Design of Comparative Experiments

346 Pages·2008·3.092 MB·English

by  R A Bailey

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Design of Comparative Experiments
Thisbookshouldbeontheshelfofeverypractisingstatisticianwhodesignsexperi-
ments.
Gooddesignconsidersunitsandtreatmentsfirst,andthenallocatestreatmentsto
units. It does not choose from a menu of named designs. This approach requires a
notationforunitsthatdoesnotdependonthetreatmentsapplied.Moststructureonthe
setofobservationalunits,oronthesetoftreatments,canbedefinedbyfactors.This
bookdevelopsacoherentframeworkforthinkingaboutfactorsandtheirrelationships,
includingtheuseofHassediagrams.Theseareusedtoelucidatestructure,calculate
degreesoffreedomandallocatetreatmentsubspacestoappropriatestrata.Basedon
a one-term course the author has taught since 1989, the book is ideal for advanced
undergraduateandbeginninggraduatecourses.Examples,exercisesanddiscussion
questionsaredrawnfromawiderangeofrealapplications:fromdrugdevelopment,
toagriculture,tomanufacturing.
r. a. baileyhasbeenProfessorofStatisticsatQueenMary,UniversityofLondon
since 1994. She is a fellow of the Institute of Mathematical Statistics and a past
presidentoftheInternationalBiometricSociety,BritishRegion.Thisbookreflectsher
extensiveexperienceteachingdesignofexperimentsandadvisingonitsapplication.
HerbookAssociationSchemeswaspublishedbyCambridgeUniversityPressin2004.
CAMBRIDGE SERIES IN STATISTICAL AND
PROBABILISTIC MATHEMATICS
EditorialBoard
R.Gill(DepartmentofMathematics,UtrechtUniversity)
B.D.Ripley(DepartmentofStatistics,UniversityofOxford)
S.Ross(DepartmentofIndustrialandSystemsEngineering,UniversityofSouthernCalifornia)
B.W.Silverman(St.Peter’sCollege,Oxford)
M.Stein(DepartmentofStatistics,UniversityofChicago)
This series of high-quality upper-division textbooks and expository monographs covers all
aspectsofstochasticapplicablemathematics.Thetopicsrangefrompureandappliedstatistics
toprobabilitytheory,operationsresearch,optimization,andmathematicalprogramming.The
books contain clear presentations of new developments in the field and also of the state of
theartinclassicalmethods.Whileemphasizingrigoroustreatmentoftheoreticalmethods,the
booksalsocontainapplicationsanddiscussionsofnewtechniquesmadepossiblebyadvances
incomputationalpractice.
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20. RandomGraphDynamics,byRickDurrett
21. Networks,byPeterWhittle
22. SaddlepointApproximationswithApplications,byRonaldW.Butler
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Design of Comparative Experiments
R. A. Bailey
QueenMary,UniversityofLondon
CAMBRIDGEUNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521865067
© R. A. Bailey 2008
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2008
ISBN-13 978-0-511-39340-2 eBook (EBL)
ISBN-13 978-0-521-86506-7 hardback
ISBN-13 978-0-521-68357-9 paperback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents
Preface pagexi
1 Forwardlook 1
1.1 Stagesinastatisticallydesignedexperiment 1
1.1.1 Consultation 1
1.1.2 Statisticaldesign 2
1.1.3 Datacollection 2
1.1.4 Datascrutiny 3
1.1.5 Analysis 4
1.1.6 Interpretation 5
1.2 Theidealandthereality 5
1.2.1 Purposeoftheexperiment 5
1.2.2 Replication 5
1.2.3 Localcontrol 6
1.2.4 Constraints 6
1.2.5 Choice 7
1.3 Anexample 7
1.4 Definingterms 8
1.5 Linearmodel 14
1.6 Summary 15
Questionsfordiscussion 16
2 Unstructuredexperiments 19
2.1 Completelyrandomizeddesigns 19
2.2 Whyandhowtorandomize 20
2.3 Thetreatmentsubspace 21
2.4 Orthogonalprojection 23
2.5 Linearmodel 24
2.6 Estimation 24
2.7 Comparisonwithmatrixnotation 26
2.8 Sumsofsquares 26
2.9 Variance 28
2.10 Replication: equalorunequal? 30
v
vi Contents
2.11 Allowingfortheoverallmean 30
2.12 Hypothesistesting 33
2.13 Sufficientreplicationforpower 35
2.14 Amoregeneralmodel 38
Questionsfordiscussion 41
3 Simpletreatmentstructure 43
3.1 Replicationofcontroltreatments 43
3.2 Comparingnewtreatmentsinthepresenceofacontrol 44
3.3 Othertreatmentgroupings 47
Questionsfordiscussion 52
4 Blocking 53
4.1 Typesofblock 53
4.1.1 Naturaldiscretedivisions 53
4.1.2 Continuousgradients 55
4.1.3 Choiceofblockingfortrialmanagement 55
4.1.4 Howandwhentoblock 56
4.2 Orthogonalblockdesigns 57
4.3 Constructionandrandomization 59
4.4 Modelsforblockdesigns 59
4.5 Analysiswhenblockshavefixedeffects 61
4.6 Analysiswhenblockshaverandomeffects 67
4.7 Whyuseblocks? 68
4.8 Lossofpowerwithblocking 69
Questionsfordiscussion 71
5 Factorialtreatmentstructure 75
5.1 Treatmentfactorsandtheirsubspaces 75
5.2 Interaction 77
5.3 Principlesofexpectationmodels 84
5.4 Decomposingthetreatmentsubspace 87
5.5 Analysis 90
5.6 Threetreatmentfactors 92
5.7 Factorialexperiments 97
5.8 Constructionandrandomizationoffactorialdesigns 98
5.9 Factorialtreatmentspluscontrol 99
Questionsfordiscussion 99
6 Row–columndesigns 105
6.1 Doubleblocking 105
6.2 Latinsquares 106
6.3 Constructionandrandomization 108
6.4 Orthogonalsubspaces 110
6.5 Fixedrowandcolumneffects: modelandanalysis 110
Contents vii
6.6 Randomrowandcolumneffects: modelandanalysis 112
Questionsfordiscussion 116
7 Experimentsonpeopleandanimals 117
7.1 Introduction 117
7.2 Historicalcontrols 118
7.3 Cross-overtrials 118
7.4 Matchedpairs,matchedthrees,andsoon 119
7.5 Completelyrandomizeddesigns 120
7.6 Bodypartsasexperimentalunits 120
7.7 Sequentialallocationtoanunknownnumberofpatients 121
7.8 Safeguardsagainstbias 122
7.9 Ethicalissues 124
7.10 Analysisbyintentiontotreat 126
Questionsfordiscussion 127
8 Smallunitsinsidelargeunits 131
8.1 Experimentalunitsbiggerthanobservationalunits 131
8.1.1 Thecontext 131
8.1.2 Constructionandrandomization 132
8.1.3 Modelandstrata 132
8.1.4 Analysis 132
8.1.5 Hypothesistesting 135
8.1.6 Decreasingvariance 137
8.2 Treatmentfactorsindifferentstrata 138
8.3 Split-plotdesigns 146
8.3.1 Blockingthelargeunits 146
8.3.2 Constructionandrandomization 147
8.3.3 Modelandstrata 148
8.3.4 Analysis 149
8.3.5 Evaluation 152
8.4 Thesplit-plotprinciple 152
Questionsfordiscussion 154
9 MoreaboutLatinsquares 157
9.1 UsesofLatinsquares 157
9.1.1 Onetreatmentfactorinasquare 157
9.1.2 Moregeneralrow–columndesigns 158
9.1.3 Twotreatmentfactorsinablockdesign 159
9.1.4 Threetreatmentfactorsinanunblockeddesign 161
9.2 Graeco-Latinsquares 162
9.3 UsesofGraeco-Latinsquares 166
9.3.1 Superimposeddesigninasquare 166
9.3.2 Twotreatmentfactorsinasquare 166
9.3.3 Threetreatmentfactorsinablockdesign 166
viii Contents
9.3.4 Fourtreatmentfactorsinanunblockeddesign 167
Questionsfordiscussion 167
10 Thecalculusoffactors 169
10.1 Introduction 169
10.2 Relationsonfactors 169
10.2.1 Factorsandtheirclasses 169
10.2.2 Aliasing 170
10.2.3 Onefactorfinerthananother 171
10.2.4 Twospecialfactors 171
10.3 Operationsonfactors 171
10.3.1 Theinfimumoftwofactors 171
10.3.2 Thesupremumoftwofactors 172
10.3.3 Uniformfactors 175
10.4 Hassediagrams 175
10.5 Subspacesdefinedbyfactors 178
10.5.1 Onesubspaceperfactor 178
10.5.2 Fittedvaluesandcrudesumsofsquares 178
10.5.3 Relationsbetweensubspaces 178
10.6 Orthogonalfactors 178
10.6.1 Definitionoforthogonality 178
10.6.2 Projectionmatricescommute 179
10.6.3 Proportionalmeeting 180
10.6.4 Howreplicationcanaffectorthogonality 181
10.6.5 Achainoffactors 181
10.7 Orthogonaldecomposition 182
10.7.1 Asecondsubspaceforeachfactor 182
10.7.2 Effectsandsumsofsquares 184
10.8 CalculationsontheHassediagram 185
10.8.1 Degreesoffreedom 185
10.8.2 Sumsofsquares 187
10.9 Orthogonaltreatmentstructures 189
10.9.1 Conditionsontreatmentfactors 189
10.9.2 Collectionsofexpectationmodels 190
10.10 Orthogonalplotstructures 193
10.10.1 Conditionsonplotfactors 193
10.10.2 Varianceandcovariance 194
10.10.3 Matrixformulation 195
10.10.4 Strata 196
10.11 Randomization 196
10.12 Orthogonaldesigns 197
10.12.1 Desirableproperties 197
10.12.2 Generaldefinition 198
10.12.3 Locatingtreatmentsubspaces 198
10.12.4 Analysisofvariance 200