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The Theory of the Design of Experiments The Theory of the Design of Experiments D.R. COX Honorary Fellow Nuffield College Oxford, UK AND N. REID Professor of Statistics University of Toronto, Canada CHAPMAN & HALL/CRC Boca Raton London New York Washington, D.C. C195X/disclaimer Page 1 Friday, April 28, 2000 10:59 AM Library of Congress Cataloging-in-Publication Data Cox, D. R. (David Roxbee) The theory of the design of experiments / D. R. Cox, N. Reid. p. cm. — (Monographs on statistics and applied probability ; 86) Includes bibliographical references and index. ISBN 1-58488-195-X (alk. paper) 1. Experimental design. I. Reid, N. II.Title. III. Series. QA279 .C73 2000 001.4 '34 —dc21 00-029529 CIP This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trade- marks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com © 2000 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-58488-195-X Library of Congress Card Number 00-029529 Printed in the United States of America 2 3 4 5 6 7 8 9 0 Printed on acid-free paper Contents Preface 1Somegeneralconcepts 1.1Typesofinvestigation 1.2Observationalstudies 1.3Somekeyterms 1.4Requirementsindesign 1.5Interplaybetweendesignandanalysis 1.6Keystepsindesign 1.7Asimplifiedmodel 1.8Abroaderview 1.9Bibliographicnotes 1.10Furtherresultsandexercises 2Avoidanceofbias 2.1Generalremarks 2.2Randomization 2.3Retrospectiveadjustmentforbias 2.4Somemoreonrandomization 2.5Moreoncausality 2.6Bibliographicnotes 2.7Furtherresultsandexercises 3Controlofhaphazardvariation 3.1Generalremarks 3.2Precisionimprovementbyblocking 3.3Matchedpairs 3.4Randomizedblockdesign 3.5Partitioningsumsofsquares 3.6Retrospectiveadjustmentforimprovingprecision 3.7Specialmodelsoferrorvariation © 2000 by Chapman & Hall/CRC 3.8Bibliographicnotes 3.9Furtherresultsandexercises 4Specializedblockingtechniques 4.1Latinsquares 4.2Incompleteblockdesigns 4.3Cross-overdesigns 4.4Bibliographicnotes 4.5Furtherresultsandexercises 5Factorialdesigns:basicideas 5.1Generalremarks 5.2Example 5.3Maineffectsandinteractions 5.4Example:continued 5.5Twolevelfactorialsystems 5.6Fractionalfactorials 5.7Example 5.8Bibliographicnotes 5.9Furtherresultsandexercises 6Factorialdesigns:furthertopics 6.1Generalremarks 6.2Confoundingin2kdesigns 6.3Otherfactorialsystems 6.4Splitplotdesigns 6.5Nonspecificfactors 6.6Designsforquantitativefactors 6.7Taguchimethods 6.8Conclusion 6.9Bibliographicnotes 6.10Furtherresultsandexercises 7Optimaldesign 7.1Generalremarks 7.2Somesimpleexamples 7.3Somegeneraltheory 7.4Otheroptimalitycriteria 7.5Algorithmsfordesignconstruction 7.6Nonlineardesign © 2000 by Chapman & Hall/CRC 7.7Space-fillingdesigns 7.8Bayesiandesign 7.9Optimalityoftraditionaldesigns 7.10Bibliographicnotes 7.11Furtherresultsandexercises 8Someadditionaltopics 8.1Scaleofeffort 8.2Adaptivedesigns 8.3Sequentialregressiondesign 8.4Designsforone-dimensionalerrorstructure 8.5Spatialdesigns 8.6Bibliographicnotes 8.7Furtherresultsandexercises AStatisticalanalysis A.1Introduction A.2Linearmodel A.3Analysisofvariance A.4Moregeneralmodels;maximumlikelihood A.5Bibliographicnotes A.6Furtherresultsandexercises BSomealgebra B.1Introduction B.2Grouptheory B.3Galoisfields B.4Finitegeometries B.5Differencesets B.6Hadamardmatrices B.7Orthogonalarrays B.8Codingtheory B.9Bibliographicnotes B.10Furtherresultsandexercises CComputationalissues C.1Introduction C.2Overview C.3RandomizedblockexperimentfromChapter3 C.4AnalysisofblockdesignsinChapter4 © 2000 by Chapman & Hall/CRC C.5ExamplesfromChapter5 C.6ExamplesfromChapter6 C.7Bibliographicnotes References Listoftables © 2000 by Chapman & Hall/CRC Preface This book is an account of the major topics in the design of experiments, with particular emphasis on the key concepts involved and on the statistical structure associated with these concepts. While design of experiments is in many ways a very well developed area of statistics, it often receives less emphasis than methods of analysis in a programme of study in statistics. We have written for a general audience concerned with statistics in experimental fields and with some knowledge of and interest in theoretical issues. The mathematical level is mostly elementary; occasional passages using more advanced ideas can be skipped or omitted without inhibiting understanding of later passages. Some specialized parts of the subject have extensive and specialized lit- eratures, a few examples being incomplete block designs, mixture designs, designs for large variety trials, designs based on spatial stochastic models and designs constructed from explicit optimality requirements. We have aimed to give relatively brief introductions to these subjects eschewing technical detail. To motivate the discussion we give outline Illustrations taken from a range of areas of application. In addition we give a limited number of Examples, mostly taken from the literature, used for the different purpose of showing detailed methods of analysis without much emphasis on specific subject-matter interpretation. We have written a book about design not about analysis, al- though, as has often been pointed out, the two phases are inex- orably interrelated. Therefore it is, in particular, not a book on the linear statistical model or that related but distinct art form the analysis of variance table. Nevertheless these topics enter and there is a dilemma in presentation. What do we assume the reader knows about these matters? We have solved this problem uneasily by somewhat downplaying analysis in the text, by assuming what- ever is necessary for the section in question and by giving a review as an Appendix. Anyone using the book as a basis for a course of lectures will need to consider carefully what the prospective stu- dents are likely to understand about the linear model and to sup- plement the text appropriately. While the arrangement of chapters © 2000 by Chapman & Hall/CRC represents a logical progression of ideas, if interest is focused on a particular field of application it will be reasonable to omit certain parts or to take the material in a different order. If defence of a book on the theory of the subject is needed it is this. Successful application of these ideas hinges on adapting gen- eral principles to the special constraints of individual applications. Thus experience suggests that while it is useful to know about spe- cial designs, balanced incomplete block designs for instance, it is rather rare that they can be used directly. More commonly they need some adaptation to accommodate special circumstances and to do this effectively demands a solid theoretical base. This book has been developed from lectures given at Cambridge, Birkbeck College London, Vancouver, Toronto and Oxford. We are grateful to Amy Berrington, Mario Cortina Borja, Christl Don- nelly, Peter Kupchak, Rahul Mukerjee, John Nelder, Rob Tibshi- rani and especially Grace Yun Yi for helpful comments on a pre- liminary version. D.R. Cox and N. Reid Oxford and Toronto January 2000 © 2000 by Chapman & Hall/CRC List of tables 3.1Strengthindexofcotton 3.2Analysisofvarianceforstrengthindex 4.15 5Graeco-Latinsquare 4.2Co×mpleteorthogonalsetof5 5Latinsquares 4.3Balancedincompleteblockdesi×gns 4.4Twospecialincompleteblockdesigns 4.5Youdensquare 4.6Intrablockanalysisofvariance 4.7Interblockanalysisofvariance 4.8Analysisofvariance,generalincompleteblock design 4.9Logdryweightofchickbones 4.10Treatmentmeans,logdryweight 4.11Analysisofvariance,logdryweight 4.12Estimatesoftreatmenteffects 4.13Expansionindexofpastrydough 4.14Unadjustedandadjustedtreatmentmeans 4.15Exampleofintrablockanalysis 5.1Weightsofchicks 5.2Meanweightsofchicks 5.3Twofactoranalysisofvariance 5.4Analysisofvariance,weightsofchicks 5.5Decompositionoftreatmentsumofsquares 5.6Analysisofvariance,22factorial 5.7Analysisofvariance,2kfactorial 5.8Treatmentfactors,nutritionandcancer 5.9Data,nutritionandcancer 5.10Estimatedeffects,nutritionandcancer 5.11Data,Exercise5.6 5.12Contrasts,Exercise5.6 © 2000 by Chapman & Hall/CRC 6.1Example,doubleconfounding 6.2Degreesoffreedom,doubleconfounding 6.3TwoorthogonalLatinsquares 6.4Estimationofthemaineffect 6.51/3fraction,degreesoffreedom 6.6Asymmetricorthogonalarray 6.7Supersaturateddesign 6.8Example,split-plotanalysis 6.9Data,tensilestrengthofpaper 6.10Analysis,tensilestrengthofpaper 6.11Tableofmeans,tensilestrengthofpaper 6.12Analysisofvarianceforareplicatedfactorial 6.13Analysisofvariancewitharandomeffect 6.14Analysisofvariance,quality/quantityinteraction 6.15Design,electronicsexample
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