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(cid:2) Statistics and the Evaluation of Evidence for Forensic Scientists (cid:2) (cid:2) (cid:2) (cid:2) WILEYSERIESINSTATISTICSINPRACTICE AdvisoryEditor,MarianScott,UniversityofGlasgow,Scotland,UK FoundingEditor,VicBarnett,NottinghamTrentUniversity,UK StatisticsinPracticeisanimportantinternationalseriesoftextswhichprovidedetailedcoverageofstatistical concepts,methods,andworkedcasestudiesinspecificfieldsofinvestigationandstudy. Withsoundmotivationandmanyworkedpracticalexamples,thebooksshowindown-to-earthtermshowtoselect anduseanappropriaterangeofstatisticaltechniquesinaparticularpracticalfieldwithineachtitle’sspecialtopic area. Thebooksprovidestatisticalsupportforprofessionalsandresearchworkersacrossarangeofemploymentfieldsand researchenvironments.Subjectareascoveredincludemedicineandpharmaceutics;industry,finance,andcommerce; publicservices;theearthandenvironmentalsciences,andsoon. Thebooksalsoprovidesupporttostudentsstudyingstatisticalcoursesappliedtotheaboveareas.Thedemandfor graduatestobeequippedfortheworkenvironmenthasledtosuchcoursesbecomingincreasinglyprevalentat universitiesandcolleges. Itisouraimtopresentjudiciouslychosenandwell-writtenworkbookstomeeteverydaypracticalneeds.Feedbackof viewsfromreaderswillbemostvaluabletomonitorthesuccessofthisaim. Acompletelistoftitlesinthisseriesappearsattheendofthevolume. WILEYSERIESINSTATISTICSINPRACTICE AdvisoryEditor,MarianScott,UniversityofGlasgow,Scotland,UK FoundingEditor,VicBarnett,NottinghamTrentUniversity,UK HumanandBiologicalSciences BrownandPrescott⋅AppliedMixedModelsinMedicine Ellenberg,FlemingandDeMets⋅DataMonitoringCommitteesinClinicalTrials:APracticalPerspective Lawson,BrowneandVidalRodeiro⋅DiseaseMappingWithWinBUGSandMLwiN Lui⋅StatisticalEstimationofEpidemiologicalRisk *MarubiniandValsecchi⋅AnalysingSurvivalDatafromClinicalTrialsandObservationStudies Parmigiani⋅ModelinginMedicalDecisionMaking:ABayesianApproach (cid:2) Senn⋅Cross-overTrialsinClinicalResearch,SecondEdition (cid:2) Senn⋅StatisticalIssuesinDrugDevelopment Spiegelhalter,AbramsandMyles⋅BayesianApproachestoClinicalTrialsandHealth-CareEvaluation Turner⋅NewDrugDevelopment:Design,Methodology,andAnalysis Whitehead⋅DesignandAnalysisofSequentialClinicalTrials,RevisedSecondEdition Whitehead⋅Meta-AnalysisofControlledClinicalTrials Zhou,Zhou,LiuandDing⋅AppliedMissingDataAnalysisintheHealthSciences EarthandEnvironmentalSciences Buck,CavanaghandLitton⋅BayesianApproachtoInterpretingArchaeologicalData Cooke⋅UncertaintyModelinginDoseResponse:BenchTestingEnvironmentalToxicity Gibbons,Bhaumik,andAryal⋅StatisticalMethodsforGroundwaterMonitoring,SecondEdition GlasbeyandHorgan⋅ImageAnalysisintheBiologicalSciences Helsel⋅NondetectsandDataAnalysis:StatisticsforCensoredEnvironmentalData Helsel⋅StatisticsforCensoredEnvironmentalDataUsingMinitab®andR,SecondEdition McBride⋅UsingStatisticalMethodsforWaterQualityManagement:Issues,ProblemsandSolutions Ofungwu⋅StatisticalApplicationsforEnvironmentalAnalysisandRiskAssessment WebsterandOliver⋅GeostatisticsforEnvironmentalScientists Industry,CommerceandFinance AitkenandTaroni⋅StatisticsandtheEvaluationofEvidenceforForensicScientists,SecondEdition Brandimarte⋅NumericalMethodsinFinanceandEconomics:AMATLAB-BasedIntroduction,SecondEdition BrandimarteandZotteri⋅IntroductiontoDistributionLogistics ChanandWong⋅SimulationTechniquesinFinancialRiskManagement,SecondEdition Jank⋅StatisticalMethodsineCommerceResearch JankandShmueli⋅ModelingOnlineAuctions LehtonenandPahkinen⋅PracticalMethodsforDesignandAnalysisofComplexSurveys,SecondEdition Lloyd⋅DataDrivenBusinessDecisions OhserandMücklich⋅StatisticalAnalysisofMicrostructuresinMaterialsScience Rausand⋅RiskAssessment:Theory,Methods,andApplications (cid:2) (cid:2) Statistics and the Evaluation of Evidence for Forensic Scientists (cid:2) (cid:2) Third Edition Colin Aitken, Franco Taroni and Silvia Bozza (cid:2) (cid:2) Thiseditionfirstpublished2021 ©2021JohnWiley&SonsLtd EditionHistory JohnWiley&Sons,Ltd(2e,2004) Allrightsreserved.Nopartofthispublicationmaybereproduced,storedinaretrieval system,ortransmitted,inanyformorbyanymeans,electronic,mechanical,photocopying, recordingorotherwise,exceptaspermittedbylaw.Adviceonhowtoobtainpermissionto reusematerialfromthistitleisavailableathttp://www.wiley.com/go/permissions. TherightofColinAitken,FrancoTaroniandSilviaBozzatobeidentifiedastheauthorsof thisworkhasbeenassertedinaccordancewithlaw. RegisteredOffices JohnWiley&Sons,Inc.,111RiverStreet,Hoboken,NJ07030,USA JohnWiley&SonsLtd,TheAtrium,SouthernGate,Chichester,WestSussex,PO198SQ,UK EditorialOffice 9600GarsingtonRoad,Oxford,OX42DQ,UK Fordetailsofourglobaleditorialoffices,customerservices,andmoreinformationabout Wileyproductsvisitusatwww.wiley.com. Wileyalsopublishesitsbooksinavarietyofelectronicformatsandbyprint-on-demand. Somecontentthatappearsinstandardprintversionsofthisbookmaynotbeavailablein otherformats. LimitofLiability/DisclaimerofWarranty Whilethepublisherandauthorshaveusedtheirbesteffortsinpreparingthiswork,they makenorepresentationsorwarrantieswithrespecttotheaccuracyorcompletenessofthe contentsofthisworkandspecificallydisclaimallwarranties,includingwithoutlimitation anyimpliedwarrantiesofmerchantabilityorfitnessforaparticularpurpose.Nowarranty maybecreatedorextendedbysalesrepresentatives,writtensalesmaterialsorpromotional statementsforthiswork.Thefactthatanorganization,website,orproductisreferredtoin (cid:2) thisworkasacitationand/orpotentialsourceoffurtherinformationdoesnotmeanthatthe (cid:2) publisherandauthorsendorsetheinformationorservicestheorganization,website,or productmayprovideorrecommendationsitmaymake.Thisworkissoldwiththe understandingthatthepublisherisnotengagedinrenderingprofessionalservices.The adviceandstrategiescontainedhereinmaynotbesuitableforyoursituation.Youshould consultwithaspecialistwhereappropriate.Further,readersshouldbeawarethatwebsites listedinthisworkmayhavechangedordisappearedbetweenwhenthisworkwaswritten andwhenitisread.Neitherthepublishernorauthorsshallbeliableforanylossofprofitor anyothercommercialdamages,includingbutnotlimitedtospecial,incidental, consequential,orotherdamages. LibraryofCongressCataloging-in-PublicationData Names:Aitken,Colin,author.|Taroni,Franco,author.|Bozza,Silvia, author. Title:Statisticsandtheevaluationofevidenceforforensicscientists/ ColinAitken,FrancoTaroni,SilviaBozza. Description:Thirdedition.|Hoboken,NJ:Wiley,[2020]|Series: Statisticsinpractice|Includesindex. Identifiers:LCCN2020021384(print)|LCCN2020021385(ebook)|ISBN 9781119245223(cloth)|ISBN9781119245254(adobepdf)|ISBN 9781119245414(epub) Subjects:LCSH:Forensicsciences–Statisticalmethods.|Forensic statistics.|Evidence,Expert. Classification:LCCHV8073.A552020(print)|LCCHV8073(ebook)|DDC 363.25/60727–dc23 LCrecordavailableathttps://lccn.loc.gov/2020021384 LCebookrecordavailableathttps://lccn.loc.gov/2020021385 CoverDesign:Wiley CoverImage:©vetalaka/Shutterstock Setin13/15.5pt,PhotinaMTStdbySPiGlobal,Chennai,India 10 9 8 7 6 5 4 3 2 1 (cid:2) (cid:2) Toourfamilies (cid:2) (cid:2) (cid:2) (cid:2) [T]herevisionofopinioninthelightofnewinforma- tion...isoneofthemostimportanthumanintellectual activities.(p.290) –WardEdwards Edwards, W. (2009). Divide and conquer: how to use likelihood and value judgments in decision making (1973). In: A Science of Decision Making: TheLegacyofWardEdwards(ed.J.W.WeissandD.J. Weiss),287–300.Oxford:OxfordUniversityPress. (cid:2) (cid:2) (cid:2) (cid:2) Contents Foreword xvii PrefacetoThirdEdition xxi PrefacetoSecondEdition xxx PrefacetoFirstEdition xxxvii (cid:2) (cid:2) 1 Uncertainty in Forensic Science 1 1.1 Introduction 1 1.2 StatisticsandtheLaw 3 1.3 UncertaintyinScientificEvidence 11 1.3.1 TheFrequentistMethod 15 1.3.2 StainsofBodyFluids 17 1.3.3 GlassFragments 21 1.4 Terminology 29 1.5 TypesofData 34 1.6 Populations 36 1.7 Probability 41 1.7.1 Introduction 41 1.7.2 AStandardforUncertainty 46 1.7.3 Events 55 1.7.4 ClassicalandFrequentistDefinitions ofProbabilityandTheir Limitations 57 1.7.5 SubjectiveDefinitionofProbability 60 vii (cid:2) (cid:2) viii Contents 1.7.6 TheQuantificationofProbability ThroughaBettingScheme 64 1.7.7 ProbabilitiesandFrequencies:The RoleofExchangeability 69 1.7.8 LawsofProbability 78 1.7.9 DependentEventsandBackground Information 82 1.7.10 LawofTotalProbability 91 1.7.11 UpdatingofProbabilities 96 2 The Evaluation of Evidence 101 2.1 Odds 101 2.1.1 ComplementaryEvents 101 2.1.2 Examples 104 2.1.3 DefinitionofOdds 105 2.2 Bayes’Theorem 108 2.2.1 StatementoftheTheorem 109 (cid:2) 2.2.2 Examples 109 (cid:2) 2.3 TheOddsFormofBayes’Theorem 121 2.3.1 LikelihoodRatio 121 2.3.2 Bayes’FactorandLikelihoodRatio 125 2.3.3 Three-WayTables 130 2.3.4 LogarithmoftheLikelihoodRatio 134 2.4 TheValueofEvidence 138 2.4.1 EvaluationofForensicEvidence 138 2.4.2 JustificationoftheUseofthe LikelihoodRatio 154 2.4.3 SingleValuefortheLikelihood Ratio 158 2.4.4 RoleofBackgroundInformation 161 2.4.5 SummaryofCompeting Propositions 163 2.4.6 QualitativeScalefortheValueofthe Evidence 168 2.5 ErrorsinInterpretation 180 2.5.1 FallacyoftheTransposed Conditional 186 (cid:2) (cid:2) Contents ix 2.5.2 SourceProbabilityError 190 2.5.3 UltimateIssueError 194 2.5.4 DefenceAttorney’sFallacy 194 2.5.5 Probability(AnotherMatch)Error 196 2.5.6 NumericalConversionError 199 2.5.7 FalsePositiveFallacy 202 2.5.8 ExpectedValueFallacy 203 2.5.9 Uniqueness 206 2.5.10 OtherDifficulties 209 2.5.11 EmpiricalEvidenceofErrorsin Interpretation 220 2.6 Misinterpretations 233 2.7 ExplanationofTransposedConditional, DefenceAttorney’sandFalsePositive Fallacies 236 2.7.1 ExplanationoftheFallacyofthe TransposedConditional 236 (cid:2) 2.7.2 ExplanationoftheDefence (cid:2) Attorney’sFallacy 239 2.7.3 ExplanationoftheFalsePositive Fallacy 241 2.8 MakingCoherentDecisions 245 2.8.1 ElementsofStatisticalDecision Theory 246 2.8.2 DecisionAnalysis:AnExample 249 2.9 GraphicalProbabilisticModels:Bayesian Networks 254 2.9.1 ElementsoftheBayesianNetworks 256 2.9.2 TheConstructionofBayesian Networks 261 2.9.3 BayesianDecisionNetworks (InfluenceDiagrams) 272 3 Historical Review 279 3.1 EarlyHistory 279 3.2 TheDreyfusCase 286 (cid:2) (cid:2) x Contents 3.3 StatisticalArgumentsbyEarlyTwentieth- CenturyForensicScientists 293 3.4 Peoplev.Collins 299 3.5 DiscriminatingPower 307 3.5.1 Derivation 307 3.5.2 EvaluationofEvidenceby DiscriminatingPower 310 3.5.3 FiniteSamples 316 3.5.4 CombinationofIndependent Systems 319 3.5.5 CorrelatedAttributes 321 3.6 SignificanceProbabilities 325 3.6.1 CalculationofSignificance Probabilities 326 3.6.2 RelationshiptoLikelihoodRatio 333 3.6.3 CombinationofSignificance Probabilities 338 3.7 CoincidenceProbabilities 342 (cid:2) (cid:2) 3.7.1 Introduction 342 3.7.2 ComparisonStage 346 3.7.3 SignificanceStage 347 3.8 LikelihoodRatio 351 4 Bayesian Inference 359 4.1 Introduction 359 4.2 InferenceforaProportion 368 4.2.1 IntervalEstimation 374 4.2.2 EstimationwithZeroOccurrences inaSample 381 4.2.3 UncertaintyonSensitivityand Specificity 387 4.3 Sampling 392 4.3.1 ChoiceofSampleSizeinLarge Consignments 398 4.3.2 ChoiceofSampleSizeinSmall Consignments 413 4.4 BayesianNetworksforSamplingInspection 420 (cid:2)