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Marginalized Regression Models for Longitudinal Categorical Data

Permanent Link: http://ufdc.ufl.edu/UFE0021244/00001

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Title: Marginalized Regression Models for Longitudinal Categorical Data
Physical Description: 1 online resource (105 p.)
Language: english
Creator: Lee, Keunbaik
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: generalized, qol, random, transition
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: Generalized linear models with serial dependence are often used for short longitudinal series. Heagerty (1999,2002) has proposed marginalized regression models for the analysis of longitudinal binary data. The focus of our research is on marginalized regression models for longitudinal ordinal response data. We propose likelihood-based Markov models and marginalized random effects models for the analysis of longitudinal ordinal data. Both models allow direct estimation of marginal regression parameters and the interpretation of these parameters is invariant to specification of the dependence. We show estimation of covariate effects is reasonably robust to mis-specification of dependence. Fisher-scoring and Quasi-Newton algorithms are developed for estimation. Methods are illustrated on quality of life data from a recent colorectal cancer clinical trial. We outline a selection model approach to deal with nonignorable dropout in this data. Many of the dropouts were due to tumor progression or death which further complicates the analysis. We propose two approaches to handle this type of dropout. The first uses a mixture model approach and is straightforward, but the causal interpretation of treatment effects via randomization is compromised. The second approach is more complex, using the tools of principal stratification, but preserves the causal interpretation of treatment effects.
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 Keunbaik Lee.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Daniels, Michael J.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-08-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021244:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021244/00001

Material Information

Title: Marginalized Regression Models for Longitudinal Categorical Data
Physical Description: 1 online resource (105 p.)
Language: english
Creator: Lee, Keunbaik
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: generalized, qol, random, transition
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: Generalized linear models with serial dependence are often used for short longitudinal series. Heagerty (1999,2002) has proposed marginalized regression models for the analysis of longitudinal binary data. The focus of our research is on marginalized regression models for longitudinal ordinal response data. We propose likelihood-based Markov models and marginalized random effects models for the analysis of longitudinal ordinal data. Both models allow direct estimation of marginal regression parameters and the interpretation of these parameters is invariant to specification of the dependence. We show estimation of covariate effects is reasonably robust to mis-specification of dependence. Fisher-scoring and Quasi-Newton algorithms are developed for estimation. Methods are illustrated on quality of life data from a recent colorectal cancer clinical trial. We outline a selection model approach to deal with nonignorable dropout in this data. Many of the dropouts were due to tumor progression or death which further complicates the analysis. We propose two approaches to handle this type of dropout. The first uses a mixture model approach and is straightforward, but the causal interpretation of treatment effects via randomization is compromised. The second approach is more complex, using the tools of principal stratification, but preserves the causal interpretation of treatment effects.
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 Keunbaik Lee.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Daniels, Michael J.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021244:00001


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IgratefullyacknowledgethenancialsupportprovidedbyNIHgrantCA85295.TheQOLdataweregenerouslyprovidedbyDr.DanielSargentandMs.ErinGreenoftheMayoclinic.Myadvisor,AssociateProfessorMichaelJ.Daniels,hasbeenapatienttutorforseveralyearsandhasprovidedinvaluableknowledgeandsupportduringthisprocess.Onmanyoccasions,ProfessorMalayGhoshoeredtimelysuggestionsthatallowedmetosurmountmanyproblems.CommitteemembersAssociateProfessorBrettPresnellandProfessorAbrahamHartzemahavealsoprovidedgoodsuggestionsatcrucialtimes.Mysuccessovertheyearswouldnothavebeenpossiblewithoutthededicationandcommitmentofmyhard-workingwife,JunghwaKimandourlovelydaughter,Charlene,whohaveprovidedunwaveringloveandsupportthroughoutmystudies.Myparentshavebeenunyieldingintheircondenceinme.Thisworkwouldnothavebeenpossiblewithoutallthesepeople.Iamreallygratefulforthem. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 10 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 12 1.1GeneralizedLinearModelsforLongitudinalCategoricalData ........ 12 1.1.1ReviewofMarginalizedRegressionModelsforLongitudinalBinaryData ................................... 13 1.1.2ReviewofOrdinalModels ....................... 15 1.2Dropout ..................................... 16 1.3DropoutDuetoDeath ............................. 18 1.4SummaryandOutlineofDissertation ..................... 20 2QUALITYOFLIFEDATAINCLINICALTRIALS ............... 22 2.1MotivatingDataset ............................... 22 2.1.1QOLoutcomes .............................. 23 2.1.2Dropouts ................................. 23 3MARGINALIZEDTRANSITIONMODELSFORLONGITUDINALORDINALDATA ......................................... 25 3.1Introduction ................................... 25 3.2MarginalizedTransitionModelsforLongitudinalOrdinalData ....... 25 3.2.1FirstOrderModels:OMTM(1)andIOMTM(1) ........... 27 3.2.2MaximumLikelihoodfortheFirstOrderModels ........... 28 3.2.3SecondOrderModels:OMTM(2)andIOMTM(2) .......... 29 3.2.4MaximumLikelihoodfortheSecondOrderModels .......... 30 3.2.5MissingData ............................... 31 3.3SimulationStudy ................................ 31 3.3.1Completedata .............................. 32 3.3.2MARDropout .............................. 34 3.4Example ..................................... 35 3.4.1ModelsFit ................................ 37 3.4.1.1GraphicalAssessmentoftheDependenceModel ...... 38 3.4.1.2InferenceonRegressionCoecients ............. 38 3.4.2GoodnessofFit ............................. 41 3.5Summary .................................... 41 5

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................................... 43 4.1Introduction ................................... 43 4.2MarginalizedRandomEectsModelsforLongitudinalOrdinalData .... 43 4.2.1Models .................................. 43 4.2.2MaximumLikelihoodEstimation .................... 45 4.3SimulationStudy ................................ 47 4.4Example ..................................... 48 4.5Summary .................................... 50 5NONIGNORABLEDROPOUT ........................... 51 5.1Introduction ................................... 51 5.2SelectionModelsforIOMTMandOMREM ................. 51 5.2.1MaximumLikelihoodAlgorithm .................... 54 5.2.2IdenticationinSelectionModels ................... 54 5.3Example ..................................... 55 5.4Summary .................................... 56 6MIXTUREMODELSFORLONGITUDINALORDINALDATAANDTUMORPROGRESSIONORDEATHTIMES ....................... 58 6.1Introduction ................................... 58 6.2ModelsforHandlingDropoutDuetoProgression/Death .......... 58 6.3DataAnalysis .................................. 59 6.3.1IOMTMAnalysis ............................ 60 6.3.2OMREMAnalysis ............................ 60 6.4Summary .................................... 62 7CAUSALEFFECTSOFTREATMENTSFORINFORMATIVEMISSINGDATADUETOPROGRESSION/DEATH ......................... 64 7.1Introduction ................................... 64 7.2PrincipalStraticationApproach ....................... 64 7.2.1DeningtheCausalEects ....................... 65 7.2.2AssumptionsforIdentiabilityofaCausalEect ........... 66 7.2.3EstimationofCausalEects ...................... 68 7.2.3.1EstimationofSACEk(1;0)andSACEk(2;0) ....... 68 7.2.3.2EstimationofSACEk(2;1) ................. 71 7.2.4EstimationofgT(Tx)andhT;Tx(k) .................. 72 7.2.5StandardErrorsforSACE 73 7.2.6SensitivityAnalysis ........................... 73 7.3AnalysisoftheQOLData ........................... 74 7.4Summary .................................... 76 6

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...................... 79 8.1Summary .................................... 79 8.2FutureWork ................................... 79 APPENDIX AMARGINALIZEDTRANSITIONMODELSFORLONGITUDINALORDINALDATA ......................................... 81 A.1ProofofTheorem 1 ............................... 81 A.2ProofofCorollary 1 .............................. 83 A.3ProofofTheorem 2 ............................... 83 A.4DetailedCalculationsforFisher-ScoringforOMTM(1) ........... 86 A.5DetailedCalculationsforFisher-ScoringforOMTM(2) ........... 87 BMARGINALIZEDRANDOMEFFECTSMODELSFORLONGITUDINALORDINALDATA ................................... 91 B.1ProofofTheorem 3 ............................... 91 B.2DetailedCalculationsofQuasi-NewtonunderMAR ............. 92 CNONIGNORABLEDROPOUT ........................... 94 C.1DetailedCalculationsfortheQuasi-NewtonAlgorithmfortheIOMTMSelectionModel ................................. 94 C.2DetailedCalculationsfortheQuasi-NewtonAlgorithmfortheOMREMSelectionModel ................................. 94 DCAUSALEFFECTSOFTREATMENTSFORINFORMATIVEMISSINGDATADUETOPROGRESSION/DEATH ......................... 96 BIOGRAPHICALSKETCH ................................ 105 7

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Table page 2-1Dropoutfrequency(proportion)bytreatmentgroup.ntand^tarethenumberandtheproportionofdropout. ........................... 23 2-2Completersanddropoutsbytreatmentgroups.Proportionsinparentheses ... 24 3-1BiasesofOMTMandIOMTMforsamplesizesof200and80for1000simulateddatasetsundermis-specicationofdependencemodel.Displayedaretheaverageregressioncoecientestimatesandtherelativebias,100(^)= 33 3-2BiasofIOMTM(1)maximumlikelihoodestimatorswhendataaremissingatrandom(MAR).Displayedistheaverageregressioncoecientestimatesandthepercentrelativebias,100(^)= 35 3-3BiasofIOMTM(2)maximumlikelihoodestimatorswhennodataaremissing.Displayedistheaverageregressioncoecientestimatesandthepercentrelativebias,100(^)= 36 3-4BiasofIOMTM(2)maximumlikelihoodestimatorswhendataaremissingatrandom(MAR).Displayedistheaverageregressioncoecientestimatesandthepercentrelativebias,100(^)= 36 3-5MaximumlikelihoodestimatesforOMTMandIOMTM ............. 40 3-6Adjustedobserved,andttedmarginalprobabilities.TherstrowforeachvisitareobservedmarginalprobabilitieswiththettedvaluesforthemissingdatabasedonconditionalprobabilitiesandthesecondrowarethettedmarginalprobabilitiesfromIOMTM(2). ........................... 42 4-1BiasofOMREMmaximumlikelihoodestimators.Displayedaretheaverageregressioncoecientestimatesandthepercentrelativebiases,100(^)= 4-2Maximumlikelihoodestimatesformarginalizedrandomeectsmodelsunderignorablemissingness ................................. 49 5-1MaximumlikelihoodestimatesofmarginalmeanparametersandselectionmodelparametersforIOMTMunderMARandMNAR ................. 55 5-2MaximumlikelihoodestimatesofmarginalmeanparametersandselectionmodelparametersforOMREMunderMARandMNAR ................. 57 6-1BreakdownofProgession/DeathwindowsbytreatmentgroupsforQOLdata.Proportionsinparentheses .............................. 60 6-2Estimatedtargetprobabilities^P(Yt>cjS>5;Tx)(standarderrors)forIOMTMwhereSisprogression/deathtime. ......................... 61 8

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.................... 62 7-1Breakdownofsubjectswhowerealivebytreatmentgroups.Proportionsareinparentheses ...................................... 74 9

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Figure page 3-1ConditionalprobabilitiesforcategorykgivenpreviousresponseunderOMTM. 41 6-1^P(Yt>cjS>5)and95%condenceintervalsforIOMTMwherec=1;2;3andSistumorprogressionordeathtime. ........................ 61 6-2^P(Yt>3jS>5;Tx)and95%condenceintervalsforOMREMwhereSistumorprogressionordeathtime. .......................... 63 7-1SACEK1(1;0)asafunctionof(Solidline).Dashedlinesare95%condenceintervals. ........................................ 75 7-2EstimatedSACEK1(2;0)asafunctionof(1)and(2). ............. 77 7-3Z-statisticsforestimatedSACEK1(2;0)asafunctionof(1)and(2). ..... 77 7-4EstimatedSACEK1(2;1)afunctionof(1)and(2). ............... 78 7-5Z-statisticsforestimatedSACEK1(2;1)asfunctionof(1)and(2). ...... 78 10

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Generalizedlinearmodelswithserialdependenceareoftenusedforshortlongitudinalseries.Heagerty(1999,2002)hasproposedmarginalizedregressionmodelsfortheanalysisoflongitudinalbinarydata.Thefocusofourresearchisonmarginalizedregressionmodelsforlongitudinalordinalresponsedata.Weproposelikelihood-basedMarkovmodelsandmarginalizedrandomeectsmodelsfortheanalysisoflongitudinalordinaldata.Bothmodelsallowdirectestimationofmarginalregressionparametersandtheinterpretationoftheseparametersisinvarianttospecicationofthedependence.Weshowestimationofcovariateeectsisreasonablyrobusttomis-specicationofdependence.Fisher-scoringandQuasi-Newtonalgorithmsaredevelopedforestimation.Methodsareillustratedonqualityoflifedatafromarecentcolorectalcancerclinicaltrial.Weoutlineaselectionmodelapproachtodealwithnonignorabledropoutinthisdata.Manyofthedropoutswereduetotumorprogressionordeathwhichfurthercomplicatestheanalysis.Weproposetwoapproachestohandlethistypeofdropout.Therstusesamixturemodelapproachandisstraightforward,butthecausalinterpretationoftreatmenteectsviarandomizationiscompromised.Thesecondapproachismorecomplex,usingthetoolsofprincipalstratication,butpreservesthecausalinterpretationoftreatmenteects. 11

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Myresearchdevelopsandevaluateslikelihoodbasedmethodsfortheanalysisofcategoricaldatawithignorableornon-ignorablemissingdata.Suchmodelsareinpartmotivatedbyqualityoflife(QOL)datainclinicaltrialswhichwillbeusedtoillustratemostofthemodelsinthisdissertation.ToproperlyanalyzetheQOLdatawealsoproposemethodstodealwithdropoutduetodiseaseprogressionordeaththatpreservethecausalinterpretationoftreatmenteects. logit(cit)=xTit+bi; 12

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Ourreviewwillfocusonlikelihoodbasedmodels.WewillbeginbyreviewingworkbyHeagerty(1999,2002)onwhichourworkwillbuild. 13

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DeneMit=E(YitjXit).WeassumethattheregressionmodelproperlyspeciesthefullcovariateconditionalmeansuchthatE(YitjXi1;;Xini)=E(YitjXit). meanmodel:logitfMitg=xTit; dependencemodel:logitfcitg=4it+pXj=1itjyitj; wherecit=E(YitjYit1;;Yitp;Xit),itj=zTitjaredependenceparametersthatcharacterizeserialdependencebyquantifyinghowstronglytheimmediatepastpredictsthepresent.zitisasubsetofxit,andthesubject/timespecicinterceptintheconditionalmodel,4it,isdeterminedimplicitlybyxit,and. Themeanparameterdescribeschangesintheaverageresponseasafunctionofcovariates.Usingalogisticlinkforthetransitionmodel,( 1{3 )impliesthattheparametersjareunconstrained.Foragivenmeanmodel( 1{2 )anddependencemodel( 1{3 ),theintercept4itisfullyconstrainedviathefollowingidentity:Mit=Xj1;;jpcit(t)ij1;;jp; Thismodelhasseveraldesirablefeatures.First,themeanmodelisspeciedseparatelyfromthedependencemodel.Asaresult,theinterpretationoftheregressionparameterisinvariantaswemodifyassumptionsregardingthedependenceinequation( 1{3 ).Thisisnottrueforclassicaltransitionmodels,whichparameterizecitdirectlyasa 14

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meanmodel:logitMit=xit; dependencemodel:logitcit(bit)=4it+bit; whereisthep1vectorofregressioncoecientsandcit(bit)=E(Yitjbit;xit).WeassumethattheresponsevectorYiisconditionallyindependentgivenbi=(bi1;;bini)Tandthat ThecovariancematrixAisassumedtohaveasimplestructure.Conditionalonxiandbi,theresponsesYitareassumedtobeconditionallyindependent.ParametersinAprovidemeasuresofrandomvariationbothacrossindividualsandovertime. Theparameters4itin( 1{5 )arefunctionsofboththemarginalmeanparametersandtherandomeectsvarianceandcanbeobtainedusingthefollowingidentity wherefistheunivariatenormaldensityfunction.FurtherdetailsonbothmodelsaregiveninHeagerty(1999,2002). 15

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Ingeneral,formissingdata,theresponsevector,Yi,canbedividedintotwovectors,basedonwhethervaluesareobserved(Yoi)ormissing(Ymi).StartingfromRubin(1976)andLittleandRubin(1987),weclassifymissingdata/dropoutintothreecategories.Ifdropoutdistributiondoesnotdependontheresponsevectororcovariates,themissingness 16

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Amissingdatamechanismisignorableif(1)themissingdatamechanismisMAR;(2)thefulldataparameter!canbedecomposedas!=(;),whereindexesthefull-dataresponsemodelf(yj)andindexesthemissingdatamechanismf(djy;).Otherwise,themissingdatamechanismiscallednonignorable.Whenthemissingnessmechanismisignorable,avalidlikelihood-basedanalysiscanignorethemissingdatamechanism.Thus,inferenceaboutandcanbebasedonalikelihoodfunctionthatisproportionalinandtotheobserved-dataresponsemodel,f(yo;dj;)=Zf(yo;ymj)f(djyo;ym;)dymMAR=f(yoj)f(djyo;): CommonclassesofmodelstoaccommodatelongitudinaldatawithdropoutaresummarizedinHoganetal.(2004).Recallthefulldatamodel,f(d;yjx).Inselectionmodels,thejointdistributionisfactoredastheproductofthefull-dataresponsemodelandthemissingdata(selection)mechanismmodel(DiggleandKenward,1994;Fitzmauriceetal.,1996;Molenberghsetal.,1997;Kenward,1998;Rotnitzkyetal.,1998; 17

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Inthepatternmixturemodels,thefulldataismodelledasamixtureoverdropoutpatterns(Little,1993,1994,1995;LittleandWang,1996;HoganandLaird,1997;FitzmauriceandLaird,2000;DanielsandHogan,2000;BirminghamandFitzmaurice,2002;Birminghametal.,2003),f(y;djx)=f(yjd;x)f(djx): Toconductthistypeofanalyses,thejointdistributionofthelongitudinalresponsesandprogression/deathtimes(HoganandLaird,1997;Pauleretal.,2003;KurlandandHeagerty,2005)needstobeconstructed.HoganandLaird(1997)proposedalikelihood-basedmixturemodelapproachtomodelthejointdistributionforincomplete 18

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Toobtainthecausaleectoftreatment,westartbydeningpotentialoutcomes.Potentialoutcomesarealltheoutcomesthatwouldbeobservedhadeachofthetreatmentsbeenappliedtoeachofthesubjects(Rubin,1974,1978;Holland,1986).FrangakisandRubin(2002)usedtheconceptofpotentialoutcomesinanapproachcalled`principalstratication'.Principalstraticationpartitionssubjectsintosetswithrespecttoposttreatmentvariables.Forexample,considertwotreatments,Tx=0;1andabinaryresponseY(Tx).ThecausaleectofthetreatmentonresponseisY(1)Y(0).However,weonlyobserveeitherY(1)orY(0)foreachsubject.LetD(Tx)beabinaryposttreatmentvariable.Inthiscase,therearefourprincipalstratadenedbythepairsofpotentialvaluesofD(Tx),(i)(fijD(0)=0;D(1)=0g),(ii)(fijD(0)=0;D(1)=1g),(iii)(fijD(0)=1;D(1)=0g),and(iv)(fijD(0)=1;D(1)=1g).Theprincipalstrataarenotaectedbytreatmentassignmentandthereforecanbeusedasanypretreatmentcovariate.Causaleectsaredenedwithintheseprincipalstrata. FrangakisandRubin(2002)discussedcausaleectsusingstudieswheretheoutcomewasrecordedandunobservedduetodeath.Rubin(2000)andHaydenetal.(2005)referredtotheestimandinFrangakisandRubin(2002)as`thesurvivorsaveragecausaleect(SACE)'.Eglestonetal.(2006)proposedassumptionstoidentifytheSACEandimplementedasensitivityanalysisforsomeofthoseassumptions.Recently,Rubin(2006)introducedthecausaleectofatreatmentonaoutcomethatiscensoredbydeathinQualityofLifestudy. 19

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2 ,wedescribeourmotivatingdataset,alongitudinalcancerclinicaltrialinwhichourmaininterestwasqualityoflife(QOL)data.Thetrialhadalotofdropouts. InChapter 3 ,weproposetwomarginalizedtransitionmodelsforlongitudinalordinaldataanddescribeaFisher-scoringalgorithmtondmaximumlikelihoodestimatesfortheparameters.Weprovetheorthogonalityofthemarginalmeananddependenceparametersandtheconsistencyofthemarginalmeanparametersregardlessofthespecicationofthedependenceparameters.ThemodelsareillustratedontheQOLdata. InChapter 4 ,weproposesmarginalizedrandomeectsmodelsforlongitudinalordinaldata.AnalgorithmformaximumlikelihoodestimationisproposedwhichutilizesaQuasi-NewtonalgorithmwithMonteCarlointegrationfortherandomeects.WealsoillustratetheproposedmodelwiththeQOLdata. InChapter 5 ,weoutlineanapproachtohandlenonignorablemissingnessusingparametricselectionmodelsusingthemarginalizedregressionmodelswhichareproposedinChapter 3 and 4 forfulldataresponsemodel.WeapplythismodeltotheQOLdatainwhichdropoutoccurredatahighrate. InChapter 6 ,weproposemodelstodealwithdropoutsduetoprogression/deathbasedonmodelingthejointdistributionofthelongitudinalQOLresponsesandthedeathtimes(HoganandLaird,1997).WeapplythisapproachtotheQOLdata. 20

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7 ,wedescribeanapproachthatcanbeusedtoobtainthecausaleectoftreatmentinthepresenceofdeath.TheframeworkisprincipalstraticationoriginallyproposedinFrangakisandRubin(2002). Finally,wesummarizethisdissertationandproposefutureworkinChapter 8 21

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Asasecondaryoutcome,manyclinicaltrialsmeasureaspectsofapatient'ssenseofwell-beingandabilitytoperformvarioustaskstoassesstheeectsthatcanceranditstreatmenthaveonthepatient.Suchqualityoflife(QOL)outcomesinclinicaltrialsaregenerallyobtainedrepeatedlyovertimeusingquestionairesthatcanbecompletedbythepatientwithoutdirectsupervision.Thesestandardizedquestionnairesratesuchfactorsaspain,treatmentside-eects,mood,energylevel,familyandsocialinteractions,sexualfunction,abilitytowork,andabilitytokeepupwithroutinedailyactivities.WedescribetheQOLdatasetthatwewillworkwithnext. Atotalof795patientswithcolorectalcancerwererandomlyassignedtooneofthreetreatmentsbetweenMay1999andApril2001.ThepatientsreceivedirinotecanandbolusuorouracilplusleucovorinforIFLarm,oxaliplatinandinfuseduorouracilplusleucovorinforFOLFOXarm,andirinotecanandoxaliplatinforIROXarm,respectively.Basedontheprimaryendpointsoftimetoprogressionandoverallsurvival,theyconcludedthattheFOLFOXarmwasbestandcomparativelysaferelativetotheotherarms. However,giventhatthetoxicityproleswerequitedierentonthethreetreatmentarms,itwasofinteresttoseeiftherewasanegativeimpactof`better'treatmentsonpatientsQOL.Inthispaperwecomparethequalityoflifeonthethreetreatments. 22

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Dropoutfrequency(proportion)bytreatmentgroup.ntand^tarethenumberandtheproportionofdropout. IFLIROXFOLFOXTotalVisitnt^tnt^tnt^tnt^t 6=stillintrialaftervisit5 Forillustration,wefocusononeQOLmeasure,fatigue.Fatigueismeasuredona5pointordinalscale(1:Iamusuallynottiredatall;2:Iamoccasionallyrathertired;3:TherearefrequentlyperiodswhenIamquitetired;4:Iamusuallyverytired;5:Ifeelexhaustedmostofthetime). 2-1 givestheproportionsofdropoutsforthethreetreatments.Theoveralldropoutratesweresimilarinthethreetreatmentgroups.About95%ofdropoutsoccurredbetweenbaselineandthefourthvisit.Dropoutwasrelatedtowhethercancerhadprogressed,death,toxicity,andotherfactors. Table 2-2 summarizesthereasonsfordropoutinthethreetreatmentarms.Alargenumberofsubjectsdroppedoutduetotumorprogressionordeath(40.5%).Thedropout 23

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Completersanddropoutsbytreatmentgroups.Proportionsinparentheses Reasonfordropout TreatmentCompleterProgression/DeathOtherTotal IFL10(0.043)117(0.498)108(0.460)235IROX4(0.017)101(0.423)134(0.561)239FOLFOX4(0.017)68(0.292)161(0.691)233Total18(0.025)286(0.405)403(0.570)707 ratesduetoprogression/deathweremarginallyhigherinIFLandIROXarmsthaninFOLFOXarm.Aconvenientfeatureofthisdataset(whoseconveniencewillbemoreclearlater)isthatwehadtheactualprogression/deathtimesforallsubjectsincludingthosewhodroppedoutforotherreasons. Wewillusethisdatatoillustratethemodelsandmethodsintroducedinthesubsequentchapters. 24

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Thesimpleextensionofmarginalizedtransitionmodeloforderp,OMTM(p),isspeciedusingthefollowingtworegressions, meanmodel:logP(Yitkjxit) 1P(Yitkjxit)=0k+xTit; dependencemodel:logP(Yit=kjYit1;;Yitp;xit) whereisther1vectorofregressioncoecients,01<02<<0K1,(k)itml=zTit(k)ml,zitisaq1vectorofsubsetofxit,(k)ml=((k)ml1;;(k)mlq)T,k=1;;K1,andYi;tm;listhesetofK1indicatorswithYi;tm;l=1ifYi;tm=l;Yi;tm;l=0otherwiseforl=1;;K1;m=1;;p.Model( 3{1 )isacumulativelogitmodel,acommonmodelforordinaldata.However,thelinkfunctionintheconditionalprobabilities( 3{2 )isamultinomiallogitsinceweexpecttheK1conditionalprobabilitiestohavedierent 25

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3{1 ),themonotonicitypropertyofthecumulativeconditionalprobabilitieswouldbediculttosatisfyasitinvolves4itk. SincethismodelisastraightforwardextensionoftheMTMforbinarydata,itsharesmanyofthesameadvantagesincludinghavingtheattractivecharacterizationofserialdependencethatatransitionmodelprovidescombinedwithamarginalregressionstructure.However,itdoeshavesomedrawbacks.First,therearealotofdependenceparameters.Forexample,ifp=2,K=4,r=2,andq=1(zit=1),weneedp(K1)2q=18dependenceparameters.Second,theorderingofthecategoriesisnotexploitedintheconditionalprobabilities( 3{2 ).Forthesetworeasons,weproposeamodiedversioninwhich( 3{2 )isreplacedwith logP(Yit=kjYit1;;Yitp;xit) wherec()isasmoothfunctionofyit1;;yitpparameterizedby(k).Forourmethods,wechooseaquadraticpolynomialforc(;(k))sinceitshouldbeadequatetoexplainthesmoothchangeoftheconditionalprobabilitiesandwehypothesizethatthereisnodramaticchangeinconditionalprobabilitiesthatwouldnecessitatehigherorderpolynomialsorothersmoothfunctions.Thus,( 3{3 )becomes logP(Yit=kjYi;t1;;Yi;tp;xit) where(k)itml=zTit(k)mland=(T11;T12;;Tp1;Tp2)T.Wecallthemodelgivenbyequations( 3{1 )and( 3{4 )aimprovedextensionoftheMTMoforderp,anIOMTM(p).TheIOMTM(p)hasfewerdependenceparametersthantheOMTM.Forexample,OMTMhas18dependenceparametersforK=4,p=2andq=1,whereasIOMTMneedsp2(K1)=12dependenceparameters.Ingeneral,forOMTM(p),thenumberofdependenceparametersincreasesquadraticallywithK,thenumberofcategories,whereas 26

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3{4 ). Clearly,formodels( 3{1 )-( 3{4 )toformacoherentprobabilitymodel,the4itkwillbeconstrained.Constraintsareneededfor( 3{1 ),( 3{2 ),and( 3{4 )tobeavalidprobabilitymodel.Specically,4itkisfoundusingthefollowingidentity ForOMTM(1):logP(Yit=kjYit1;xit) ForIOMTM(1):logP(Yit=kjYi;t1;xit) ForbothIOMTM(1)andOMTM(1),meananddependenceparametersareorthogonalasgiveninthefollowingTheoremandCorollary. Proof. A WealsohaveorthogonalityintheIOMTMfromthefollowingcorollary. Proof. A 27

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Proof. A logL(;y)=NXi=1niXt=2(K1Xk=1yitk4itk+(k)it11yit11++(k)it1K1yit1K1+logPcitK)+NXi=1K1Xk=1fRi1ki1kRi1k+1g(i1k)glet=logL(2)(;y)+logL(1)(;y); 28

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Toevaluatethelikelihood,weneedtoevaluateboththecontributionfromtheinitialstate,L(1),andthesubsequentcontributionfromeachtransitionprobability,L(2).Let=(0;;)beanarbitraryparametervector.Weobtain@logL @=@logL(2) A .TheFisher-scoringmethodcanbeusedtosolvethelikelihoodequations,using(c+1)=(c)+E@2logL @@T1@logL @(c); @@T=E@2logL(2) Fortheexplicitformsoftheseexpectations,seeAppendix A .Thederivatives@4itk 3{5 );detailedcalculationsaregiveninAppendix A .4itareafunctionof0,,and.Estimatesof4itcanbeobtainedusingNewton-Raphson.DetailedcalculationsarealsogiveninAppendix A EstimationfortheIOMTM(1)issimilar. 3{1 )andthedependencemodels( 3{10 )and 29

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3{11 )fortheOMTM(2)andtheIOMTM(2),respectively,aregivenby ForOMTM(2):logP(Yit=kjYit1;Yit2;xit) ForIOMTM(2):logP(Yit=kjYi;t1;Yit2;xit) UnliketoOMTM(1)andIOMTM(1),theorthogonalityofmarginalanddependenceparametersdosenotholdanymore.Thus,consistentestimationofmarginalmeanparametersrequiresappropriatedependencemodeling. WeproposeaFisher-scoringalgorithmtondtheMLEoftheparametersofinterestintheOMTM(2).Forestimation,werewritetheloglikelihoodas

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@=@logL(3) A .TheFisher-scoringmethodcanbeusedtosolvethelikelihoodequations,using( 3{9 ).TheexpectationsofthenegativesecondderivativesofloglikelihoodaregivenbyE@2logL @@T=E@2logL(3) A .Thederivatives@4itk 3{5 );detailedcalculationsaregiveninAppendix A ThealgorithmformaximumlikelihoodestimationfortheIOMTM(2)issimilar. 31

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1P(Yitkjxit)=0k+1timeit+2group1i+3group2i;0=(01;02;03)=(1:0;0:7;2:0);=(1;2;3)=(0:5;0:1;0:5);

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BiasesofOMTMandIOMTMforsamplesizesof200and80for1000simulateddatasetsundermis-specicationofdependencemodel.Displayedaretheaverageregressioncoecientestimatesandtherelativebias,100(^)= 3-1 presentstheaverageestimatesandthepercentrelativebias.Inthesamplesizeof200,relativebiasinthecoecientswassmall(from0.0to2.0%)inbothOMTMandIOMTM.Inthesamplesizeof80,therelativebiasintheregressioncoecientswasalsosmallrangingfrom0.0to4.7%.Thesimulationstudyillustratestherobustnessofthemarginalmeanparameterstomis-specicationofdependenceinsmallsamples.Inlargesamples,byTheorem 2 ,weknowtheyarerobust. 33

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3{1 ),onewithanunstructuredmeanovertimewithdesignvectoroftheform,xTit=(0;:::;1;:::;0;group1i;group2i),wherethe1isinthetthslot,andonewithalineartrendovertime,xTit=(Visitit;group1i;group2i),wheregroupjiisanindicatorofwhetherthesubjectwasingroupj.Thetwodierentmeanmodelswereincludedtoseeifanybiasinthepresenceofdropoutdieredbetweentheunstructuredandstructured(overtime)means. Foreachsimulation,weconsideredthefollowingMARdropoutmodel,logitP(dropout=tjdropoutt)=2:5+0:6Yit1+0:3Yit2: 3{7 )with(k)it1m=m0+m1Visitit,m=1;2,(10;11;20;21)=(1:0;0:5;0:5;0:1)andtthemodelassuming(k)it1m=m0. Table 3-2 presentsthepointestimatesandthepercentrelativebias.Percentrelativebiasesinthecoecientsweresmall,thelargestbeingabout3%.Thecoecientsoftime-varyingcovariatesinthestructuredmeanmodelappearedtobemorerobusttomisspecicationofdependenceandMARmissingnessthanintheunstructuredmeanmodel. Weperformedseveralsimilarsimulationsusingthesamemodelsforthemarginalmeans,butnowgeneratingdataunderthefollowingIOMTM(2),logP(Yit=kjYit1;Yit2;xit) 34

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BiasofIOMTM(1)maximumlikelihoodestimatorswhendataaremissingatrandom(MAR).Displayedistheaverageregressioncoecientestimatesandthepercentrelativebias,100(^)= int1-1.00-1.000.0int1-1.00-1.000.0int20.700.700.0int20.700.700.0int32.002.010.5int32.002.000.0grp10.100.100.0grp10.100.100.0grp2-0.50-0.500.0grp2-0.50-0.500.0vis10.100.100.0vis-0.50-0.49-2.0vis2-1.50-1.510.7vis31.201.200.0vis4-0.40-0.39-2.5vis50.800.800.0vis6-0.30-0.29-3.3vis7-1.00-1.033.0 ((1)220;(2)220;(3)220)=(0:4;0:5;0:6).WethentanIOMTM(1)withzit=1.TheresultswithnodropoutappearinTable 3-3 andthosewith(MAR)dropoutappearinTable 3-4 Inthepresenceofdependencemis-specicationandMARdropout,wesawconsiderablebias,withrelativebiasesaslargeas74%fortheunstructuredmean.Underdependencemis-specicationwithnodropout,thebiaseswerequitesmalldespitetheorthogonalityofthemeananddependenceparametersbeinglostintheIOMTM(2).Thesesimulationsfurtheremphasizetheimportanceofcorrectlyspecifyingthedependenceinthepresenceofmissingdata. 3.2 toanalyzeQOLdatafromtheclinicaltrialofmetastaticcolorectalcancerintroducedinChapter 2 .Atotalof795patientswithcolorectalcancerwererandomlyassignedtooneofthreetreatments(FOLFOX,IFL(Control),IROX)betweenMay1999andApril2001(Goldbergetal.,2004).WefocusononeQOLmeasure,fatigue.Fatigueismeasuredona5pointordinalscale(1:Iamusuallynottiredatall;2:Iamoccasionallyrathertired;3:Therearefrequentlyperiods 35

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BiasofIOMTM(2)maximumlikelihoodestimatorswhennodataaremissing.Displayedistheaverageregressioncoecientestimatesandthepercentrelativebias,100(^)= int1-1.00-1.011.0int1-1.00-1.000.0int20.700.700.0int20.700.700.0int32.002.000.0int32.002.000.0grp10.100.100.0grp10.100.100.0grp2-0.50-0.500.0grp2-0.50-0.500.0vis10.100.100.0vis-0.50-0.512.0vis2-1.50-1.500.0vis31.201.210.8vis4-0.40-0.400.0vis50.800.800.0vis6-0.30-0.300.0vis7-1.00-1.000.0 Table3-4. BiasofIOMTM(2)maximumlikelihoodestimatorswhendataaremissingatrandom(MAR).Displayedistheaverageregressioncoecientestimatesandthepercentrelativebias,100(^)= int1-1.00-0.94-5.6int1-1.00-1.021.8int20.700.7912.6int20.700.711.0int32.002.063.0int32.001.97-1.6grp10.100.100.0grp10.100.100.0grp2-0.50-0.49-1.6grp2-0.50-0.49-1.2vis10.100.03-74.0vis-0.50-0.36-28.2vis2-1.50-1.574.9vis31.201.16-3.3vis4-0.40-0.39-2.5vis50.800.76-5.1vis6-0.30-0.313.0vis7-1.00-1.066.0 36

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Weassumedthemissingresponsesweremissingatrandom(MAR).WeusedtheAkaikeInformationCriteria(AIC)(Akaike,1974)asthemodelselectioncriterion. TheFisher-scoringalgorithmisnottrivialcomputationallyduetotheneedtoobtainanestimateofitforallsubjectsandatalltimes,withineachFisher-scoringstep.However,inourdataset,with795patients,afourcategoryordinalresponses,andsixvisittimes,thecomputationalburdenwasnothigh.EachFisher-scoringstep(inwhichalltheitarecomputed)onaPentiumwitha1.6GHzprocessortookabout10and30secondsfortheIOMTM(1)andtheIOMTM(2),respectively.Inaddition,usinggoodinitialvaluesbasedonttinganindependentproportionaloddsmodelinstandardsoftwareresultsinaminimalnumberofiterationsuntilconvergence. Table 3-5 presentsmaximumlikelihoodestimatesforallsevenmodels.PointestimatesandstandarderrorformarginalmeanparametersfortheOMTMweresimilartothosefortheIOMTM(1)andIOMTM(2).Tocomparethetofthemodels,inparticular, 37

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3-1 presentsconditionalprobabilitiesgivenpreviousvaluesfortheOMTM(1)givenin( 3{6 ).Wearbitrarilychoseonesubjectandvisitbecausetheshapesofconditionalprobabilitiesarethesameacrosssubjects(theintercepts4itkonlyshiftthetrajectoriesupordown).Thetrajectoriesofconditionalprobabilitieschangequadraticallywithpreviousresponses.Thus,thequadraticmodelgivenin( 3{7 )appearsadequatetoexplainthechangeintheconditionalprobabilitiesasafunctionofthepreviousresponse.WewereunabletogetanOMTM(2)toconvergetoobtainasimilarplotforrstandsecondorderdependence. 38

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39

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MaximumlikelihoodestimatesforOMTMandIOMTM IOMTM(2)-1IOMTM(2)-2IOMTM(2)-3 OMTM Marg.para.Coef.(SE)Coef.(SE)Coef.(SE) Coef.(SE)Coef.(SE)Coef.(SE) Coef.(SE) Intercept11:099(0.124)1:094(0.122)1:104(0.124) 0:849(0.120)0:841(0.117)0:832(0.119) 0:844(0.121)Intercept32:318(0.143)2:316(0.142)2:301(0.142) 2:319(0.143)2:317(0.141)2:293(0.143) 2:319(0.143)Visit0:290(0.675)0:427(0.683)0:235(0.678) 0:025(0.163)0:055(0.160)0:036(0.162) 0:037(0.166)Visit*Arm10:647(1.013)0:458(1.027)0:767(1.006) Max.loglike.1924:1531916:2251916:690 3840:6603825:9103861:820 3880:700

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ConditionalprobabilitiesforcategorykgivenpreviousresponseunderOMTM. 3-6 indicatesthattheIOMTM2-2tsthedataquitewell,validatingtheproportionaloddsassumptionin( 3{1 ). 41

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Adjustedobserved,andttedmarginalprobabilities.TherstrowforeachvisitareobservedmarginalprobabilitieswiththettedvaluesforthemissingdatabasedonconditionalprobabilitiesandthesecondrowarethettedmarginalprobabilitiesfromIOMTM(2). 00.250.470.190.090.250.440.200.120.280.420.230.080.250.450.210.090.240.450.220.100.260.450.210.0910.230.450.240.070.210.440.230.120.210.440.230.110.250.450.220.090.220.440.230.110.220.440.230.1020.230.450.230.090.200.440.240.130.180.450.260.110.240.440.220.100.200.430.250.120.190.430.260.1230.230.430.240.100.190.420.250.140.160.410.280.150.230.440.220.100.190.430.260.130.160.410.280.1540.220.450.230.100.180.410.270.140.130.400.300.170.230.440.230.100.170.420.270.140.140.390.300.1750.230.440.230.110.160.410.280.160.120.360.320.200.220.440.230.110.150.400.290.150.120.360.320.20 3{6 )whentherewasnomissingdata.Simulationstudiesindicatedthatmarginalmeanparameterestimateswererobusttothedependencemodelbeingincorrectlyspeciedin(I)OMTM(1)'sunderMARdropout.However,IOMTM(2)'sweremuchlessrobusttodependencemodelmis-specicationunderMARdropout. Calculationsandanalysesinthischapterarebasedona1storderand2ndordermodels.Extensiontohigherordersisalsopossible.However,suchmodelsintroducemanymoredependenceparametersandmorecomplexcomputingandconstraints.Analternativeformulationwouldbetointroducerandomeectsinsteadof,orinadditionto,previousresponsetomodelthelongitudinaldependence(seeChapter 4 ). 42

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4.2.1Models meanmodel:logP(Yitkjxit) 1P(Yitkjxit)=0k+xTit; dependencemodel:logP(Yitkjbi;xit) 1P(Yitkjbi;xit)=4itk+bit;; wherebTi=(bi1;;bini)N(0;i)fori=1;;Nandk=1;;K1.Weassumethevariance-covariancematrixiofbiisanautoregressivecovariancestructure

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Intheconditionalmodel,( 4{2 ),weneedtoensurethemonotonicityof4inkbecausetoguaranteesthatvalidityofthecumulativeoddsmodelin( 4{2 ). 4{1 )and( 4{2 ),if01<<0K1,then4it1<<4itK1. Proof. B Themarginalmeanmodelisacumulativeoddsmodel.Serialdependenceiscapturedbytherandomeects.Theregressionparametersin( 4{1 )haveamarginalinterpretationsunlikegeneralizedlinearmixedmodels(Diggleetal.,2002).Anadvantageofthismodel( 4{2 )istheabilitytouseconditionalmodelsforassociationwhilestillstructuringthemarginalmeandirectly,usingaregressionmodel.Asaresult,theinterpretationoftheregressioncoecients,(0;)doesnotdependonthespecicationofthedependencemodelin( 4{2 ). Forlongitudinaldatawithrandomeects,bit,themarginalprobabilitycapturesthesystematicvariationinthemarginalprobabilitythatisduetoxit,whereasparametersincov(bi)provideameasureofrandomvariationbothacrossindividualsandovertime.HeagertyandKurland(2001)investigatedtheimpactontheestimatesofregressioncoecientsfromincorrectassumptionsregardingtherandomeectsingeneralizedlinearmixedmodelsandmarginalizedmodelsandfoundthatmarginalizedregressionmodelsaremuchlesssusceptibletobiasresultingfromrandomeectsmodelmisspecication.UnliketheIOMTM(1)fromChapter 3 ,theOMREMdoesnothaveorthogonalityof(marginal)meananddependenceparameters. Themarginalandconditionalprobabilitiesin( 4{1 )and( 4{2 )arerelatedasfollows 44

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4{4 )tosolvefor4itgiven0,,and. 2aiwhere1 2,alowertriangularmatrixwithpositivediagonalelements,istheCholeskyfactoroftheTTmatrix(GibbonsandBock,1987)andaiisani1vectorofindependentstandardnormals.ThereparameterizedconditionalmodelisthengivenbylogP(Yitkjai;xit) 1P(Yitkjai;xit)=4itk+is(t)ai;aiN(0;I); 2andIistheidentitymatrixoforderT.ThistransformationallowsustoestimatetheCholeskyfactor1 2insteadofthecovariancematrix.SincetheCholeskyfactoristhesquarerootofthecovariancematrix,itallowsmorestableestimationofnear-zerovarianceterms(HedekerandGibbons,1994). wherePcitk(ai)=P(Yitkjai;xit),()isamultivariatenormaldensitywithmeanvector0andvariance-covariancematrixI,=(0;;;),andyitk=Ifyit=kg.Themarginalizedlikelihoodin( 4{5 )isnotavailableinclosedform.Thereareseveralapproachesto(numerically)integrateouttherandomeects.Gauss-Hermitequadratureispopularforsimplemodelssuchasrandominterceptmodels(HedekerandGibbons,1994); 45

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4{5 )sincethedimensionofaiishigh. Maximizingthelog-likelihoodwithrespecttoyieldsthelikelihoodequationNXi=1@logL(;yi) where The(K1+p+c+1)-dimensionallikelihoodequationsaregiveninAppendix B Thematrixofsecondderivativesoftheobserveddatalog-likelihoodhasaverycomplexform.Fortunately,thesampleempiricalcovariancematrixoftheindividualscoresinanycorrectlyspeciedmodelisaconsistentestimatoroftheinformationandinvolvesonlytherstderivatives.SotheQuasi-Newtonmethodcanbeusedtosolvethelikelihoodequations,using(c+1)=(c)+Ie(c);y1@logL @(c); 46

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4{4 ),seeAppendix B Theintercepts4itkareafunctionof0k,,,andandmustbeobtainedwithintheQuasi-Newtonalgorithm.Letf(4itk)=RPcitk(bit)(bit)dbitPMitk.Estimatesof4itkcanbeobtainedusingNewton-Raphsonasfollows,4(n+1)itk=4(n)itk@f(4itk) 1P(Yitkjxit)=0k+1timeit+2groupi;0=(01;02;03)=(1:0;0:7;2:0);=(1;2)=(0:8;0:2); 4{2 )and( 4{3 )withi=1:5and=0:2(correlation=exp(0:2)=0:819).Wesimulated500datasetseachwithasamplesizeof200.WethentanOMREMwithindependentrandomeects,bit=bi0N(0;2). FortheMARmissingness,wespeciedthefollowingMARdropoutmodel,logitP(dropout=tjdropoutt)=2:5+0:3Yit1+0:1Yit2:

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BiasofOMREMmaximumlikelihoodestimators.Displayedaretheaverageregressioncoecientestimatesandthepercentrelativebiases,100(^)= int1-1.00-1.011.0-0.99-1.0int20.700.69-1.40.722.9int32.002.000.02.031.5vis0.800.822.50.52-35.0grp-0.20-0.200.0-0.200.0 Table 4-1 presentsthepointestimatesandthepercentrelativebias.Whentherewasnomissingdata,thepercentrelativebiasesinthecoecientsweresmall.InthepresenceofMARdropout,however,wesawconsiderablebias,withrelativebiasaslargeas35%forthecoecientofvisit(time).Thesesimulationsemphasizetheimportanceofcorrectlyspecifyingthedependenceinthepresenceofmissingdata.TheresultshereareverysimilartothoseinthesimulationstudyfortheIOMTM(2)inChapter 3 2 .AsweindicatedinSection 3.2 ,wefocusononeQOLmeasure,fatigue. TheQuasi-Newtonalgorithmisnottrivialcomputationallyduetotheneedtoobtainanestimateofitforallsubjectsandatalltimes,withineachQuasi-Newtonstep.EachQuasi-Newtonstep(inwhichalltheitneedtobecomputed)onaPentiumwitha1.6GHzprocessortookabout10minutesfortheMREMwithMCsamplesizeof20,000.Usinggoodinitialvaluesbasedonttinganindependentproportionaloddsmodelinstandardsoftwareresultsinaminimalnumberofiterationsuntilconvergence. WettedthreeOMREM'sunderanassumptionofignorabledropout.MREM-1allowedthevariancecoecientstodependontreatments,logi=0+1Armi1+2Armi2.MREM-2wasasimplermodel,withatimehomogeneousvariance,logi=0.Bothhadautoregressionvariance-covariancestructuresasgivenin( 4{3 ).MREM-3 48

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Maximumlikelihoodestimatesformarginalizedrandomeectsmodelsunderignorablemissingness MREM-1MREM-2MREM-3 MarginalParametersInt11:107(0.121)1:089(0.125)1:109(0.123)Int20:837(0.122)0:851(0.125)0:830(0.122)Int32:302(0.144)2:322(0.146)2:292(0.145)Visit0:293(0.553)0:365(0.544)0:206(0.455)Arm10:080(0.164)0:089(0.167)0:081(0.163)Arm20:049(0.164)0:035(0.167)0:055(0.162)Visit*Arm10:554(0.817)0:461(0.826)0:519(0.697)Visit*Arm21:758(0.885)1:660(0.911)1:900(0.786) ConditionalParametersInt0:746(0.233)0:975(0.204)0:717(0.120)Arm10:232(0.396)Arm20:271(0.366)0:092(0.027)0:131(0.027) Max.LogL1909:6311911:1741914:253 wastheMREM-2withi=I(independencestructure).Table 4-2 presentsmaximumlikelihoodestimatesforallthreemodels. PointestimatesandstandarderrorformarginalmeanparametersfortheMREM-1andMREM-2weresimilar.TocomparethetofthetwomodelsunderMAR,wecomputedthelikelihoodratiotest.ComparisonofdeviancesforMREM-2andMREM-1whichwerenestedyielded4D12=2(1911:1741909:631)=3:086,pvalue=0:214on2d.f.ThiscomparisonindicatedthattheMREM-2wasabetterttingmodelthantheMREM-1.WealsocomputedthelikelihoodratiotesttocompareMREM-2andMREM-3.Thedeviancedierencewas4D23=2(1911:1741909:631)=3:086pvalue=0:014on1d.f.ThisindicatedthattheMREM-2tbetterthantheMREM-3. Thecorrelationestimatorwassignicantandimpliedacorrelationof^=exp(^)=exp(0:131)=0:8772.TheMLestimatefor(2.651)indicatedlargesubject-to-subjectvariationintheoddsofthecumulativeprobabilityoffatigue.Themarginalmeancoecientswerenotsignicantindicatingfatiguedidnotsignicantlydierbytreatment. 49

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Wearecurrentlydevelopingmoreecient(MonteCarlo)approachestoevaluatetheintractableintegralsrequiredformaximumlikelihoodestimation. 50

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KurlandandHeagerty(2004)proposedselectionmodelsforlongitudinalbinarydata.WedevelopsimilarmodelsforlongitudinalordinaldatausingtheIOMTMandOMREMasthefulldataresponsemodels. Thecontributionofobserveddatalikelihoodforsubjectiis ( 5{1 )isatypicalselectionmodelfactorization.f(dijyi)iscalledthemissingdatamechanism.Dropoutcandependoncovariatesandonbothobservedandmissingresponses.LetH(d)it=(di1;;dit1)bethehistoryfordithroughtimet1andH(y)it=(yi1;;yit1)bethehistoryforresponseyitinsubjecti.Denetdbetherst 51

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wheref(ditjH(d)it;H(y)it;Yit)=uditit(1uit)1dit; TheparameterpitisprobabilitythatYitisobservedgiventheresponsehistoryandcovariatevalues.Wecanrewritef(dijyi)asafunctionofpit,f(dijyi)=td1Yt=2pditit(1pitd)1ditdTYl=td+1(1)1dit=td1Yt=2pit!(1pitd): logit(pit)=0+1yit1+2yit+zTi; whereziisatime-invariantsubsetofxit.Here1and2areparametersassociatedwiththepreviousandthecurrentobservation,respectively.DropoutisMARwhen2=0,andMCARwhenboth1and2are0. 5{1 )canbesimpliedto {z }td1Yt=2pit!| {z }Zymi(1pitd)f(ymijyoi)dymi| {z };

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Thecontributionoftheithindividualtoobserveddataliklihood,( 5{3 )isdividedintothreeparts,L1,L2,andL3.L1isforlikelihoodofobservedresponsedata.L2isthecontributionofthedropoutmodelfortheobservedresponse(Dit=1fort=1;;td1).Finally,L3istheexpectedvalueoftheprobabilitythattheresponseismissing,(1pid)withrespecttof(ymijyoi). ( 5{3 )canbefurthersimpliedasf(yoi;di)=f(yoi)td1Yt=2pit!KXj=1(1pitd)(td)ij; =td1Yt=2pit!| {z }Zf(yoijai)Zymi(1pitd)f(ymijai)dymif(ai)dai| {z }

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5{5 )isdividedintotwoparts,L1andL2.L1isthecontributionofthedropoutmodelfortheobservedresponse(Dit=1fort=1;;td1).L2isthelikelihoodcontributionforboththeobservedresponsesandtheexpectedvalueoftheprobabilitythattheresponseismissing(1pid)similartoL3fortheIOMTMin( 5{3 ).( 5{5 )canbefurthersimpliedto where(td)ij(ai)=Pcitdj(ai)Pcitdj1(ai). @(c); andIe()isanempiricalandconsistentestimatoroftheinformationmatrixatstepc,givenbyIe(;y)=NXi=1L2(;yi)@L(;yi) C 54

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MaximumlikelihoodestimatesofmarginalmeanparametersandselectionmodelparametersforIOMTMunderMARandMNAR VariableCoef.Coef.Coef. Int.11:09(0.12)1:03(0.13)1:03(0.13)Int.20:84(0.12)0:74(0.13)0:75(0.13)Int.32:32(0.14)1:96(0.15)1:95(0.15)Vis.0:43(0.68)6:51(0.70)6:32(0.73)Arm10:07(0.16)0:01(0.17)0:00(0.17)Arm20:04(0.16)0:10(0.17)0:09(0.17)Vis.*Arm10:46(1.03)1:45(0.90)1:72(0.91)Vis.*Arm21:58(1.00)1:55(0.90)2:07(0.92) int.0:55(0.09)3.73(0.64)3:64(0.67)yit10:19(0.06)0:70(0.12)0:68(0.12)yit2:18(0.29)2:27(0.30)Arm10:28(0.16)Arm20:56(0.13) Max.loglik.3040:092998:382998:12 5{2 )thatinformationonthedropoutparameter2,andhenceonthedropoutmechanism,dependsontheformofdistributionassumedforYmjYo,andontheformofthedropoutmodel. Wespeciedthemissingnessmechanismasin( 5{2 ).WerstttedandcomparedthreeselectionmodelsusingtheIOMTM(1)asthefulldataresponsemodel(seeTable 5-1 ).Therstmodel(IOMTM-MAR)assumedMARdropout(2=0in( 5{2 ))with=0.Theothertwomodels(IOMTM-MNAR1andIOMTM-MNAR2)assumedMNARdropoutwith=0inIOMTM-MNAR1. PointestimatesandstandarderrorsforthemarginalmeanparametersfortheIOMTM-MNAR1andIOMTM-MNAR2weresimilar.Tocomparethetofthetwomodels(IOMTM-MNAR1andIOMTM-MNAR2)underMNARwithrespecttothe 55

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5{2 )werenotsignicant.WealsocomputedthelikelihoodratiotesttocomparetheMNARmodeltotheMARmodel.Thedeviancedierencewas4D01=2(3040:092998:38)=83:43pvalue<0:001on1d.f.ThisindicatedthattheIOMTM-MNAR1tconsiderablybetterthantheIOMTM-MAR. WebaseinferenceonmodelIOMTM-MNAR1.Thettedmissingdatamechanismparameter2in( 5{2 )was-2.180whichindicatedthatprobabilitiesfordropoutincreasedascurrentfatigueincreased.ThecoecientofVisitwassignicantwithanestimateof-6.51andastandarderrorof0.70.Thisindicatedthattheleveloffatiguedecreasedovertime.However,therateofdecreasedidnotdierbytreatment. SimilartotheIOMTM,wealsotandcomparedthreeselectionmodelsusingtheOMREMasthefulldataresponsemodel(SeeTable 5-2 ).Usingthelikelihoodratiotest,wedeterminedthattheMREM-MNAR1tbest.TheestimatesofthemarginalmeanparameterswereverysimilartothoseoftheIOMTMthoughsomeweresignicant.Inaddition,thecorrelationparameter^=exp(^)=exp(0:30)=0:741wassignicant. Ofalltheselectionmodelsconsideredhere,theIOMTM2-1tbestbycomparisonoftheAIC's(6042.76forIOMTM-MNAR1and6056.26forMREM-MNAR1). 56

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MaximumlikelihoodestimatesofmarginalmeanparametersandselectionmodelparametersforOMREMunderMARandMNAR Int.11:09(0.13)1:06(0.13)1:05(0.13)Int.20:85(0.13)0:79(0.13)0:80(0.13)Int.32:32(0.15)2:08(0.15)2:10(0.15)Vis.0:37(0.54)4:99(0.62)4:75(0.62)Arm10:09(0.17)0:01(0.17)0:02(0.17)Arm20:04(0.17)0:10(0.17)0:09(0.17)Vis.*Arm10:46(0.83)1:38(0.80)1:27(0.81)Vis.*Arm21:66(0.91)1:64(0.85)1:84(0.87) int.0:54(0.09)0:86(0.18)0:76(0.19)yit10:18(0.06)1:05(0.20)1:00(0.20)yit2:65(0.36)2:53(0.36)Arm10:00(0.18)Arm20:13(0.19) Max.loglik.3019:683015:133014:75 2-2 inSection 2.1 showed,alargenumberofsubjectsdroppedoutfromthestudyduetodeath.Thus,theproposedmodelinthischaptermaynotbeappropriateforthesetypesofdropouts.Weintroduceseveralalternativemodels/approachesinChapter 6 and 7 57

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3 4 ,and 5 areproblematicforhandlingdropoutduetodeath.ThereasonisthatQOLresponsesnotmeasuredduetoparticipantdeathdonotexistwhereasscheduledmeasurementsduetodropoutforotherreasonscanbeviewedasexistingbutunobserved.AfterconsultationwithinvestigatorsattheMayoclinic,wegroupeddropoutsduetodeathortumorprogressiontogether. Inthischapter,weproposemethodsforinferenceinthepresenceofdropoutsduetoprogression/deathusingmodelsforthejointdistributionforthelongitudinalQOLresponsesandprogression/deathtimes(HoganandLaird,1997).WeapplythisapproachtotheQOLdata. logP(Ytcjxt;S=j) 1P(Ytcjxt;S=j)=0c(j)+xTt(j); 1P(Yitcjbi;xit)=4itk(j)+bit(j);andbi(j)T=(bi1(j);;bij(j)(j))N(0;i(j));forOMREM. wherextisavectorofcovariatesincludingTx,j=1;;J,andJisthenumberofpattern(JT).Withinthisframework,theprobabilityoffatigueforpatientswhoprogressed/diedbeyondtimekisgivenby 58

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6{1 )givenTxandisonlydenedfortcjS>j;Tx).Theasymptoticvarianceisgivenbyvarn^P(Yt>cjS>j;Tx)o=rfP(Yt>cjS>j;Tx)gTvar(^)rfP(Yt>cjS>j;Tx)g; 6-1 showsthenumberofsubjectsineachprogression/deathwindowbytreatmentgroup.Duetothesmallsamplesizespertreatmentgroupineachcolumnofthetable,wecreatedonlytwopatterns(J=2):S=1,subjectswhoprogressed/diedbeforetheendofthestudy,andS=2,thosewhodidnot. 59

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BreakdownofProgession/DeathwindowsbytreatmentgroupsforQOLdata.Proportionsinparentheses 123456 IFL35(0.15)32(0.14)38(0.16)26(0.11)18(0.08)86(0.37)235FOLFOX16(0.07)28(0.12)24(0.10)17(0.07)19(0.08)129(0.55)233IROX22(0.09)33(0.14)36(0.15)27(0.11)16(0.07)105(0.44)239Total73(0.103)93(0.132)98(0.139)70(0.099)53(0.075)320(0.453)707 Forouranalysishere,thetargetprobabilities( 6{2 )areevaluatedatk=5;i.e.,thosewhodidnotprogress/diebeforetheendofthestudy.WedidtheanalysisusinganIOMTMandanOMREM. 3.4 toestimateourtargetprobabilities.Table 6-2 givestheestimatedtargetprobabilitiesforthethreetreatmentarms.Figure 6-2 showstheprobabilitiesthatpatients,wholivedpasttheendofthestudy,wereverytired,^P(Y>cjS>5;Tx): 4.4 .Table 6-3 presentstheestimatedtargetprobabilitiesonthethreetreatmentarms.Figure 6-2 showsthattheestimatedtargetprobabilitiesforIROXandFOLFOXarmincreasedovertimelikethepreviousanalysisfortheIOMTM,whereasthosefortheIFLarmdidnotchange.Thiscorrespondstotheleveloffatigueincreasingovertime 60

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Estimatedtargetprobabilities^P(Yt>cjS>5;Tx)(standarderrors)forIOMTMwhereSisprogression/deathtime. Visit(t) Trtc123456 10.760.750.750.750.750.75(0.04)(0.03)(0.04)(0.05)(0.06)(0.08)IFL20.280.280.280.280.270.27(0.04)(0.03)(0.04)(0.05)(0.06)(0.08)30.080.080.080.070.070.07(0.02)(0.02)(0.02)(0.02)(0.02)(0.03) 10.740.760.770.790.800.81(0.03)(0.03)(0.03)(0.04)(0.06)(0.07)FOLFOX20.270.280.300.320.330.35(0.03)(0.03)(0.04)(0.06)(0.08)(0.10)30.070.080.080.090.100.10(0.01)(0.01)(0.02)(0.02)(0.03)(0.04) 10.740.760.780.800.810.83(0.04)(0.03)(0.04)(0.04)(0.05)(0.06)IROX20.270.290.310.330.350.38(0.03)(0.03)(0.04)(0.06)(0.08)(0.10)30.070.080.090.100.100.11(0.02)(0.02)(0.02)(0.03)(0.04)(0.05) ^P(Yt>cjS>5)and95%condenceintervalsforIOMTMwherec=1;2;3andSistumorprogressionordeathtime. 61

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Estimatedtargetprobabilities^P(Yt>cjS>5;Tx)(standarderrors)fortheOMREMwhereSisprogression/deathtime. Visit(t) Trtc123456 10.750.750.750.750.750.75(0.04)(0.03)(0.04)(0.05)(0.06)(0.07)IFL20.280.280.280.280.280.28(0.04)(0.04)(0.04)(0.05)(0.06)(0.08)30.080.080.080.080.080.07(0.02)(0.02)(0.02)(0.02)(0.02)(0.03) 10.750.760.770.790.800.81(0.03)(0.03)(0.03)(0.04)(0.05)(0.06)FOLFOX20.270.290.300.320.340.36(0.03)(0.03)(0.04)(0.06)(0.07)(0.09)30.070.080.080.090.100.10(0.02)(0.02)(0.02)(0.02)(0.02)(0.03) 10.740.760.790.810.830.85(0.04)(0.03)(0.04)(0.05)(0.06)(0.07)IROX20.260.290.320.350.380.42(0.04)(0.03)(0.05)(0.07)(0.10)(0.13)30.070.080.090.100.120.13(0.02)(0.02)(0.02)(0.03)(0.04)(0.06) ontheIROXandFOLFOXarms.However,thesedierencesbetweenarmswerenotsignicant.Thus,wehavesameconclusionasthepreviousanalysisusinganIOMTM. Inthecolorectalcancerclinicaltrial,qualityoflifeatlatertimepointsisonlyobservedforpatientswhosurvivetothosetimes.Thus,theanalysesinthischapterforanytimetisrestrictedtothesubgroupofpatientswhosurvivedpasttimetresultinginanonrandomizedcomparisonbetweenthetreatmentarms.Toaddressthisproblem,weproposeaprincipalstraticationapproach(FrangakisandRubin,2002)toestimatethecausaleectsoftreatmentinChapter 7 62

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^P(Yt>3jS>5;Tx)and95%condenceintervalsforOMREMwhereSistumorprogressionordeathtime. 63

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6 werenonrandomizedcomparisonsbetweentreatmentarmsastheyanalyzedthesubgroupofpatientswhosurvivedpastacertaintime.Onewaytoestimatethecausaleectofthetreatmentsinsuchsituationsisaprincipalstraticationapproach(FrangakisandRubin,2002).Eglestonetal(2006)introducedasetofassumptionstoidentifythecausaleectoftwotreatmentarmsforbinarydataandproposedasensitivityanalysisprocedurefordrawinginference.Weextendthisworktothreetreatmentarmswithordinaloutcomes. andletDit(Txi)betumor-progression/deathindicatorforsubjectiatvisitt,Dit(Tx)=8><>:0;alive;1;tumorprogressedordead. Obviously,ifDit(Tx)=1,thenDis(Tx)=1fors>t.Tomaintainclarity,wewillrefertothetumorprogression/deathindicatorDit()asthedeathindicatorforrestofchapter. Wedenethefollowingve(principal)strata: 64

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2. 3. 4. 5. Now,letYit(Txi)bethepotentialoutcomeforsubjectiattimet.ThefullsetofpotentialoutcomesatvisittisPt=fDt(0);(Yt(0);Dt(0)=0);Dt(1);(Yt(1);Dt(1)=0);Dt(2);(Yt(2);Dt(2)=0)g; Inthenextsection,weformallydenethecausaleectsofinterestandthenstatesomeadditionalassumptionsthatarenecessarytoestimatethem. 65

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7.2.2 )whichwewillverifylater. Wedenethefollowingthreecausaleectsofinterest, 1. SACEk(1;0)andSACEk(2;0)aretheoddsratiosbasedontheprobabilitythatresponseattimeTislargerthankforsubjectswhowouldbealiveunderallthreetreatmentarmsbetweenIROXandthecontrolarmandbetweenFOLFOXandthecontrol,respectively.SACEk(2;1)isoddsratiobasedontheprobabilitythattheresponseattimeTisgreaterthankforsubjects,whowouldbealiveeitherunderallthreetreatmentarmsoronlyunderthetwoactivetreatmentarmsbetweentreatmentIROXandFOLFOX. 66

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Assumption1assumestheactivetreatmentsareeective.Thatis,ifapatientisaliveundercontrolarm,thenthepatientwillalsobealiveunderactivearms. Assumption2issimilartoassumption1andisbasedonresultsinGoldbergetal.(2004)whichfoundthattheFOLFOXtreatmenthadsignicantlylowerratesofprogression/deaththantheIROXtreatment. wherePtisallthepotentialoutcomeuptoandincludingtimet.Assumption3statesthatthetreatmentarmisunrelatedtothesetofpotentialoutcomes. Thisassumptionstatesthatmissingnessofoutcomeisindependentofthethevalueoftheoutcomegivenallthepotentialoutcomeuptoandincludingtimet1.Itissimilartoanassumptionofsequentialmissingatrandom(Robinsetal.,1995) Thisassumptionisusedtoreducethenumberofsensitivityparameters.Thiswillbecomeapparentinwhatfollows. Thegoalistousetheobserveddata,Ot,alongwiththeseassumptionstodrawinferenceaboutSACE.ToidentifySACE,weneedtoidentifyP(YT(0)>kjAT(0)), 67

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andP(Yt(tx)>kjDt(tx)=0;Dt1(tx))=P(Yt>kjDt=0;Dt1;Tx=tx); RecallweneedtoestimateP(YT(1)>kjAT(0))andP(YT(2)>kjDT(0))toestimateSACEk(1;0)andSACEk(2;0).Todothis,weshowthatP(YT(1)>kjDT(1)=0)and 68

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7{2 )and( 7{3 )areidentied.Themixingprobabilities,P(DT(0)=0jDT(1)=0),P(DT(0)=1jDT(1)=0),P(DT(0)=0jDT(2)=0),P(DT(0)=1jDT(2)=0),andP(DT(1)=1jDT(2)=0),areidentiableby( 7{1 )sinceP(DT(0)=0jDT(1)=0)=gT(0)

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7{2 )andP(YT(2)>kjAT(0)),P(YT(2)>kjAT(1)),andP(YT(2)>kjAT(2))in( 7{3 )arenotidentied.Giventheidentiedcomponents,toidentifytheseunidentiedquantities,weonlyneedtoknowtheirratios.AllthreeratiosareidentiedviaAssumption5, andoddsP(YT(2)>kjAT(2)) From( 7{2 )and( 7{4 ),wecanidentifyP(YT(1)>kjAT(0))bysolvingthefollowingquadraticequation, wherex=P(YT(1)>kjAT(0)).When=1,P(YT(1)>kjAT(0))=hT;1(k): 2a(); 70

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7{5 ),( 7{6 ),and( 7{3 ),wecanidentifyP(YT(2)>kjAT(0))bysolvingthefollowingcubicequation, wherex=P(YT(2)>kjAT(0)).When(1)=(2)=1,P(YT(2)>kjAT(0))=hT;2(k)gT(2) 2a((1)); 2a((2)); 7{8 )numerically. ToestimateSACEk(2;1),weneedtoidentifyP(YT(1)>kjAT(0)[AT(1))andP(YT(2)>kjAT(0)[AT(1)).ByAssumptions2,3,and4,theprobabilityfromthedenominatorofSACEk(2;1)isidentiedsinceP(YT(1)>kjAT(0)[AT(1))=P(YT(1)>kjDT(1)=0)=hT;1(k):

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Here,given(1)and(2)(fromtheestimationofSACEk(2;0)),wecanidentifyP(YT(2)>kjAT(0))bysolving( 7{8 )andcancalculateP(YT(2)>kjAT(1))using( 7{5 ).Theothertermsin( 7{9 )areidentiedasP(AT(0)[AT(1))=P(DT(1)=0)=gT(1);P(AT(0))=P(DT(0)=0)=gT(0);andP(AT(1))=P(DT(0)=1;DT(1)=0)=P(DT(1)=0)P(DT(0)=0)=gT(1)gT(0); TheSACEisafunctionofgT(Tx)andhT;Tx(k).Weknowthat,forTx=0;1;2, 1DTjTx=txBin(N(tx);gT(tx)) (7{10) 72

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(7{11) =Pj>T1P(YT>kjS=j;Tx)P(S=jjTx) wherethelastequationisfrom( 6{2 )inChapter 6 andP(YT>kjS=j;Tx)isderivedbasedonanIOMTM(2)modelforP(yjS=j;Tx). 3 aregiveninAppendix D 73

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Breakdownofsubjectswhowerealivebytreatmentgroups.Proportionsareinparentheses AliveProg/DeathTotal IFL86(0.366)149(0.634)235IROX105(0.439)134(0.561)239FOLFOX129(0.554)104(0.446)233Total302387707 parameters,(1),and(2)wherearesimplefunctionsof((1)1;(2)1;(2)2)inAssumption5.ThisissimilartotheapproachinEglestonetal.(2006).ThesensitivityparameteristheoddsratiooftheprobabilityoffatigueundertheIROXarmamongthegroupthatisaliveunderthetwoactivetreatmentsbutnotaliveunderthecontroltreatmentatvisitTascomparedtothegroupthatisaliveunderallthreearms.Thesensitivityparameter(1)istheoddsratioofprobabilityoffatigueundertheFOLFOXarmamongthegroupthatisaliveunderthetwoactivetreatmentsbutnotaliveunderthecontroltreatmentatvisitTascomparedtothegroupthatisaliveunderallthreearms.Thesensitivityparameter(2)hasasimilarinterpretationto(1)butcomparingthegroupthatisaliveunderFOLFOXbutnotaliveunderthecontrolarmandIROXatvisitTcomparedtothegroupthatisaliveunderallthreearms. AlargevalueofmeansthattheprobabilityoffatigueundertheIROXarmamongpatientsinstratumAT(1)ishigherthanthatamongpatientsinstratumAT(0).Thesensitivityparameters(1)and(2)areinterpretedsimilarly. 7.2 toanalyzetheQOLdatafromChapter 2 .Inparticular,weemploythesensitivityanalysisapproachfrom 7.2.6 toestimatetheSACE. UsingthedatainTable 7-1 ,theMLE'sofgT()are^gT(0)=:366,^gT(1)=:439,and^gT(2)=:554.TheMLE'sofhT;(K1)basedontheIOMTM(2)weregiveninTable 6-2 ,^hT;0(K1)=0:073,^hT;0(K1)=0:114,and^hT;0(K1)=0:103. 74

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ToestimateSACEK1(1;0)wealsoneedtoestimateP(DT(0)=0jDT(1)=0),P(YT(0)>K1jAT(0)),andP(YT(1)>K1jAT(0)).AlltheseprobabilitiesareestimablegivenAssumptions1-4.TheestimateofP(DT(0)=0jDT(1)=0),theprobabilityofsurvivingunderthecontrolarmgivensurvivalundertheIROXarm,is0.834(95%CIis[0.822,0.846]).TheestimateofP(YT(0)>K1jAT(0))is^hT;0(K1)=0:073.Forxedbetween0.1and8.3,P(YT(1)>K1jAT(0))decreasesfrom0.123to0.067. TheestimatedvaluesofSACEK1(1;0)andassociatedcondenceintervalsoverrangingfrom0.1to8.3aregivenbyFigure 7-1 .Asincreased,theestimatedvaluesofSACEK1(1;0)decreased.TheestimatedvaluesofSACEK1(1;0)waslargerthan1until=6.Thisindicatedthattheprobabilityofapatient'sfatigueontheIROXtreatmentwaslargerthantheprobabilityofapatient'sfatigueontheIFLtreatmentwhen<6.When6,therelationreversed.However,therewasnosignicantdierence 75

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WereplicatedtheanalysesforSACEK1(2;0)andSACEK1(2;1).Figure 7-2 and 7-4 areplotsoftheestimatedvaluesofSACEK1(2;0)andSACEK1(2;1)asfunctionsofthesensitivityparameters(1)and(2),respectively.Figure 7-3 and 7-5 areplotsofthez-statisticsforestimatedvaluesofSACEK1(2;0)andSACEK1(2;1),respectively.FortheSACEK1(2;0),wehavesimilarpatterntoSACEK1(1;0).Whenvaluesof(1)and(2)weresmall,theprobabilityofapatient'sfatigueontheFOLFOXtreatmentwaslargerthantheprobabilityofapatient'sfatigueontheIFLtreatment.However,therewasnosignicantdierencebetweenthetwoarms TheestimatedSACEK1(2;1)'swerelessthan1forallvaluesof(1)and(2)considered.ThismeansthattheprobabilityofhighfatigueontheIROXtreatmentwaslargerthantheprobabilityofhighfatigueontheFOLFOXtreatment.For(1)<4:5and(2)>4:5thisdierencebetweenthetwoarmswassignicant(at95%level). Toidentifythecausaleects,wemadeveassumptions.Futureworkwillexaminesensitivityofinferencetotheseassumptions,particularly,Assumptions2and5.WewillalsobecollaboratingwithinvestigatorsatMayoclinictohelpdeterminelikelyvalues/rangeforthesensitivityparameters(,(1),and(2)). 76

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EstimatedSACEK1(2;0)asafunctionof(1)and(2). Figure7-3. Z-statisticsforestimatedSACEK1(2;0)asafunctionof(1)and(2). 77

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EstimatedSACEK1(2;1)afunctionof(1)and(2). Figure7-5. Z-statisticsforestimatedSACEK1(2;1)asfunctionof(1)and(2). 78

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Weremovedtheignorabledropoutassumptionandalsoconsideredselectionmodelsfornonignorabledropout.LikelihoodratiotestsindicatedthatthemodelsunderMNARdropouttbetterthanthoseunderMARdropout. Wealsoproposedtwoapproachestoaccommodatedropoutsduetoprogressionordeath.Basedonthemixturemodelapproachforwhichwecannotestimatethecausaleectoftreatment,weconcludedthatpatients'fatiguewasnotaectedbytreatment.However,theprincipalstraticationapproach(FrangakisandRubin,2002)forwhichwecouldestimatethecausaleects,suggestedtherewasaneectivetreatmentarm. 3 and 4 andsimilarmodelsforlongitudinalnominaldata.Wegiveabriefoverviewofsomeideasnext. WearedevelopingamodelforlongitudinalordinaldatabasedontheIOMTM,butwithrandomeects.Thedependencemodelwillallowserialdependenceviaa1storder 79

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Weplanondevelopingmarginalizedmodelsforlongitudinalnominaldata.AnyextensionoftheOMTMwillhavemanydependenceparametersand`shrinkage'ontheseparameterswillbenecessary.Wewillinvestigatevariouspriorstodothisusingideasofpartialexchangeability(EvansandSedransk,2001). Extensionsofmarginalizedrandomeectsmodels(MREM)tolongitudinalnominaldatawillinvolvehighdimensionalcovariancematrices(ofdimensionT(K1),whereKisthenumberofcategoriesandTisthenumberofobservationtimes).Wewillexplorevariouswaystostructurethismatrixbothintermsofstableestimationandintermsoffeasiblecomputations. 80

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TheappendixincludesproofsoftheTheoremsfromChapter 3 .ThesearesimilartoresultsinHeagerty(2002)andFitzmauriceandLaird(1993),buthereweprovideadditionaldetailsininterestofgreaterclarity. 1 1 ,weneedthefollowingLemma, Proof. Wealsohave From( A{1 )and( A{2 ),wehave and From( A{4 ),wehave From( A{3 )and( A{5 ),wehaveI12@2c() 1 :Usingthefactthatthelikelihoodcontributionfromeachsubjectcanbefactoredintotheproductoftheprobabilityfortherstobservationtimes 81

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wherei0T=log1+K1Xm=1ei1k!TXj=2log1+K1Xm=1e4ijm!;ijk=4ijk+log1+K1Xm=1e4ij+1m!log1+K1Xm=1e4ij+1m+(m)ij+11k!;andiTk=4iTk: A{6 )isacanonicalexponentialform,()isorthogonaltoE(yijk)byLemma 1 .NotethatE(yijk)isafunctionofonly0and.ByBarndor-NielsenandCox(1994,p50),isorthogonalto0and. 1 1 2 1 ,weobtainthejointdistributionandreexpressitas 83

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(A{8)

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A{8 )withrespecttoandexpectationwithrespecttothetrueandarbitrary,wehaveNXi=1@ @XiAiV1i11E;(yi)PMi=0: A{7 ). 85

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@0j@0l=Ey0@NXi=1niXt=2K1Xk=1K1Xg=1@Pcitk @@T=Ey0@NXi=1niXt=2K1Xk=1K1Xg=1@Pcitk @0a@=Ey0@NXi=1niXt=2K1Xk=1K1Xg=1@Pcitk @(d)1b@(f)1eT1A=Ey24NXi=1niXt=28<:K1Xk=10@K1Xg=1@Pcitk @(d)1b@0j!=Ey24NXi=1niXt=2K1Xk=18<:K1Xg=1@Pcitk @(d)1b@T!=Ey24NXi=1niXt=2K1Xk=18<:K1Xg=1@Pcitk A{9 )and( A{10 )arezeroiftherearenomissingobservations. 86

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Estimatesof4itcanbeobtainedusingNewton-Raphson.Letf(4it)=(f1(4it);;fK1(4it))); A.4 .ThecontributionofL(3),L(1)andL(2)is 87

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Nowfortheinformationmatrix,wehavethefollowingresults, 88

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Afterobtainingcomputationsfortimet,wemakecalculationsfortimet+1thatrequireanupdate(PJitkl;h(t)ilg)!PJitklaswellasupdatesforderivatives,PJit+1kl=P(Yit+1=k;Yit=l)=KXg=1P(Yit+1=kjYit=l;Yit1=g)P(Yit=l;Yit1=g)=KXg=1h(t+1)iklgPJitlg;@PJitkl

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90

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3 Wenowclaimthat4itk>4itk1forallk.Wewillshowthatifitisnottrue(i.e,4itk4itk1forsomek),thencondition( B{1 )cannotbesatised. i)If4itk=4itk1forsomek,thenthelefttermof( B{1 )iszero.Thisisacontradictionto( B{1 ). ii)If4itk<4itk1forsomek,thenlet4itk1=4itk+forsome>0.So( B{1 )isgivenby0
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TheformsofthederivativesforQuasi-Newtonalgorithmfollow@logL @0j=NXi=1L(;yi)1ZL(jyi;ai)(niXt=1K1Xk=1RitkRitk+1eitk @=NXi=1L(;yi)1ZL(jyi;ai)(niXt=1K1Xk=1RitkRitk+1eitk @=NXi=1L(;yi)1ZL(jyi;ai)"niXt=1K1Xk=1RitkRitk+1eitk @=NXi=1L(;yi)1ZL(jyi;ai)"niXt=1K1Xk=1RitkRitk+1eitk

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4{6 )andreexpressedbyL(jyi;ai)=exp"log(TXt=1K1Xk=1(RitkitkRitk+1g(itk)))#; ( 4{4 )isreexpressedgivenby where()isthestandardnormaldensityfunction.Tocomputethescorevectorandinformationmatrix,wealsoneedderivativesof4itwithrespectto0,,and.Theycanbeobtainedfromtherelationship( B{2 ),@PMitk

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5{3 ))isgivenby,logf(yo;d)=NXi=124logf(yoijai)+td1Xt=2logpit+log8<:KXyitd=1(1pitd)Pcidyitd9=;35: A.4

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4{6 ). 95

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where@P(YT(1)>kjAT(0)) 2a()(1)+(1)fb()+2c(1)g 2a2()(1)nb()+p 2a()(1)hT;1(k)+f(1)hT;1(k)gb()2a()hT;1(k) 2a()(1)gT(1)+(1)gT(1)b()2a()gT(1)

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2a((1))(1)+(1)b((1)) 2a((2))(2)+(2)b((2)) 2a((1))(1)+(1)b((1)) 2a((2))1+b((2))+2(1(2))c(2) 2a2((2))(1(2))b((2))+b((2))+2(1(2))c(2);(1)=1and(2)6=1;Numericalderivativeisgivenlater;(1)6=1and(2)6=1.

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2a((1))b((1))f((1)1)hT;2(k)1g2(1(1))c(2)2a((1))hT;2(k) 2a((2))hT;2(k)((2)1)(2)+b((2))fhT;2(k)((2)1)(2)g2a((2))hT;2(k) 2a((1))((1)1)gT(2)+((1)1)gT(2)b((1))2a((1))gT(2) 2a((2))((2)1)gT(2)+((2)1)gT(2)b((2))2a((2))gT(2) 7{8 )inChapter 7 where=(gT(0);gT(1);gT(2);hT;2(k)),andlet(i)beavectorwiththeithelementofremoved.Theithelementof,(i),isperturbedasfollows:i=c1(i) wherec1andc2aresmall(here,wesetc1=0:01andc2=0:001).Usingthecurrentestimatesoftheparameters,thederivativeof(i)is@P(YT(2)>kjAT(0)) 2i:

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KeunbaikLeewasbornonNovember11,1971,inKyungsan,Korea.In1991,heregisteredwithKyungpookNationalUniversitytopursuestudiesinstatisticsandenteredKyungpookNationalGraduateSchoolin1995.Hegraduatedwithhismaster'sdegreeandabeststudentawardin1997andcompletedhismilitaryserviceasaprivatesoliderin1999.Eagertofurtherintensifyhisknowledge,hewenttotheUniversityofMissouri-Columbiain2000.Afteroneyearofcoursework,hetransferredtotheUniversityofFlorida.Asidefromcompletingthestandardcurriculum,heworkedasateachingassistantandaresearchassistantbeforejoiningDr.MichaelDanielsasaresearchassistant.In2003,hemarriedJunghwaKim,agraduatestudentatSeoulNationalGraduateSchool.Theirdaughter,Charlene,wasborninGainesvillethenextyear.Inthefallof2002,KeunbaikLeebeganonmarginalizedregressionmodels,leadingtohisdissertationworkinmodelinglongitudinalcategoricaldata.Aftergraduating,KeunbaikLeewillbeaassistantprofessoratLouisianaStateUniversity-HealthScienceCenter. 105