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1、CHAPTER6MixedEffectsModelswithCensoring6.1IntroductionInmanylongitudinalstudies,suchasstudiesonenvironmentalpollutionandinfectiousdiseases,measurementsofsomevariablesmaybesubjecttoade-tectionlimit,i.e.,acertainthresholdvaluebeloworabovewhichthemeasure-mentsarenotquantiable.Inotherwords,somedataare
2、leftorrightcensored,sotheirvaluesarenotobserved.Inthischapter,weconsidermixedeffectsmodelsforcensoredlongitudinaldata.Insomestudies,thedetectionlimitsmayvaryfromlaboratorytolaboratoryandmaychangeovertimeasnewlaboratorymethodsareimplemented.Forexample,inHIV/AIDSstudiesviralloadsmaybesubjecttoleftcen
3、soringduetoadetectionlimitsoviralloadvaluesbelowthislimitcannotbemeasuredorquantied.Figure1.3inChapter1(page11)andFigure6.1inthischapter(page209)showviralloaddatainanAIDSstudywherethedetectionlimitis100(or2inlog10scale).Becausetheproportionofcensoreddatamaynotbesmallinmanylongitudinalstudies,failu
4、retoaccountforthecensoringinstatisticalanalysismayleadtosignicantbiasesintheparameterestimatesandinference(Hughes1999).ForAIDSdatawithleftcensoring,commonlyusednaivemethodsincludeim-putingthecensoredvaluesbythedetectionlimitorhalfthedetectionlimitorothersimpleimputationprocedures(Paxtonetal.1997).
5、Thesenaivemeth-ods,however,mayleadtobiasedresults,asdemonstratedinHughes(1999).Moreover,thesenaivesingleimputationmethodsignoretheuncertaintyofthecensoredvalues.Therefore,oneshouldnotusethesenaivemethodsinstatis-ticalinference,andmoreappropriateformalmethodsdescribedinthischaptershouldbeconsidered.
6、Censoreddatamaybeviewedasspecialmissingdatasince,althoughthetruevaluesarenotobserved,thereissomeinformationavailableforthecensored203204MIXEDEFFECTSMODELSFORCOMPLEXDATAvalues:thecensoredvaluesareknowntobesmallerorlargerthanaknownnumber.Thisisdifferentfromusualmissingdataforwhichoneknowsnoth-ingabou
7、ttheunobservedtruevalues.Censoreddatamayalsobeviewedasnon-ignorablemissingdatasincethemissingness(censoring)isrelatedtotheunobservedvalues(thecensoredvalues).Censoreddataareperhapsmorewellknowninthecontexto