Nonparametric Function Estimation for Clustered Da

NonparametricFunctionEstimationforClusteredDataWhenthePredictorisMeasuredWithout/WithErrorXihongLinandRaymondJ.Carroll∗October4,1999AbstractWeconsiderlocalpolynomialkernelregressionwithasinglecovariateforclustereddatausingestimatingequations.Weassumethatatmostm∞observationsareavailableoneachcluster.Inthecaseofrandomregressors,withnomeasurementerrorinthepredictor,weshowthatitisgenerallythebeststrategytoignoreentirelythecorrelationstructurewithineachcluster,andinsteadtopretendthatallobservationsareindependent.Inthefurtherspecialcaseoflongitudinaldataonindividualswithfixedcommonobservationtimes,weshowthatequivalenttothepooleddataapproachisthestrategyoffittingseparatenonparametricregressionsateachobservationtimeandconstructinganoptimalweightedaverage.Wealsoconsiderwhathappenswhenthepredictorismeasuredwitherror.UsingtheSIMEXapproachtocorrectformeasurementerror,weconstructanasymptotictheoryforboththepooledandweightedaverageestimators.Surprisingly,forthesameamountofsmoothing,theweightedaverageestimatorstypicallyhavesmallervariancesthanthepoolingstrategy.WeapplytheproposedmethodstotheanalysisoftheAIDSCostsandServicesUtilizationSurvey.KEYWORDS:AIDS;Asymptoticbiasandvariance;Clustereddata;Efficiency;Errorsinvariables;Estimatingequations;Generalizedlinearmodels;Kernelregression;Longitudinaldata;Measure-menterror;Nonparametricregression;Paneldata;SIMEX.Shorttitle.NonparametricRegressionforClusteredData∗XihongLinisAssociateProfessor,DepartmentofBiostatistics,UniversityofMichigan,AnnArbor,MI48109-2029.HerresearchwassupportedbyagrantfromtheNationalCancerInstitute(CA–76404).RaymondJ.CarrollisUniversityDistinguishedProfessor,DepartmentsofStatisticsandBiostatistics&Epidemiology,TexasA&MUni-versity,CollegeStationTX77843–3143.HisresearchwassupportedbyagrantfromNationalCancerInstitute(CA–57030),andbytheTexasA&MCenterforEnvironmentalandRuralHealthviaagrantfromtheNationalInstituteofEnvironmentalHealthSciences(P30-ESO9106).NonparametricFunctionEstimationforClusteredDataWhenthePredictorisMeasuredWithout/WithErrorAbstractWeconsiderlocalpolynomialkernelregressionwithasinglecovariateforclustereddatausingestimatingequations.Weassumethatatmostm∞observationsareavailableoneachcluster.Inthecaseofrandomregressors,withnomeasurementerrorinthepredictor,weshowthatitisgenerallythebeststrategytoignoreentirelythecorrelationstructurewithineachcluster,andinsteadtopretendthatallobservationsareindependent.Inthefurtherspecialcaseoflongitudinaldataonindividualswithfixedcommonobservationtimes,weshowthatequivalenttothepooleddataapproachisthestrategyoffittingseparatenonparametricregressionsateachobservationtimeandconstructinganoptimalweightedaverage.Wealsoconsiderwhathappenswhenthepredictorismeasuredwitherror.UsingtheSIMEXapproachtocorrectformeasurementerror,weconstructanasymptotictheoryforboththepooledandweightedaverageestimators.Surprisingly,forthesameamountofsmoothing,theweightedaverageestimatorstypicallyhavesmallervariancesthanthepoolingstrategy.WeapplytheproposedmethodstotheanalysisoftheAIDSCostsandServicesUtilizationSurvey.KEYWORDS:AIDS;Asymptoticbiasandvariance;Clustereddata;Efficiency;Errorsinvariables;Estimatingequations;Generalizedlinearmodels;Kernelregression;Longitudinaldata;Measure-menterror;Nonparametricregression;Paneldata;SIMEX.Shorttitle.NonparametricRegressionforClusteredData1INTRODUCTIONThereisavastliteraturedevelopedinthepastdecadeonparametricregressionforclustereddatausingestimatingequations(LiangandZeger,1986),wheregeneralizedlinearmodelsareaspecialcase.Suchparametricassumptionsmaynotalwaysbedesirable,sinceappropriatefunctionalformsofthecovariatesmaynotbeknowninadvanceandtheoutcomemaydependonthecovariatesinacomplicatedmanner.Therehasbeensubstantialinterestrecentlyinextendingtheexistingparametricmodelstoallowfornonparametriccovariateeffects(ZegerandDiggle,1994;SeveriniandStaniswalis,1994;WildandYee,1996).Suchnonparametricregressionallowsformoreflexiblefunctionaldependenceoftheoutcomevariableonthecovariatesandcanalsobeusedtoinvestigatewhetheranappropriateparametricfunctioncanbedevelopedtodescribethedatawell.Anothercomplicationintheanalysisofclustereddataisthepresenceofcovariatemeasurementerror.Forexample,ithasbeenwelldocumentedintheliteraturethatcovariatessuchasbloodpressure(Carroll,RuppertandStefanski,1995)andCD4count(Tsiatis,Degruttola,Wulfsohn,1995)areoftensubjecttomeasurementerror.WeconsiderinthispaperdatafromtheAIDSCostsandServicesUtilizationSurvey(ACSUS)(Berk,etal.,1993).TheACSUSsampled2487subjectsin10randomlyselectedUScitieswithhighestAIDSrates.Aseriesofsixinterviewswereconductedforeachrespondenteverythreemonthsfrom1991to1992.Amainoutcomeofinterestwaswhetheranintervieweehadhadhospitaladmissions(yes/no)duringthepastthreemonths.Thecollectedcovariatesincludeddemographicvariables,HIVstatus,CD4count,andtreatments.AquestionofinterestinthisstudyishowCD4countaffectstheriskofhospitalization.Theanalysisofthisdatasethastwomajorcomplications.ThefirstcomplicationisthateventhoughitisbelievedthatalowerCD4countisassociatedwithahigherriskofhospitalization,