An Introduction to Evidence Based Statistics

245 Pages·2020·5.408 MB·English

by  Peter M. B. Cahusac

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Evidence-BasedStatistics
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Evidence-Based Statistics
An Introduction to the Evidential Approach — from
Likelihood Principle to Statistical Practice
Peter M. B. Cahusac
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Thiseditionfirstpublished2021
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v
Contents
Acknowledgements xi
AbouttheAuthor xiii
AbouttheCompanionSite xv
Introduction 1
References 2
1 TheEvidenceistheEvidence 3
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1.1 Evidence-BasedStatistics 3
1.1.1 TheLiterature 4
1.2 StatisticalInference–TheBasics 6
1.2.1 DifferentStatisticalApproaches 7
1.2.2 TheLikelihood/EvidentialApproach 8
1.2.3 TypesofApproachUsingLikelihoods 11
1.2.4 ProsandConsofLikelihoodApproach 11
1.3 EffectSize–TrueIfHuge! 12
1.4 Calculations 15
1.5 SummaryoftheEvidentialApproach 16
References 18
2 TheEvidentialApproach 21
2.1 Likelihood 21
2.1.1 ThePrinciple 22
2.1.2 Support 24
2.1.3 Example–OneSample 29
2.1.4 DirectionMatters 36
2.1.5 MaximumLikelihoodRatio 37
2.1.6 LikelihoodIntervals 39
2.1.7 TheSupportFunction 42
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2.1.8 ChoosingtheEffectSize 42
2.2 MisleadingandWeakEvidence 46
2.3 AddingMoreDataandMultipleTesting 48
2.4 SequenceofCalculationsUsingt 49
2.5 LikelihoodTerminology 51
2.6 RCodeforChapter2 52
2.6.1 CalculatingtheLikelihoodFunctionforaOneSamplet 52
2.7 Exercises 53
References 53
3 TwoSamples 55
3.1 BasicsUsingthetDistribution 55
3.1.1 StepsinCalculations 56
3.2 RelatedSamples 56
3.3 IndependentSamples 59
3.3.1 IndependentSampleswithUnequalVariances 60
3.4 CalculationSimplification 62
3.5 IfVarianceIsKnown,orLargeSampleSize,Usez 63
3.6 MethodologicalandProFormaAnalyses 65
3.7 AddingMoreData 68
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3.8 EstimatingSampleSize 70
3.8.1 SampleSizeforOneSampleandRelatedSamples 71
3.8.2 SampleSizeforIndependentSamples 73
3.9 DifferencesinVariances 73
3.10 RCodeForChapter3 74
3.10.1 CalculatingtheLikelihoodFunction,theLikelihoodsandSupportfor
IndependentSamples 74
3.10.2 CreatingaGardner–AltmanEstimationPlotwithLikelihoodFunction
andInterval 76
3.11 Exercises 77
References 77
4 ANOVA 79
4.1 MultipleMeans 79
4.1.1 TheModellingApproach 79
4.1.2 ModelComplexity 80
4.2 Example–Fitness 81
4.2.1 ComparingModels 82
4.2.2 SpecificModelComparisons 84
4.2.2.1 ANon-OrthogonalContrast 88
4.2.3 UnequalSampleSizes 89
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4.3 FactorialANOVA 90
4.3.1 Example–BloodClottingTimes 91
4.3.2 SpecificAnalysesinFactorialANOVA,IncludingContrasts 93
4.4 Alertingr2 96
4.4.1 Alertingr2toCompareContrastsforEffectSize 96
4.5 RepeatedMeasuresDesigns 97
4.5.1 MixedRepeatedMeasureswithBetweenParticipantDesigns 98
4.5.2 ContrastsinMixedDesigns 100
4.6 Exercise 102
References 102
5 CorrelationandRegression 103
5.1 RelationshipsBetweenTwoVariables 103
5.2 Correlation 103
5.2.1 LikelihoodIntervalsforCorrelation 107
5.3 Regression 108
5.3.1 ObtainingEvidencefromFvalues 110
5.3.2 ExaminingNon-linearity 111
5.4 LogisticRegression 113
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References 120
6 CategoricalData 121
6.1 TypesofCategoricalData 121
6.1.1 HowIsthe𝜒2TestUsed? 122
6.2 Binomial 123
6.2.1 LikelihoodIntervalsforBinomial 125
6.2.2 ComparingDifferent𝜋 126
6.2.3 TheSupportFunction 127
6.3 Poisson 129
6.4 RateRatios 131
6.5 One-WayCategoricalData 134
6.5.1 One-WayCategoricalComparingDifferentExpectedValues 135
6.5.2 One-WaywithMorethanTwoCategories 135
6.6 2×2ContingencyTables 137
6.6.1 Paired2×2CategoricalAnalysis 139
6.6.2 DiagnosticTests 141
6.6.2.1 SensitivityandSpecificity 141
6.6.2.2 PositiveandNegativePredictiveValues 142
6.6.2.3 LikelihoodRatioandPost-testProbability 143
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6.6.2.4 ComparingSensitivitiesandSpecificitiesofTwoDiagnostic
Procedures 144
6.6.3 OddsRatio 146
6.6.3.1 LikelihoodFunctionfortheOddsRatio 149
6.6.4 LikelihoodFunctionforRelativeRiskwithFixedEntries 151
6.7 LargerContingencyTables 151
6.7.1 MainEffects 153
6.7.2 EvidenceforLinearTrend 154
6.7.3 HigherDimensions? 155
6.8 DataThatFitsaHypothesisTooWell 158
6.9 TransformationsoftheVariable 159
6.10 ClinicalTrials–ATragedyin3Acts 161
6.11 RCodeforChapter6 164
6.11.1 One-WayCategoricalDataSupportAgainstSpecifiedProportions 164
6.11.2 CalculatingtheOddsRatioLikelihoodFunctionandSupport 164
6.11.3 CalculatingtheLikelihoodFunctionandSupportforRelativeRisk
withFixedEntries 166
6.11.4 CalculatingInteractionandMainEffectsforLargerContingency
Tables 168
6.11.5 Log-LinearModellingforMulti-wayTables 169
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6.12 Exercises 171
References 172
7 NonparametricAnalyses 175
7.1 So-Called‘Distribution-Free’Statistics 175
7.2 HackingS 176
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7.3 OneSampleandRelatedSamples 176
7.4 IndependentSamples 179
7.5 MorethanTwoIndependentSamples 181
7.6 PermutationAnalyses 182
7.7 BootstrapAnalysesforOneSampleorRelatedSamples 184
7.7.1 BootstrapAnalysesforIndependentSamples 186
7.8 RCodeforChapter7 187
7.8.1 CalculatingRelativeSupportforOneSample 187
7.8.2 CalculatingRelativeSupportforDifferencesinTwoIndependent
Samples 188
7.8.3 CalculatingRelativeSupportforDifferencesinThreeIndependent
Samples 189
7.8.4 CalculatingRelativeSupportUsingPermutationsAnalysis 189
7.8.5 BootstrapAnalysesforOneSample 191
7.8.6 BootstrapAnalysesforTwoIndependentSamples 193
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Contents ix
7.9 Exercises 195
References 196
8 OtherUsefulTechniques 197
8.1 OtherTechniques 197
8.2 CriticalPriorInterval 197
8.3 FalsePositiveRisk 201
8.4 TheBayesFactorandtheProbabilityoftheNullHypothesis 205
8.4.1 Example 208
8.5 BayesiantTests 210
8.6 TheArmitageStoppingRule 212
8.7 CounternullEffectSize 214
References 217
AppendixA OrthogonalPolynomials 219
AppendixB Occam’sBonus 221
Reference 222
AppendixC ProblemswithpValues 223
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C.1 TheMisuseofpValues 223
C.1.1 pValueFallacies 225
C.2 TheUseofpValues 225
C.2.1 TwoContradictoryTraditions 226
C.2.2 WhitherthepValue? 227
C.2.3 Remedies 228
References 229
Index 231
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xi
Acknowledgements
Iwouldliketothankseveralpeoplewhoinfluencedmeduringthewritingofthis
book. I am fortunate and honoured to have made acquaintance with Professor
A.W.F. Edwards (University of Cambridge) and thank him for his suggestions
and reprints. I appreciate the replies to my questions from Professor P. Dixon
(University of Alberta) and Dr Scott Glover (University of London). This book
would have been difficult to complete without the support of my loving wife
AnnahAdero.Finally,thisbookisdedicatedtotheCollegeofMedicineatAlfaisal
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About the Author
PeterM.B.CahusacgraduatedwithBScHonsinPsychologyfromStAndrews
Universityin1980,followedbyaPhDinneuropharmacologyfromtheMedical
SchoolBristolUniversityin1984.Duringa post-doctoralatOxfordposition,he
became interested in statistics and subsequently obtained an MSc in Applied
Statistics from Oxford University in 1992. He taught statistics at Stirling
Universityfrom1990to2012.HewaselectedordinarymemberofthePhysiolog-
icalSociety(UK)since1993andthenelectedFellow(FTPS)from2018.Hehas
(cid:2) beenamemberoftheBritishPharmacologicalSocietysince2006.HeisFellow (cid:2)
of the Royal Statistical Society, and GradStat status since 2009. From 2008, he
becameparticularlyinterestedinthelikelihoodapproachtostatisticalinference
asitappearedtoavoidsomeofthedifficultiesassociatedwithotherapproaches.
In2014,alongwithDrPatriciadeWinter,hepublishedanintroductorybookon
statistics.Currently,heisAssociateProfessorinBiostatisticsandPharmacology
atAlfaisalUniversity,Riyadh,SaudiArabia.
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