Applied Survival Analysis(生存分析) lesson 2

Lesson2EstimatingtheSurvivorFunctionBIS630bYaleUniversityDepartmentofBiostatisticsCollettChapter2DateModied:1/20/2015ParametricEstimationofS(t)Kaplan-MeierMethodLife-TableMethodContents1ParametricEstimationofS(t)ExponentialModel2Kaplan-MeierMethodEmpiricalSurvivalFunctionKaplan-MeierEstimatorVarianceofKaplan-MeierEstimatorExample3Life-TableMethodLife-TableEstimatorEstimatingtheHazardFunctionEstimatingtheCumulativeHazardFunctionBIS630bLesson2EstimatingtheSurvivorFunction2/115ParametricEstimationofS(t)Kaplan-MeierMethodLife-TableMethodSurvivorFunctionSurvivalFunctionThesurvivalfunctionS(t)isprobabilitythatanindividualsurvivesatleasttotimet.SeveralwaystoestimateS(t):AssumeaparametricmodelandusemaximumlikelihoodtheorytoestimatemodelparametersandS(t),accountingforcensoringinthelikelihoodfunctionUseanonparametricmethodBIS630bLesson2EstimatingtheSurvivorFunction3/115ParametricEstimationofS(t)Kaplan-MeierMethodLife-TableMethodExponentialModelProgressthisUnit1ParametricEstimationofS(t)ExponentialModel2Kaplan-MeierMethodEmpiricalSurvivalFunctionKaplan-MeierEstimatorVarianceofKaplan-MeierEstimatorExample3Life-TableMethodLife-TableEstimatorEstimatingtheHazardFunctionEstimatingtheCumulativeHazardFunctionBIS630bLesson2EstimatingtheSurvivorFunction4/115ParametricEstimationofS(t)Kaplan-MeierMethodLife-TableMethodExponentialModelExponentialModelLikelihoodRecall,L()=nYi=1P(Ti=ti)iP(Titi)1i=nYi=1f(ti;)iS(ti;)1i=nYi=1(exp(ti))i(exp(ti))1i=rexp(ST)Wherer=Pi,theobservednumberofeventsST=Pti,totalfollow-uptimefornindividualsinstudyBIS630bLesson2EstimatingtheSurvivorFunction5/115ParametricEstimationofS(t)Kaplan-MeierMethodLife-TableMethodExponentialModelExponentialModelLikelihoodFindingtheMLEof,L()=rexp(ST)l()=logL()=rlog()STl0()=rST^=rSTThus,^S(t)=exp(^t)BIS630bLesson2EstimatingtheSurvivorFunction6/115ParametricEstimationofS(t)Kaplan-MeierMethodLife-TableMethodExponentialModelTypicalDataSurvivaldataconsistofalistofsurvivaltimes(t)foreachsubject,someofwhichmaybecensored,andaneventindicator()Inaddition,theremightbesomecovariates(x)oneachpatient(e.g.treatmentgroup,age,weight,sex,smokingstatus)Subjecttx11t11x112t22x12...ntnnx1nPti=STPi=r;TotalnumberoffailuresBIS630bLesson2EstimatingtheSurvivorFunction7/115ParametricEstimationofS(t)Kaplan-MeierMethodLife-TableMethodExponentialModelTypicalData:ExampleExample:Clinicaltrialinleukemiapatientsinremission.21subjectsrandomizedtochemotherapytreatment(6-MP);21subjectsrandomizedtoplacebo.Event(failure)ofinterest:Relapse.Timemeasuredinmonths.TreatmentGroup(x=1)PlaceboGroup(x=0)n=21n=216,6,6,6+,7,9+,10,10+,1,1,2,2,3,4,4,11+,13,16,17+,19+,20+,22,5,5,8,8,8,8,11,11,23,25+,32+,32+,34+,35+12,12,15,17,22,23Apersonisrightcensored(+)if:He/sheremainsinremissionuntiltheendofthestudy,Islosttofollow-up,OrwithdrawsbeforetheendofthestudyBIS630bLesson2EstimatingtheSurvivorFunction8/115ParametricEstimationofS(t)Kaplan-MeierMethodLife-TableMethodExponentialModelTypicalData:ExamplePuttingthisdataintotabularform:Subjecttx(group)161126113611460157116901710118100191101101311...2211023110...412210422310BIS630bLesson2EstimatingtheSurvivorFunction9/115ParametricEstimationofS(t)Kaplan-MeierMethodLife-TableMethodExponentialModelLeukemiaDataRCode:EstimatingtheRateleuk-read.csv(KMLeukemia.csv)leuk$group-factor(leuk$group,levels=c(0,1),labels=c(Placebo,6-MP))head(leuk)table(leuk$censor,leuk$group)lambdahat=sum(leuk$censor[leuk$group==6-MP])/sum(leuk$time[leuk$group==6-MP])ROutputhead(leuk)grouptimecensorlogWBC1Placebo112.802Placebo115.003Placebo214.914Placebo214.485Placebo314.016Placebo414.36ROutputtable(leuk$censor,leuk$group)Placebo6-MP00121219lambdahat[1]0.02506964BIS630bLesson2EstimatingtheSurvivorFunction10/115ParametricEstimationofS(t)Kaplan-MeierMethodLife-TableMethodExponentialModelLeukemiaSurvivalCurveRCode:PlottingSurvivalCurvet=seq(from=0,to=35,by=0.01)St=exp(-lambdahat*t)plot(t,St,type=l,xlab=Time(weeks),ylab=SurvivalProbability,lwd=2,col=blue)Figure:SurvivalCurvefromExponentialModelfor6-MPGroupBIS630bLesson2EstimatingtheSurvivorFunction11/115ParametricEstimationofS(t)Kaplan-MeierMethodLife-TableMethodExponentialModelParametricvs.Non-ParametricEstimationParametricmodelsyieldanequationfor^S(t)and^h(t)Whatifmodelspeciedisincorrect?DrawinappropriateconclusionsNon-parametricapproachesrequirefewerassumptionsandaremorerobustDonotrequirespecicationofadistributionforTBIS630bLesson2EstimatingtheSurvivorFunction12/115ParametricEstimationofS(t)Kaplan-MeierMethodLife-TableMethodExponentialModelEstimatingtheSurvivorFunctionWewilldiscusstwonon-parametricmethodsforestimatingthesurvivorfunction,S(t)1Kaplan-Meier(Product-Limit)Method,^SKM(t)2Life-Table(Actuarial)Method,^SLT(t)BIS630bLesson2EstimatingtheSurvivorFunction13/115ParametricEstimationofS(t)Kaplan-MeierMethodLife-TableMethodEmpiricalSurvivalFunctionKaplan-MeierEstimatorVarianceofKaplan-MeierEstimatorExampleProgressthisUnit1ParametricEstimationofS(t)ExponentialModel2Kaplan-MeierMethodEmpiricalSurvivalFunctionKaplan-Meier