35Introduction to Survival Analysis

IntroductiontoSurvivalAnalysisOctober19,2004BrianF.Gage,MD,MScwiththankstoBingHo,MD,MPHDivisionofGeneralMedicalSciencesPresentationgoalsSurvivalanalysiscomparedw/otherregressiontechniquesWhatissurvivalanalysisWhentousesurvivalanalysisUnivariatemethod:Kaplan-MeiercurvesMultivariatemethods:•Cox-proportionalhazardsmodel•ParametricmodelsAssessmentofadequacyofanalysisExamplesRegressionvs.SurvivalAnalysisTechniquePredictorVariablesOutcomeVariableCensoringpermitted?LinearRegressionCategoricalorcontinuousNormallydistributedNoLogisticRegressionCategoricalorcontinuousBinary(exceptinpolytomouslog.regression)NoSurvivalAnalysesTimeandcategoricalorcontinuousBinaryYesRegressionvs.SurvivalAnalysisTechniqueMathematicalmodelYieldsLinearRegressionY=B1X+Bo(linear)LinearchangesLogisticRegressionLn(P/1-P)=B1X+Bo(sigmoidalprob.)OddsratiosSurvivalAnalysesh(t)=ho(t)exp(B1X+Bo)HazardratesWhatissurvivalanalysis?Modeltimetofailureortimetoevent•Unlikelinearregression,survivalanalysishasadichotomous(binary)outcome•Unlikelogisticregression,survivalanalysisanalyzesthetimetoanevent–Whyisthatimportant?AbletoaccountforcensoringCancomparesurvivalbetween2+groupsAssessrelationshipbetweencovariatesandsurvivaltimeImportanceofcensoreddataWhyiscensoreddataimportant?Whatisthekeyassumptionofcensoring?TypesofcensoringSubjectdoesnotexperienceeventofinterestIncompletefollow-up•Losttofollow-up•Withdrawsfromstudy•Dies(ifnotbeingstudied)LeftorrightcensoredWhentousesurvivalanalysisExamples•Timetodeathorclinicalendpoint•Timeinremissionaftertreatmentofdisease•RecidivismrateafteraddictiontreatmentWhenonebelievesthat1+explanatoryvariable(s)explainsthedifferencesintimetoaneventEspeciallywhenfollow-upisincompleteorvariableRelationshipbetweensurvivorfunctionandhazardfunctionSurvivorfunction,S(t)definestheprobabilityofsurvivinglongerthantimet•thisiswhattheKaplan-Meiercurvesshow.•Hazardfunctionisthederivativeofthesurvivorfunctionovertimeh(t)=dS(t)/dt–instantaneousriskofeventattimet(conditionalfailurerate)SurvivorandhazardfunctionscanbeconvertedintoeachotherApproachtosurvivalanalysisLikeotherstatisticswehavestudiedwecandoanyofthefollowingw/survivalanalysis:•Descriptivestatistics•Univariatestatistics•MultivariatestatisticsDescriptivestatisticsAveragesurvival•Whencanthisbecalculated?•Whattestwouldyouusetocompareaveragesurvivalbetween2cohorts?Averagehazardrate•Total#offailuresdividedbyobservedsurvivaltime(unitsaretherefore1/tor1/pt-yrs)•Anincidencerate,withahighervaluesindicatingmoreeventspertimeUnivariatemethod:Kaplan-MeiersurvivalcurvesAlsoknownasproduct-limitformulaAccountsforcensoringGeneratesthecharacteristic“stairstep”survivalcurvesDoesnotaccountforconfoundingoreffectmodificationbyothercovariates•Whenisthataproblem?•WhenisthatOK?0.00.10.20.30.40.50.60.70.80.91.00100200300400500600700800900DaysSinceIndexHospitalizationWarfASANoRxAge76YearsandOlder(N=394)TimetoCardiovascularAdverseEventinVIGORTrialComparingKaplan-MeiercurvesLog-ranktestcanbeusedtocomparesurvivalcurves•Less-commonlyusedtest:Wilcoxon,whichplacesgreaterweightsoneventsneartime0.Hypothesistest(testofsignificance)•H0:thecurvesarestatisticallythesame•H1:thecurvesarestatisticallydifferentComparesobservedtoexpectedcellcountsTeststatisticwhichiscomparedto2distributionComparingmultipleKaplan-MeiercurvesMultiplepair-wisecomparisonsproducecumulativeTypeIerror–multiplecomparisonproblemInstead,compareallcurvesatonce•analogoustousingANOVAtocompare2cohorts•Thenusejudiciouspair-wisetestingLimitofKaplan-MeiercurvesWhathappenswhenyouhaveseveralcovariatesthatyoubelievecontributetosurvival?Example•Smoking,hyperlipidemia,diabetes,hypertension,contributetotimetomyocardialinfarctCanusestratifiedK-Mcurves–for2ormaybe3covariatesNeedanotherapproach–multivariateCoxproportionalhazardsmodelismostcommon--formanycovariates•(thinkmultivariateregressionorlogisticregressionratherthanaStudent’st-testortheoddsratiofroma2x2table)Multivariatemethod:CoxproportionalhazardsNeededtoassesseffectofmultiplecovariatesonsurvivalCox-proportionalhazardsisthemostcommonlyusedmultivariatesurvivalmethod•EasytoimplementinSPSS,Stata,orSAS•Parametricapproachesareanalternative,buttheyrequirestrongerassumptionsabouth(t).CoxproportionalhazardmodelWorkswithhazardmodelConvenientlyseparatesbaselinehazardfunctionfromcovariates•Baselinehazardfunctionovertime–h(t)=ho(t)exp(B1X+Bo)•Covariatesaretimeindependent•B1isusedtocalculatethehazardratio,whichissimilartotherelativeriskNonparametricQuasi-likelihoodfunctionCoxproportionalhazardsmodel,continuedCanhandlebothcontinuousandcategoricalpredictorvariables(think:logistic,linearregression)Withoutknowingbaselinehazardho(t),canstillcalculatecoefficientsforeachcovariate,andthereforehazardratioAssumesmultiplicativerisk—thisistheproportionalhazardassumption•CanbecompensatedinpartwithinteractiontermsLimitationsofCoxPHmodelDoesnotaccommodatevariablesthatchangeovertime•Luckilymostvariables(e.g.gender,ethnicity,orcongenitalcondition)areconstant–Ifnecessary,onecanprogramtime-depende