Analyzing Incomplete Discrete Longitudinal Clinica

arXiv:math/0606497v1[math.ST]20Jun2006StatisticalScience2006,Vol.21,No.1,52–69DOI:10.1214/088342305000000322cInstituteofMathematicalStatistics,2006AnalyzingIncompleteDiscreteLongitudinalClinicalTrialDataIvyJansen,CarolineBeunckens,GeertMolenberghs,GeertVerbekeandCraigMallinckrodtAbstract.Commonlyusedmethodstoanalyzeincompletelongitudi-nalclinicaltrialdataincludecompletecaseanalysis(CC)andlastobservationcarriedforward(LOCF).However,suchmethodsrestonstrongassumptions,includingmissingcompletelyatrandom(MCAR)forCCandunchangingprofileafterdropoutforLOCF.Suchassump-tionsaretoostrongtogenerallyhold.Overthelastdecades,anumberoffulllongitudinaldataanalysismethodshavebecomeavailable,suchasthelinearmixedmodelforGaussianoutcomes,thatarevalidun-derthemuchweakermissingatrandom(MAR)assumption.Suchamethodisuseful,evenifthescientificquestionisintermsofasin-gletimepoint,forexample,thelastplannedmeasurementoccasion,anditisgenerallyconsistentwiththeintention-to-treatprinciple.Thevalidityofsuchamethodrestsontheuseofmaximumlikelihood,un-derwhichthemissingdatamechanismisignorableassoonasitisMAR.Inthispaper,wewillfocusonnon-Gaussianoutcomes,suchasbinary,categoricalorcountdata.Thissettingislessstraightforwardsincethereisnounambiguouscounterparttothelinearmixedmodel.Wefirstprovideanoverviewofthevariousmodelingframeworksfornon-Gaussianlongitudinaldata,andsubsequentlyfocusongeneralizedlinearmixed-effectsmodels,ontheonehand,ofwhichtheparameterscanbeestimatedusingfulllikelihood,andongeneralizedestimatingequations,ontheotherhand,whichisanonlikelihoodmethodandhencerequiresamodificationtobevalidunderMAR.Webrieflycom-mentonthepositionofmodelsthatassumemissingnessnotatrandomandarguetheyaremostusefultoperformsensitivityanalysis.Ourdevelopmentsareunderscoredusingdatafromtwostudies.Whilethecasestudiesfeaturebinaryoutcomes,themethodologyappliesequallywelltootherdiscrete-datasettings,hencethequalifier“discrete”inthetitle.Keywordsandphrases:Completecaseanalysis,ignorability,gener-alizedestimatingequations,generalizedlinearmixedmodels,lastob-servationcarriedforward,missingatrandom,missingcompletelyatrandom,missingnotatrandom,sensitivityanalysis.IvyJansenisPostdoctoralResearcher,CenterforStatistics,HasseltUniversity,AgoralaanGebouwD,B-3590Diepenbeek,Belgium(e-mail:[e-mail]).CarolineBeunckensisResearchAssistant,CenterforStatistics,HasseltUniversity,AgoralaanGebouwD,B-3590Diepenbeek,Belgium(e-mail:[e-mail]).GeertMolenberghsisProfessor,CenterforStatistics,HasseltUniversity,AgoralaanGebouwD,B-3590Diepenbeek,Belgium(e-mail:[e-mail]).12JANSENETAL.1.INTRODUCTIONDatafromlongitudinalstudies,ingeneral,andfromclinicaltrials,inparticular,arepronetoincom-pleteness.Dropoutisaspecialcaseofincomplete-ness.Sinceincompletenessusuallyoccursforreasonsoutsidethecontroloftheinvestigatorsandmayberelatedtotheoutcomemeasurementofinterest,itisgenerallynecessarytoaddresstheprocessthatgov-ernsincompleteness.Onlyinspecialbutimportantcasesisitpossibletoignorethemissingnessprocess.Whenreferringtothemissing-value,ornonre-sponse,process,wewillusetheterminologyofLittleandRubin(2002,Chapter6).Anonresponseprocessissaidtobemissingcompletelyatrandom(MCAR)ifthemissingnessisindependentofbothunobservedandobserveddata,andsaidtobemissingatran-dom(MAR)if,conditionalontheobserveddata,themissingnessisindependentoftheunobservedmeasurements.AprocessthatisneitherMCARnorMARistermednonrandom(MNAR).Inthecon-textoflikelihoodinference,andwhentheparame-tersthatdescribethemeasurementprocessarefunc-tionallyindependentoftheparametersthatdescribethemissingnessprocess,MCARandMARareig-norable,whileanonrandomprocessisnonignorable.Earlyworkregardingmissingnessfocusedontheconsequencesoftheinducedlackofbalanceofdevia-tionsfromthestudydesign(AfifiandElashoff,1966;HartleyandHocking,1971).Later,algorithmicde-velopmentstookplace,suchastheexpectation–maxi-mizationalgorithm(EM;Dempster,LairdandRu-bin,1977)andmultipleimputation(Rubin,1987).Theseadvanceshavebroughtlikelihood-basedig-norableanalysiswithinreachforalargeclassofdesignsandmodels.However,theyusuallyrequireextraprogramminginadditiontoavailablestandardstatisticalsoftware.GeertVerbekeisProfessor,BiostatisticalCentre,KatholiekeUniversiteitLeuven,Kapucijnenvoer35,B-3000Leuven,Belgium(e-mail:[e-mail]).CraigMallinckrodtisSeniorResearchFellow,EliLilly&Company,Indianapolis,Indiana46285,USA(e-mail:[e-mail]).ThisisanelectronicreprintoftheoriginalarticlepublishedbytheInstituteofMathematicalStatisticsinStatisticalScience,2006,Vol.21,No.1,52–69.Thisreprintdiffersfromtheoriginalinpaginationandtypographicdetail.Inthemeantime,however,clinicaltrialpracticehasputastrongemphasisonsuchmethodsascom-pletecaseanalysis(CC),whichrestrictstheanal-ysistothosesubjectsforwhichallinformationhasbeenmeasuredaccordingtoprotocol,andlastobser-vationcarriedforward(LOCF),forwhichthelastobservedmeasurementissubstitutedforvaluesatlaterpointsintimethatarenotobserved,orothersimpleformsofimputation.Claimedadvantagesin