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1、CHAPTER7SurvivalMixedEffects(Frailty)Models7.1IntroductionInpracticeweareofteninterestedinmodelingthetimetoaneventofinterest.Forexample,inalongitudinalstudysomesubjectsmaydropoutbeforetheendofthestudy,soonemaybeinterestedinndinganypossiblerelationshi
2、pbetweendropouttimesandcovariatessuchasageandgender(e.g.,areyoungersubjectsmorelikelytodropoutearlier?).Othercommoneventsofinterestincludetimetodeath,timetoinfectionofadisease,timetoacaraccident,timetocompletionofatask,etc.Thesetypesofdataarecalledeve
3、nt-timedataorsurvivaldata.Theanalysisofevent-timedataorsurvivaldataiscalledsurvivalanalysis.Survivaldataoreventtimedataareverycommoninpractice.Forsimplicity,inthischapterwewilloftentreattheeventas“death”,buttheeventcanbedenedinamuchbroadersense(e.g.,
4、dropout,infection,accident,etc.).Therearesomespecialcharacteristicsofsurvivaldata:survivaldatamaybecensored.Forexample,theeventofinterestmaynotbeobservedforsomesubjectsinthestudy,possiblyduetosubjects'dropouts,lossoffollow-up,orearlyterminationofthes
5、tudy.survivaldataareoftenskewed,andmanydataareskewedtotheright.Sosurvivaldatamaynotfollowsymmetricdistributionssuchasthenormaldistributions.survivaldatamayhaveunequalfollow-uptimes.Forexample,subjectsmayenterthestudyatdifferenttimes.Intheanalysisofs
6、urvivaldataorevent-timedata,thesespecialfeaturesmustbeincorporated,sospecialstatisticalmethodsareoftenrequired.Figure7.1showstypicalsurvivaldataforfoursubjectsandhowthedatamaybecensored:oneeventtimeiscensoredduetoearlyterminationofthestudy,theother229
7、230MIXEDEFFECTSMODELSFORCOMPLEXDATASurvivalDatawithCensoring10eventtimeobservedendofstudyeventtimecensored86individual420246810timeFigure7.1Survivaldataforfourindividuals.Theeventtimesfortwoindividualsareobserved,whiletheeventtimesfortheothertwoindivi
8、dualsarecensored(onecensor-ingisduetotheterminationofthestudywhiletheothercensoringispossiblyduetodropout).eventtimeiscensoredpossiblyduetodropoutorlossoffollow-up,whiletheothertwoeventtimesareobserved.Sotheobservedsurvivaldataforthesefoursubj
