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Bayesian Model Selection and Fit for Incomplete Longitudinal Data

Permanent Link: http://ufdc.ufl.edu/UFE0045176/00001

Material Information

Title: Bayesian Model Selection and Fit for Incomplete Longitudinal Data
Physical Description: 1 online resource (97 p.)
Language: english
Creator: Chatterjee, Arkendu S
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: bayes -- incomplete -- longitudinal -- model -- posterior -- predictive -- selection
Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this dissertation, we explore Bayesian approaches for model selection and fit in models for incomplete  longitudinal based on the posterior predictive distribution. We first explore the use of a posterior predictive loss criterion for model selection for incomplete longitudinal data.  We begin by identifying a property all model selection criteria for incomplete data should have.  We then show that a straightforward extension of the Gelfand and Ghosh (1998) criterion to incomplete data  has two problems.   First, it introduces an extra term (in addition to the goodness of fit and penalty terms) that compromises the criterion.  Second, it does not satisfy the aforementioned property. We propose an alternative and explore its properties via simulations and on a real dataset.  The new criterion appears to work well and is very easy to compute. We examine the operating characteristics of two approaches to assess model fit for incomplete longitudinal data.  The first approach assesses fit based on replicated complete data as advocated in Gelman et. al. (2005). We propose an alternative that assesses fit based on replicated observed data.  Pros and cons of each approach are discussed and simulations and analytical results are presented that compare the power under each approach. Finally we review different methods for model diagnostics for incomplete longitudinal data that have been proposed in the literature. Most of the literature focuses on on assessing fit for fully identified parametric models. We briefly describe the local influence  method by Verbeke et. al. (2001)  and index of local sensitivity to nonignorability by Troxel et. al. (2004)  and show their  connection. We then review graphical approach as proposed by Dobson and Henderson (2003) for assessment of continuous time dropout based on residuals  and connect this  to the local influence method. We then connect these to the  Bayesian approaches introduced in Chapter 3.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Arkendu S Chatterjee.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Daniels, Michael Joseph.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-05-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045176:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045176/00001

Material Information

Title: Bayesian Model Selection and Fit for Incomplete Longitudinal Data
Physical Description: 1 online resource (97 p.)
Language: english
Creator: Chatterjee, Arkendu S
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: bayes -- incomplete -- longitudinal -- model -- posterior -- predictive -- selection
Statistics -- Dissertations, Academic -- UF
Genre: Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this dissertation, we explore Bayesian approaches for model selection and fit in models for incomplete  longitudinal based on the posterior predictive distribution. We first explore the use of a posterior predictive loss criterion for model selection for incomplete longitudinal data.  We begin by identifying a property all model selection criteria for incomplete data should have.  We then show that a straightforward extension of the Gelfand and Ghosh (1998) criterion to incomplete data  has two problems.   First, it introduces an extra term (in addition to the goodness of fit and penalty terms) that compromises the criterion.  Second, it does not satisfy the aforementioned property. We propose an alternative and explore its properties via simulations and on a real dataset.  The new criterion appears to work well and is very easy to compute. We examine the operating characteristics of two approaches to assess model fit for incomplete longitudinal data.  The first approach assesses fit based on replicated complete data as advocated in Gelman et. al. (2005). We propose an alternative that assesses fit based on replicated observed data.  Pros and cons of each approach are discussed and simulations and analytical results are presented that compare the power under each approach. Finally we review different methods for model diagnostics for incomplete longitudinal data that have been proposed in the literature. Most of the literature focuses on on assessing fit for fully identified parametric models. We briefly describe the local influence  method by Verbeke et. al. (2001)  and index of local sensitivity to nonignorability by Troxel et. al. (2004)  and show their  connection. We then review graphical approach as proposed by Dobson and Henderson (2003) for assessment of continuous time dropout based on residuals  and connect this  to the local influence method. We then connect these to the  Bayesian approaches introduced in Chapter 3.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Arkendu S Chatterjee.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Daniels, Michael Joseph.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045176:00001


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BAYESIANMODELSELECTIONANDFITFORINCOMPLETELONGITUDIN ALDATA By ARKENDUSEKHARCHATTERJEE ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2013

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c r 2013ArkenduSekharChatterjee 2

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Tomyfamily 3

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ACKNOWLEDGMENTS IamreallythankfultomyresearchadvisorProfessorMichae lJ.Danielsforgiving metheopportunitytoworkwithhim.Itwasimpossibletowrit ethisthesiswithout hishelp.ProfessorDanielsisakindheartedenthusiasticp erson.Itwasanamazing experienceandpleasure,IhadduringmyPhDlife. Ialsowanttothankmycommitteemembers,ProfessorMalayGh osh,Professor AndrewRosalsky,andDr.RohnShorr,whohaveprovidedmethe irinsightsthroughout thewritingprocess.IwouldliketothankProfessorGhoshfo rhisguidancewhichhelped mealottonishmydissertation. Iamgratefultomyparentsforalltheirsupportandlove.Wit houtthemitwas impossiblealso.Alsoalotofthankstomyin-lawsforallthe irsupport.It'stimetothank mylovelywifeHiyaBanerjee,whostoodbymeformyPh.D.life andinspiringmein everysituation. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 1.1MissingDataMechanism ........................... 11 1.1.1Identiability ............................... 12 1.1.2Ignorabilityvs.Nonignorability ..................... 12 1.2InferencewithMissingData .......................... 13 1.3ModelSelection ................................ 15 1.4PosteriorPredictiveAssessment ....................... 17 2BAYESIANMODELSELECTIONFORINCOMPLETELONGITUDINALDAT A 20 2.1OverviewofModelSelection ......................... 20 2.1.1BayesFactors .............................. 20 2.1.2LikelihoodBasedPenalizedCriteria .................. 21 2.1.3PosteriorPredictiveDistributionBasedCriteria ........... 21 2.1.4IssueswithBayesianModelSelectionwithIncomplete Data .... 23 2.2PosteriorPredictiveLoss:AQuickReview .................. 24 2.3PPLforIncompleteLongitudinalData .................... 26 2.3.1ARe-formulation ............................ 27 2.3.2Choicesfor T ( r r y ) ......................... 28 2.3.3Computations .............................. 28 2.4DataExample .................................. 29 2.4.1ModelsConsidered ........................... 29 2.4.2Results ................................. 31 2.5Simulations ................................... 31 2.6Discussion ................................... 34 3POSTERIORPREDICTIVEASSESSMENTFORLONGITUDINALDATA ... 41 3.1OverviewofPosteriorPredictiveAssessment ................ 41 3.1.1CompleteDataModelFit ........................ 41 3.1.2IncompleteDataModelFit ....................... 43 3.2PosteriorPredictiveChecksforIncompleteData .............. 44 3.2.1ObservedandCompleteReplication ................. 44 3.2.2Issues .................................. 45 3.3AnalyticResults ................................ 46 3.4DataExample .................................. 48 5

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3.5SimulationModelsandResults ........................ 49 3.6SummaryandDiscussion ........................... 51 4REVIEWONMODELDIAGNOSTICSFORINCOMPLETEDATA ........ 58 4.1BackgroundandMotivation .......................... 58 4.2DiagnosticsBasedonFullyParametrizedIdentiedMode ls ........ 61 4.2.1LocalInuence ............................. 62 4.2.2ApplicationofLocalInuencemethodinSelectionMod els ..... 63 4.2.3ConnectionBetweenILSNandLocalInuence ........... 63 4.2.4ApplicationofILSNinSelectionModels ............... 64 4.3DiagnosticsforLongitudinalandContinuousTimeDropo ut ........ 64 4.3.1JointModelforRandomEffects .................... 65 4.3.2ClassicationsofDropout ....................... 65 4.3.3RelationbetweenDropoutandResiduals .............. 66 4.3.4RelationBetweenResidualAnalaysisandLocalInuen ceMethod 67 4.3.5ResidualDiagnosticsforMixtureModels ............... 68 4.4BayesianDiagnosticsBasedonthePosteriorPredictive Distribution ... 69 4.5Summary .................................... 71 5CONCLUSIONSANDEXTENSIONS ....................... 72 5.1SummaryofChapters ............................. 72 5.2ExtensionsforModelSelection ........................ 73 5.3ExtensionsforModelFit ............................ 73 APPENDIX APPLCRITERIAANDTHEOREM .......................... 74 BGROWTHHORMONEDATAANALYSIS ...................... 77 CCLOSEDFORMEXPECTATIONS ......................... 78 DSIMULATIONRESULTFORPPLCRITERION .................. 79 EANALYTICCALCULATIONOFPOSTERIORPREDICTIVEPROBABILIT IES 85 REFERENCES ....................................... 93 BIOGRAPHICALSKETCH ................................ 97 6

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LISTOFTABLES Table page 2-1PPLforGHData ................................... 37 2-2DIC o forGHData ................................... 37 2-3NumberofTimesthePPLandDIC o CriterionChoosetheTrueModel ..... 38 2-4AveragePPLCriteriaforDataSimulatedunderMM1,Sampl eSize2000 .... 39 2-5SimulationforAverageofPPLCriteriaunderSM0,Sample Size100 ...... 39 2-6DataSimulatedunderMM2,SampleSize2000withAverageo fPPLCriteria 40 3-1GrowthHormoneTrial:SampleMeans(StandardDeviation s). ......... 52 3-2PosteriorPredictiveProbabilitiesunderWrongModel( 21 =0 ). ........ 52 3-3WrongModel( ( k ) 1 = 1 ( k ) 1 = ):PosteriorPredictiveProbablities. ...... 53 3-4UnderTrueModel:NatureofPosteriorPredictiveProbab ilities. ......... 53 3-5SimulationforCompleteReplicationunderTrueMM1andF ittedMM2 ..... 54 3-6CompleteReplicatedDataunderTrueMM1andFittedMM1 ........... 54 3-7SimulatedObservedReplicatedDataunderMM1andFitted MM2 ....... 55 3-8UnderMM1SimulatedObservedReplicateddataandFitted MM1 ....... 55 3-9CompleteandObservedReplicationunderTrueMM1andFit tedMM1 ..... 56 3-10ForTrueMM1andFittedWrongModel:CompleteandObserv edReplication 57 B-1GrowthHormoneTrial:SampleMeans(StandardDeviation s). .......... 77 B-2DICBasedonObservedDataLikelihoodfortheGHDataAnal ysis ....... 77 D-1SimulatingModelMARSM0,MM1andMM2 ................... 79 D-2AveragePPL:UnderTrueSM0andSampleSize50: .............. 80 D-3SimulatedDataunderModelMARSM0withSampleSize100 ......... 80 D-4DataSimulatedunderMARSM0withSampleSize2000 ............. 81 D-5UnderMARMM1SimulateddatawithSampleSize50 .............. 81 D-6DataofSampleSize100underModelMARMM1 ................ 82 D-7AveragePPL:SimulatingModelMARMM1,SampleSize2000: ........ 82 7

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D-8UnderModelMARMM2,withSampleSize50 .................. 83 D-9SimulationofMARMM2,SampleSize100:AveragePPL ............ 83 D-10SimulatedDataunderMARMM2,SampleSize2000:Average PPL ...... 84 8

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy BAYESIANMODELSELECTIONANDFITFORINCOMPLETELONGITUDIN ALDATA By ArkenduSekharChatterjee May2013 Chair:MichaelJ.DanielsMajor:Statistics Inthisdissertation,weexploreBayesianapproachesformo delselectionandt inmodelsforincompletelongitudinalbasedontheposterio rpredictivedistribution. Werstexploretheuseofaposteriorpredictivelosscriter ionformodelselectionfor incompletelongitudinaldata.Webeginbyidentifyingapro pertyallmodelselection criteriaforincompletedatashouldhave.Wethenshowthata straightforwardextension ofthe GelfandandGhosh ( 1998 )criteriontoincompletedatahastwoproblems.First, itintroducesanextraterm(inadditiontothegoodnessoft andpenaltyterms)that compromisesthecriterion.Second,itdoesnotsatisfythea forementionedproperty.We proposeanalternativeandexploreitspropertiesviasimul ationsandonarealdataset. Thenewcriterionappearstoworkwellandisveryeasytocomp ute. Weexaminetheoperatingcharacteristicsoftwoapproaches toassessmodelt forincompletelongitudinaldata.Therstapproachassess estbasedonreplicated completedataasadvocatedin Gelmanetal. ( 2005 ).Weproposeanalternativethat assessestbasedonreplicatedobserveddata.Prosandcons ofeachapproachare discussedandsimulationsandanalyticalresultsareprese ntedthatcomparethepower undereachapproach. Finallywereviewdifferentmethodsformodeldiagnosticsf orincompletelongitudinal datathathavebeenproposedintheliterature.Mostoftheli teraturefocuseson onassessingtforfullyidentiedparametricmodels.Webr ieydescribethelocal 9

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inuencemethodby Verbekeetal. ( 2001 )andindexoflocalsensitivitytononignorability by Troxeletal. ( 2004 )andshowtheirconnection.Wethenreviewgraphicalapproa chas proposedby DobsonandHenderson ( 2003 )forassessmentofcontinuoustimedropout basedonresidualsandconnectthistothelocalinuencemet hod.Wethenconnect thesetotheBayesianapproachesintroducedinChapter3. 10

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CHAPTER1 INTRODUCTION Longitudinalstudiescollectrepeatedobservationsovert imeonseveralindividuals. Itiscommonthatallthedesireddataarenotcollected,andt husoneneedstoaddress themissingdata.Themissingnessmayoccurduetolosstofol lowingup,mortalityor otherreasons.Drawinginferenceaboutthistypeofdataisa challenge. Inthisdissertationthefollowingnotationwillbeusedthr oughout, Y :fulldataresponse,R:observeddataindicators: R ij = I f y ij isobserved g ,( Y obs r ):observeddata, Y mis :missingdata, Y rep com :replicatedfulldataresponse, Y repobs : replicatedobserveddataresponse, :parameterindexingthefulldatamodel 1.1MissingDataMechanism Asimplewaytomakeinferenceinthepresenceofincompleted ataistoonly usethecompletecases.Thisresultsarevalidforinference onlyifthemissingnessis independentoftheresponsewhichiscalledmissingcomplet elyatrandom(MCAR).We reviewtheclassicationofmissingdataasdenedby Rubin ( 1976 ). LittleandRubin'smissingnesstaxonomy( Rubin 1976 ; LittleandRubin 1987 )is denedas, Denition1.1. Missingcompletelyatrandom (MCAR)mechanismisdened conditionedforallxand as, p ( r j Y x )= p ( r j x ), (1–1) i.e., p ( r j Y x ) isconstantfunctionofY. Denition1.2. Missingatrandom (MAR)mechanismisdenedas, p ( r j Y obs Y mis x )= p ( r j Y obs x ), (1–2) forall Y obs ,xand ,i.e., p ( r j Y obs Y mis x ) isconstantfunctionof Y mis 11

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MARholdsifandonlyif p ( y mis j y obs r )= p ( y mis j y obs ). (1–3) Denition1.3. Missingnotatrandom (MNAR)mechanismisdenedas, p ( r j Y obs Y mis x ) 6 = p ( r j Y obs Y 0mis x ), (1–4) forsome Y mis and Y 0mis ,i.e.,Rdependsonsomepartof Y mis ,evenafterconditioning on Y obs 1.1.1Identiability Denition1.4. The extrapolationfactorization ( DanielsandHogan 2008 )is p ( y r ; )= p ( y mis j y obs r ; E ) p ( y obs r ; O ). (1–5) Theobserveddatadistribution p ( y obs r ; O ) ,isidentiableandisindexedby O .The extrapolationdistribution p ( y mis j y obs r ; E ) ,isindexedby E whichisnotidentied withoutmodelingassumptions.1.1.2Ignorabilityvs.Nonignorability InlikelihoodorBayesianinference,oftenmorenaturaltoc lassifymissingnessas ignorableornonignorable. Denition1.5. Themissingdatamechanismis ignorable if,( Rubin 1976 ) 1.ThemissingdatamechanismisMAR.2. istheparameteroffulldataresponsemodelanditcanbedeco mposedinas =( ) ,where isthefulldataresponsemodelparameterand isassociatedwith themissingdatamechanism p ( r j y ) 3.Theparameters and areaprioriindependent; p( )=p( )p( ) (1–6) 12

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ThethirdconditionisnecessaryfortheBayesianinference .Ifanyofthethreeconditions donothold,missingnessisnonignorable.Underignorabili ty,posteriorinferenceon parameterarebasedonobserveddataresponselikelihood. L( j y obs ) / Z p(y obs ,y mis j )dy mis (1–7) Thiscanbeseenvia( DanielsandHogan 2008 ), L ( j y obs r )= L ( j y obs r ) = Z p ( y obs y mis r j ) d y mis MAR = Z p ( r j y obs ) p ( y obs y mis j ) d y mis = p ( r j y obs ) Z p ( y obs y mis j ) d y mis = p ( r j y obs ) p ( y obs j ) = L ( j r y obs ) L ( j y obs ). (1–8) p( j y obs ,r) / p( )L( j y obs ,r)=p( )L( j r,y obs )p( )L( j y obs ). (1–9) Thusposteriorinferencefor basedonobserveddataresponselilkelihood p ( j y obs ) / L ( j y obs ) p ( ). 1.2InferencewithMissingData ForBayesiannonignorableinferenceweneedtomodelthejoi ntdistributionof missingnessandtheresponsevariable.Therearethreegene ralapproacheswhich dependonhowfulldatamodelisfactoredandparametrized. Denition1.6. Pattern-mixturemodel :Pattern-mixturemodelsusethefactorization, p ( y r j )= p ( y j r ) p ( r j ) (1–10) Little ( 1993 ), Little ( 1994 ), LittleandWang ( 1996 )hasdoneconsiderableworkon patternmixturemodels.Heproposedseveralidentifyingre strictionstoidentifythe full-datamodel.Forcontinuousdropouttimes, Hoganetal. ( 2004 )proposedmixture 13

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modelwithresponsegivendropoutbyvaryingacoefcientmo del. HoganandLaird ( 1997 )introducedrandomeffectsmixturemodelsfortimetoevent datawhichhave informativecensoring.Thisapproachwasgeneralizedby FitzmauriceandLaird ( 2000 ) forcategorical,ordinalandcountdataviageneralizedlin earmixturemodels.Anintuitive parameterizationofthepatternmixturemodelwasintroduc edby DanielsandHogan ( 2000 )forcontinuousdatadesignedforsensitivityanalysis. Amajoradvantageofpatternmixturemodelsisthat,missing nessistreatedas sourceofvariationinthefulldatadistribution.Inspiteo fthedifcultiesofspecifying differentdistributionforeachmissingdatapattern,itha sadvantagethatitcanclearly expresstheparameterswhicharenotidentiedbyobservedd ata.Theyhavebeen criticizedgiventhefactthatthehazardofmissingnessata particulartimeoftendepends onfutureobservations.However,theycanbeconstrainedin awaythat,conditioned onpastresponses,hazardofdropoutatparticulartimepoin tdependsonthepast andcurrentresponse,andconditionallyonthose,donotdep endonfutureresponses ( Kenwardetal. 2003 ). Denition1.7. Selectionmodel :Selectionmodelsusethefactorizationoffulldata distributionas, p ( y r j )= p ( r j y ) p ( y j ). (1–11) Selectionmodelswereintroducedby Heckman ( 1976 )forabivariateresponse.The workbyHeckmanwasextendedby DiggleandKenward ( 1994a )tothegeneral Longitudinaldata.Theattractivenessofselectionmodels isthat,theyspecifythe full-dataresponsedirectlyandenableeasycharacterizat ionofthemissingdata mechanismunlikemixturemodels.Disadvantagesincludeth eirsensitivitytomodel specication,thenon-existenceofsensitivityparameter s,andcomputationofBayesian modelselectioncriteriae.g.,DIC. 14

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Denition1.8. Sharedparametermodel :Sharedparametermodelsusethe factorizationas, p(y,r j )= Z p(y,r j b, )p(b j )db, (1–12) where y and r areindependentconditionedonb, p ( y r j b )= p ( y j b ) p ( r j b ) (1–13) Sharedparameterrandomeffectsmodelforcontinuousrespo nsesandinformative censoringwasintroducedby WuandCarroll ( 1988 ).Theyusedtheindividualeffectsas interceptsandslopesinthemissingdataprocess. Hendersonetal. ( 2000 )proposeda jointmodelingapproachfortwocorrelatedGaussianrandom processes.Advantagesof sharedparametermodelsarethattheyprovideasimplespeci cationfortheresponse andmissingnessprocessesconditionalonrandomeffects.T hroughtheuseofrandom effects,sharedparametermodelscanbeusedtohandlehighdimentionalresponse data.Theyhavesamedisadvantagesasselectionmodels. 1.3ModelSelection BayesianmodelselectionisthetopicfortheChapter2ofthi sdissertation. Inpresenceofdifferentparametricmodelsasdiscussedint heprevioussection researchersareinterestedinndingwhichmodeltsthedat abest.Thereareseveral waystoselectthemodels,butthemostcommonmethodsareBay esFactor(BF), likelihoodbasedpenalizedcriteria,andposteriorpredic tivedistributionbasedcriteria. Bayesfactor(BF)isthestandardBayesianapproachforcomp aringthemodels, whichisbasedontheratioofmarginallikelihoods.Themarg inallikelihoodformodel m isdenedas, p(y j m)= Z p(y j (m) ,m)p( (m) j m)d (m) (1–14) TheproblemwithBayesfactorarecomputingthemarginallik elihoodofthemodelsand thechoiceofpriors.Chibandcolleagues( CarlinandChib 1995 ;& ChibandJeliazkov 15

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2001 )inaseriesofpapershaveproposedcomputationallyefcie ntwaystocompute BayesFactorsusingMCMCoutput.RecentworkbyJohnsonandc olleagues( Johnson ( 2005 ); JohnsonandHu ( 2009 ))haveproposedBayesFactorsbasedonteststatistics. WeconnectJohnson'sworktoourapproachinChapter2. AnotherapproachisLikelihoodbasedpenalizedcriteria. Spiegelhalteretal. ( 2002 ) proposedthedevianceinformationcriteria(DIC)whichisc omprisedofagoodnessoft termandapenaltyterm.Theydeneitsas, DIC ( y ; )= D ( E [ j y ])+ 2p D (1–15) where D (.) isthedevianceand p D = D ( E [ j y ]) E [ D j y ] ,theeffectivenumberof parameters.ThedrawbackofDICislackofinvariancetothep arameterizationandthe choiceoflikelihoodinhierarchical/multilevelmodels. Celeuxetal. ( 2006 )proposed severalversionsofDICforsettingswithmissingdata,butt heirrecommendations werebasedonlatentdata. DanielsandHogan ( 2008 )and WangandDaniels ( 2011 ) recommendedconstructingtheDICbasedontheobserveddata likelihoodfor comparisonofmodelsbasedonincompletedatawiththelatte rexaminingitsperformance withsimulationstudies. Criteriabasedontheposteriorpredictivedistributionha vebeenproposedby, ( IbrahimandLaud 1994 ; Ibrahimetal. 2001 ; GelfandandGhosh 1998 ; Chenetal. 2004 ).Theposteriorpredictivedistributionforthereplicate ddata y rep undermodel m is givenby p ( y rep j y m )= Z p ( y rep j ( m ) m ) ( ( m ) j y m ) d ( m ) (1–16) IbrahimandLaud ( 1994 )denedtheircriterionastheexpectedsquaredEuclidean distancebetween y and y rep L = E [( y rep y ) 0 ( y rep y )], (1–17) 16

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wheretheexpectationwastakenwithrespecttotheposterio rpredictivedistribution, p ( y rep j y ) .Lcanbere-expressedas L = n X i =1 Var ( y rep j y )+( E [ y rep j y ] y i ) 2 (1–18) GelfandandGhosh ( 1998 )proposedamoregenerallossfunction(PPL) L ( y rep a ; y )= L ( y rep ; a )+ kL ( y a ), k > 0. (1–19) Foramodel m theyminimizedE [ L ( y rep a ; y ) j y ] ,theposteriorpredictiveexpectationof thelosswithrespecttoanaction, a ThegoalofChapter2istoextendtheposteriorpredictivecr iteriaproposedby GelfandandGhoshtoincompletelongitudinaldata. 1.4PosteriorPredictiveAssessment Parameterdependentstatistics(ordiscrepancystatistic s)wereintroducedin aBayesiansettingby Zellner ( 1975 ). Rubin ( 1984 )usedtheposteriorpredictive distributionofastatistictocalculatethetail-areaprob abilitycorrespondingtothe observedvalueofthestatistic. Meng ( 1994 )calledthisprobabilityaposteriorpredictive p-value.Infollowing,wewillrefertoitasposteriorpredi ctiveprobabilityduetoits problematicinterpretationasap-value( Robinsetal. 2000 ).Thisprobabilityisa measureofdiscrepancybetweentheobserveddataandthepos itedassumptions asmeasuredbythesummaryquantity T ( ) .Theposteriorpredictivedistributionof T ( y rep ) shouldidentifythedeparturewhenthewrongmodelisttedo nthedataand comparedwithdistributionof T ( y ) .FortheassumedBayesianmodel,theposterior predictiveapproachprovidesareferencedistribution.Co nditionedontheparameter thedata y andthereplicateddata y rep areindependent.Thetofthemodeltothedata isdeterminedbycomparingtheposteriorpredictivedistri butionof T ( y rep ) with T ( y ) 17

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Meng ( 1994 )formallydenedthisprobabilityas, P ( T ( y rep ) > T ( y ) j m y )= ZZ I [ T ( y rep ) > T ( y )] p ( y rep j ) p ( j y ) d y rep d IntheBayesianformulationthisapproachallowstheuseofa parameterdependenttest statistic,whichiscalledadiscrepancystatisticsForadi screpancy, D ( y ; ) ,thereference distributioncanbederivedfromthejointdistributionof ( y rep ) P ( y rep j m y )= P ( y rep j m ) P ( j m y ). Howeverlocatingtherealizedvalueof D ( y ; ) withinthereferencedistributionisnot feasiblesince D ( y ; ) dependsontheunknown .Thiscomplicationhasledauthorsto usethetailareaprobabilityof D underitsposteriorreferencedistribution. Aprobabilitywasconstructedby Gelmanetal. ( 1996 )asin( 3–1 )byeliminating thedependenceonunknown .Theyconsideredminimumdiscrepancy( D min ( y ) )or averagediscrepancy( D avg ( y ) )statistic, T ,overtheparameter .Forthelatter,thetail areaprobabilityofthediscrepancystatisticisgivenby: P ( D ( y rep ; ) D ( y ; ) j m y ). Thisisanalogoustotheposteriorpredictiveprobabilityi n( 3–1 ).Fortheformer,the correspondingposteriorpredictiveprobabilityisdened as, P ( D min ( y rep ) D min ( y ) j m y )= Z P ( D min ( y rep ) D min ( y ) j m ) P ( j m y ) d Thechoiceofdiscrepancyisclearlyveryimportantandofte nreectstheinferential interests.Ingeneral,suchchecksarecalledposteriorpre dictivechecks. Gelmanetal. ( 2005 )proposedanextensionoftheposteriorpredictiveapproac hto thesettingofmissingdata(andlatentdata).Similartothe completedatasettingsthe complicationforthesesituationsisconstructingtherefe rencedistributions.Toassess thetofthemodeltheydeneateststatistic T ( ) ,whichisafunctionofthecomplete 18

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data.The'missing'dataeithermissingorlatentislledin ateachiterationusingdata augmentation.Theteststatisticiscomparedtotheteststa tisticcomputedbasedon replicatedcompletedata.Graphicalapproacheswereimplementedformodelcheckingf orbothmissingandlatent datasetupalongwiththecalculationofposteriorpredicti vep-valuesfortestvariables. Wewillexploreposteriorpredictiveassessmentforincomp letedatainChapter3. InChapter2weexploretheuseofposteriorpredictivelossc riterionformodel selectionforincompletelongitudinaldata.Weshowanexte nsionofcriterionproposed by GelfandandGhosh ( 1998 )toincompletedata.Weapplytheproposedcriterion andexploreitspropoertyviasimulationsandonrealdatase t.Chapter3reviewsthe operatingcharacteristicsoftwoapproachesbasedonrepli catedcompletedataand replicatedobserveddatatoassessthemodeltforincomple telongitudinaldata.We showsomeanalyticalresultsforeachapproachandsimulati onsaredonetocompare thethepowerundereachapproach.Simulationresultsshowt hesamebehavioras theanalyticalresults.Chapter4focusesonthemodeldiagn osticsforincomplete longitudinaldataandthesensitivityanalysisoffullyide ntiedparametricmodels.We describethelocalinuencemethodintroducedby Verbekeetal. ( 2001 )andexplainits relationwithILSNmethodby Troxeletal. ( 2004 ).Wefocusonthegraphicalapproaches introducedby DobsonandHenderson ( 2003 )andtrytomakeconnectionbetweentheir workandthelocalinuenceapproach.Wereviewthemodelass essmentbasedon observedreplicationasoneoftheBayesianapproachesfort hemodeldiagnosticswhich havethepropertyofinvariancetothetheextrapolationdis tribution. 19

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CHAPTER2 BAYESIANMODELSELECTIONFORINCOMPLETELONGITUDINALDATA 2.1OverviewofModelSelection Whenseveralparametricmodelsareunderconsideration,it isoftenofinterestto determinewhichonetsthedatathebest.Morespecically, choosingaprobability modelfortheobserved Y ,indexedby m ,conditionedonaparametervector ( m ) p(y j m, (m) ), m 2 M (m) 2 (m) (2–1) where M isthemodelspaceand ( m ) istheparameterspace.Wechoosethemodel withthebestvalueforthechosencriterion. InthecontextofBayesianinference,therehavebeenmanycr iteriaproposedfor modelselection.Wewillbrieyreviewthreepopularchoice s:BayesFactors(BF), likelihoodbasedpenalizedcriteria,andposteriorpredic tivedistributionbasedcriteria. Wewillthendiscussissuesinusingthesedifferentcriteri aforincompletelongitudinal data.2.1.1BayesFactors ThestandardBayesianapproachtocomparemodelsisbasedon theratioof marginallikelihoods,ortheBayesFactor(foranexcellent review,see KassandRaftery 1995 ).Themarginallikelihoodformodel m isdenedas p(y j m)= Z p(y j (m) ,m)p( (m) j m)d (m) (2–2) ThemainissueswithBayesFactorsarerelatedtocomputatio n(i.e.,ofthemarginal likelihoodsofthemodelsunderconsideration)andtheneed touseproperpriorsforthe parametersbeing'compared'acrossmodels.However,anatt ractivefeatureofBayes Factorsistheirconnectiontoposteriormodelprobabiliti es;amongotherthings,this providesagoodwaytocalibratethem. Chibandcolleagues( CarlinandChib 1995 ;& ChibandJeliazkov 2001 and 20

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ChibandJeliazkov 2005 )inaseriesofpapershaveproposedcomputationallyefcie nt waystocomputeBayesFactorsusingMCMCoutput.Recentwork byJohnsonand colleagues( Johnson ( 2005 ), JohnsonandHu ( 2009 ))haveproposedBayesFactors basedonteststatistics.WewillconnectJohnson'sworktoo urapproachlater. 2.1.2LikelihoodBasedPenalizedCriteria Giventhepopularityofsamplingbasedapproachestocomput eposteriordistributions, themostcommonlikelihoodbasedpenalizedcriterionisthe 'easytocompute'Deviance informationcriterion(DIC). Spiegelhalteretal. ( 2002 )proposedthiscriterionwhich iscomposedoftwoterms,agoodnessofttermandacomplexit y/penaltyterm.The goodnessofttermisthedevianceevaluatedatasummaryoft heposteriordistribution oftheparameters(oftentheposteriormean).Thecomplexit ypenaltyisdenedasthe posteriormeandeviance( D )minusthedevianceevaluatedattheposteriormeanof theparameters;thisisrelatedtotheideaofresidualinfor mation.Twoofthedrawbacks ofthiscriterionarethelackofinvariancetotheparameter izationofthemodeland thechoiceofthelikelihoodinhierarchical/multilevelmo dels.Theseminalpaperby Spiegelhalteretal.hasbeenfollowedbynumerouspapersex aminingtheDICinmore complexsettings. Celeuxetal. ( 2006 )proposedseveralversionsofDICforsettings withmissingdata.However,theirrecommendationswerebas edonlatentdata,not responsesthatcouldbeobserved.Wefocusonthelatter. DanielsandHogan ( 2008 ) and WangandDaniels ( 2011 )recommendedconstructingtheDICbasedonthe observeddatalikelihoodforcomparisonofmodelsbasedoni ncompletedatawiththe latterexaminingitsperformancewithsimulationstudies. Treatingthemissingresponses as'latent'dataandusingtherecommendationsinCeleuxeta l.willresultincriteriathat domatchsatisfydesiredproperties,includingtheonetobe introducedinSection 2.1.4 2.1.3PosteriorPredictiveDistributionBasedCriteria NumerouspapershaveproposedBayesiancriteriabasedonth eposterior predictivedistribution( GeisserandEddy 1979 ; IbrahimandLaud 1994 ; Laudand 21

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Ibrahim 1995 ; Ibrahimetal. 2001 ; Gelmanetal. 1996 ; GelfandandGhosh 1998 ; Chenetal. 2004 ).Theposteriorpredictivedistributionforthereplicate ddata y rep under model m isgivenby p ( y rep j y m )= Z p ( y rep j ( m ) m ) ( ( m ) j y m ) d ( m ) (2–3) Inwhatfollows,forclaritywedropdependenceonthemodel m .Ibrahimandcolleagues haveproposedgeneralBayesiancriteriafromtheposterior predictivedistributionof thedata.Ingeneral,goodmodelsshouldmakepredictions, y rep closetowhatwas observed, y IbrahimandLaud ( 1994 )denedtheircriterionastheexpectedsquared Euclideandistancebetween y and y rep L = E [( y rep y ) 0 ( y rep y )], (2–4) wheretheexpectationwastakenwithrespecttotheposterio rpredictivedistribution, p ( y rep j y ) .Lcanbere-expressedas L = n X i =1 Var ( y rep j y )+( E [ y rep j y ] y i ) 2 (2–5) TheycalltheproposedpredictivecriteriontheL-measure. Theyexaminedthe L-measureindetailforavarietyofmodels.Theyalsosugges tapproachesforcalibration ofthecriterionandexploreavarietyofweightingstrategi es. GelfandandGhosh ( 1998 ) proposedamoregenerallossfunction L ( y rep a ; y )= L ( y rep ; a )+ kL ( y a ), k > 0. (2–6) Foramodel m theyminimizedE [ L ( y rep a ; y ) j y ] ,theposteriorpredictiveexpectationof thelosswithrespecttoanaction, a .Weprovidesomemoredetailsonthisapproachin Section 2.2 andusethisasthestartingpointforourproposal. Chenetal. ( 2004 )later usedthislossfunctioninthecontextofcategoricalregres sionmodels. 22

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Modelcomparisonisanimportantpartofinferentialstatis tics.Wehavebriey reviewedthemostrelevantliteratureonBayesianmethodsf ormodelcomparison.We nowdiscussissuesspecictoincompletedata.2.1.4IssueswithBayesianModelSelectionwithIncomplete Data ForBayesianinferencewithincompletedata,weoftenwantt ocomparethetof selectionmodels( Heckman 1976 ; DiggleandKenward 1994b ; Fitzmauriceetal. 1995 )sharedparametermodels( WuandCarroll 1988 ; Rizopoulosetal. 2008 );and mixturemodels( Little 1994 ; DanielsandHogan 2000 ; Kenwardetal. 2003 ).Fora goodreviewofmodels,seetextsby MolenberghsandKenward ( 2007 )and DanielsandHogan ( 2008 ).Herewewillfocusonincompletelongitudinaldata. Modelselectioncriteriaforincompletedatashouldhaveac ertainproperty. Beforeweintroduceit,werstintroducesomenotationandr eviewtheextrapolation factorization(DanielsandHogan,2008).Let R bethevectorofobserveddata indicators;i.e., R ij = I f Y ij isobserved g and Y obs as f Y ij : r ij =1 g .Thefulldatais givenas ( y r ) ;theobserveddataas ( y obs r ) .Theextrapolationfactorizationis p ( y r ; )= p ( y mis j y obs r ; E ) p ( y obs r ; O ), (2–7) where p ( y obs r ; O ) istheobserveddatamodeland p ( y mis j y obs r ; E ) isthe (extrapolation)distributionofthemissingdatagiventhe observeddata.Thereisno informationintheobserveddataabouttheextrapolationdi stribution. PropertyI(invariancetoextrapolationdistribution):Tw omodelsforthefulldatawith thesamemodelspecicationfortheobserveddata, p ( y obs r ; O ) andsamepriorfor p ( O ) shouldgivethesamevalueoftheBayesianmodelselectioncr iterion. Thedevianceinformationcriterionbasedontheobservedda talikelihoodhasthis property( DanielsandHogan 2008 ; WangandDaniels 2011 ). Amaincomplicationwithcriteriaforincompletedataiscom putational.Forexample, boththeDICandBayesFactorsrequirecomputationofobserv eddatalikelihoodwhich 23

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isverydifcultformostselectionmodelsandsharedparame termodels.Approaches basedontheposteriorpredictivedistributionbasedcrite riaingeneraldonotneedto useaclosedformfortheobserveddatalikelihood.Ourpropo salwillbesimpleand computationallyattractiveandwillsatisfyPropertyI.Ou rultimateobjectivewillbeto choosethemodelunderconsiderationthatprovidesthebest t,andthentoproceed withasensitivityanalysis(DanielsandHogan,2008). InSection 2.2 ,wereviewthePosteriorPredictiveLoss(PPL)modelselect ion criterionproposedbyGelfandandGhosh(andIbrahimandLau dandcolleagues)and proposeasimplemodicationforcompletelongitudinaldat a.InSection 2.3 ,wepropose extensionsforincompletelongitudinaldatapointingoutp roblemsusingthecriterion basedonastraightforwardgeneralization.InSection 2.4 ,weapplyourcriterionto incompletelongitudinaldatafromarecentclinicaltrial. FinallyinSection 2.5 weconduct somesimulationstoexaminetheoperatingcharacteristics ofthiscriterionandcompare itsperformancetotheDIC.Weofferconclusionsandextensi onsinSection 2.6 2.2PosteriorPredictiveLoss:AQuickReview PosteriorPredictiveLoss(PPL),isthemodelselectioncri terionproposedby GelfandandGhosh ( 1998 )PPLquantiesthetofthemodelbycomparingfeatures oftheposteriorpredictivedistribution, p ( y rep j y ) toequivalentfeaturesoftheobserved data.Thecomparisonisbasedonalossfunction L ( y rep a ; y j y ) ,whereaischosento minimizetheexpectationofthelosswithrespecttothepost eriorpredictivedistribution E [ L ( y rep a ; y j y )] .GelfandandGhosh[GG](amongothers)proposedthefollowi ngloss function L ( y rep a ; y )= L ( y rep ; a )+ kL ( y a ), k > 0. (2–8) When L ( ) ischosenassquarederrorlosstheyshowedthat, min[ E ( L ( y rep a ; y ) j y ))]= n X i =1 Var [ Y rep i j y ]+ k k +1 n X i =1 ( E [ Y rep i j y ] y i ) 2 = PenaltyTerm + GoodnessOfFitTerm (2–9) 24

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Theexpectationiswithrespecttotheposteriorpredictive distributionassociated with y rep .AsthemodelsbecomeincreasinglycomplextheGoodnessofF ittermwill decreasebutthepenaltytermwillbegintoincrease.Overt tingofmodelresultsin largepredictivevariances 2 i andlargevaluesofthepenaltyfunction.Thechoiceofk determineshowmuchweightisplacedonthegoodnessoftter mrelativetothepenalty term.Askgoestoinnity,equalweightisplacedonthesetwo terms;andcorresponds totheoriginalLcriterionin IbrahimandLaud ( 1994 ).Thecriterioniseasytocalculate usingsamplesfromtheposteriorpredictivedistribution. Asimplemodicationfor(complete)longitudinaldata:Now let y i bea T 1 vector oflongitudinalresponsesobservedattimes t 1 ,..., t T .OneissueinapplyingaPPL criteriontomultivariateobservationsisthelackofindep endenceofcomponentsof y i Weightingeachofthecomponentsofthe y i vectorequallymaynotbeagoodchoice. Toaddressthis,optionsincludeamultivariatelossfuncti on(e.g.,deviancebased lossormultivariateweightedsquarederrorloss)orusinga univariatesummary.The multivariatelossalternativehascomplicationsincludin gtheintractabilityoftheobserved datalikelihoodandweightedmultivariatenormallosstype measuressuchas ( IbrahimandLaud 1994 ; Chenetal. 2004 )requireknowingtheweightmatrix(i.e., theinverseofthecovariancematrix).Hereweproposerepla cing y inthecriterionbya univariatesummaryof y h ( y ) ,possiblyof(inferential)interest.Theresultingcriter ion canbeshowntobe, C k ( h )= X i Var ( h ( y i rep ) j y )+ k 1+ k X i ( h ( y i ) E ( h ( y i rep ) j y )) 2 (2–10) AderivationcanbefoundinAppendixA. Choosingasummarymeasureaswedoabove,issimilar,tosome extenttothe approachofJohnsonwhocomputesBayesFactorsbasedonates tstatistic( Johnson 2005 ; JohnsonandHu 2009 ).However,usingthestatisticashedoescreatesseveral complicationsinoursetting.First,wewilltypicallynotb eabletoobtainclosedforms 25

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fortheBayesfactorsbasedontheteststatisticsinthesett ingofmodelsforincomplete dataandthedistributionsoftheteststatisticswilllikel ybecomplex.Second,mostofthe modelswecomparearenotnestedmodelsandthelikelihoodis notavailableinclosed formsotheapproachtomodelselectionin JohnsonandHu ( 2009 )cannotbereadily adaptedtooursetting. 2.3PPLforIncompleteLongitudinalData Theobviousextensionfromthecompletelongitudinaldatac aseistojusttake expectationswithrespectto p ( y rep j y obs r ) (insteadof p ( y rep j y ) ).Thecriterioncan thenbeshowntohavethefollowingform(seeWebAppendixAfo rthederivation), C k ( h )= X i Var ( h ( y i rep ) j y obs r )+ k X i Var ( h ( y i ) j y obs r ) + k 1+ k X i [ E [ h ( y i ) j y obs r ] E ( h ( y i rep ) j y obs r )] 2 (2–11) Theresultingcriterionhasanextraterm, k Var ( h ( y ) j y obs r ) .Thisistheconditional varianceof h ( y ) withrespectto p ( y j y obs r ) ;notethatVar ( y j y obs r ) Var ( y mis j y obs r ) Thistermisproblematicformodelselectioncriteriawhich weshowinthefollowing theorem.However,notethatwhenthereisnomissingness,th istermiszeroand( 2–11 ) simpliesto( 2–10 ). TheoremI:Fortwomodelswith 1. thesameobserveddatamodel, p ( y obs r ; O ) 2. thesameprior, p ( ) ,and 3. thesameconditionalexpectation,E [ y mis j y obs r ; E ] fortheextrapolation distribution, thecriterionin( 2–11 )(for k > 0 )isminimizedwhentheextrapolationdistribution, p ( y mis j y obs r ; E ) isdegenerate. SeeAppendixAforaproof. Thetheoremimpliesthatthiscriterionwillalwayspicka's ingleimputationtype' procedurethatgivesthesamevaluesforE ( h ( y rep ) j y obs r ) asacorrespondingmultiple 26

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imputationtypeprocedure.Obviouslythisisbadpracticea ndthecriterionisawedas itfavors not allowinguncertaintyaboutthe'lled-in'missingdata(an d penalizes extra uncertaintyaboutit).Inaddition,thecriteriondoesnots atisfyPropertyI.Sotheformof theextrapolationdistributionimpactsthemodelselectio ncriterioneventhoughthedata providenoinformationaboutit. Awaytoavoidthisproblemwouldbetoallow k tobeunit-specic,i.e., k i andset k =0 if h ( y i ) is not observed;GGsuggestthisasanoption(topofp.4).However, thisalternativedoesnotuseallthedataaspartof y i willbeobservedandthisoption 'throws'awaytheentirevector y i ifitisincomplete;inaddition,itwilllikelyintroducebi as inmodelselectionasitwouldbedoneon'completersonly'. Inthenextsection,weprovideanalternativeformulationt hatavoidstheproblemsof ( 2–11 ). 2.3.1ARe-formulation ThecomplicationwithadirectextensionofthePPLtoincomp letelongitudinaldata arisingfromthefactthat h ( y ) isnotalwaysobservedandthisresultsinanextratermin thecriterion.Astraightforwardandnaturalwaytoovercom ethiscomplicationistouse anewunivariatefunctionofthedatathatisonlyafunctiono f observables ,i.e., ( r r y ) where ( r y )=( r 1 y 1 r 2 y 2 ,..., r T y T ) .Toderivethecriterionhere,wejustreplace h (.) by T ( r r y ) fromthepreviousderivationandobtain C k ( T )= X i Var [ T ( r i rep r i rep y i rep ) j y obs r ] + k 1+ k X i f T ( r i r i y i ) E [ T ( r i rep r i rep y i rep ) j y obs r ] g 2 (2–12) Thisnolongerhastheproblematicextraterm.Wediscussthe choiceof T ( ) and somecomputationalissuesinthenexttwosectionsandthene valuatethecriterionvia simulations.Notethatthecriterionassesses replicatedobserved datahere(asopposed toreplicatedfull(orcomplete)data).Thisversionofthec riterionsatisesPropertyI,i.e., 27

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itisinvarianttotheextrapolationdistributionandwillo nlygiveinformationaboutthetof p ( y obs r ) 2.3.2Choicesfor T ( r r y ) Wediscusssomechoicesofthesummaryfunction T ( ) inthefollowing.Functions of r relatetohowwellwemodelthemissingness.Functionsof r y relatetohowwellwe modeltheobservedy'sincludinghowlikelythat y wasobservedunderthemodel.Some possiblechoicesfor T ( r r y ) follow. T 1 ( r r y )= r T y T r 1 y 1 ;differenceinmeanofobservedatendofstudyand observedatbeginningofstudy T 2 ( r r y )= r T ( r T y T r 1 y 1 ) ;observedchangefrombaseline T 3 ( r r y )= P Tj =1 r j ;numberofobservedcomponentsof y T 4 ( r r y )= P Tj =1 r j y j P Tj =1 r j ;themeanoftheobservedresponses. T 5 ( r r y )= P Tj =1 t j r j y j P j r j t j ;theobservedleastsquareslopes T 6 ( r r y )= P Tj =1 I f r j =1, r j +1 =0 g r j y j ) I f r 2 =1 g r 1 y 1 ;changefrombaselineto lastobservedresponseundermonotonemissingness. T 7 ( r r y )=[ r T ( r T y T r 1 y 1 )] 2 ;secondmomentofdifferenceinmeanofobserved atendofstudyandobservedatbeginningofstudy. T 8 ( r r y )= h P Tj =1 I f r j =1, r j +1 =0 g r j y j ) I f r 2 =1 g r 1 y 1 i 2 ;secondmomentof changefrombaselinetolastobservedresponseundermonoto nemissingness. Inthedataanalysisandsimulations,wefocuson T 1 ( ) T 2 ( ) T 6 ( ) and T 8 ( ) 2.3.3Computations Assumethemodelisparameterizedviaavectorofparameters .Computation ofthePPLcriterionherecanbedonemoreefcientlyusingou tputfromanMCMC algorithmwhenthefollowingexpectationscanbeexpressed inclosedform,E [ T p ( r rep r rep y rep ) j ]: p =1,2 .Thisexpectationcorrespondstothefollowingintegral, ZZ T p ( r rep r rep y rep ) p ( r rep y rep j ) d r rep d y rep (2–13) 28

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Theavailabilityoftheexpectationinclosedformdependso nboththemodelandthe choiceof T ( ) 2.4DataExample WeusethePPLcriterioninSection 2.3.1 toselectamongmodelsfordatafroma randomizedclinicaltrialconductedtoexaminetheeffects ofrecombinanthumangrowth hormonetherapyforbuildingandmaintainingmusclestreng thintheelderly.Thestudy, whichwewillrefertoasGH,enrolled161participantsandra ndomizedthemtooneof fourtreatmentsarms.Theresponseofinterestherewasmean quadricepsstrength, measuredasthemaximumfoot-poundsoftorquethatcanbeexe rtedagainstresistance providedbyamechanicaldevice,whichwasrecordedatbasel ine,6months,and12 months.Werestrictouranalysestoonlytwoofthetreatment groups,Exercise+Growth Hormone(EG)andExercise+Placebo(EP).Ofthe78randomize dtothesetwoarms, only53hadcompletefollow-up(andthemissingnesswasmono tone);seeTable B-1 in AppendixB. Dene Y =( Y 1 Y 2 Y 3 ) T tobequadstrengthmeasuredatmonths0,6,and 12withcorrespondingobserveddataindicators, R =( R 1 R 2 R 3 ) T .Inthisdata,the baselinequadstrengthisalwaysobserved,so P ( R 1 =1)=1 .Giventhatthedropout ismonotone,withoutanylossofinformation,inspecifying ourmodelswereplace R with S = P 3j =1 R j (thenumberofquadstrengthmeasuresobserved). 2.4.1ModelsConsidered Weconsideredbothpatternmixturemodelsandselectionmod elstojointlymodel thedistributionofthefulldata, ( y r ) .Themixturemodelweconsiderforeachtreatment isspeciedas Y 1 j S = k N ( ( k ) 1 ( k ) 1 ): k =1,2,3 Y 2 j Y 1 S = k N ( 2 + 21 Y 1 2 ): k =1,2,3 Y 3 j Y 1 Y 2 S = k N ( 3 + 31 Y 1 + 32 Y 2 3 ): k =1,,2,3 29

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S Mult ( ). (2–14) Themultinomialparameteris =( 1 2 3 ) ,where s =P(S=s)and P s s =1 RecallthatthePPLisinvarianttotheextrapolationdistri bution,i.e.,thedistributions p ( y 2 j y 1 S =1) and p ( y 3 j y 1 y 2 S =1) and p ( y 3 j y 1 y 2 S =2) .Intheabove,withoutloss ofgenerality,wehavesettheparametersoftheextrapolati ondistributiontotheirvalues underMAR. Wealsoconsideramoreparsimoniousversionsofthemixture model,MM2which allowssomeequalityofparametersbetweentreatments.MM2 assumestheconditional distributions [ Y 3 j Y 1 Y 2 S = j ] and [ Y 2 j Y 1 S = j ] aresameoverthebothtreatments(i.e., theparameters ( 3 31 32 3 2 21 2 ) ). Fortheselectionmodel,foreachtreatment,thefulldatare sponsemodelis speciedas Y N ( ,) R 2 j y Ber ( 2 ) R 3 j R 2 =1, y Ber ( 3 ), (2–15) wherelogit ( 2 )= 02 + 1 Y 1 + 2 Y 2 andlogit ( 3 )= 03 + 1 Y 2 + 2 Y 3 .Inthemissing datamechanismintheselectionmodelabove,wehaveimplici tlyassumednon-future dependence( Kenwardetal. 2003 )andrstorderMarkovdependence(constant overtime).Theformercorrespondstothemissingnessatmon thjdependingonthe pastandthepotentialresponseatmonthj,butnotresponses aftermonthj.Thelatter correspondstothedependenceonlydependingontheimmedia tepast(theprevious visittime). Forboththemixtureandselectionmodels,weusediffusepri orsformostofthe parameters.Inparticular,forthemean/regressionparame ters( )inthemixture modelsweusenormalpriorswithvariances, (10 6 = 10 4 ) .Forthevariances( ),weuse 30

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uniformpriorswithupperboundof 100 .Fortheselectionmodel,themarginalmean hasanormalpriorwithvariance 10 6 1 hasaWishartprior,andtheparametersinthe logisticmodel( )formissingnesshavediffusenormalpriorsspeciedasthe priorfor exceptfor 2 whichwasgivenanormalpriorwithmean0andvariance 5 (notethat inferenceswerenotsensitivetochoicesofthevariancebet ween1and10).Wechosea somewhatinformativepriorfor 2 forstability. 2.4.2Results WerantheGibbssamplingalgorithminWinBUGSfor100Kitera tions.Trace plotssuggestedgoodmixing(notshown).WecomputedthePPL criterionforthe fourchoicesof T ( ) : T 1 ( r r y )= r 3 y 3 r 1 y 1 T 2 ( r r y )= r T ( r T y T r 1 y 1 ) T 6 ( r r y )= P Tj =1 I f r j =1, r j +1 =0 g r j y j ) I f r 2 =1 g r 1 y 1 ,and T 8 ( r r y )= h P Tj =1 I f r j =1, r j +1 =0 g r j y j ) I f r 2 =1 g r 1 y 1 i 2 .NotethatinWebAppendixC,we deriveexplicitformsfor( 2–13 )forthesomeofthechoicesof T ( ) consideredherein thecontextofthemodelgivenin( 3–5 ).Therearenotclosedformsavailableforthe selectionmodelin( 2–15 ). Table 2-1 givesthePPLcriterionvaluesforthethreemodelsttotheG Hdata foreachofthefourchoicesof T ( ) .Allfavortheselectionmodeloverthetwomixture models.Theselectionmodelalsohadthesmallestcomplexit y(penalty)andasimilart tothemostcomplexmixturemodel(MM1). WealsocomputedDICbasedontheobserveddatalikelihood(s ee( 2–16 )in Section 2.5 )forthethreemodels.TheresultsarepresentedinTable B-2 .DICbasedon theobserveddatalikelihoodalsofavorstheselectionmode l. 2.5Simulations ToassesstheabilityofthePPLtoselectthebestmodel,weco nductedseveral simulations.Wesimulated200datasetsbasedontheparamet ervaluesgivenin Table D-2 inAppendixD(thesevaluesarepartiallybasedontheGHdata ).Wet threemodelstodatasimulatedunderthesesamethreemodels withsamplesizes 31

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pertreatmentof50,100,and2000.Thethreetruemodelswere MM1andMM2from Section 2.4 andtheselectionmodelfrom( 2–15 )with 2 =0 .Wedenotethisnal modelasSM0.TocomparethemodelsweusedtheproposedPPLcr iteriawiththefour differentchoicesfor T ( r r y ) consideredinSection 2.3.2 WealsocomputedtheDICbasedontheobserveddatalikelihoo d, L( j y obs ,r) to comparetotheproposedcriterion.WeexpecttheDICtobemor epowerfulsinceituses theentirelikelihood,butformanymodels,suchasselectio nmodels,itscomputation isquiteburdensome,whichdiscouragesitsuse.Theobserve ddatalikelihoodDICis denedas DIC O = 4 E j y r ( log L ( j y obs r ) ) +2log L ( E j y r ( ) j y obs r ). (2–16) Weputtherestrictionthat 2 =0 fromtheselectionmodelsothattheDICwouldbe availableinclosedform. ThepercentagesoftimesthePPLandDIC o criterionchoosethetruemodelare presentedinTable 2-3 .TheaveragePPLvaluesofseveralscenariosarepresentedi n Tables 2-4 2-6 .ThedetailedPPLandDIC o resultsarereportedinAppendixD,Tables D-1 D-10 WhenMM1wasthetruemodel,allthechoicesof T ( ) didwellandasthesample sizeincreased,theprobabilityofchoosingthecorrectmod elapproachedone,withthe leastpowerfor T 1 ( ) andmuchhigherpowersfortheotherchoices. WhenMM2wasthetruemodel,itwaschosenwithprobabilityof around 50% for thesmallandmediumsamplesforallchoicesof T ( ) expectfor T 6 ( ) forwhichitwas chosenwithprobabilityaround 60% .Forthelargestsamplesize( n =2000 ),itwas pickedapproximately 50 = 50 withMM1.Forallthesamplesizes,thecriteriongave verysimilarvaluesunderbothmixturemodels(seeTable 2-6 andTables D-8 D-10 inAppendixD).NotethatwhenMM2isthetruemodel,botharec orrectsinceMM2is nestedinMM1.Wediscussthisfurtherinthenextsection. 32

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Whentheselectionmodelwastrue,itwasselectedwithhighp robabilityin non-largesamples( n =50,100 pertreatmentarm),withprobabilities > 80% (Table 2-3 ) forallchoicesof T ( ) T 2 ( ) appearedtobethebestdiscriminatoramongmodelsfor thissetting,pickingSM0withprobability > 80% forallsamplesizes. TheDICbasedontheobserveddatalikelihooddoesverywelli nallsituations thoughforcomparingMM2toMM1undertrueMM2,theprobabili tyofchoosingMM2 doesnotappeartobeapproachingone.Theoverallbehaviori snotsurprisingasituses thedatainthemostefcientwayintermsofcomparingfullpr obabilitymodels.However, asstatedearlier,itisoftenacomputationalburdentoimpl ementitgiventheneedto evaluatetheobserveddatalikelihood. Simulationconclusions:Innon-largesamples( n =50,100 ),thecriteriondoesa verygoodjobselectingthebestmodelwiththespecicperfo rmancedependingonthe choiceof T ( ) (Table 2-3 ). Forlargersamplesizes( n =2000 ),inmostcases,theprobabilityofselectingthe correctmodelapproachesonewithanappropriatelychosen T ( ) .However,fornested models,thecriteriontakesthesamevalueforlargersample s.Assuch,inthiscase,one mightchoosethemoreparsimoniousmodelfornalinference .UnderSM0,whenthe wrongmodelwaschosenwithhighprobability,thePPLvalues wereverysimilar(see Table D-4 ). Wealsonotethatcertainchoicesof T ( ) doconsiderablybetterhere,e.g., T 2 ( ) for trueSM0ortrueMM1.Ingeneral,werecommendsimilarchoice sforcomparingSM's andMM's. Wealsopointoutthatforcertainchoicesof T ( ) ,thewrongmodelisselectedin thelargersamplesizes.However,thisisarguablyoflessim portanceif T ( ) ischosen asafunctionofinterestandthe'wrong'modelprovidesabet ter(orequivalent)'t'to thisfunction,whichisthecasewhenthishappens.Insuchca sesinthesimulations,the actualPPLvalueswere(essentially)thesame. 33

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Forsmalltomediumsizesamples,thePPLdoesagoodjobincho osingthecorrect model.Inlargersamplesizes(e.g.,n=2000pertreatmentar m),thecomputationally intensiveDICmightsometimesbeabetterchoice.Inallthes imulations,asthesample sizeincreased,theprobabilityoftheDICchoosingthecorr ectmodelwasapproaching one(notingthatwhenMM2isthetruemodel,bothMM1andMM2ar ethecorrect model).Oncethe'best'modelischosen,theuserwouldthenc onductasensitivity analysis(DanielsandHogan,2008)usingthechosenmodel. 2.6Discussion Wehaveproposedacomputationallyconvenientwaytocompar emodelsfor incompletelongitudinaldatathatsatisesthepropertyof beinginvarianttothe specicationoftheextrapolationdistribution(Property I).Viasimulations,theproposed criterionappearstoworkwell,especiallyfortypicalsamp lesizesof50to100subjects pertreatmentarm.Clearly,thechoiceofthesummary T ( ) affectsthepowerand discriminativeabilityofthecriterion.Careshouldbetak eninchoosinganappropriate summary T ( ) (ideallybasedonafeatureofthedataofinterest);however ,theabilityto chooseafeatureofinterestallowsmorefocusedandtargete dmodelselectionbasedon aspecicquantityofinterestforinference.Infuturework ,wewillbeexploringinmore detailthebestchoicesfor T ( ) forcomparingdifferenttypesofmodelforincomplete data. ItisalsopossibletouseaDeviancebasedloss(Chenetal.,2 004);however,the probleminourcaseistheintractabilityoftheobserveddat alikelihoodformanymodels forincompletedataandthesamecomputationalproblemswou ldariseaswithDIC.The criteriaproposedhereisinthespiritofIbrahimandLaudin thatitmeasuresdiscrepacy fromtheobserveddata(whichhereis ( r r y ) ). Oneissuewithourapproachisaliasing,i.e.,smallvalueso f y beingsimilarto ry when r =0 .However,wetypicallydonotexpectthistobeamajorissue, especiallyfor continuousresponses.Forbinaryresponses,codingtheres ponseas 1 and 1 (and 34

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similarlyforcategoricaldataingeneral)willalleviatep roblems;inaddition,weighted versionsofthesecriteriacouldalsohelp( Chenetal. 2004 ).Moreover,itwouldbeof interesttoexploresummarystatisticssuch T ( r r y )= a 1 t 1 ( r )+ a 2 t 2 ( r r y ) .However, thesewouldneedtobeappropriatelycalibratedtoensureon eofthetwotermsdoesnot inadvertentlydominatethecriterion. Ageneralissueswithposteriorpredictivebasedcriteriai scalibration.Calibration requiresadditionalstraightforwardcomputations(see,e .g., Chenetal. 2004 )and requiresproper(informative)priors.However,thestrate gyfromChenetal.couldbe implementedinoursettingwithanappropriatechoiceofpri ors.Forthesimulation scenarioofcomparingthetwomixturemodelswherethemorep arsimoniousmodel istrue,calibrationcouldbeusedtochoosethesimplerone. However,aspointedout earlier,inthissetting,forlargersamplesizes,weobtain (essentially)thesamevalueof thecriterion.Andwealsorecallthatintheseincompletese ttings,theultimategoalis tochooseamodelandthendosensitivityanalysisonthismod el.Sotosomeextent, pickingagoodmodel(intermsofprovidingagood't'totheq uantityofinterest, T ( ) ), butnotnecessarilythecorrectmodel,canbesufcient. Provingconsistencyofposteriorpredictivebasedcriteri aisdifcultandspecic tothemodelsetting;Ibrahimetal.(2001)provesomeresult sforlinearmodels.For anappropriatechoiceof T ( ) thePPLcriterionappearstopickthecorrectmodelwith probabilitygoingtooneincertaincases.Wearecurrentlyw orkingonanalyticalresults toverifyandbetterunderstandthebehaviorseenhere;howe ver,suchderivationsare verycomplexexceptforthesimplestmodelsettings.Inpart icular,exploringthelarge samplebehaviorofthepenaltyterminthesesituationswoul dbeofmajorinterest.It wouldalsobeofinteresttoexaminemoreformallythelarges amplebehavioroftheDIC, inparticularfornestedmodelsettings. Ageneralissueinmodelselectionforincompletelongitudi naldataiscomparing ignorableandnon-ignorablemodels;fortheformer p ( r j y )= p ( r j y obs ) isnotexplicitly 35

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modeled.Itisnotclearthatsuchmodelcomparisonscanbema debasedonacriterion thatsatisesPropertyI.Thisisalsorelatedtoposteriorp redictivechecksbasedon replicatedobserveddataversusreplicatedcompletedatat helatterwhichwasexplored in Gelmanetal. ( 2005 ). DobsonandHenderson ( 2003 )proposedexploratoryresiduals fortheresponseconditionalonnotdroppingout.However,b othoftheseapproaches focusongraphicalandexploratorymodelchecking,notform almodelcomparison. InSection 2.1 ,wedescribehowmodelselectioncriterionforincompleted atashould satisfyPropertyI.However,theremaybesituationsthatex ternalinformationisavailable aboutthedistributionofthefulldataresponsesuchthatth ispropertymightbecome lessimportant. Ibrahimetal. ( 2008 )recentlyconsideredfrequentistmethodsforthe computationofmodelselectioncriteriainmissing-datapr oblemsbasedonoutputof theEMalgorithminafrequentistsetting.Theydevelopedac lassofinformationcriteria formissing-dataproblems.Thegeneralformsatisesthepr opertyofbeinginvariant tothedistributionofthemissingdataconditionalontheob serveddata(moredetailin Section 2.3 ).However,theyneedananalyticapproximationtocomputet his(similar problemtonothavingclosedformfortheobserveddatalikel ihood).Thesimplerform theyproposethatdoesnotrequiretheapproximationdoesno tsatisfythefrequentist version(nopriors)ofPropertyI.Wediscussthisissuesabo utmodelcheckandanalytic approximationinChapter3. 36

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Table2-1.PPLforGHData ModelGOFComplexity C 1 T ( r r y )= r T y T r 1 y 1 SelectionModel2960.22907.65867.8MixtureMM12961.73958.66920.3MixtureMM23058.33498.56556.8T ( r r y )= r T ( r T y T r 1 y 1 ) SelectionModel390.7425.2815.9MixtureMM1390.2517.8907.9MixtureMM2484.7605.71090.3T ( r r y )= P Tj =1 I f r j =1, r j +1 =0 g r j y j ) I f r 2 =1 g r 1 y 1 SelectionModel1670.51759.73430.2MixtureMM11670.02211.43881.4MixtureMM21768.12606.34374.4T ( r r y )= h P Tj =1 I f r j =1, r j +1 =0 g r j y j ) I f r 2 =1 g r 1 y 1 i 2 SelectionModel155630391165506427218103MixtureMM1157122942347246739184760MixtureMM2157604692204355537804025 Table2-2.DIC o forGHData Model p D DevianceDIC o SelectionModel23.45.629.1MixtureMM130.710.641.3MixtureMM225.426.451.8 37

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Table2-3.NumberofTimesthePPLandDIC o CriterionChoosetheTrueModel TrueModelSizeModel T 1 T 2 T 6 T 8 DIC o SM050SM0193198192194200 MM170230MM202630 SM0100SM0161197158168199 MM13932201MM20040120 SM02000SM01616547113200 MM1184351740MM200152130 MM150SM0117421210 MM183196179179200MM200000 MM1100SM01110200 MM189200198200200MM200000 MM12000SM0790000 MM1121200200200200MM200000 MM250SM0290570 MM19198987840MM28010297115160 MM2100SM050000 MM187901007246MM2108110100128154 MM22000SM000000 MM110111010610357MM299909497143 38

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Table2-4.AveragePPLCriteriaforDataSimulatedunderMM1 ,SampleSize2000 ModelGOFComplexity C 1 T ( r r y )= r T y T r 1 y 1 SM01114.31114.12228.4MixtureMM11113.21113.72226.8MixtureMM21601.51644.63246.1T ( r r y )= r T ( r T y T r 1 y 1 ) SM0270.3291.6561.9MixtureMM1270.0270.2540.2MixtureMM2758.6873.71632.3T ( r r y )= P Tj =1 I f r j =1, r j +1 =0 g r j y j ) I f r 2 =1 g r 1 y 1 SM0418.9445.6864.5MixtureMM1417.5417.7835.2MixtureMM21074.91354.42429.3T ( r r y )= h P Tj =1 I f r j =1, r j +1 =0 g r j y j ) I f r 2 =1 g r 1 y 1 i 2 SM0299940350146650086MixtureMM1298273298677596950MixtureMM2101900229086653927667 Table2-5.SimulationforAverageofPPLCriteriaunderSM0, SampleSize100 ModelGOFComplexity C 1 T ( r r y )= r T y T r 1 y 1 SM041.441.783.2MixtureMM141.543.284.7MixtureMM241.947.689.5T ( r r y )= r T ( r T y T r 1 y 1 ) SM08.78.917.6MixtureMM18.79.418.1MixtureMM29.210.019.2T ( r r y )= P Tj =1 I f r j =1, r j +1 =0 g r j y j ) I f r 2 =1 g r 1 y 1 SM030.530.861.4MixtureMM130.533.163.6MixtureMM231.131.762.8T ( r r y )= h P Tj =1 I f r j =1, r j +1 =0 g r j y j ) I f r 2 =1 g r 1 y 1 i 2 SM0212221404263MixtureMM1212425664690MixtureMM2212726114739 39

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Table2-6.DataSimulatedunderMM2,SampleSize2000withAv erageofPPLCriteria ModelGOFComplexity C 1 T ( r r y )= r T y T r 1 y 1 SM01669.61699.43369.0MixtureMM11668.41668.33336.7MixtureMM21668.41668.53337.0T ( r r y )= r T ( r T y T r 1 y 1 ) SM0511.4552.01063.3MixtureMM1511.0511.21022.3MixtureMM2511.1511.21022.3T ( r r y )= P Tj =1 I f r j =1, r j +1 =0 g r j y j ) I f r 2 =1 g r 1 y 1 SM0409.3462.0871.3MixtureMM1409.1409.1818.3MixtureMM2409.1409.3818.4T ( r r y )= h P Tj =1 I f r j =1, r j +1 =0 g r j y j ) I f r 2 =1 g r 1 y 1 i 2 SM04973195809961078315MixtureMM1494065494774988839MixtureMM2494143494568988711 40

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CHAPTER3 POSTERIORPREDICTIVEASSESSMENTFORLONGITUDINALDATA 3.1OverviewofPosteriorPredictiveAssessment Bayesianinferenceischaracterizedbyspecifyingamodela ndpriordistributionfor theparametersofthemodel.Theposteriordistributionoft heparametersisobtained byupdatingthepriorinformationontheparametersusingth emodelandthedata.The likelihoodsummarizestheinformationthedatahasaboutth eparameters.Foramodel m andparameter ,thelikelihood L ( ; y m ) isafunctionproportionalin tothejoint densityevaluatedattheobservedsample.Givenpriordistr ibutionfortheparameter as ( m ) ,andalikelihood L ( ; y m ) theposteriordistributionof isproportionalto L ( ; y m ) ( m ) Theposteriorpredictivedistributionforthereplicatedd ata y rep undermodelmis givenby p ( y rep j y m )= Z p ( y rep j ( m ) m ) ( ( m ) j y m ) d ( m ) Samplesfromtheposteriorpredictivedistributionarerep licatesofthedatagenerated bythemodel.InthispaperwewilldiscussapproachesforBay esianmodelcheckingfor modelsforincompletedatabasedontheposteriorpredictiv edistribution.Wereviewthe relevantliteratureonposteriorpredictivechecksnext.3.1.1CompleteDataModelFit Parameterdependentstatistics(ordiscrepancystatistic s)wereintroducedin aBayesiansettingby Zellner ( 1975 ). Rubin ( 1984 )usedtheposteriorpredictive distributionofastatistictocalculatethetail-areaprob abilitycorrespondingtothe observedvalueofthestatistic. Meng ( 1994 )calledthisprobabilityaposteriorpredictive p-value.Infollowing,wewillrefertoitasposteriorpredi ctiveprobabilityduetoits problematicinterpretationasap-value( Robinsetal. 2000 ).Thisprobabilityisa 41

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measureofdiscrepancybetweentheobserveddataandthepos itedassumptions asmeasuredbythesummaryquantity T ( ) .Theposteriorpredictivedistributionof T ( y rep ) shouldidentifythedeparturewhenthewrongmodelisttedo nthedataand comparedwithdistributionof T ( y ) .FortheassumedBayesianmodel,theposterior predictiveapproachprovidesareferencedistribution.Co nditionedontheparameter thedata y andthereplicateddata y rep areindependent.Thetofthemodeltothedata isdeterminedbycomparingtheposteriorpredictivedistri butionof T ( y rep ) with T ( y ) Meng ( 1994 )formallydenedthisprobabilityas, P ( T ( y rep ) > T ( y ) j m y )= ZZ I [ T ( y rep ) > T ( y )] p ( y rep j ) p ( j y ) d y rep d (3–1) IntheBayesianformulationthisapproachallowstheuseofa parameterdependenttest statistic,whichiscalledadiscrepancystatisticsForadi screpancy, D ( y ; ) ,thereference distributioncanbederivedfromthejointdistributionof ( y rep ) P ( y rep j m y )= P ( y rep j m ) P ( j m y ). Howeverlocatingtherealizedvalueof D ( y ; ) withinthereferencedistributionisnot feasiblesince D ( y ; ) dependsontheunknown .Thiscomplicationhasledauthorsto usethetailareaprobabilityof D underitsposteriorreferencedistribution.Aprobability wasconstructedby Gelmanetal. ( 1996 )asin( 3–1 )byeliminatingthedependenceon unknown .Theyconsideredminimumdiscrepancy( D min ( y ) )oraveragediscrepancy ( D avg ( y ) )statistic, T ,overtheparameter .Forthelatter,thetailareaprobabilityofthe discrepancystatisticisgivenby: P ( D ( y rep ; ) D ( y ; ) j m y ). 42

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Thisisanalogoustotheposteriorpredictiveprobabilityi n( 3–1 ).Fortheformer,the correspondingposteriorpredictiveprobabilityisdened as, P ( D min ( y rep ) D min ( y ) j m y )= Z P ( D min ( y rep ) D min ( y ) j m ) P ( j m y ) d Thechoiceofdiscrepancyisclearlyveryimportantandofte nreectstheinferential interests.Ingeneral,suchchecksarecalledposteriorpre dictivechecks. 3.1.2IncompleteDataModelFit Tointroduceposteriorpredictivechecksforincompletelo ngitudinaldata,weneed torstintroducesomenotation.Let R bethevectorofobserveddataindicators;i.e., R ij = I f Y ij isobserved g andlet Y obs be f Y ij : r ij =1 g .Thefulldataisgivenas ( y r ) ;the observeddataas ( y obs r ) .Theextrapolationfactorizationofthefulldatamodelis, p ( y mis y obs r j )= p ( y mis j y obs r E ) p ( y obs r j O ), (3–2) where E indexestheconditionaldistributionofmissingresponses givenobserveddata (theextrapolationdistribution)and O indexesthedistributionoftheobserveddata. Inferenceaboutthefulldatadistributionrequiresunveri ableassumptionsaboutthe extrapolationdistribution p ( y mis j y obs r E ) forwhichtheobserveddataprovidesno information.Sensitivityparametersarefunctionof E ( DanielsandHogan 2008 )and areusedtointroduceexternalinformationaboutthemissin gdatamechanism. Anextensionwasproposedby Gelmanetal. ( 2005 )oftheposteriorpredictive approachtothesettingofmissingdata(andlatentdata).Si milartothecompletedata settingsthecomplicationforthesesituationsisconstruc tingthereferencedistributions. Toassessthetofthemodeltheydeneateststatistic T ( ) ,whichisafunction ofthecompletedata.The'missing'dataeithermissingorla tentislledinateach iterationusingdataaugmentation.Theteststatisticisco mparedtotheteststatistic computedbasedonreplicatedcompletedata.Atiteration l thecompletedatais, ( y obs y ( l ) mis ) ,wheresample y ( l ) mis issampledfrom p(y ( l ) mis j y obs ( l ) ) T ( y repcom ) are 43

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sampledfromfulldataresponsemodel, p(y repcom j ( l ) ) ,conditionalonparametervalues atiteration l .Graphicalapproacheswereimplementedformodelchecking alongwith thecalculationofposteriorpredictiveprobabilities.In oursettingofmissingdatain particular,non-ignorable,suchchecksareproblematicwh ichwediscussinSection 3.2 InSection 3.2 ,wereviewtheposteriorpredictivechecksforincompletel ongitudinal data.InSection 3.3 ,weprovideanalyticalresultsforcalculatingtheposteri orpredictive probabilitiesforcompleteandobservedreplicationswith respecttodenedtest statistics.InSection 3.4 weapplyouranalyticalresultstoincompletelongitudinal datafromarecentclinicaltrial.Finally,inSection 3.5 weconductsomesimulationsto examinethechecksforreplicatedcompleteandobserveddat a.Weprovideconclusions andextensionsinSection 3.6 3.2PosteriorPredictiveChecksforIncompleteData Inthissectionwefurtherexploreposteriorpredictiveche cksforincomplete longitudinaldata.Toimplementthesechecks,wewillsampl efromtheposterior predictivedistribution, p ( y rep r rep j y obs r ) .Whenwemodelthemissingdata mechanism,asinnonignorablemissingnesswecancomputeda tasummariesbased onreplicatesofobserveddataandreplicatesofcompleteda ta.Gelmanet.al.(2005) proposeddoingchecksusingcompletedata.HoweverGelmanc onsideredmoregeneral settingsthatincludelatentvariables(asmissingdata),i gnorablemissingnessand nonignorablemissingness.Wefocusonthenonignorablecla ssforwhichweargue completedatacheckarenotoptimal.Moredetailsareinthef ollowingsection. 3.2.1ObservedandCompleteReplication Forcompletereplicationsatiteration k ,thecompletedatawouldbe, y com = ( y obs y ( k ) mis ) ,where y ( k ) mis issampledfrom p ( y ( k ) mis j y obs ( k ) ) withinthedataaugmented Gibbssampler.Thethereplicateddataaresampledfromthef ulldataresponsemodel, 44

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p ( y rep ( k ) com j ( k ) ) andforagiventeststatistics, T ( ) ,itsexpectationis E ( T ( Y repcom ) j Y )= ZZ T ( y repcom ) p ( y repcom j ) ( j y obs r ) d y repcom d (3–3) Implicitly,inanonignorablemodel,wehaveaverageoverth ereplicatedmissing dataindicators, r rep .Wenowintroducedreplicatedobserveddata.Wedenethese replicatesas y rep obs = f y rep j : r rep j =1 g i.e.,thecomponentsofthe y rep j ofthereplicatedcompletedatasetsforwhichthe correspondingreplicatedmissingdataindicatoris1.Weco mputecorresponding statisticsinwaythat,ateachiterationwecomputethechec ksusingthecomponentsof y obs and y repobs (forwhich r rep ij =1 ).Foragiventeststatistics, T ( ) ,itsexpectationis E ( T rep Y obs j Y obs R )= ZZZ T f y repobs ( y rep r rep ) g p ( y rep j r rep y obs r ) p ( r rep j y obs r ) ( j y obs r ) d y rep d r rep d (3–4) 3.2.2Issues Posteriorpredictivecheckbasedonreplicationsofcomple tedatahavesome advantages.Forexample,underignorablemissingness,whi chdoesnotrequireexplicit specicationofthemissingdatamechanism,itisnotaprobl emtocreatereplicationsof thecompletedata.Butofcourse,inthatsituation,itisnot possibletoassessthejoint toftheobservedresponses and themissingnessindicators.Obviously,thisapproach losessomepowerversusasettingwithnomissingdatasincet hemissingdataislled in(viadataaugmentation)undertheassumedmodel;thus,sl ightlybiasingthechecks infavorofthemodel.Inaddition,inthesettingofnonignor ablemissingness,thechecks haveafatalawastheyare not invarianttotheextrapolationdistribution( Danielsetal. 2012 )andthuscannotdirectlyassessthe(joint)tof ( y obs r ) andarenotinthespiritof sensitivityanalysis(anessentialpartoftheanalysisofm issingdataasdocumentedina recentNRCreport).Inotherwords,twomodelswiththesame ttotheobserveddata, 45

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butdifferentimplicitextrapolationdistributionscangi vedifferentconclusionsonmodel t. Checksbasedonreplicatedobserveddatasatisfytheproper tyofinvariancetothe extrapoloationdistributionandprovidethesamevaluesas sensitivityparametersare varied(unlikechecksbasedonreplicatedcompletedata),i .e.,differentnonignorable modelswiththesamettotheobserveddata.Inaddition,the ycanassessany featuresofthejointdistributionof ( y obs r ) asdesired.However,onewouldsurmise thatsomepowerislostrelativetocheckswithnomissingdat agivenalackofone toonecorrespondencebetweentheobserveddataresponsesa ndthereplicated observeddataresponses;forexample, Y ij mightbeobservedintheactualdata,butnot necessarily'observed'inthereplicateddata. Wewillexploreboththesechecksfurtherinpracticeinthen extfewsections. 3.3AnalyticResults Inthissectionweexamineanalyticallythebehaviorofthep osteriorpredictive probabilitybasedonobserveddataandcompletedatareplic ationsunderasimple model,amixturemodelforabivariateresponse,givenby Y 1 j S = k N ( 1 + I f k =2 g ( k ) 1 ): k =1,2 Y 2 j Y 1 S =1 N ( 2 ++ 21 Y 1 2 ) Y 2 j Y 1 S =2 N ( 2 + 21 Y 1 2 ) S Mult ( ) (3–5) where S = P 2j =1 R j (numberofobservedresponses),thereisonlymissingnessi n Y 2 and isthesensitivityparameter(measuresdeparturefromMAR) Tosimplifythebelowderivation,weuseadiffusenormalpri orontheregression parameter, 2 andassumetheremainingparametersareknown.Theteststat istics, T ( y ) isdenedasthemeanof Y 2 .Fortheobservedreplicateddata,itis Y 2, obs ,the 46

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marginalmeanofobserveddataattime 2 .Fortheobservedreplicateddataitis Y rep 2, obs themeanofreplicatedobserveddataattime2.Thecorrespon ding(posteriorpredictive) tailareaprobabilitycanbeapproximatedas, P ( T ( y repobs ) T ( y obs ) j y obs r )= P ( y rep 2 obs > y 2 obs j y obs S ) 1 p n y 2 obs ( ZB + 21 ( 1 + )) ( 2 + B + 221 2 2 ) 1 2 (3–6) where Z = 1 2 P ni = n 1 +1 y 2 obs 21 P ni = n 1 +1 y 1 obs n 1 isthenumberofobservationswith R 2 = 0(assumethedataaresortedsothatalltheunitswithmissin g Y 2 areattheend),and 1 B = ( n n 1 ) + 2 2 .Fordetailsonthederivation,seetheappendix. Forcompletereplication,wedenethecorrespondingtests tatisticsare Y 2, com and Y rep 2, com themeanofthecompletedataandthemeanofreplicatedcompl etedataattime 2.Thecorrespondingtailareaprobabilitycanbeapproxima ted,buthasamorecomplex formthattheobservedreplicateddata.Itsformandadetail edderivationcanbefoundin theappendix. Behavioroftheposteriorpredictiveprobabilities:Inthi ssectionweexaminethe behavioroftheposteriorpredictiveprobabilityasafunct ionofthetruepercentageof missingnessandthenonignorabilitysensitivityparamete r, Underthetruemodelasthepercentagemissingincreaseswee xpecttheposterior predictiveprobabilitywillmoveawayfrom 0.50 .Toseetherateatwhichthishappens, wecomputethegradientoftheposteriorpredictiveprobabi lityundercomplete replicationwithrespectto .Theformofthiscanbefoundintheappendix. ThedegreeofdeparturefromMARismeasuredbythesensitivi typarameter Checksbasedontheobservedreplicateddataarenotafuncti onof .Toexaminethe rateofchangeoftheposteriorpredictiveprobabilityfort hereplicatedcompletedatawe 47

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computethegradientwithrespecttotheparameter .Theformofthiscanbefoundin theappendix. 3.4DataExample Weobtainnumericalresultsfortheprobabilitiesandgradi entsinsection3, usingdatafromaclinicaltrial.Theobjectiveofthetrialw astoexaminetheeffects ofrecombinanthumangrowthhormonetherapyforbuildingan dmaintainingmuscle strengthintheelderly.Thestudy,whichwewillrefertoasG H,enrolled161participants andrandomizedthemtooneoffourtreatmentsarms.Variousm usclestrength measureswererecordedatbaseline,6months,12months.Wef ocusonmean quadricepsstrength,measuredasthemaximumfoot-poundso ftorquethatcanbe exertedagainstresistanceprovidedbymechanicaldevice. Wewillfocusontwoofthe treatmentgroups,Exercise+GrowthHormone(EG)andExerci se+Placebo(EP). Ofthe78randomizedtothesetwoarms,only53hadcompletefo llow-up(andthe missingnesswasmonotone);seeTable B-1 Dene Y =( Y 1 Y 2 Y 3 ) T tobequadstrengthmeasuredatmonths0,6,and12 andthecorrespondingobserveddataindicatorsas R =( R 1 R 2 R 3 ) T .Inthisdata, thebaselinequadstrengthisalwaysobserved,so P ( R 1 =1)=1 .Insection3we consideredabivariatesetting,sofortheGHdataweonlycon sidermonths 0 and 12 Results:WeusedtheGHdatatonumericallyassesstheprobabilitiesd erivedinSection3. Inwhatfollows,wedenepowerasmoreextremeposteriorpre dictiveprobabilities(i.e., toward0or1)undertheincorrectmodel.Forallsituationsc onsidered,thebehaviorof thechecksfollowedourintuition(Tables 3-2 3-4 ).Forexample,whenthetruemodel isttothedata,wewouldexpectposteriorpredictiveproba bilitiestobecloseto 0.5 So,asthepercentagemissingnessincreases,wewouldexpec ttheposteriorpredictive probabilitiestomoveawayfrom 0.5 ;thisiswhatwasobservedforbothcompleteand observedreplications.Underanincorrectmodel,weobserv etheprobabilitiestomove 48

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towardseither 1.0 or 0.0 asthepercentageofmissingnessdecreases.Similarbehavi or isseenasthesamplesizeincreases,butwithrelativelymor epoweronthechecks (basedontheposteriorpredictiveprobabilities). However,therelativebehaviorofthechecksislessclearfr omourintuition.Under MAR,thethecompletereplications( =0 )seemtohaveslightlymorepower(todetect modeldepartures)thanobservedreplications.However,un derMNAR( 6 =0 ),the observedreplicationsseemtohaveslightlymorepower;see Tables 3-2 3-4 ).We explorethisfurtherviasimulationsinthenextsection.Al so,were-iteratethatchecks shouldnotdependon as doesnotimpactthetofthemodeltotheobserveddata. Underthetruemodelweexaminetherateofchangeoftheposte riorpredictive probabilitybycomputingthegradientwithrespecttoperce ntagemissingnessparameter andthenonignorabilitysensitivityparameter .Undertruemodelforaxedsample size,weobservenumericallythatpredictiveprobabilityd ecreasesfrom 0.5 as increases.Thismakessensebecauseasthemissingnessincr easesposteriorpredictive probabilityshouldmovefarfrom 0.5 .Underthetruemodeltheposteriorpredictive probabilitiesareadecreasingfunctionof ,i.e.,completereplicationlosespower relativetoobservedreplicationasthemodeldepartsfromM AR. 3.5SimulationModelsandResults Weassesstheproposedchecksusingmorecomplexmodelsthan themodels consideredinSection3viasimulations.Foreachscenario, wesimulated100datasets, withparametervaluesbasedonGHdata.Wettwomodels(ment ionedbelow)tothe datasimulatedunderthesesamemodelswithsamplesizesper treatmentof30,100, and300.Therstmixturemodel,whichwedenoteasMM1,isgiv enby Y 1 j S = k N ( ( k ) 1 ( k ) 1 ) Y 2 j Y 1 S = k N ( 2 + 21 Y 1 I f trt =2 g 2 ) 49

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Y 3 j Y 1 Y 2 S = k N ( 3 + 31 Y 1 + 32 Y 2 I f trt =2 g 3 ) S Mult ( ). (3–7) Themultinomialparameteris =( 1 2 3 ) ,where s = P ( S = s ) WeconsidertwodeviationsfromMM1toassesstheoperatingc haracteristicsof theproposedchecks.Therstinvolvesthevariability.Inm odelMM1weset =16 and inthealternativemodelMM2,weset =1 .Toseehowwellthechecksassessthis departure,weconsiderthefollowingteststatisticsforea chtimepoint, T ( y y rep )= log SD ( y rep ) 1 2 [ SD trt =1 ( y )+ SD trt =2 ( y )] (3–8) basedoneithertheobserved(replicated)dataorthecomple te(replicated)data. SD (.) isthestandarddeviationcalculatedfortheobserved(repl icated)orcomplete(replicated) data. Thesecondinvolvesdeparturesintheconditionalmeanstru cture.Inthealternative modelweset 31 = 0 32 = 0 (and = 1 ).Theteststatisticconsideredis T ( y )= y 3 (3–9) basedoneithertheobserved(replicated)dataorthecomple te(replicated)dataattime point3. Simulationresults:WerantheGibbssamplingalgorithminW inBUGSfor100K iterationsforeachmodelandeachsimulateddataset.Wecom putedvaluesofthetest statistics, T ,withcorrespondingposteriorpredictiveprobabilities. Theresultsrelatedtothedepartureinvolvingthevariabil ity(equation 3–8 )are reportedinTables 3-5 3-8 .UnderthetruemodelMM1(Tables 3-6 and 3-8 ),we expecttheposteriorpredictiveprobabilitytomoveawayfr om 0.5 asthepercentage ofmissingnessincreases,butitshouldmovetowards 0.5 assamplesizeincreases; weobservebothhere.FortrueMM1orMM2,ifthewrongmodelsa ret(Tables 3-5 50

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and 3-7 ),weexpectourcheckwilldetectthedeparturefromtruemod el(basedonthe variabilitycomponent)andaspredictedattimes2and3thep redictiveprobabilities movetowards 0.5 aspercentagemissingincreases,andmoveawayfrom 0.5 toward 0 or 1 assamplesizeincreases. Ifwecomparethechecksbasedonobservedreplicationwithc ompletereplication, weseecompletereplicationwithMARassumptionappearstoh avemorepowerthan observedreplicationtodetectthedepartureinvariabilit yfromtruemodel(Tables 3-5 and 3-7 ).Theresultsherearesimilartothosecomputedusingthean alyticresultsin Section 3.4 WealsoimplementedfurtherchecksunderMNARassumptions, usingthemean structureandtheresultsareshowninTable 3-9 forthedatasimulatedundertruemodel (MM1)andttedtotruemodel.Table 3-10 showsthesimulationresultsforthedata simulatedundertruemodelwherethewrongmodel( 31 =0, 32 =0)ist.Weobserve thesimilarbehaviorfortheobservedreplicationandforco mpletereplication(MARand MNAR)andtheresultsaresimilartoanalyticresultscomput edinSection 3.4 .Under thetruemodelweobservetheposteriorpredictiveprobabil itymovesfarfrom 0.5 asthe percentagemissingincreasesandmovestowards 0.5 assamplesizeincreases.Under MNAR,thechecksbasedontheobservedreplicationsseemtoh aveslightlymore powerthanthosebasedonthecompletereplications.Overal l,thesimulationresults supporttheanalyticalresults. 3.6SummaryandDiscussion WehaveproposedaconvenientwaytoassessthetoftheBayes ianmodelsin thepresenceofincompletedatausingposteriorpredictive checks;suchcheckscan easilybeimplementedinWinBUGS.Viasimulations,theprop osedchecksappear toworkwell.Thesimulationresultsweresupportedbythean alyticalresults.Both approaches(completeandobservedreplication)notsurpri singlyresultinlesspower thanifweactuallyhadcompletedata.Itappearstobesettin g-speciconwhether 51

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theobservedorcompletereplicationsprovidemorepowerth oughinourlimited simulations,weobservedmorepowerfortheobservedreplic ationsunderMNAR andmorepowerforthecompletereplicationsunderMAR.Howe ver,theobserved replicationssatisfytheinvariancetotheextrapolationd istributionunlikethecomplete replicationswhicharguably,isanecessaryproperty( Danielsetal. 2012 )andwould leadustorecommendchecksbasedontheobservedreplicated dataasthepreferred approachfornonignorablemissingness.Clearly,furtherw orkneedstobedonetobetter understandthebehaviorofthechecksinothersettings. Table3-1.GrowthHormoneTrial:SampleMeans(StandardDev iations). Month Treatment sn s 0612 EG11258(26) 2457(15)68(26)32278(24)90(32)88(32) All3869(25)87(32)88(32) EP1765(32) 2287(52)86(51)33165(24)81(25)73(21) All4066(26)82(26)73(21) Table3-2.PosteriorPredictiveProbabilitiesunderWrong Model( 21 =0 ). complete Sample Size Proportion Missing Obs =0 =5 =10 =-5 =-10 1/30.130.090.160.200.170.21 n=301/20.160.120.190.210.190.23 2/30.200.160.210.230.220.253/40.240.210.260.280.280.291/30.080.070.110.170.130.19 n=1001/20.110.090.160.190.170.21 2/30.180.140.190.210.210.233/40.210.170.220.250.230.271/30.070.040.090.130.110.15 n=3001/20.090.070.120.160.120.17 2/30.150.110.160.180.170.183/40.190.150.200.220.210.23 52

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Table3-3.WrongModel( ( k ) 1 = 1 ( k ) 1 = ):PosteriorPredictiveProbablities. complete Sample Size Proportion Missing Obs =0 =5 =10 =-5 =-10 1/30.090.070.110.160.120.18 n=301/20.140.100.160.190.180.21 2/30.170.110.180.220.190.233/40.210.180.240.250.240.251/30.070.050.100.130.120.14 n=1001/20.100.070.140.160.150.17 2/30.150.090.170.190.180.193/40.180.160.210.230.210.241/30.060.040.050.080.070.10 n=3001/20.070.060.060.110.080.12 2/30.110.100.080.130.100.153/40.160.130.140.180.150.18 Table3-4.UnderTrueModel:NatureofPosteriorPredictive Probabilities. complete Sample Size Proportion Missing Obs =0 =5 =10 1/30.430.460.410.39 n=301/20.410.440.390.38 2/30.350.390.370.363/40.330.370.350.341/30.460.480.420.40 n=1001/20.430.460.400.39 2/30.380.420.380.373/40.360.400.360.351/30.470.510.440.43 n=3001/20.440.470.430.41 2/30.400.440.410.403/40.380.420.380.37 53

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Table3-5.SimulationforCompleteReplicationunderTrueM M1andFittedMM2. TreatmentTime1Time2Time3 =(1 = 4,1 = 4,1 = 2) Trt1-0.26(0.41)-0.71(0.16)-0.73(0.13)Trt20.30(0.58)0.80(0.79)0.86(0.81) n =30 =(1 = 3,1 = 3,1 = 3) Trt1-0.29(0.38)-0.68(0.19)-0.71(0.16)Trt20.32(0.59)0.78(0.74)0.81(0.78) =(1 = 2,1 = 4,1 = 4) Trt1-0.30(0.37)-0.64(0.23)-0.68(0.20)Trt20.33(0.61)0.75(0.71)0.77(0.74) =(1 = 4,1 = 4,1 = 2) Trt1-0.24(0.44)-0.74(0.12)-0.76(0.10)Trt20.28(0.55)0.83(0.81)0.88(0.84) n =100 =(1 = 3,1 = 3,1 = 3) Trt1-0.28(0.41)-0.72(0.16)-0.73(0.14)Trt20.31(0.57)0.80(0.77)0.85(0.81) =(1 = 2,1 = 4,1 = 4) Trt1-0.29(0.39)-0.67(0.19)-0.71(0.17)Trt20.33(0.58)0.78(0.74)0.81(0.77) =(1 = 4,1 = 4,1 = 2) Trt1-0.20(0.47)-0.78(0.08)-0.81(0.06)Trt20.23(0.53)0.86(0.83)0.90(0.86) n =300 =(1 = 3,1 = 3,1 = 3) Trt1-0.23(0.43)-0.74(0.11)-0.77(0.09)Trt20.28(0.55)0.83(0.79)0.86(0.83) =(1 = 2,1 = 4,1 = 4) Trt1-0.25(0.42)-0.71(0.15)-0.74(0.12)Trt20.30(0.56)0.80(0.77)0.82(0.80) Table3-6.CompleteReplicatedDataunderTrueMM1andFitte dMM1. TreatmentTime1Time2Time3 =(1 = 4,1 = 4,1 = 2) Trt1-0.24(0.41)-0.29(0.39)-0.28(0.38)Trt20.30(0.58)0.56(0.55)0.58(0.59) n =30 =(1 = 3,1 = 3,1 = 3) Trt1-0.28(0.37)-0.31(0.38)-0.32(0.37)Trt20.33(0.59)0.59(0.58)0.60(0.61) =(1 = 2,1 = 4,1 = 4) Trt1-0.30(0.36)-0.32(0.36)-0.33(0.34)Trt20.33(0.61)0.63(0.60)0.64(0.62) =(1 = 4,1 = 4,1 = 2) Trt1-0.23(0.44)-0.24(0.42)-0.25(0.40)Trt20.28(0.54)0.54(0.53)0.55(0.57) n =100 =(1 = 3,1 = 3,1 = 3) Trt1-0.28(0.41)-0.26(0.40)-0.28(0.39)Trt20.31(0.57)0.57(0.55)0.58(0.58) =(1 = 2,1 = 4,1 = 4) Trt1-0.29(0.38)-0.27(0.38)-0.29(0.38)Trt20.33(0.58)0.58(0.56)0.59(0.60) =(1 = 4,1 = 4,1 = 2) Trt1-0.20(0.46)-0.22(0.45)-0.23(0.43)Trt20.23(0.53)0.53(0.51)0.55(0.56) n =300 =(1 = 3,1 = 3,1 = 3) Trt1-0.23(0.43)-0.23(0.43)-0.26(0.40)Trt20.28(0.55)0.55(0.54)0.56(0.57) =(1 = 2,1 = 4,1 = 4) Trt1-0.25(0.41)-0.26(0.40)-0.27(0.40)Trt20.30(0.56)0.57(0.54)0.58(0.58) 54

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Table3-7.SimulatedObservedReplicatedDataunderMM1and FittedMM2. TreatmentTime1Time2Time3 =(1 = 4,1 = 4,1 = 2) Trt1-0.26(0.41)-0.64(0.21)-0.66(0.19)Trt20.30(0.58)0.58(0.68)0.62(0.72) n =30 =(1 = 3,1 = 3,1 = 3) Trt1-0.29(0.38)-0.61(0.25)-0.62(0.23)Trt20.32(0.59)0.56(0.65)0.59(0.68) =(1 = 2,1 = 4,1 = 4) Trt1-0.30(0.37)-0.58(0.29)-0.59(0.26)Trt20.33(0.61)0.55(0.63)0.57(0.65) =(1 = 4,1 = 4,1 = 2) Trt1-0.24(0.44)-0.66(0.20)-0.68(0.17)Trt20.28(0.55)0.61(0.71)0.64(0.73) n =100 =(1 = 3,1 = 3,1 = 3) Trt1-0.28(0.41)-0.64(0.23)-0.65(0.20)Trt20.31(0.57)0.58(0.67)0.61(0.69) =(1 = 2,1 = 4,1 = 4) Trt1-0.29(0.39)-0.61(0.26)-0.62(0.23)Trt20.33(0.58)0.57(0.66)0.58(0.66) =(1 = 4,1 = 4,1 = 2) Trt1-0.20(0.47)-0.67(0.17)-0.69(0.14)Trt20.23(0.53)0.64(0.74)0.66(0.75) n =300 =(1 = 3,1 = 3,1 = 3) Trt1-0.23(0.43)-0.66(0.19)-0.67(0.17)Trt20.28(0.55)0.61(0.69)0.63(0.72) =(1 = 2,1 = 4,1 = 4) Trt1-0.25(0.42)-0.62(0.21)-0.64(0.19)Trt20.30(0.56)0.59(0.67)0.61(0.69) Table3-8.UnderMM1SimulatedObservedReplicateddataand FittedMM1. TreatmentTime1Time2Time3 =(1 = 4,1 = 4,1 = 2) Trt1-0.26(0.40)-0.28(0.37)-0.29(0.38)Trt20.30(0.58)0.36(0.61)0.38(0.60) n =30 =(1 = 3,1 = 3,1 = 3) Trt1-0.29(0.38)-0.31(0.32)-0.32(0.34)Trt20.32(0.59)0.38(0.63)0.41(0.64) =(1 = 2,1 = 4,1 = 4) Trt1-0.30(0.37)-0.32(0.29)-0.35(0.31)Trt20.33(0.61)0.39(0.65)0.44(0.66) =(1 = 4,1 = 4,1 = 2) Trt1-0.24(0.44)-0.26(0.41)-0.27(0.40)Trt20.28(0.56)0.34(0.58)0.35(0.58) n =100 =(1 = 3,1 = 3,1 = 3) Trt1-0.28(0.40)-0.27(0.38)-0.29(0.39)Trt20.30(0.57)0.36(0.60)0.38(0.61) =(1 = 2,1 = 4,1 = 4) Trt1-0.29(0.39)-0.28(0.37)-0.30(0.36)Trt20.33(0.58)0.38(0.61)0.39(0.62) =(1 = 4,1 = 4,1 = 2) Trt1-0.20(0.46)-0.23(0.43)-0.24(0.42)Trt20.23(0.53)0.32(0.56)0.34(0.56) n =300 =(1 = 3,1 = 3,1 = 3) Trt1-0.23(0.43)-0.25(0.40)-0.25(0.41)Trt20.28(0.55)0.33(0.57)0.35(0.58) =(1 = 2,1 = 4,1 = 4) Trt1-0.25(0.41)-0.26(0.39)-0.27(0.40)Trt20.30(0.56)0.36(0.58)0.38(0.59) 55

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Table3-9.CompleteandObservedReplicationunderTrueMM1 andFittedMM1. Sample Size Proportion Missing TrtObs =0 =5 =10 1/310.410.470.420.40 20.560.540.580.60 n=301/210.380.430.380.37 20.590.570.610.61 2/310.360.370.350.33 20.610.600.620.63 3/410.320.350.310.30 20.620.610.640.64 1/310.430.490.430.42 20.540.520.570.58 n=1001/210.420.460.390.38 20.560.560.580.60 2/310.400.410.380.37 20.570.570.590.61 3/410.360.390.400.34 20.600.580.600.62 1/310.450.490.430.43 20.530.510.540.56 n=1001/210.440.470.420.40 20.550.530.560.57 2/310.410.460.400.38 20.570.540.570.69 3/410.390.420.380.37 20.590.560.590.60 56

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Table3-10.ForTrueMM1andFittedWrongModel:Completeand Observed Replication. Sample Size Proportion Missing TrtObs =0 =5 =10 1/310.190.160.200.23 20.700.740.700.69 n=301/210.210.190.210.24 20.670.710.660.64 2/310.280.210.290.31 20.610.680.600.58 3/410.290.240.300.33 20.590.640.560.54 1/310.130.110.150.18 20.730.750.700.69 n=1001/210.170.150.180.21 20.700.740.670.65 2/310.230.180.240.26 20.640.690.620.60 3/410.240.200.270.28 20.610.660.580.55 1/310.100.080.140.16 20.740.770.710.70 n=1001/210.150.120.170.19 20.720.750.690.68 2/310.190.150.210.23 20.660.710.630.62 3/410.210.160.230.25 20.640.690.610.60 57

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CHAPTER4 REVIEWONMODELDIAGNOSTICSFORINCOMPLETEDATA 4.1BackgroundandMotivation Longitudinalstudiescollectrepeatedobservationsovert imeonseveralindividuals. Itiscommonthatallthedesireddataarenotcollected,andt husoneneedstoaddress themissingdata.Themissingnessmayoccurduetolosstofol lowingup,mortality, orotherreasons.Drawinginferenceaboutthistypeofdatai sachallenge.Asimple waytomakeinferenceinthepresenceofincompletedataisto onlyusethecomplete cases.Thisresultsarevalidforinferenceonlyifthemissi ngnessisindependentofthe responsewhichiscalledmissingcompletelyatrandom(MCAR ).Webrieyreviewthe classicationofmissingdataasdenedby Rubin ( 1976 ).Beforedoingthiswedene thefollowingnotationY :fulldataresponse, R :observeddataindicators: R ij = I f y ij isobserved g ,( Y obs r ):observeddata, Y mis :missingdata, Y repcom :replicatedfulldataresponse, Y repobs : replicatedobserveddataresponse, :parameterindexingthefulldatamodel. LittleandRubin'smissingnesstaxonomy( Rubin 1976 ; LittleandRubin 1987 )is denedas, Denition4.1. Missingcompletelyatrandom (MCAR)mechanismisdened conditionedforallxand as, p ( r j Y x )= p ( r j x ), (4–1) i.e., p ( r j Y x ) isconstantfunctionofY. Denition4.2 Missingatrandom (MAR)mechanismisdenedas, p ( r j Y obs Y mis x )= p ( r j Y obs x ), (4–2) forall Y obs ,xand ,i.e., p ( r j Y obs Y mis x ) isconstantfunctionof Y mis 58

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MARholdsifandonlyif p ( y mis j y obs r )= p ( y mis j y obs ). (4–3) Denition4.3. Missingnotatrandom (MNAR)mechanismisdenedas, p ( r j Y obs Y mis x ) 6 = p ( r j Y obs Y 0mis x ), (4–4) forsome Y mis and Y 0mis ,i.e.,Rdependsonsomepartof Y mis ,evenafterconditioning on Y obs Denition4.4. :The extrapolationfactorization ( DanielsandHogan 2008 )is p ( y r ; )= p ( y mis j y obs r ; E ) p ( y obs r ; O ). Thisfactorizationmakesclearwhattheobserveddatacanid entify.Theobserved datadistribution p ( y obs r ; O ) ,isidentiableandisindexedby O .Theextrapolation distribution p ( y mis j y obs r ; E ) ,isindexedby E andisnotidentiedfromtheobserved datawithoutmodelingassumptions. InvariancetoExtrapolationDistribution :Twomodelsforthefulldatawiththe samemodelspecicationfortheobserveddata, p ( y obs r ; O ) andsamepriorfor p ( O ) shouldgivethesamevalueoftheBayesianmodelselectioncr iterion.Observed replicationssatisfytheinvariancetotheextrapolationd istributionunlikethecomplete replications.Soitisrecommendedchecksbasedontheobser vedreplicateddataasthe preferredapproachfornonignorablemissingness. Ignorabilityvs.Nonignorability: InlikelihoodorBayesianinference,oftenmore naturaltoclassifymissingnessasignorableornonignorab le.Wedenethisformally next. Denition4.5 Themissingdatamechanismis ignorable if,( Rubin 1976 ) 1 .ThemissingdatamechanismisMAR. 59

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2 istheparameteroffulldataresponsemodelanditcanbedeco mposedinas =( ) ,where isthefulldataresponsemodelparameterand isassociatedwith themissingdatamechanism p ( r j y ) 3 .Theparameters and areaprioriindependent; p ( )= p ( ) p ( ) (4–5) ThethirdconditionisnecessaryfortheBayesianinference .Ifanyofthethreeconditions donothold,missingnessisnonignorable.Underignorabili ty,posteriorinferenceon parameterisbasedonobserveddataresponselikelihood. Denition4.6. Sensitivityparameter :Itisverycommonthatparametersassociated withtheextrapolationdistribution, p ( y mis j y obs r ; E ) arenotidentiedby y obs .The fulldatamodelisidentiedifthoseparametersarexedwit hsomevalues.Those parametersarecalledsensitivityparameters( DanielsandHogan 2008 ).Thisisformally denedby DanielsandHogan ( 2008 ). Letthefulldatamodelis p ( y r j ) andtheextrapolationdistributionis p ( y r ; )= p ( y mis j y obs r ; E ) p ( y obs r ; O ). Considerareparameterization ( )=( S M ) : 1 S isanon-constantfunctionof E 2 .theobservedlikelihood L ( S M j y obs r ) isconstantfunctionof S 3 .foraxed S ,thelikelihood L ( S M j y obs r ) isnon-constantfunctionof M then S isthe sensitivityparameter ( DanielsandHogan 2008 ). Sensitivityanalysisisdonetoexplorethesensitivityofi nferencesaboutthe parameterofinterestaboutthefulldataresponsemodeltou nveriableassumptions abouttheextrapolationmodel( DanielsandHogan 2008 ). InSection 4.2 wediscusalocalinuenceapproach( Verbekeetal. 2001 )toassess themodeltandprovideconnectionstoindexoflocalsensit ivitytononignorability 60

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( Troxeletal. 2004 ).Section 4.3 reviewsexploratoryapproachesforcontinuous timedropout.InSection 4.4 wereviewBayesianapproachesbasedontheposterior predictivedistribution. 4.2DiagnosticsBasedonFullyParametrizedIdentiedMode ls Diagnosticsforincompletedatawasdiscussedby Verbekeetal. ( 2001 )inthe settingofselectionmodel( DiggleandKenward 1994b )forcontinuouslongitudinaldata subjecttodropout. N subjectsarestudied,andtheresponses Y ij aremeasuredattime j = 1 ,..., n i .Dropoutoccursattimepoint t i istheparameterforthemeasurement modelandand istheparameterizationforthedropoutmodel.Alinearmixe dmodel isassumedfortheresponseprocess.Logisticregressionar edonefortheprobabilityof dropout.Thisprobabilityisdenotedby g ( h ij y ij ) ,where h ij isthehistoryofresponses observeduptotheobservationattime j 1 .Themodelis logit ( g ( h ij y ij ))= logit ( P ( t i = j j t i j y i ) = h ij + y ij (4–6) equalszeroimpliesthedropoutisMAR.Todealwiththesensi tivityofestimation Verbekeetal. ( 2001 )consideredaperturbedversionof( 4–6 ): logit ( g ( h ij y ij ))= logit ( P ( t i = j j t i j y i ) = h ij + i y ij (4–7) Verbekeetal. ( 2001 )dened i astheindividualspecicperturbationsaroundaMAR modelandusedtocalculateaninuencemeasure( Cook 1986 ),whichwedeneinthe nextsection. 61

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4.2.1LocalInuence Thegoalof Verbekeetal. ( 2001 )wastoassesswhethersmallperturbationsaway fromMARcanhaveeffectoninferenceamodel.Theycheckwhet her E ( Y mis j Y obs ) deviatesfromMARornot. Thelog-likelihoodwasdenedby Verbekeetal. ( 2001 )correspondingtoa particularmodelby, l ( j )= N X i =1 l i ( j i ) wherecontributionofaparticularindividualis l i ( j i ) and = ( ) istheparameter ofthemeasurementmodelanddropout.ForMARmodel Verbekeetal. ( 2001 )dened l ( j 0 ) where 0 isthevectorconsistingof 0 foreachelement. Dene ^ tobetheMLEfor l ( j 0 ) and ^ istheMLEof for l ( j ) forany xed Cook ( 1986 )suggestedtoobtainthe likelihooddisplacement anddenedas LD ( )=2( l (^ j 0 ) l (^ j )). (4–8) Thegraphisformedbyplotting LD ( ) versus andisgivenby, ( )= 0B@ LD ( ) 1CA Cook ( 1986 )dened i as i = @ 2 l i ( j i ) @! i @ j = ^ i =0 andthelocalinuencemeasureas, C h = 2 j h 0 0 L 1 h j ,where L isthesecondorder derivativeof l ( j 0 ) withrespectto C h canbeusedtostudy ( ) ,fordifferentchoices of h .Onecommonchoiceforthevector h i is 1 correspondingtothe i th positionand 0 in theotherplaces.Then C i =2 j 0i L 1 i j 62

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4.2.2ApplicationofLocalInuencemethodinSelectionMod els Thelocalinuencemethoddenedby Verbekeetal. ( 2001 ),canbeappliedtothe selectionmodelby DiggleandKenward ( 1994b )and i forcompletersisgivenby, @ 2 l i @@! i j i =0 =0, (4–9) @ 2 l i @ @! i j i =0 = n i X j =2 h ij y ij g ( h ij )(1 g ( h ij )) and, @ 2 l i @@! i j i =0 =(1 g ( h it )) @ ( y it j h it ) @ (4–10) @ 2 l i @ @! i j i =0 = t 1 X j =2 h ij y ij g ( h ij )(1 g ( h ij )) h it ( y it j h it ) g ( h it )(1 g ( h it )) forindividualswhodropout,and g ( h ij ) = g ( h ij y ij ) j i =0 correspondstotheMAR versionofdropoutmodel( Verbekeetal. 2001 ). Thijsetal. ( 2000 )dened V i ,11 be thecovariancematrixfor y obs and V i ,21 bethecovariancevectorbetween y obs and y mis .Theconditionalexpectationisgivenby ( y it j h it )= ( y it )+ V i ,21 V 1 i ,11 ( h it ( h it )). Thelocalinuencemeasurecanbecalculatedby C h ( )= 2 h 0 [ @ 2 l i @@! i j i =0 ] 0 L 1 ( )[ @ 2 l i @@! i j i =0 ] h C h ( )= 2 h 0 [ @ 2 l i @ @! i j i =0 ] 0 L 1 ( )[ @ 2 l i @ @! i j i =0 ] h where istheparameterofinterestand( 4–9 )and( 4–10 )demonstratetheinuenceon fortheindividualswhodropout. 4.2.3ConnectionBetweenILSNandLocalInuence Alocalsensitivityindexwasproposedby Troxeletal. ( 2004 )thatisrelatedtothe approachof Verbekeetal. ( 2001 ). Troxeletal. ( 2004 )alsofocusesonthesensitivity aroundtheMARmodel.Toevaluatetheirdiagnostic,anindex oflocalsensitivityto 63

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nonignorability(ILSN)wasproposedwithoutestimatingal ltheparametersinafull identiednonignorablemodel.ILSNiseasytocomputeasito nlyrequiresonetotthe MARmodelandtondtheMLEfromthatMARmodel.ILSNingenera lisapplicableto theselectionmodelsby DiggleandKenward ( 1994b ). TheILSNmeasureshowtheMLEof foragivenvalueof of( ^ ( ) )dependson and isdenedas, ILSN = @ ^ ( ) @! j 0 4.2.4ApplicationofILSNinSelectionModels Assumetheoutcome Y i havingmultivariateGaussiandistributionwithpredictor s X i andthedropoutprobabilityisspeciedas( Thijsetal. 2000 ; Troxeletal. 2004 ), P ( R ij =1 j Y i = y i X ij = x ij R i j 1 =1)= g ( x ij 01 + y i j 1 02 + y i j ) where x ij isthecurrentpredictor, y i j 1 isthepreviousoutcome, g isthelinkfunction. Let, L bethelikelihoodfunction.Then, ILSN = @ ^ ( ) @! = @ 2 L @ @ 0 j ^ 0 ,^ 0 =0 1 @ 2 L @ @! j ^ 0 ,^ 0 =0 TheILSNmeasuringtheextenttowhichtheMLEsgiventhevalu eofnonignorability parameter departsfromitsMARvalue.Localinuencemethodby Verbekeetal. ( 2001 )andtheILSNby Troxeletal. ( 2004 )bothexamine`local'sensitivityaroundMAR infullyidentiednonignorablemodels. 4.3DiagnosticsforLongitudinalandContinuousTimeDropo ut Anassessmentofdropoutclassicationsbasedontheresidu alswithoutnonignorable dropoutwasconsideredby DobsonandHenderson ( 2003 ).Themainpurposeisto differentiatethesubjectsonthebasisofdifferentdropou treasons.Theyalsoproposed conditionalresidualanalysis.Theirframeworkfocuseson continuoustimedropout. 64

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4.3.1JointModelforRandomEffects Assumethefollowingmodelforresponses Y ( t ) attimet, Y ( t )= 0 1 x 1 ( t )+ W 1 ( t )+ Z ( t ), (4–11) where x 1 ( t ) istheexplanatoryvariable, W 1 ( t ) isanunobservedzero-meanGaussian process,and Z ( t ) isthezeromeanGaussianmeasurementerror.Themeasuremen t errorsaremutuallyindependent.Denethedropouttimeas T and ascategorical randomvariablethatidentiesthereasonforthewithdrawa lfromstudy.Aconditional proportionalhazardmodelisassumedfortimetononignorab ledropoutandisdened as, ( t j x 2 W 2 )= 0 ( t ) exp f 0 2 x 2 ( t )+ W 2 ( t ) g (4–12) where 0 ( t ) isthebaselinehazardand x 2 ( t ) istheobservedexplanatoryvariables. W 2 ( t ) isspeciedas W 2 ( t )= 1 U 1 + 2 U 2 + 3 W 1 ( t ). (4–13) U 1 and U 2 aretherandomeffectsandareassumedtobemultivariateGau ssian.The conditionalexpectationof h ( U ) canbewrittenas, E [ h ( U ) j Y T ]= R h ( U ) p ( t j U ) p ( U j Y ) d U R p ( t j U ) p ( U j Y ) d U (4–14) where h ( U ) isafunctionoftherandomeffects. p ( t j U ) isobtainedfrom( 4–12 )and p ( U j Y ) istheconditionaldistributionoftherandomeffects.Thei ntegralsareevaluated usingMonteCarlointegration.4.3.2ClassicationsofDropout Theobjectivehereistondadiagnostictodiscriminatesub jectsthathave nonignorabledropoutversusthosewhohaveignorabledropo ut.Thisisdoneby 65

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examiningtheresidualsafterttingthemodel.Theresidua l R i forsubject i iscalculated fromthetasfollows R i ( t )= Y i ( t ) ^ 0 1 i x 1 i ( t ). Where ^ 1 istheMLE.Toexaminetheeffectofthenonignorabilityonth eresidualsthe individualscanbedividedingroupsasfollows = 8><>: 1 ifthesubjectshavenonignorabledropoutatnextmeasureme nttime. 2 ifthesubjectshasignorabledropout. Themodelgivenin( 4–11 )isttotheresponsevariable Y DobsonandHenderson ( 2003 )dened R t betherisksetwhichconsistsofsubjectshavingatleast t 1 responsesanddropoutattime t ,and S t tobethesubsetconsistingofthesubjectswho dropoutbeforetime t Adiscriminationrulewassuggestedby DobsonandHenderson ( 2003 )andthey used P t = twodimensionalsummaryofresidualsateachtimepointbefo retimet. TheycalculatetheMahalanobisdistance, M jk =( P jt P kt ) 0 ^ Var ( P it ) 1 ( P jt P kt ), (4–15) where P jt isthesamplemeanof P t forsubjectsinaparticulardenedgroup. M jk isused asthediscriminationruleanditisusedindrawingscatterp lotsforeachgrouphaving ignorableandnonignorabledropout4.3.3RelationbetweenDropoutandResiduals Residualsarealsoexaminedafterttingthejointmodelsde nedin( 4–11 )( 4–13 )( DobsonandHenderson 2003 ).Theconditionalmeanofresidual, R ( t ) given nonignorabledropout( = 1 )attime T is E [ R ( t ) j T =1]= R W 1 ( t ) p ( t j U ) p ( U ) dU R p ( t j U ) p ( U ) d U (4–16) 66

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where p ( t j U )= 0 ( t ) exp ( 0 2 x 2 ( t )+ W 2 ( t )) exp ( Z t 0 0 ( s ) exp ( 0 2 x 2 ( s )+ W 2 ( s )) ds ), and p ( U ) ismultivariateGaussiandensityfor U .Forignorabledropout ( 6 =1) E [ R ( t ) j T 6 =1]= R W 1 ( t ) S ( t j U ) p ( U ) d U R S ( t j U ) p ( U ) d U (4–17) where S ( t j U ) issurvivorfunctionforthenonignorabledropoutand S ( t j U )= exp ( Z t 0 0 ( s ) exp ( 0 2 x 2 ( s )+ W 2 ( s )) d U ). TheconditionalresidualdistributionisGaussianif W 1 ( t ) and W 2 ( t ) areindependent. Itisexpectedtolookatthebehavioroftherstandsecondor dermomentsof theresidualsovertime t .Agraphicalpresentationof E ( R ( t ) j T = t =1) and E ( R ( t ) j T = t 6 =1) canbedonetoseethechangewithtime t andalsotoseethe effectofdropouttime t 4.3.4RelationBetweenResidualAnalaysisandLocalInuen ceMethod Thediscriminationruledenedin( 4–15 )canbepresentedastheCookdistances consideringthecompleteregressiondenedin( 4–11 ).Forsimplicationweconsider theparameterestimate ( ^ 0 ) underthemodelwithignorabledropoutand ( ^ ) fortheother grouphavingnonignorabledropout,andwecanreformulatet heMahalanobisDistance asfollows, M 01 =( R 0 t R 1 t ) 0 ^ Var ( R it ) 1 ( R 0 t R 1 t ), where R 0 t isthemeanof R it forsubjectsinthegrouphavingindividualswithignorable dropoutand R 1 t isthemeanof R it forsubjectsinthegrouphavingindividualswith nonignorabledropout.Itcanbeshownifweplugintheparame tersestimatesforeach groupintheaboveexpressioninsteadoftheresiduals,thed iscriminationruledening 67

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statisticis ( ^ 0 ^ 1 ) 0 ^ Var ( ^ ) 1 ( ^ 0 ^ 1 ), whichisanalogoustoCook-distancemeasures.4.3.5ResidualDiagnosticsforMixtureModels Itispossibletoapplytheresidualdiagnosticsof DobsonandHenderson ( 2003 ) tomixturemodelsinthecontinuoustimedropoutsettings.T hefulldatamodelcanbe factoredas( DanielsandHogan 2008 ), p ( y j )= p ( y j ) p ( j ) = p ( y mis j y obs E ) p ( y obs j o ) Thefulldataresponsemodelisobtainedfrom, p ( y j )= Z p ( y j ) p ( j ) d Conditionedondropoutattime = ,themeanofthefulldataresponsebeforethe dropouttime andafterthedropouttime areassumedequivalenti.e., E ( Y ( t ) j = t )= E ( Y ( t ) j = t ) or, E ( Y ( t ) j = t )= q ( t ) E ( Y ( t ) j = t ), where q ( t ) isasensitivityparameter.Inthenormalresponsecase( Hoganetal. 2004 ),thefulldataresponseprocessis Y ( t ) attime t andthemean E ( Y ( t )) are obtainedasfollows, E ( Y ( t ))= Z ( t j ) p ( ) d Thefunction ( t j ) isafunction(known)of t forgiven .Theseformulationscanbe foundin DanielsandHogan ( 2008 ). 68

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Weproposetheresidualanalysisafterttingthemodelment ionedabove, conditionedondropouttime = .Wecandenotetheresidualsby R ( t j )= Y ( t ) ^ ( t j ) .Wecaninvestigatetheconditionalmomentpropertiesof R ( t ) asproposedby DobsonandHenderson ( 2003 ).Theconditionalmeanof R ( t ) giventhedropouttime = isgivenby, E ( R ( t ) j = t < )= Z 0 R ( t j ) p ( y ( t ) j ) d y ( t ) (4–18) = q ( t ) Z T R ( t j ) p ( y ( t ) j ) d y ( t ) = q ( t ) E ( R ( t ) j = t > ) and E ( R ( t ))= ZZ R ( t j ) p ( y ( t ) j ) p ( ) d y ( t ) d (4–19) Toassesstheassociationbetweentheresidualsandthedrop outtime,theconditional residualprolescanbeshownovertime t foreachgroup( ( t > d ) and ( t < d ) ).Inother wordsforthecontinuoustimedropoutthegraphicalpresent ationof E ( R ( t ) j = t < ) and E ( R ( t ) j = t > ) versusthetime t areshowntoseetheeffectofdropouttime onthechangeoftheexpectedresidualprole. WenowreviewBayesianapproachesfordiagnostics. 4.4BayesianDiagnosticsBasedonthePosteriorPredictive Distribution Theposteriorpredictivedistribution, p ( y rep r rep j y obs r ) isusedtoassessthet andisdenedas p(y rep ,r rep j y obs ,r)=p(y rep j r rep ,y obs ,r)p(r rep j ,y obs ,r). Whenweassumethemissingdatamechanism,isnonignorablem issingnesswecan computedatasummariesbasedonreplicatesofobserveddata andreplicatesof 69

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completedata.Gelmanet.al.(2005)proposeddoingchecksu singcompletedata. HoweverGelmanconsideredmoregeneralsettingsthatinclu delatentvariables(as missingdata),ignorablemissingnessandnonignorablemis singness.Toassess thetofthemodeltheydeneateststatistic T ( ) ,whichisafunctionofcomplete data.The'missing'dataeithermissingorlatentislledin ateachiterationusing dataaugmentation.Thisteststatisticiscomparedtorepli catedcompletedata.At iteration l thecompletedatais, ( y obs y ( l ) mis ) ,wheresample y ( l ) mis issampledfrom p ( y ( l ) mis j y obs ( l ) ) T ( y repcom ) aresampledfromfulldataresponsemodel, p ( y repcom j ( l ) ) conditionalonparametervaluesatiteration l .Graphicalapproacheswereimplemented formodelcheckingalongwiththecalculationofposteriorp redictiveprobabilities. Wenowprovidemoredetailsonobservedreplication.Wenowi ntroducereplicated observeddata.Wedenethesereplicatesas y rep obs = f y rep j : r rep j =1 g i.e.,thecomponentsofthe y rep j ofthereplicatedcompletedatasetsforwhichthe correspondingreplicatedmissingdataindicatoris1.Then ateachiterationwecompute checksusingthecomponentsof y obs and y repobs forwhich r rep ij =1 .Forexample, E ( T rep Y obs j Y obs R )= ZZZ T f y repobs ( y rep r rep ) g p ( y rep j r rep y obs r ) p ( r rep j y obs r ) ( j y obs r ) d y rep d r rep d (4–20) InChapter3weexaminedtheoperatingcharacteristicsofth etwoapproachestoassess modeltforincompletelongitudinaldata.Bothapproaches (completeandobserved replication)notsurprisinglyresultinlesspowerthanifw eactuallyhadcompletedata. However,theobservedreplicationssatisfytheinvariance totheextrapolationdistribution unlikethecompletereplicationswhichisanimportantprop erty( Danielsetal. 2012 ) andwouldleadustorecommendchecksbasedontheobservedre plicateddataasthe preferredBayesianapproachfornonignorablemissingness 70

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4.5Summary InthisChapter,wehavegivenareviewofmodeldiagnosticsf orincompletedata. Werstfocusedonassessingthetoffullyidentiedparame tricmodels.Onepossible waytoassessthesensitivityisthemethodoflocalinuence introducedby Verbeke etal. ( 2001 ).Theirmethodisbasedonindividualspecicperturbation saroundMAR modelandisveryeasytoincorporatetotheselectionmodelw ithdiscretetimedropout in DiggleandKenward ( 1994b )framework. Troxeletal. ( 2004 )proposedILSNasatool toassessthesensitivitytononignorability,andmeasured sensitivityinneighborhoodof MARmodelsimilarto Verbekeetal. ( 2001 ). DobsonandHenderson ( 2003 )introduced graphicalapproachesfordiagnosticassessmentforcontin uoustimedropoutbased onresidualsconditionedondropoutinformation.Finallyw ereviewtheBayesian approachesintroducedinChapter3andultimatelyrecommen dthosebasedontheir invariancetotheextrapolationdistribution. 71

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CHAPTER5 CONCLUSIONSANDEXTENSIONS 5.1SummaryofChapters Inthisdissertation,wehaveexploredBayesianmethodsfor modelselectionandt forincompletelongitudinaldata.Therstpartofthedisse rtationdescribestheBayesian modelselectionforincompletelongitudinaldata.Weexplo retheuseposteriorpredictive loss(PPL)criterionformodelselectionandextendedthe GelfandandGhosh ( 1998 ) criteriontoincompletedata.Weproposedaconvenientwayt ocomparemodelsfor incompletelongitudinaldataanditisinvarianttotheextr apolationdistribution.Wehave shownviasimulationsthatourcriteriondoesagoodjobsele ctingthebestmodels dependingonthechoiceofunivariatefunction T (.) .ComputationallyintensiveDIC basedontheobserveddatalikelihooddoesthebestjobselec tingthebestmodel.Soit ispreferabletochoosewhenitcanbecalculated. Inthesecondpartofthedissertationweproposedawaytoass essthetofthe Bayesianmodelsforincompletelongitudinaldatausingpos teriorpredictivechecks.The checksbasedonobservedandcompletereplication(MARandM NAR)showssimilar behaviorforaxedsamplesizebutcompletereplicationund eranMARassumption demonstratedmorepowerthanobservedreplicationtodetec tthelackoftinourlimited simulations.Howeverwerecommendthecheckbasedonobserv edreplicationdueto theirbeinginvarianttotheextrapolationdistribution. Inthethirdpartofthedissertationwereviewmodeldiagnos ticsforincomplete longitudinaldata.Wemakeconnectionswithdifferentmeth odsforassessingthetof fullyidentiedparametricmodels.Wediscussedthelocali nuencemethodby Verbeke etal. ( 2001 )andtheILSNmethodby Troxeletal. ( 2004 )andshowedtheirconnectionin thecontextofaselectionmodel( DiggleandKenward 1994b ).Theexploratoryresiduals methodsproposedby DobsonandHenderson ( 2003 )forcontinuoustimedropoutwere alsodiscussedalsoandconnectedwiththelocalinuenceme thod. 72

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5.2ExtensionsforModelSelection Itispossibletouseadeviancebasedloss( Chenetal. 2004 )asateststatisticin posteriorpredictivelosscriterion.However,itsharesth esameproblemtotheDICwhen theobserveddatalikelihoodisnotavailableinclosedform .ForCategoricaldatawecan consideraweightedversionofthecriteriaproposedby Chenetal. ( 2004 ).Wealsowant tofurtherexplorethelargesamplebehaviorofthepenaltyt erminoursettings. 5.3ExtensionsforModelFit AnotherapproachforBayesianmodelchecking(notspecica llyinthesettingof incompletedata)wasproposedby Deyetal. ( 1998 ).Thisapproachismorecomputationally intensive.Extendingthisworktooursettingsofobservedr eplicateddataandcomparing thepowerforbothapproacheswouldbeofinterest. 73

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APPENDIXA PPLCRITERIAANDTHEOREM WederivethePPLfor h ( y ) withfulldataandconsiderthelossfunction Lf h ( y rep ), a ; y g = L f h ( y rep ), a g + kL f h ( y ), a g (A–1) When L fg issquarederrorloss,theposteriorpredictiveexpectatio nof L minimizedwith respecttotheaction, a ,is C k ( h )= n X i min a i E h L f h ( y rep i ), a i g + kL f h ( y i ), a i g i = n X i min a i E h f h ( y rep i ) a i g 2 + k f h ( y i ) a i g 2 ] j y i = n X i =1 Var f h ( y rep i ) j y g + n X i =1 min a i h E f h ( y rep i ) j y g a i g 2 + k f h ( y i ) a i g 2 i (A–2) Thevalueof a i thatminimizesis a i = 1 1+ k h E f h ( y rep i ) j y obs r g + k E f h ( y i ) j y obs r g i (A–3) Aftersubstitutingthevalueof a ,weobtain C h ( k )= n X i =1 Var f h ( y rep i ) j y obs r g + k n X i =1 Var f h ( y i ) j y obs r g + k 1+ k n X i =1 h E f h ( y i ) j y obs r g E f h ( y rep i ) j y obs r g i 2 (A–4) ProofofTheoremI:ToproveTheoremI,wewillexamineeachofthethreetermsind ividually.Firstwenote thattheconditionthat p ( ) and p ( y obs r ; O ) arethesameensures p( j y obs ,r) isthe sameforbothmodels.Forclarifyandconcisenessinthebelo w,weleto = obsand m = mis. 74

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Term3:Clearly,undertheconditionsintheTheorem, E f h ( y ) j y o r g isthesamefora degenerateandnon-degeneratemodel.Wenowshowthat E f h ( y rep ) j y obs r g isequal aswell. E f h ( y rep ) j y o r g = Z h ( y rep ) p ( y rep j y o r ) d y rep = ZZZZ h ( y rep ) p ( y rep m j y rep o r rep ) p ( y rep o j r rep ) p ( r rep j ) p ( j y o r ) d d r rep d y rep o d y rep m = ZZZ E f h ( y rep ) j y rep o r rep g p ( y rep o j r rep ) p ( r rep j ) p ( j y o r ) d d r rep d y rep o = ZZZ E n h ( y rep o ,E( y rep m j y rep o r rep )) j y rep o r rep o p ( y rep o j r rep ) p ( r rep j ) p ( j y o r ) d d r rep d y rep o (A–5) Theconditionalexpectation, E( y rep m j y rep o r rep ) isthesameunderthedegenerate andnon-degeneratemodelsandtheothertermsintheintegra ndarethesameunder bothmodels.Term1: GiventhatweshowedforTerm3that E f h ( y rep ) j y obs r g isthesameforboth cases,wejustneedtoshowthat E f h 2 ( y rep ) j y obs r g isthesameorlargerinthe non-degenerate(ND)modelasthedegeneratemodel.Usingas imilardevelopment toTerm3,forthedegenerate(D)modelwehavethefollowinge xpectation(takenwith respectto p ( y rep m j y rep o r rep ) ), E D f h 2 ( y rep ) j y rep o r rep g =E D f h 2 ( y rep o y rep m ) j y rep o r rep g 75

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=E D n h 2 ( y rep o ,E( y rep m j y rep o r rep )) j y rep o r rep o =Var D n h ( y rep o ,E( y rep m j y rep o r rep )) j y rep o r rep o + h E D n h ( y rep o ,E( y rep m j y rep o r rep )) j y rep o r rep oi 2 =0+ h E D n h ( y rep o ,E( y rep m j y rep o r rep )) j y rep o r rep oi 2 Var ND f h ( y rep o y rep m ) j y rep o r rep g + h E D f h ( y rep o ,E( y rep m j y rep o r rep )) j y rep o r rep g i 2 =E ND f h 2 ( y rep ) j y rep o r rep g (A–6) Theexpressionfollowingthelastinequalityholdsgiventh eequalityoftheexpectations E f h ( y rep ) j y rep o r rep g betweenthedegenerateandnon-degeneratemodels. Term2:Clearly, Var f h ( y ) j y obs r g isequaltozeroif p ( y mis j y obs r ) isdegenerate.If p ( y mis j y obs r ) isnotdegenerate,thenthistermwillbepositive. 76

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APPENDIXB GROWTHHORMONEDATAANALYSIS TableB-1.GrowthHormoneTrial:SampleMeans(StandardDev iations). Month Treatment sn s 0612 EG11258(26) 2457(15)68(26)32278(24)90(32)88(32) All3869(25)87(32)88(32) EP1765(32) 2287(52)86(51)33165(24)81(25)73(21) All4066(26)82(26)73(21) TableB-2.DICBasedonObservedDataLikelihoodfortheGHDa taAnalysis Model p D DevianceDIC o SM23.45.629.1 MM130.710.641.3MM225.426.451.8 77

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APPENDIXC CLOSEDFORMEXPECTATIONS ClosedformexpectationsforthemixturemodelinSection4f orseveralchoicesof T ( ) Recall S = P 3j =1 r j .First,wederivetheexpectationsfor T ( r r y )= r 3 y 3 r 1 y 1 E y rep r rep j ( r 3 y 3 r 1 y 1 )=E r rep j ( r 3 E y rep j r rep y 3 r 1 E y rep j r rep y 1 ) =E r rep j ( r 3 ( s ) 3 r 1 ( s ) 1 ) = 3 (3)3 1 (C–1) where, 1 = P 3s =1 s ( s ) 1 and (3)3 =E( Y 3 j S =3) = 3 + 2 32 +( 31 + 32 21 ) (3)1 (C–2) E y rep r rep j f ( r 3 y 3 r 1 y 1 ) 2 g =E r rep j E y rep j r rep f ( r 3 y 3 r 1 y 1 ) 2 g =E r rep j E y rep j r rep f ( r 2 3 y 2 3 )+( r 2 1 y 2 1 ) 2 r 1 r 3 y 1 y 3 g =E r rep j n r 2 3 E y rep j r rep ( y 2 3 )+ r 2 1 E y rep j r rep ( y 2 1 ) 2 r 1 r 3 E y rep j r rep ( y 1 y 3 ) o = 3 ( V 33 + (3) 2 3 )+ 3 X s =1 s ( ( s ) 2 1 + ( s ) 2 1 ) 2 3 f ( 31 + 32 21 ) (3) 2 1 + (3)1 (3)3 g (C–3) where V 33 =Var( Y 3 j S =3)=( 231 + 232 221 +2 31 32 21 ) (3) 2 1 + 232 2 2 + 2 3 .Nowwe derivetheexpectationsfor T ( r r y )= r 3 ( r 3 y 3 r 1 y 1 ) E y rep r rep j f r 3 ( r 3 y 3 r 1 y 1 ) g = 3 ( (3)3 (3)1 ) (C–4) E y rep r rep j f r 3 ( y 3 y 1 ) g 2 =E y rep r rep j ( r 3 y 2 3 + r 3 y 2 1 2 r 3 y 3 y 1 ) = 3 ( V 33 + (3) 2 3 )+ 3 ( (3) 2 1 + (3) 2 1 ) 2 3 f ( 31 + 32 21 ) (3) 2 1 + (3)1 (3)3 g (C–5) 78

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APPENDIXD SIMULATIONRESULTFORPPLCRITERION TableD-1.Simulating(True)ModelMARSelectionModel(SM0 ),MixtureModel1 (MM1)andMixtureModel2(MM2). Model p D DevianceDIC o TrueModel:SM0,SampleSize:50 SM023.61245.81269.4MM142.01253.51295.6MM235.81285.81321.6 TrueModel:SM0,SampleSize:100 SM023.82493.92517.7MM134.42501.62536.0MM228.72567.82596.5 TrueModel:SM0,SampleSize:2000 SM024.049801.649825.6MM134.449944.749979.1MM229.451238.451267.8 TrueModel:MM1,SampleSize:50 SM022.31372.01394.3MM139.91261.01300.9MM235.51575.31610.8 TrueModel:MM1,SampleSize:100 SM023.22745.42768.6MM134.02516.52550.4MM228.63146.63175.2 TrueModel:MM1,SampleSize:2000 SM023.954889.654913.5MM134.150274.550308.6MM229.362820.462849.7 TrueModel:MM2,SampleSize:50 SM022.21380.51402.7MM140.21262.61302.8MM233.61265.11298.7 TrueModel:MM2,SampleSize:100 SM023.22760.92784.1MM133.92517.02550.9MM228.22520.12548.3 TrueModel:MM2,SampleSize:2000 SM024.055232.555256.4MM134.250253.850287.9MM229.450256.950286.2 79

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TableD-2.Under(true)MARSelectionModel(SM0)Simulated DataofSampleSize50: AveragePPLCriteriaandSD(inParenthesis). ModelGOFComplexity C 1 T ( r r y )= r T y T r 1 y 1 SM041.2(4.7)41.6(4.2)82.8(8.7)MM141.2(4.7)49.6(6.8)90.8(10.0)MM241.8(4.8)53.8(7.8)95.6(11.2)T ( r r y )= r T ( r T y T r 1 y 1 ) SM08.4(1.3)8.9(1.3)17.3(2.5)MM18.4(1.3)10.0(1.5)18.5(2.7)MM29.0(1.6)10.5(1.8)19.6(3.3)T ( r r y )= P Tj =1 I f r j =1, r j +1 =0 g r j y j ) I f r 2 =1 g r 1 y 1 SM030.6(5.0)31.0(4.8)61.6(9.6)MM130.6(5.0)37.0(5.1)67.5(9.6)MM231.3(5.1)35.7(4.7)67.0(9.0)T ( r r y )= h P Tj =1 I f r j =1, r j +1 =0 g r j y j ) I f r 2 =1 g r 1 y 1 i 2 SM02151(660)2162(527)4313(1104)MM12179(652)7038(12083)9217(12052)MM22185(648)8370(18109)10555(18059) TableD-3.SimulationunderSelectionModel(SM0)andSampl eSize100:AveragePPL CriteriaandSD(inParenthesis). ModelGOFComplexity C 1 T ( r r y )= r T y T r 1 y 1 SM041.4(3.7)41.7(3.1)83.2(6.5)MM141.5(3.7)43.2(3.5)84.7(7.1)MM241.9(3.7)47.6(4.2)89.5(7.6)T ( r r y )= r T ( r T y T r 1 y 1 ) SM08.7(0.9)8.9(0.9)17.6(1.7)MM18.7(0.9)9.4(0.9)18.1(1.8)MM29.2(1.1)10.0(1.1)19.2(2.1)T ( r r y )= P Tj =1 I f r j =1, r j +1 =0 g r j y j ) I f r 2 =1 g r 1 y 1 SM030.5(4.0)30.8(3.6)61.4(7.5)MM130.5(4.0)33.1(3.9)63.6(7.9)MM231.1(4.0)31.7(3.6)62.8(7.5)T ( r r y )= h P Tj =1 I f r j =1, r j +1 =0 g r j y j ) I f r 2 =1 g r 1 y 1 i 2 SM02122(537)2140(397)4263(877)MM12124(537)2566(607)4690(1132)MM22127(536)2611(616)4739(1139) 80

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TableD-4.Simulating(True)ModelMARSelectionModel(SM0 )andSampleSize2000: AveragePPLCriteriaandSD(inParenthesis). ModelGOFComplexity C 1 T ( r r y )= r T y T r 1 y 1 SM041.7(0.8)41.8(0.7)83.4(1.5)MM141.7(0.8)41.0(0.9)82.7(1.7)MM242.0(0.8)45.2(1.0)87.2(1.8)T ( r r y )= r T ( r T y T r 1 y 1 ) SM08.7(0.2)8.8(0.2)17.5(0.4)MM18.8(0.2)8.8(0.2)17.6(0.4)MM29.0(0.3)9.3(0.2)18.4(0.4)T ( r r y )= P Tj =1 I f r j =1, r j +1 =0 g r j y j ) I f r 2 =1 g r 1 y 1 SM030.8(1.0)30.8(0.9)61.6(1.8)MM130.8(1.0)31.5(1.0)62.4(1.9)MM231.2(1.0)30.1(1.0)61.3(1.9)T ( r r y )= h P Tj =1 f I( r j =1, r j +1 =0) r j y j g I( r 2 =1) r 1 y 1 i 2 SM02116(127)2126(103)4242(216)MM12117(127)2145(130)4262(254)MM22120(127)2186(133)4305(257) TableD-5.Simulating(True)ModelMixtureModel1(MM1)and SampleSize50: AveragePPLCriteriaandSD(inParenthesis). ModelGOFComplexity C 1 T ( r r y )= r T y T r 1 y 1 SM01095.9(67.9)1097.9(63.2)2193.8(129.6)MM11094.4(68.1)1105.7(67.4)2200.1(135.2)MM21662.4(243.7)1852.6(657.9)3515.0(690.0)T ( r r y )= r T ( r T y T r 1 y 1 ) SM0266.1(19.1)289.9(23.8)556.0(42.2)MM1265.7(19.0)271.6(18.8)537.3(37.7)MM2833.8(251.4)1071.7(314.9)1905.6(480.9)T ( r r y )= P Tj =1 f I( r j =1, r j +1 =0) r j y j g I( r 2 =1) r 1 y 1 SM0413.0(47.6)443.2(48.0)856.2(94.1)MM1411.3(47.3)421.6(45.4)832.9(92.1)MM21184.5(312.0)1388.6(270.3)2573.2(460.6)T ( r r y )= h P Tj =1 f I( r j =1, r j +1 =0) r j y j g I( r 2 =1) r 1 y 1 i 2 SM0296118(32710)356876(49052)652994(78363)MM1294008(32332)328074(73249)622082(93555)MM21722024(1986146)5537388(4892633)7259412(6795196) 81

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TableD-6.Simulating(True)ModelMixtureModel1(MM1)and SampleSize100: AveragePPLCriteriaandSD(inParenthesis). ModelGOFComplexity C 1 T ( r r y )= r T y T r 1 y 1 SM01104.2(50.7)1105.8(46.2)2210.0(96.2)MM11102.8(50.9)1107.0(50.1)2209.9(100.9)MM21607.3(207.1)1734.7(459.2)3342.0(456.6)T ( r r y )= r T ( r T y T r 1 y 1 ) SM0267.6(13.3)290.0(15.7)557.5(28.7)MM1267.3(13.3)270.1(12.9)537.4(26.1)MM2771.0(204.6)963.8(205.7)1734.8(330.4)T ( r r y )= P Tj =1 f I( r j =1, r j +1 =0) r j y j g I( r 2 =1) r 1 y 1 SM0415.0(32.8)444.1(33.7)859.1(65.3)MM1413.4(32.5)417.5(32.3)830.9(64.5)MM21092.3(270.7)1214.1(233.7)2306.4(410.1)T ( r r y )= h P Tj =1 f I( r j =1, r j +1 =0) r j y j g I( r 2 =1) r 1 y 1 i 2 SM0297395(23936)352048(31864)649442(53299)MM1295418(23730)303460(22823)598878(46048)MM21151272(1048539)3579020(2478676)4730292(3481461) TableD-7.Simulating(True)ModelMixtureModel1(MM1)and SampleSize2000: AveragePPLCriteriaandSD(inParenthesis). ModelGOFComplexity C 1 T ( r r y )= r T y T r 1 y 1 SM01114.3(11.4)1114.1(10.4)2228.4(21.4)MM11113.2(11.4)1113.7(11.6)2226.8(22.7)MM21601.5(195.4)1644.6(374.1)3246.1(183.6)T ( r r y )= r T ( r T y T r 1 y 1 ) SM0270.3(3.0)291.6(3.5)561.9(6.4)MM1270.0(3.0)270.2(3.1)540.2(5.9)MM2758.6(194.6)873.7(60.4)1632.3(143.2)T ( r r y )= P Tj =1 f I( r j =1, r j +1 =0) r j y j g I( r 2 =1) r 1 y 1 SM0418.9(7.6)445.6(7.7)864.5(14.8)MM1417.5(7.5)417.7(7.6)835.2(14.9)MM21074.9(267.2)1354.4(114.8)2429.3(160.3)T ( r r y )= h P Tj =1 f I( r j =1, r j +1 =0) r j y j g I( r 2 =1) r 1 y 1 i 2 SM0299940(5617)350146(7537)650086(12618)MM1298273(5586)298677(5512)596950(10869)MM21019003(348422)2908665(176399)3927667(480104) 82

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TableD-8.Simulating(True)ModelMixtureModel2(MM2)and SampleSize50: AveragePPLCriteriaandSD(inParenthesis). ModelGOFComplexity C 1 T ( r r y )= r T y T r 1 y 1 SM01626.4(91.9)1657.1(85.7)3283.5(176.2)MM11624.2(92.3)1635.7(90.7)3259.9(182.8)MM21624.8(92.2)1635.3(91.0)3260.1(182.5)T ( r r y )= r T ( r T y T r 1 y 1 ) SM0499.7(26.4)541.2(32.2)1040.9(57.7)MM1499.1(26.3)505.8(25.8)1004.9(51.9)MM2499.7(26.2)505.1(25.4)1004.8(50.9)T ( r r y )= P Tj =1 f I( r j =1, r j +1 =0) r j y j g I( r 2 =1) r 1 y 1 SM0394.5(44.0)448.6(45.2)843.1(86.7)MM1393.9(43.8)403.3(43.7)797.2(86.8)MM2394.6(43.8)402.7(43.5)797.3(86.3)T ( r r y )= h P Tj =1 f I( r j =1, r j +1 =0) r j y j g I( r 2 =1) r 1 y 1 i 2 SM0484041(46637)585615(69484)1069657(111360)MM1480156(46125)515598(55962)995754(99057)MM2482642(46208)512871(56387)995514(97830) TableD-9.Simulating(True)ModelMixtureModel2(MM2)and SampleSize100: AveragePPLCriteriaandSD(inParenthesis). ModelGOFComplexity C 1 T ( r r y )= r T y T r 1 y 1 SM01655.1(66.9)1683.5(59.6)3338.6(125.5)MM11653.6(67.3)1657.3(66.8)3310.9(133.9)MM21653.8(67.3)1656.6(66.8)3310.4(133.9)T ( r r y )= r T ( r T y T r 1 y 1 ) SM0506.1(18.9)546.2(22.2)1052.3(40.5)MM1505.7(18.8)508.6(18.4)1014.3(37.0)MM2506.0(18.9)507.9(18.4)1014.0(37.0)T ( r r y )= P Tj =1 f I( r j =1, r j +1 =0) r j y j g I( r 2 =1) r 1 y 1 SM0408.7(32.3)460.2(30.3)868.8(61.2)MM1408.2(32.1)411.3(31.2)819.5(62.9)MM2408.5(32.2)410.9(31.4)819.5(63.2)T ( r r y )= h P Tj =1 f I( r j =1, r j +1 =0) r j y j g I( r 2 =1) r 1 y 1 i 2 SM0495264(33281)585498(43762)1080763(74380)MM1491650(32949)504644(32542)996295(64911)MM2492923(33312)502432(33131)995355(65514) 83

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TableD-10.Simulating(True)ModelMixtureModel2(MM2)an dSampleSize2000: AveragePPLCriteriaandSD(inParenthesis). ModelGOFComplexity C 1 T ( r r y )= r T y T r 1 y 1 SM01669.6(14.0)1699.4(13.5)3369.0(27.1)MM11668.4(14.0)1668.3(15.1)3336.7(28.7)MM21668.4(14.0)1668.5(14.8)3337.0(28.5)T ( r r y )= r T ( r T y T r 1 y 1 ) SM0511.4(4.0)552.0(5.0)1063.3(8.8)MM1511.0(4.0)511.2(4.3)1022.3(8.1)MM2511.1(4.0)511.2(4.2)1022.3(8.0)T ( r r y )= P Tj =1 f I( r j =1, r j +1 =0) r j y j g I( r 2 =1) r 1 y 1 SM0409.3(6.6)462.0(7.3)871.3(13.4)MM1409.1(6.5)409.1(7.0)818.3(13.2)MM2409.1(6.5)409.3(6.7)818.4(13.0)T ( r r y )= h P Tj =1 f I( r j =1, r j +1 =0) r j y j g I( r 2 =1) r 1 y 1 i 2 SM0497319(7482)580996(10900)1078315(17886)MM1494065(7360)494774(7606)988839(14647)MM2494143(7369)494568(7702)988711(14752) 84

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APPENDIXE ANALYTICCALCULATIONOFPOSTERIORPREDICTIVEPROBABILITI ES Derviationoftheposteriorpredictiveprobabilityforthe observedreplicationfrom Chapter3. Wederivetheanalyticalresultforndingtheposteriorpre dictiveprobabilityfor theobservedreplicationusingthemodelMM1insection3.We onlyconsidera priordistribution[Prioron 2 : 2 N (0, ) ]ontheregressionparameter 2 (for simplication)andkeeptheotherregressionparameter 21 xed.Firstwederivethe posteriordistributionof 2 p ( 2 j y obs S ) / exp [ 1 2 22 ]. exp [ 1 2 2 n 2 X i = n 1 +1 ( y 2 obs 2 21 y 1 obs ) 2 ] / exp 264 1 2 ( n 2 n 1 ) 2 + 1 0B@ 2 ( P n 2 i = n 1 +1 y 2 obs 2 21 2 P n 2 i = n 1 +1 y 1 obs ) ( n 2 n 1 ) 2 + 1 1CA 2 375 p ( 2 j y obs S ) / exp 1 2 B f 2 BZ g 2 Where B 1 = ( n 2 n 1 ) + 2 2 and Z = 1 2 P n 2 i = n 1 +1 y 2 obs 21 P n 2 i = n 1 +1 y 1 obs Thejointposteriorpredictivedistributionof ( Y rep 1 obs Y rep 2 obs S rep ) ,marginalizingoverthe posteriorof 2 p ( y rep 2 obs y rep 1 obs j S rep =2, y obs S ). p ( S rep =2 j S y obs ) = Z p ( 2 j y rep 2 obs y rep 1 obs y obs S ) p ( y rep 2 obs y rep 1 obs j 2 S rep =2, y obs S ). p ( S rep =2 j 2 S y obs ) d 2 = Z exp [ 1 2 ( 1 2 + 1 B )[ 2 ( y rep 2 obs 2 + Z 21 2 y rep 1 obs ) ( 1 2 + 1 B ) ] 2 ]. exp [ 1 2 f ( y rep 2 obs 2 + Z 21 2 y rep 1 obs ) 2 ( 1 2 + 1 B ) + y rep 2 2 obs 2 + 221 2 y rep 2 1 obs 2 21 2 y rep 1 obs y rep 2 obs + y rep 2 1 obs 2 2 2 (2)1 2 2 y rep 1 obs g ] d 2 85

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/ exp [ 1 2 f y rep 2 1 obs ( 1 2 2 + 221 2 221 B 2 ( 2 + B ) ) 2 y rep 1 obs ( (2)1 2 2 + 21 2 y rep 2 obs Z 21 B ( 2 + B ) 21 y rep 2 obs B 2 ( 2 + B ) )] exp [ 1 2 f y rep 2 2 obs B 2 ( 2 + B ) 2 Zy rep 2 obs B ( 2 + B ) + y rep 2 2 obs 2 g ] = exp 26664 1 2 ( 1 2 2 + 221 2 + B ) 0BB@ y rep 1 obs ( (2)1 2 2 Z 21 B ( 2 + B ) + 21 y rep 2 obs 2 + B ) ( 1 2 2 + 221 2 + B ) 1CCA 2 37775 exp 24 1 2 f y rep 2 2 obs B 2 ( 2 + B ) 2 Zy rep 2 obs B ( 2 + B ) + y rep 2 2 obs 2 g 35 exp 2664 1 2 f ( (2)1 2 2 Z 21 B ( 2 + B ) + 21 y rep 2 obs 2 + B ) 2 ( 1 2 2 + 221 2 + B ) g 3775 Thenalstepistomarginalizeover Y rep 1 obs toobtainthejointposteriorpredictive distributionof Y rep 2 obs and S rep =2 p ( y rep 2 obs j S rep =2, y obs S ). p ( S rep =2 j S y obs ) = Z p ( y rep 1 obs j y rep 2 obs S rep =2, y obs S ). p ( y rep 2 obs j S rep =2, y obs S ). p ( S rep =2 j S y obs ) dy rep 1 obs / exp 1 2 ( 1 2 + B + 221 2 2 )( y rep 2 obs ( ZB + 21 (2)1 )) 2 # (E–1) Nowdene (2)1 = 1 + and y rep 2 obs = 1 P I ( S rep i =2) P ni =1 y rep 2 i I ( S rep i =2). Wenotethat P I ( S rep i =2) n P and n P I ( S rep i =2) P 1 y rep 2 i I ( S rep i =2)= X N (( ZB + 21 ( 1 + )), 2 + B + 221 2 2 ), n P I ( S rep i =2) y rep 2 i I ( S rep i =2) d 1 X where X N (( ZB + 21 ( 1 + )), 2 + B + 221 2 2 ). (E–2) 86

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Thus, E ( Y rep 2, obs )=( ZB + 21 ( 1 + )) .Usingtheaboveresultsweobtain, P ( y rep 2 obs > y 2 obs j y obs ) 1 p n y 2 obs ( ZB + 21 ( 1 + )) ( 2 + B + 221 2 2 ) 1 2 Derivationoftheposteriorpredictiveprobabilityforthe completereplicationfrom Chapter3. Herewederivetheposteriorpredictiveprobabilityforthe completereplication usingthemodelMM1.Asforobservedreplicationpriordistr ibutionontheregression parameter 2 (forsimplication)is 2 N (0, ) Theposteriorfor 2 issameasbefore.Conditionedondataand 2 ,withtheother parameterxedthejointdistributionof ( Y rep 2 com Y 1 S rep ) is, p ( y rep 2 com j y 1 2 S rep y obs S ) p ( y 1 j S rep y obs S ) p ( S rep j y obs S ) / 1 p 2 exp 1 2 2 2 f y rep 2 com 2 21 y 1 g 2 1 2 exp 1 2 2 2 f y 1 (2)1 g 2 I ( S rep =2) .(1 ) 1 p 2 exp 1 2 2 2 f y rep 2 com 2 21 y 1 g 2 1 1 exp 1 2 2 1 f y 1 (1)1 g 2 I ( S rep =1) Nowweintegrateout S rep p ( y rep 2 com j y 1 2 y obs S ) p ( y 1 j y obs S ) / 1 2 p 2 exp 24 1 2 f y 2 1 ( 1 2 2 + 221 2 ) 2 y 1 ( (2)1 2 2 2 21 2 + 21 y rep 2 com 2 )+ y rep 2 2 com 2 + 2 2 2 2 2 2 y rep 2 com g 35 +(1 ) 1 1 p 2 exp [ 1 2 f y 2 1 ( 1 2 1 + 221 2 ) 2 y 1 ( (1)1 2 1 2 21 2 + 21 y rep 2 com 2 21 2 ) 2 2 y rep 2 com + 2 2 +2 2 2 + y rep 2 2 com 2 + 2 2 2 2 2 2 y rep 2 com g ]. Nowweintegrateout Y 1 toobtaintheconditionaldistributionof Y rep 2 com j 2 Y obs S p ( y rep 2 com j 2 y obs S ) / 1 2 p 2 exp [ 1 2 f y rep 2 2 com 2 + 221 2 2 2 y rep 2 com ( (2)1 21 2 + 221 2 2 + 2 2 2 221 2 2 2 ( 2 + 221 2 2 ) )+2 2 (2)1 21 2 + 221 2 2 87

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+ 2 2 2 2 2 221 2 2 2 ( 2 + 221 2 2 ) g ]+(1 ) 1 1 p 2 exp [ 1 2 f y rep 2 2 com 2 + 221 2 2 2 y rep 2 com ( (1)1 21 2 + 221 2 1 + 2 221 2 1 2 ( 2 + 221 2 2 ) )+ 2 2 2 2 2 221 2 1 2 ( 2 + 221 2 1 ) 2 2 ( 2 2 + y rep 2 com 2 + 221 2 1 + 221 2 1 2 ( 2 + 221 2 1 ) 21 (1)1 ( 2 + 221 2 1 ) ) g ]. Ultimatelytoobtainaformfortheposteriorpredictivepro bability,weneedthejoint distributionof Y rep 2 com and Y 2 mis .NoteforMM1, y 2 mis j 2 N ( 2 ++ 21 y 1 obs 2 ) ,so p ( y 2 mis j 2 y obs S ) / exp [ 1 2 2 ( y 2 miss 2 21 y 1 obs ) 2 ]. Thusthejointdistributionof ( Y rep 2 com Y 2 mis 2 ) conditionedon ( Y obs S ) is, p ( y rep 2 com j 2 y obs S ) p ( y 2 mis j 2 y obs S ) p ( 2 j y obs S ) / 1 2 p 2 exp [ 1 2 f 2 2 ( 1 2 + 1 B + 1 2 + 221 2 2 ) 2 2 ( Z 2 21 y 1 obs 2 + y rep 2 com 2 + 221 2 2 21 (2)1 2 + 221 2 2 + y 2 mis 2 )+ y rep 2 2 com 2 + 221 2 2 2 y rep 2 com 21 (2)1 2 + 221 2 2 2 y 2 mis 2 + y 2 2 mis 2 2 21 y 2 mis y 1 obs 2 g ] +(1 ) 1 1 p 2 exp [ 1 2 f 2 2 ( 1 2 + 1 B + 1 2 + 221 2 1 ) 2 2 ( Z 2 21 y 1 obs 2 + y rep 2 com 2 + 221 2 1 2 + 221 2 1 21 (1)1 2 + 221 2 1 + y 2 mis 2 )+ y rep 2 2 com 2 + 221 2 1 2 y rep 2 com 21 (1)1 2 + 221 2 1 2 y 2 mis 2 + y 2 2 mis 2 2 y 2 mis 2 2 21 y 2 mis y 1 obs 2 g ] Let, A 1 = 1 2 + 1 B + 1 2 + 221 2 2 and A 2 = 1 2 + 1 B + 1 2 + 221 2 1 Nowweintegrateovertheposteriorof 2 p ( y rep 2 com y 2 mis j y obs S ) / 1 2 p 2 exp [ 1 2 f y rep 2 2 com ( 1 2 + 221 2 2 1 ( 2 + 221 2 2 ) 2 A 1 ) 2 y rep 2 com ( Z ( 2 + 221 2 2 ) A 1 ( 2 + 221 2 2 ) 2 A 1 21 y 1 obs ( 2 + 221 2 2 ) 2 A 1 21 (2)1 ( 2 + 221 2 2 ) 2 A 1 + 21 (2)1 ( 2 + 221 2 2 ) 2 )+ y 2 2 mis ( 1 2 1 2 2 A 1 ) 88

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2 y 2 mis ( Z 2 A 1 2 2 A 1 21 y 1 obs 2 2 A 1 21 (2)1 ( 2 + 221 2 2 ) 2 A 1 + 2 + 21 y 1 obs 2 ) 2 y rep 2 com y 2 mis ( 1 ( 2 + 221 2 2 ) 2 A 1 ) g ]+(1 ) 1 1 p 2 exp [ 1 2 f y rep 2 2 com ( 1 2 + 221 2 1 1 ( 2 + 221 2 1 ) 2 A 2 ) 2 y rep 2 com ( Z ( 2 + 221 2 1 ) A 2 ( 2 + 221 2 1 ) 2 A 2 ( 2 + 221 2 1 ) 2 A 2 21 y 1 obs ( 2 + 221 2 1 ) 2 A 2 21 (1)1 ( 2 + 221 2 1 ) 2 A 2 + 21 (1)1 ( 2 + 221 2 1 ) 2 + ( 2 + 221 2 1 ) ) + y 2 2 mis ( 1 2 1 2 2 A 2 ) 2 y 2 mis ( Z 2 A 2 2 2 A 2 2 A 2 ( 2 + 221 2 1 ) 21 y 1 obs 2 2 A 2 21 (1)1 ( 2 + 221 2 1 ) 2 A 2 + 2 + 21 y 1 obs 2 ) 2 y rep 2 com y 2 mis ( 1 ( 2 + 221 2 1 ) 2 A 2 ) g ] Tosimplifytheabovedene a 1 = ( 1 2 + 221 2 2 1 ( 2 + 221 2 2 ) 2 A 1 ) b 1 = ( 1 2 1 2 2 A 1 ) c 1 = ( Z ( 2 + 221 2 2 ) A 1 ( 2 + 221 2 2 ) 2 A 1 21 y 1 obs ( 2 + 221 2 2 ) 2 A 1 21 (2)1 ( 2 + 221 2 2 ) 2 A 1 + 21 (2)1 ( 2 + 221 2 2 ) 2 ) d 1 = ( Z 2 A 1 2 2 A 1 21 y 1 obs 2 2 A 1 21 (2)1 ( 2 + 221 2 2 ) 2 A 1 + 2 + 21 y 1 obs 2 ) e 1 = ( 1 ( 2 + 221 2 2 ) 2 A 1 ) p ( y rep 2 com y 2 mis j y obs S ) / 1 p 2 1 2 exp [ 1 2 f y 2 2 mis a 1 + y rep 2 2 com b 1 2 y 2 mis c 1 2 y rep 2 com d 1 2 y rep 2 com y 2 mis e 1 g ] +(1 ). 1 p 2 1 1 exp [ 1 2 f y 2 2 mis a 2 + y rep 2 2 com b 2 2 y 2 mis c 2 2 y rep 2 com d 2 2 y rep 2 com y 2 mis e 2 g ]. Wecanwritetheposteriorpredictiveprobabilityofintere stas, P ( y rep 2 com > y 2 com j y obs S ) = P ( y rep 2 com > 1 n f n 1 X 1 y 2 mis + n X n 1 +1 y 2 obs gj y obs S ) = P ( y rep 2 com 1 n n 1 X 1 y 2 mis > 1 n n X n 1 +1 y 2 obs j y obs S ) = P ( 1 n ( X y rep 2 com I ( S i =1)+ X y rep 2 com I ( S i =2)) 1 n n 1 X 1 y 2 mis > 1 n n X n 1 +1 y 2 obs j y obs S ) = P ( 1 n ( X y rep 2 com I ( S i =1) n 1 X 1 y 2 mis )+ 1 n X y rep 2 com I ( S i =2) > 1 n n X n 1 +1 y 2 obs j y obs S ). 89

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Wenowdene, U = y rep 2 com y 2 mis and V = y rep 2 com + y 2 mis ,so, y rep 2 com = U + V 2 and y 2 mis = V U 2 Thejointdistributionof U and V is p ( U V j y obs S ) / 1 p 2 exp [ 1 2 f ( U + V ) 2 4 a 1 + ( V U ) 2 4 b 1 2 ( U + V ) 2 c 1 2 V U 2 d 1 2 V 2 U 2 4 e 1 g ] +(1 ) 1 p 2 exp [ 1 2 f ( U + V ) 2 4 a 2 + ( V U ) 2 4 b 2 2 ( U + V ) 2 c 2 2 V U 2 d 2 2 V 2 U 2 4 e 2 g ] / 1 p 2 exp [ 1 2 f V 2 ( a 1 4 + b 1 4 e 1 2 ) 2 V ( Ub 1 4 Ua 1 4 + c 1 2 + d 1 2 ) 2 U ( d 1 2 c 1 2 ) + U 2 ( a 1 4 + b 1 4 + e 1 2 ) g ]+(1 ) 1 p 2 exp [ 1 2 f V 2 ( a 2 4 + b 2 4 e 2 2 ) 2 V ( Ub 2 4 Ua 2 4 + c 2 2 + d 2 2 ) 2 U ( d 2 2 c 2 2 )+ U 2 ( a 2 4 + b 2 4 + e 2 2 ) g ]. IntegratingoutV, p ( U j y obs S ) / 1 p 2 1 2 exp [ 1 2 f U 2 ( a 1 4 + b 1 4 + ( a 1 b 1 ) 2 4( a 1 + b 1 2 c 1 ) + e 1 2 ) 2 U ( d 1 2 c 1 2 + c 1 ( a 1 b 1 ) 2( a 1 + b 1 2 c 1 ) + d 1 ( a 1 b 1 ) 2( a 1 + b 1 2 c 1 ) ) g ]+(1 ). 1 p 2 1 1 exp [ 1 2 f U 2 ( a 2 4 + b 2 4 + ( a 2 b 2 ) 2 4( a 2 + b 2 2 c 2 ) + e 2 2 ) 2 U ( d 2 2 c 2 2 + c 2 ( a 2 b 2 ) 2( a 2 + b 2 2 c 2 ) + d 2 ( a 2 b 2 ) 2( a 2 + b 2 2 c 2 ) g ]. Usingalltheseresultsweobtain, P ( y rep 2 com > y 2 com j y obs S ) 1 0B@ 1 n P nn 1 +1 y 2 obs E 1 E 2 [ V 1 ] 1 = 2 p ( n (1 )) + [ V 2 ] 1 = 2 p ( n ( )) 1CA where E 1 = ( d 1 2 c 1 2 + c 1 ( a 1 b 1 ) 2( a 1 + b 1 2 c 1 ) + d 1 ( a 1 b 1 ) 2( a 1 + b 1 2 c 1 ) ) ( a 1 4 + b 1 4 + ( a 1 b 1 ) 2 4( a 1 + b 1 2 c 1 ) + e 1 2 ) + (1 ) ( d 2 2 c 2 2 + c 2 ( a 2 b 2 ) 2( a 2 + b 2 2 c 2 ) + d 2 ( a 2 b 2 ) 2( a 2 + b 2 2 c 2 ) ) ( a 2 4 + b 2 4 + ( a 2 b 2 ) 2 4( a 2 + b 2 2 c 2 ) + e 2 2 ) E 2 = (2)1 21 2 2 ( 2 + B ) 1 Z 221 B ( 2 + B ) 2 1 + ZB ( 2 + B ) ( 1 2 + B 221 ( 2 + B ) 2 1 ) +(1 ). (1)1 21 2 1 ( 2 + B ) 2 Z 221 B ( 2 + B ) 2 2 + ZB ( 2 + B ) 221 ( 2 + B ) 2 2 B 2 ( 2 + B ) ( 1 2 + B 221 ( 2 + B ) 2 2 ) V 1 = [ f ( d 1 2 c 1 2 + c 1 ( a 1 b 1 ) 2( a 1 + b 1 2 c 1 ) + d 1 ( a 1 b 1 ) 2( a 1 + b 1 2 c 1 ) ) ( a 1 4 + b 1 4 + ( a 1 b 1 ) 2 4( a 1 + b 1 2 c 1 ) + e 1 2 ) E 1 g 2 +( a 1 4 + b 1 4 + ( a 1 b 1 ) 2 4( a 1 + b 1 2 c 1 ) + e 1 2 ) 1 ] 90

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+ (1 )[ f ( d 2 2 c 2 2 + c 2 ( a 2 b 2 ) 2( a 2 + b 2 2 c 2 ) + d 2 ( a 2 b 2 ) 2( a 2 + b 2 2 c 2 ) ) ( a 2 4 + b 2 4 + ( a 2 b 2 ) 2 4( a 2 + b 2 2 c 2 ) + e 2 2 ) E 1 g 2 +( a 2 4 + b 2 4 + ( a 2 b 2 ) 2 4( a 2 + b 2 2 c 2 ) + e 2 2 ) 1 ] Formofthegradientoftheposteriorpredictiveprobabilit yundercompletereplication withrespectto fromChapter3. @ @ P ( y rep 2 com > y 2 com j y obs S ) @ @ 0@ 1 n P nn 1 +1 y 2 obs ( k 1 + k 3 ) (1 )( k 2 + k 4 ) [ f (1 )( k 1 k 2 ) 2 + k 5 g +(1 ) f ( k 2 k 1 ) 2 + k 6 g ] 0.5 p n (1 ) + [ f (1 )( k 3 k 4 ) 2 + k 7 g +(1 ) f ( k 4 k 3 ) 2 + k 8 g ] 0.5 p n ( ) 1A = 0B@ P nn 1 +1 y 2 obs ( k 1 + k 3 ) (1 )( k 2 + k 4 ) (2( k 1 k 2 ) 2 (1 )+ k 5 +(1 ) k 6 ) 0.5 p (1 ) n + (2( k 3 k 4 ) 2 (1 )+ k 7 +(1 ) k 8 ) 0.5 p ( ) n 1CA f f ( k 2 + k 4 ) ( k 1 + k 3 ) g ( (2( k 1 k 2 ) 2 (1 )+ k 5 +(1 ) k 6 ) 0.5 p (1 ) n + (2( k 3 k 4 ) 2 (1 )+ k 7 +(1 ) k 8 ) 0.5 p ( ) n ) f P nn 1 +1 y 2 obs ( k 1 + k 3 k 2 k 4 ) ( k 2 + k 4 ) g n f (1 2 )(1 ))2( k 1 k 2 ) 2 +(1 2 ) k 5 2(1 ) k 6 g ( (2( k 1 k 2 ) 2 (1 )+ k 5 +(1 ) k 6 ) 0.5 p (1 ) n + (2( k 3 k 4 ) 2 (1 )+ k 7 +(1 ) k 8 ) 0.5 p ( ) n ) 2 g where k 1 = ( d 1 2 c 1 2 + c 1 ( a 1 b 1 ) 2( a 1 + b 1 2 c 1 ) + d 1 ( a 1 b 1 ) 2( a 1 + b 1 2 c 1 ) ) ( a 1 4 + b 1 4 + ( a 1 b 1 ) 2 4( a 1 + b 1 2 c 1 ) + e 1 2 ) k 2 = ( d 2 2 c 2 2 + c 2 ( a 2 b 2 ) 2( a 2 + b 2 2 c 2 ) + d 2 ( a 2 b 2 ) 2( a 2 + b 2 2 c 2 ) ) ( a 2 4 + b 2 4 + ( a 2 b 2 ) 2 4( a 2 + b 2 2 c 2 ) + e 2 2 ) k 3 = (2)1 21 2 2 ( 2 + B ) 1 Z 221 B ( 2 + B ) 2 1 + ZB ( 2 + B ) ( 1 2 + B 221 ( 2 + B ) 2 1 ) k 4 = (1)1 21 2 1 ( 2 + B ) 2 Z 221 B ( 2 + B ) 2 2 + ZB ( 2 + B ) 221 ( 2 + B ) 2 2 B 2 ( 2 + B ) ( 1 2 + B 221 ( 2 + B ) 2 2 ) Formofthegradientoftheposteriorpredictiveprobabilit yundercompletereplication withrespectto fromChapter3 @ @ P ( y rep 2 com > y 2 com j y obs S ) @ @ 0B@ 1 n P nn 1 +1 y 2 obs E 1 E 2 [ V 1 ] 1 = 2 p ( n (1 )) + [ V 2 ] 1 = 2 p ( n ( )) 1CA = 0B@ 1 n P nn 1 +1 y 2 obs E 1 E 2 [ V 1 ] 1 = 2 p ( n (1 )) + [ V 2 ] 1 = 2 p ( n ( )) 1CA ( [ V 1 ] 1 = 2 p ( n (1 )) + [ V 2 ] 1 = 2 p ( n ( )) )( @ E 1 @ @ E 2 @ ) ( 1 n n X n 1 +1 y 2 obs E 1 E 2 )( V 3 = 2 1 2 p n (1 ) @ V 1 @ V 3 = 2 2 2 p n ( ) @ V 2 @ ), where @ c 1 @ = 1 ( 2 + 221 2 1 ) 2 A 1 @ d 1 @ = 1 2 2 A 1 + 1 2 @ c 2 @ = 1 ( 2 + 221 2 2 ) 2 A 2 @ d 2 @ = 1 2 2 A 2 + 1 2 91

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@ E 2 @ = (1 ) f 221 2 1 ( 2 + B )( 2 + B + 221 2 1 ) + B 2 ( 2 + B ) 1 ( 2 + B + 221 2 1 ) g = (1 )( B + 221 2 1 ) 2 @ E 1 @ = (( a 1 4 + b 1 4 + ( a 1 b 1 ) 2 4( a 1 + b 1 2 c 1 ) + e 1 2 ) f @ c 1 @ (2( a 2 1 b 2 1 )+4 d 1 ( a 1 b 1 ) 1 2 ) g ( d 1 2 c 1 2 + c 1 ( a 1 b 1 ) 2( a 1 + b 1 2 c 1 ) + d 1 ( a 1 b 1 ) 2( a 1 + b 1 2 c 1 ) )(8( a 1 b 1 ) 2 @ c 1 @ ))+(1 )( a 2 4 + b 2 4 + ( a 2 b 2 ) 2 4( a 2 + b 2 2 c 2 ) + e 2 2 ) f @ c 2 @ (2( a 2 2 b 2 2 )+4 d 2 ( a 2 b 2 ) 1 2 ) g ( d 2 2 c 2 2 + c 2 ( a 2 b 2 ) 2( a 2 + b 2 2 c 2 ) + d 2 ( a 2 b 2 ) 2( a 2 + b 2 2 c 2 ) )(8( a 2 b 2 ) 2 @ c 2 @ ) @ V 1 @ = 2 f ( d 1 2 c 1 2 + c 1 ( a 1 b 1 ) 2( a 1 + b 1 2 c 1 ) + d 1 ( a 1 b 1 ) 2( a 1 + b 1 2 c 1 ) ) ( a 1 4 + b 1 4 + ( a 1 b 1 ) 2 4( a 1 + b 1 2 c 1 ) + e 1 2 ) E 1 g [( a 1 4 + b 1 4 + ( a 1 b 1 ) 2 4( a 1 + b 1 2 c 1 ) + e 1 2 ) f @ c 1 @ (2( a 2 1 b 2 1 )+4 d 1 ( a 1 b 1 ) 1 2 ) g ( d 1 2 c 1 2 + c 1 ( a 1 b 1 ) 2( a 1 + b 1 2 c 1 ) + d 1 ( a 1 b 1 ) 2( a 1 + b 1 2 c 1 ) )(8( a 1 b 1 ) 2 @ c 1 @ ) @ E 1 @ ]+ 8( a 1 b 1 ) 2 @ c 1 @ +2(1 ). f ( d 2 2 c 2 2 + c 2 ( a 2 b 2 ) 2( a 2 + b 2 2 c 2 ) + d 2 ( a 2 b 2 ) 2( a 2 + b 2 2 c 2 ) ) ( a 2 4 + b 2 4 + ( a 2 b 2 ) 2 4( a 2 + b 2 2 c 2 ) + e 2 2 ) E 1 g [( a 2 4 + b 2 4 + ( a 2 b 2 ) 2 4( a 2 + b 2 2 c 1 ) + e 2 2 ) f @ c 2 @ (2( a 2 2 b 2 2 )+4 d 2 ( a 2 b 2 ) 1 2 ) g ( d 2 2 c 2 2 + c 2 ( a 2 b 2 ) 2( a 2 + b 2 2 c 2 ) + d 2 ( a 2 b 2 ) 2( a 2 + b 2 2 c 2 ) )(8( a 2 b 2 ) 2 @ c 2 @ ) @ E 1 @ ]+8( a 2 b 2 ) 2 @ c 2 @ @ V 2 @ = 2 f ( 1 + ) 21 + ZB E 2 g ( @ E 2 @ ) (1 ) ( B + 221 2 1 ) 2 92

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REFERENCES Carlin,B.andChib,S.(1995).BayesianmodelchoiceviaMar kovChainMonteCarlo methods. JournalOfRoyalStatisticalSociety,SeriesB 57, 473–484. Celeux,G.,Forbes,F.,Robert,C.,andTitterington,M.(20 06).DevianceInformation Criteriaformissingdatamodels. BayesianAnalysis 1, 651–674. Chen,M.,Dey,D.,andIbrahim,J.(2004).Bayesiancriterio nbasedmodelassessment forcategoricaldata. Biometrika 91, 45–63. Chib,S.andJeliazkov,I.(2001).Marginallikelihoodfrom theMetropolis-Hastings output. JournalOftheAmericanStatisticalAssociation 96, 270–281. Chib,S.andJeliazkov,I.(2005).AcceptrejectMetropolis Hastingssamplingand marginallikelihoodestimation. StatisticaNederlandica 59, 30–44. Cook,R.D.(1986).Assessmentoflocalinuence. JournalOfRoyalStatisticalSociety, SeriesB 48, 133–169. Daniels,M.andHogan,J.(2000).Reparameterizingthepatt ernmixturemodelfor sensitivityanalysesunderinformativedropout. Biometrics 56, 1241–1248. Daniels,M.andHogan,J.(2008). MissingDatainLongitudinalStudies:Strategiesfor BayesianModelingandSensitivityAnalysis .Chapman&Hall. Daniels,M.J.,Chatterjee,A.S.,andWang,C.(2012).Bayes ianmodelselectionfor incompletedatausingtheposteriorpredictivedistributi on. Biometrics 68, 1055– 1063. Dey,D.,Gelfand,A.,Swartz,T.,andVlachos,P.(1998).Asi mulation-intensive approachforcheckinghierarchicalModels. Test 7, 325–346. Diggle,D.andKenward,M.(1994a).Informativedropoutlon gitudinaldataanalysis. AppliedStatistics 43, 49–93. Diggle,P.andKenward,M.(1994b).Informativedrop-outin longitudinaldataanalysis. AppliedStatistics 43, 49–93. Dobson,A.andHenderson,R.(2003).Diagnosticsforjointl ongitudinalanddropout timemodeling. Biometrics 59, 741–751. Fitzmaurice,G.andLaird,N.(2000).Generalizedlinearmi xturemodelsforhandling nonignorabledrop-outsinlongitudinalStudies. Biostatistics 1, 141–156. Fitzmaurice,G.,Molenberghs,G.,andLipsitz,S.(1995).R egressionmodelsfor longitudinalbinaryresponseswithinformativedrop-outs JournalOfRoyalStatistical Society,SeriesB 57, 691–704. 93

PAGE 94

Geisser,S.andEddy,W.(1979).Apredictiveapproachtomod elselection. Journalof theAmericanStatisticalAssociation 74, 153–160. Gelfand,A.andGhosh,S.(1998).Modelchoice:Aminimumpos teriorpredictiveloss approach. Biometrika 85, 1–11. Gelman,A.,Mechelen,I.,Verbeke,G.,Heitjan,D.,andMeul ders,M.(2005).Multiple imputationformodelchecking:completed-dataplotswithm issingandlatentData. Biometrics 61, 74–85. Gelman,A.,Meng,X.,andStern,H.(1996).Posteriorpredic tiveassessmentofmodel fitnessviarealizeddiscrepancies. StatisticalSinica 6, 733–807. Heckman,J.(1976).TheCommonstructureofStatisticalmod elsoftruncation,sample selectionandlimitedindependentvariablesandasimplees timatorforsuchmodels. AnnalsofEconomicandSocialMeasurement 5, 120–137. Henderson,R.,Diggle,P.,andDobson,A.(2000).Jointmode lingoflongitudinal measurementsandeventtimedata. Biostatistics 1, 465–480. Hogan,J.andLaird,N.(1997).Mixturemodelsforthejointd istributionofreapted measuresandeventtimes. StatisticsinMedicine 16, 239–257. Hogan,J.,Lin,X.,andHerman,B.(2004).Mixturesofvaryin gcoefcientmodelsfor longitudinaldatawithdiscreteorcontinuousnonignorabl eDropout. Biometrics 60, 854–864. Ibrahim,J.,Chen,M.,andSinha,D.(2001).Criterion-base dmethodsforBayesian modelassessment. StatisticalSinica 11, 419–443. Ibrahim,J.andLaud,P.(1994).Apredictiveapproachtothe analysisofdesigned Expreiments. JournalOftheAmericanStatisticalAssociation 89, 309–319. Ibrahim,J.,Zhu,H.,andTang,N.(2008).Modelselectioncr iteriaformissing-data problemsusingtheEMalgorithm. JournalOftheAmericanStatisticalAssociation 103, 1648–1658. Johnson,V.(2005).BayesfactorsbasedonTestStatistics. JournalOfRoyalStatistical Society,SeriesB 67, 689–701. Johnson,V.andHu,J.(2009).Bayesianmodelselectionusin gTestStatistics. Journal OfRoyalStatisticalSociety,SeriesB 71, 143–158. Kass,R.andRaftery,A.(1995).BayesFactors. JournalOftheAmericanStatistical Association 90, 773–795. Kenward,M.,Molenberghs,G.,andThijs,H.(2003).Pattern -mixturemodelswithproper timedependence. Biometrika 90, 53–71. 94

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Laud,P.andIbrahim,J.(1995).Predictivemodelselection JournalOfRoyalStatistical Society,SeriesB 57, 247–262. Little,R.(1993).Pattern-mixturemodelsformultivariat eincompletedata. Journalofthe AmericanStatisticalAssociation 88, 125–134. Little,R.(1994).Aclassofpattern-mixturemodelsfornor malincompleteData. Biometrika 81, 471–483. Little,R.andRubin,D.(1987). StatisticalAnalysiswithMissingData .Wiley. Little,R.andWang,Y.(1996).Pattern-mixturemodelsform ultivariateincompletedata withcovariates. Biometrics 95, 98–111. Meng,X.L.(1994).PosteriorpredictiveP-values. AnnalsofStatistics 22, 1142–1160. Molenberghs,G.andKenward,M.(2007). MissingDatainClinicalTrials .Wiley. Rizopoulos,D.,Verbeke,G.,andMolenberghs,G.(2008).Sh aredparametermodels underrandomeffectsmisspecication. Biometrika 95, 63–74. Robins,J.,Vaart,A.V.D.,andVentura,V.(2000).Asymptot icdistributionofP-values incompositenullmodels. JournaloftheAmericanStatisticalAssociation 95, 1143– 1156. Rubin,D.(1976).Inferenceandmissingdata. Biometrika 63, 581–592. Rubin,D.(1984).Bayesianlyjustiableandrelevantfrequ encycalculationsforthe appliedstatistician. AnnalsofStatistics 12, 1151–1172. Spiegelhalter,D.,Best,N.,Carlin,B.,andVanDerLinde,A .(2002).Bayesianmeasures ofmodelcomplexityandfit. JournalOfRoyalStatisticalSociety,SeriesB 64, 583–639. Thijs,H.,Molenberghs,G.,andVerbeke,G.(2000).Themilk proteintrial:inuence analysisofthedropoutprocess. BiometricalJournal 42, 617–646. Troxel,A.B.,Ma,G.,andHeitjan,D.F.(2004).Anindexoflo calsensitivityto nonignorability. StatisticaSinica 14, 1221–1237. Verbeke,G.,Molenberghs,G.,Thijs,H.,Lesaffre,E.,andK enward,M.(2001). AnalyzingincompletelongitudinalClinicalTrialdata. Biometrics 57, 7–14. Wang,C.andDaniels,M.(2011).ANoteonMAR,identifyingre strictions,andsensitivity analysisinpatternmixturemodelswithandwithoutcovaria tesforincompletedata. Biometrics 67, 810–818. Wu,M.andCarroll,R.(1988).Estimationandcomparisonofc hangesinthepresence ofinformativerightcensoringbymodelingthecensoringpr ocess. Biometrics 44, 175 –188. 95

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Zellner,A.(1975).Bayesiananalysisofregressionerrort erms. JournaloftheAmerican StatisticalAssociation 70, 138–144. 96

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BIOGRAPHICALSKETCH ArkenduSekharChatterjeehasreceivedhisBachelorofScie nceinstatisticsfrom St.XaviersCollege,UniversityofCalcutta,IndiaandMast erofScienceinstatisticsfrom UniversityofCalcutta,India.AfternishingMasterofSci enceArkendujoinedUniversity ofFlorida,withmajorstatistics.Arkenduworkedasasumme rinternatmethodology groupofNovartisPharmaceuticalinsummer,2010andatAmge nINCinsummer, 2011.Arkendu'sresearchisfocusedonBayesianmodelselec tionandtforincomplete longitudinaldata.Thesetypeofworkhasabroadapplicatio ninclinicaltrialsandhuman diseases.ArkenduisamemberofAmericanStatisticalAssoc iationandEasternNorth AmericanRegion/InternationalBiometricSociety. 97