Table Of ContentPrelims 17.11.2003 5:26pm page 1
Bayesian Approaches to Clinical Trials
and Health-Care Evaluation
Prelims 17.11.2003 5:26pm page 2
STATISTICSIN PRACTICE
Advisory Editor
Stephen Senn
University of Glasgow,UK
Founding Editor
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NottinghamTrentUniversity, UK
StatisticsinPracticeisanimportantinternationalseriesoftextswhichprovide
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With sound motivation and many worked practical examples, the books
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A completelist of titles inthis series appears atthe end of the volume.
Prelims 17.11.2003 5:26pm page 3
Bayesian Approaches
to Clinical Trials and
Health-Care Evaluation
David J. Spiegelhalter
MRC Biostatistics Unit, Cambridge, UK
Keith R. Abrams
University of Leicester, UK
Jonathan P. Myles
Cancer Research UK, London, UK
Prelims 17.11.2003 5:26pm page 4
TheworkisbasedonanoriginalNHSHealthTechnologyAssessmentfundedproject(93/50/05).
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Contents
Preface xi
List of examples xiii
1 Introduction 1
1.1 WhatareBayesianmethods? 1
1.2 Whatdowemeanby‘health-careevaluation’? 2
1.3 ABayesianapproachtoevaluation 3
1.4 Theaimofthisbookandtheintendedaudience 3
1.5 Structureofthebook 4
2 Basic Concepts from Traditional Statistical Analysis 9
2.1 Probability 10
2.1.1 Whatisprobability? 10
2.1.2 Oddsandlog-odds 12
2.1.3 Bayestheoremforsimpleevents 13
2.2 Randomvariables,parametersandlikelihood 14
2.2.1 Randomvariablesandtheirdistributions 14
2.2.2 Expectation,variance,covarianceandcorrelation 16
2.2.3 Parametricdistributionsandconditionalindependence 17
2.2.4 Likelihoods 18
2.3 Thenormaldistribution 20
2.4 Normallikelihoods 22
2.4.1 Normalapproximationsforbinarydata 23
2.4.2 Normallikelihoodsforsurvivaldata 27
2.4.3 Normallikelihoodsforcountresponses 30
2.4.4 Normallikelihoodsforcontinuousresponses 31
2.5 Classicalinference 31
2.6 Acatalogueofusefuldistributions* 34
2.6.1 BinomialandBernoulli 34
2.6.2 Poisson 35
2.6.3 Beta 36
2.6.4 Uniform 38
2.6.5 Gamma 39
2.6.6 Root-inverse-gamma 40
2.6.7 Half-normal 41
v
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vi Contents
2.6.8 Log-normal 42
2.6.9 Student’st 43
2.6.10 Bivariatenormal 44
2.7 Keypoints 46
Exercises 46
3 An Overview of the Bayesian Approach 49
3.1 Subjectivityandcontext 49
3.2 Bayestheoremfortwohypotheses 51
3.3 Comparingsimplehypotheses:likelihoodratiosandBayesfactors 54
3.4 Exchangeabilityandparametricmodelling* 56
3.5 Bayestheoremforgeneralquantities 57
3.6 Bayesiananalysiswithbinarydata 57
3.6.1 Binarydatawithadiscretepriordistribution 58
3.6.2 Conjugateanalysisforbinarydata 59
3.7 Bayesiananalysiswithnormaldistributions 62
3.8 Pointestimation,intervalestimationandintervalhypotheses 64
3.9 Thepriordistribution 73
3.10 HowtouseBayestheoremtointerprettrialresults 74
3.11 The‘credibility’ofsignificanttrialresults* 75
3.12 SequentialuseofBayestheorem* 79
3.13 Predictions 80
3.13.1 PredictionsintheBayesianframework 80
3.13.2 Predictionsforbinarydata* 81
3.13.3 Predictionsfornormaldata 83
3.14 Decision-making 85
3.15 Design 90
3.16 Useofhistoricaldata 90
3.17 Multiplicity,exchangeabilityandhierarchicalmodels 91
3.18 Dealingwithnuisanceparameters* 100
3.18.1 Alternativemethodsforeliminatingnuisanceparameters* 100
3.18.2 Profilelikelihoodinahierarchicalmodel* 102
3.19 Computationalissues 102
3.19.1 MonteCarlomethods 103
3.19.2 MarkovchainMonteCarlomethods 105
3.19.3 WinBUGS 107
3.20 SchoolsofBayesians 112
3.21 ABayesianchecklist 113
3.22 Furtherreading 115
3.23 Keypoints 116
Exercises 117
4 Comparison of Alternative Approaches to Inference 121
4.1 Astructureforalternativeapproaches 121
4.2 Conventionalstatisticalmethodsusedinhealth-careevaluation 122
4.3 Thelikelihoodprinciple,sequentialanalysisandtypesoferror 124
4.3.1 Thelikelihoodprinciple 124
4.3.2 Sequentialanalysis 126
4.3.3 TypeIandTypeIIerror 127
4.4 P-valuesandBayesfactors* 127
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Contents vii
4.4.1 CriticismofP-values 127
4.4.2 BayesfactorsasanalternativetoP-values:simplehypotheses 128
4.4.3 BayesfactorsasanalternativetoP-values:compositehypotheses 130
4.4.4 Bayesfactorsinpreferencestudies 133
4.4.5 Lindley’sparadox 135
4.5 Keypoints 136
Exercises 136
5 Prior Distributions 139
5.1 Introduction 139
5.2 Elicitationofopinion:abriefreview 140
5.2.1 Backgroundtoelicitation 140
5.2.2 Elicitationtechniques 141
5.2.3 Elicitationfrommultipleexperts 142
5.3 Critiqueofpriorelicitation 147
5.4 Summaryofexternalevidence* 148
5.5 Defaultpriors 157
5.5.1 ‘Non-informative’or‘reference’priors: 157
5.5.2 ‘Sceptical’priors 158
5.5.3 ‘Enthusiastic’priors 160
5.5.4 Priorswithapointmassatthenullhypothesis
(‘lump-and-smear’priors)* 161
5.6 Sensitivityanalysisand‘robust’priors 165
5.7 Hierarchicalpriors 167
5.7.1 Thejudgementofexchangeability 167
5.7.2 Theformfortherandom-effectsdistribution 168
5.7.3 Thepriorforthestandarddeviationoftherandomeffects* 168
5.8 Empiricalcriticismofpriors 174
5.9 Keypoints 176
Exercises 177
6 Randomised Controlled Trials 181
6.1 Introduction 181
6.2 Useofalossfunction:isaclinicaltrialforinferenceordecision? 182
6.3 Specificationofnullhypotheses 184
6.4 Ethicsandrandomisation:abriefreview 187
6.4.1 Israndomisationnecessary? 187
6.4.2 Whenisitethicaltorandomise? 187
6.5 Samplesizeofnon-sequentialtrials 189
6.5.1 Alternativeapproachestosample-sizeassessment 189
6.5.2 ‘Classicalpower’:hybridclassical-Bayesianmethods
assumingnormality 193
6.5.3 ‘Bayesianpower’ 194
6.5.4 Adjustingformulaefordifferenthypotheses 196
6.5.5 Predictivedistributionofpowerandnecessarysamplesize 201
6.6 Monitoringofsequentialtrials 202
6.6.1 Introduction 202
6.6.2 Monitoringusingtheposteriordistribution 204
6.6.3 Monitoringusingpredictions:‘interimpower’ 211
6.6.4 Monitoringusingaformallossfunction 220
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viii Contents
6.6.5 FrequentistpropertiesofsequentialBayesianmethods 221
6.6.6 Bayesianmethodsanddatamonitoringcommittees 222
6.7 Theroleof‘scepticism’inconfirmatorystudies 224
6.8 Multiplicityinrandomisedtrials 227
6.8.1 Subsetanalysis 227
6.8.2 Multi-centreanalysis 227
6.8.3 Clusterrandomisation 227
6.8.4 Multipleendpointsandtreatments 228
6.9 Usinghistoricalcontrols* 228
6.10 Data-dependentallocation 235
6.11 Trialdesignsotherthantwoparallelgroups 237
6.12 Otheraspectsofdrugdevelopment 242
6.13 Furtherreading 244
6.14 Keypoints 245
Exercises 247
7 Observational Studies 251
7.1 Introduction 251
7.2 Alternativestudydesigns 252
7.3 Explicitmodellingofbiases 253
7.4 Institutionalcomparisons 258
7.5 Keypoints 262
Exercises 263
8 Evidence Synthesis 267
8.1 Introduction 267
8.2 ‘Standard’meta-analysis 268
8.2.1 ABayesianperspective 268
8.2.2 SomedelicateissuesinBayesianmeta-analysis 274
8.2.3 Therelationshipbetweentreatmenteffectandunderlyingrisk 278
8.3 Indirectcomparisonstudies 282
8.4 Generalisedevidencesynthesis 285
8.5 Furtherreading 298
8.6 Keypoints 299
Exercises 299
9 Cost-effectiveness, Policy-Making and Regulation 305
9.1 Introduction 305
9.2 Contexts 306
9.3 ‘Standard’cost-effectivenessanalysiswithoutuncertainty 308
9.4 ‘Two-stage’andintegratedapproachestouncertaintyincost-effectiveness
modelling 310
9.5 Probabilisticanalysisofsensitivitytouncertaintyaboutparameters:
two-stageapproach 312
9.6 Cost-effectivenessanalysesofasinglestudy:integratedapproach 315
9.7 Levelsofuncertaintyincost-effectivenessmodels 320
9.8 Complexcost-effectivenessmodels 322
9.8.1 Discrete-time,discrete-stateMarkovmodels 322
9.8.2 Micro-simulationincost-effectivenessmodels 323
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Contents ix
9.8.3 Micro-simulationandprobabilisticsensitivityanalysis 324
9.8.4 Comprehensivedecisionmodelling 328
9.9 Simultaneousevidencesynthesisandcomplexcost-effectivenessmodelling 329
9.9.1 Generalisedmeta-analysisofevidence 329
9.9.2 ComparisonofintegratedBayesianandtwo-stageapproach 335
9.10 Cost-effectivenessofcarryingoutresearch:paybackmodels 335
9.10.1 Researchplanninginthepublicsector 335
9.10.2 Researchplanninginthepharmaceuticalindustry 336
9.10.3 Valueofinformation 337
9.11 Decisiontheoryincost-effectivenessanalysis,regulationandpolicy 341
9.12 Regulationandhealthpolicy 343
9.12.1 Theregulatorycontext 343
9.12.2 Regulationofpharmaceuticals 343
9.12.3 Regulationofmedicaldevices 344
9.13 Conclusions 344
9.14 Keypoints 345
Exercises 345
10 Conclusions and Implications for Future Research 349
10.1 Introduction 349
10.2 GeneraladvantagesandproblemsofaBayesianapproach 349
10.3 Futureresearchanddevelopment 350
A Websites and Software 353
A.1 Thesiteforthisbook 353
A.2 Bayesianmethodsinhealth-careevaluation 353
A.3 Bayesiansoftware 354
A.4 GeneralBayesiansites 355
References 357
Index 381
