chiang, Kok-Leong, "Confidence intervals for functions of variance components " (2000). Retrospective Theses and Dissertations. 13890. https://lib.dr.iastate.edu/rtd/13890 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while ottiers may be f^ any type of computer printer. The quality of this reproduction is dependent upon the quality of ttw copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author dkl not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g.. maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overiaps. Photographs Included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additkmal charge. Contact UMI directly to order. Bell & Howell Information and Learning 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 Confidence intervals for functions of variance components by Kok-Leong Chiang A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major; Statistics Major Professor; Stephen B. Vardeman Iowa State University Ames, Iowa 2000 UMI Number 9962806 UMI' UMI Microfomi9962806 Copyright 2000 by Bell & Howell Infomiation and Leaming Company. All rights reserved. This microfomn edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Leaming Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 ii Graduate College Iowa State University This is to certify that the Doctoral dissertation of Kok-Leong Chiang has met the dissertation requirements of Iowa State University Signature was redacted for privacy. Major ProfessorPr< Signature was redacted for privacy. For the Major Program Signature was redacted for privacy. For thfi^Sraouate College iii TABLE OF CONTENTS ACKNOWLEDGEMENTS iv CHAPTER 1. GENERAL INTRODUCTION 1 CHAPTER 2. A SIMPLE GENERAL METHOD FOR CONSTRUCTING CONFIDENCE INTERVALS FOR FUNCTIONS OF VARIANCE COMPONENTS 7 CHAPTER 3. AN APPLICATION OF THE SURROGATE VARIABLES METHOD OF CONSTRUCTING CONFIDENCE INTERVALS FOR FUNCTIONS OF VARIANCE COMPONENTS IN THE TWO-WAY RANDOM EFFECTS MODEL WITH INTERACTION 39 CHAPTER 4. CONFIDENCE INTERVALS FOR FUNCTIONS OF VARIANCE COMPONENTS IN THE TWO-FOLD NESTED RANDOM EFFECTS MODEL 70 CHAPTER 5. CONFmENCE INTERVALS FOR FUNCTIONS OF VARLANCE COMPONENTS IN THE THREE-FACTOR CROSSED- CLASSIFICATION RANDOM EFFECTS MODEL 96 CHAPTER 6. A COMPARISON BETWEEN THE SURROGATE VARIABLES CONFIDENCE INTERVALS AND THE BAYESLAN POSTERIOR INTERVALS (USING INDEPENDENT JEFFREYS PRIORS FOR EXPECTED MEAN SQUARES) FOR FUNCTIONS OF VARLANCE COMPONENTS IN THE ONE-FOLD NESTED RANDOM EFFECTS MODEL 116 CHAPTER 7. ON THE APPLICATION OF THE GUI, GRAYBILL, BURDICK, AND TING (1995) AND THE SATTERTHWAITE APPROXIMATE CONFIDENCE INTERVALS FOR A PARTICULAR RATIO OF VARIANCE COMPONENTS 127 CHAPTER 8. CONCLUSION 146 iv ACKNOWLEDGEMENTS I wish to tha-nlc my Ph.D. advisor, Professor Stephen Vardeman, for valuable advice and encouragement. More than being a great advisor on research matters related to this dissertation, he has been a mentor and a role model whom I look up to for the greater aspects of life. I also wish to thank my wife, Betsy, for her understanding, patience and support since we began our academic life here in Ames. Further, I wish to thank my undergraduate advisor. Dr. Tin-Chiu Chua, who is an ISU aliunnus, for having encouraged me to pursue my graduate education in Statistics at Iowa State. Finally, the privilege of unrestricted access to the Snedecor Hall Research Computers (consisting of nine DEC Alphastation 500 workstations) at the Department of Statistics is gratefully acknowledged. Without which, the duration of the various simulation projects would have definitely been extended nine-fold (literally). 1 CHAPTER 1. GENERAL INTRODUCTION 1. bitroduction Since their introduction in the 1930's, variance components models have been extensively used to model experiments in various fields ranging from animal breeding, agricultural and biological sciences to engineering and industrial manufacturing. The primary methods of inference in these models are point estimation and interval estimation of the variance components as well as functions of these. The focus of this dissertation is on confidence intervals for functions of variance components in balanced data normal theory random effects models. Specifically, a new, simple and general method of constructing confidence intervals is proposed, which can be used to estimate arbitrary functions of variance components in balanced data normal theory random effects models. 2. Dissertation Organization This dissertation is organized using the "alternative format" of compiling together several manuscripts prepared for submission to journals. The proposed method of constructing confidence intervals can be applied to an extremely wide range of parametric functions in random effects models. This dissertation takes a systematic approach to evaluating the effectiveness of the proposed method by organizing our discussion according to various standard models. 2 For each model, we consider a niunber of commonly studied functions of variance components for that model. Chapter 2 introduces the proposed method in general terms. It also considers the two-way random effects model without interaction. The two-way random effects model with interaction, two-fold nested random effects and three-way crossed- classification random effects model are considered in Chapters 3, 4 and 5 respectively. Chapter 6 compares the proposed method with that of a Bayesian posterior analysis (assvuning independent Jeffireys priors on the expected mean squares) in the context of a one-fold nested random effects model. Not related to the discussion of the proposed method. Chapter 7 describes a problem with two existing methods for estimating a particular ratio of variance components in the two-way model. The same ratio is discussed in Chapter 3 where it is shown that the proposed method is effective for estimating it. Chapter 8 contains a general conclusion for the dissertation and some remarks about possible future research. 3. Litraature Review Over the years, various authors have written reviews on the development of estimation methods for variance components models, notably, Crump (1951), Sahai (1979), Sahai, Khuri, and Kapadia (1985), Khuri and Sahai (1985), Burdick and Graybill (1988), and Burdick Guid Graybill (1992, Chapters 2 wd 3). Depending on the particular function and model of interest, various methods of constructing approximate confidence intervals have been proposed. Two of the 3 earliest and most popular general methods for estimating linear combinations of variance components are those proposed by Satterthwaite (1946) and Welch (1956). However, these should only be used for estimating nonnegative linear combinations of expected mean squares when the dataset is relative large. Perhaps, the most successful and systematic development of approximate interval estimators for functions of variance components can be found in the works of Burdick, Graybill and associates. These can be generally described as modified large-sample (MLS) methods, and they form a large collection of formulas and procedures for estimating various functions in different models. One of the earliest MLS pajjers is that of Graybill and Wang (1980) whose method can be used for nonnegative linear combinations of varieince components. Various MLS methods have been developed for estimating particular types of parametric functions in selected random effects models. Some of these are in Graybill and Wang (1979), Wang and Graybill (1981), Arteaga and Graybill (1982), Leiva and Graybill (1986), Lu, Graybill and Burdick (1987,1988,1989). One of the two most important breakthroughs in the MLS approach in the ten years since its inception was the introduction of the MLS method of Ting, Burdick, Graybill, Jeyaratnam and Lu (1990) for constructing confidence intervals for linear combinations of variance components that are unrestricted in sign. The other is the method of Ting, Burdick suid Graybill
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