23An Introduction to Survival Analysis

AnIntroductiontoSurvivalAnalysisMarkStevensonEpiCentre,IVABS,MasseyUniversityJune4,2009Contents1Generalprinciples31.1Describingtimetoevent.....................................3Instantaneousfailurerate....................................3Survival..............................................3Hazard...............................................31.2Censoring.............................................42Non-parametricsurvival62.1Kaplan-Meiermethod......................................62.2Lifetablemethod.........................................72.3Nelson-Aalenestimator.....................................82.4Workedexamples.........................................8Kaplan-Meiermethod......................................9Flemington-Harringtonestimator................................10Instantaneoushazard.......................................10Cumulativehazard........................................113Parametricsurvival113.1Theexponentialdistribution..................................113.2TheWeibulldistribution.....................................123.3Workedexamples.........................................13Theexponentialdistribution..................................13TheWeibulldistribution.....................................134Comparingsurvivaldistributions144.1Thelog-ranktest.........................................154.2Othertests............................................154.3Workedexamples.........................................15NotesforMVScourse195.721AnalysisandInterpretationofAnimalHealthData.t......................................205.5Workedexamples.........................................21Selectionofcovariates......................................21Fitmultivariablemodel.....................................23Checkscaleofcontinuouscovariates(method1)........................24Checkscaleofcontinuouscovariates(method2)........................24Interactions............................................25Testingtheproportionalhazardsassumption.........................25Residuals.............................................26Overallgoodnessoft......................................27Dealingwithviolationoftheproportionalhazardsassumption...............275.6Poissonregression........................................276Parametricmodels286.1Exponentialmodel........................................286.2Weibullmodel...........................................286.3Acceleratedfailuretimemodels.................................286.4Workedexamples.........................................30ExponentialandWeibullmodels................................30Acceleratedfailuretimemodels.................................317Timedependentcovariates3121GeneralprinciplesSurvivalanalysisisthenameforacollectionofstatisticaltechniquesusedtodescribeandquantifytimetoeventdata.Insurvivalanalysisweusetheterm`failure'todenetheoccurrenceoftheeventofinterest(eventhoughtheeventmayactuallybea`success'suchasrecoveryfromtherapy).Theterm`survivaltime'speciesthelengthoftimetakenforfailuretooccur.Situationswheresurvivalanalyseshavebeenusedinepidemiologyinclude:Survivalofpatientsaftersurgery.Thelengthoftimetakenforcowstoconceiveaftercalving.Thetimetakenforafarmtoexperienceitsrstcaseofanexoticdisease.1.1DescribingtimetoeventInstantaneousfailurerateWhenthevariableunderconsiderationisthelengthoftimetakenforaneventtooccur(e.g.death)afrequencyhistogramcanbeconstructedtoshowthecountofeventsasafunctionoftime.Acurvettedtothishistogramproducesaplotoftheinstantaneousfailureratef(t),asshowninFigure1.Ifwesettheareaunderthecurveofthedeathdensityfunctiontoequal1thenforanygiventimettheareaunderthecurvetotheleftoftrepresentstheproportionofindividualsinthepopulationwhohaveexperiencedtheeventofinterest.TheproportionofindividualswhohavediedasafunctionoftiscalledthefailurefunctionF(t).SurvivalConsideragaintheplotofinstantaneousfailurerateshowninFigure1.Theareaunderthecurvetotherightoftimetistheproportionofindividualsinthepopulationwhohavesurvivedtotimet,S(t).S(t)canbeplottedasafunctionoftimetoproduceasurvivalcurve,asshowninFigure2.Att=0therehavebeennofailuressoS(t)=1.Byday15allmembersofthepopulationhavefailedandS(t)=0.Becauseweusecountsofindividualspresentatdiscretetimepoints,survivalcurvesareusuallypresentedinstepformat.HazardTheinstantaneousrateatwhicharandomly-selectedindividualknowntobealiveattime(t-1)willdieattimetiscalledtheconditionalfailurerateorinstantaneoushazard,h(t).Instantaneoushazar