76survival analysis

A“Survivable”IntroductiontoSurvivalAnalysisStephenD.KachmanDepartmentofBiometryUniversityofNebraska–Lincoln“Survivable”IntroductiontoSurvivalAnalysis1Introduction•Lengthoftimebetweentwoevents–Define:∗Startpoint(birth,enterproduction,...)·Timet=0∗Endpoint(death,sale,illness,...)•Incompleterecord–Endpointhasn’toccurredyet–Animalisremovedfromtheherdbeforetheend•DistributionisheavilyskewedA“Survivable”IntroductiontoSurvivalAnalysis2Survivalanalysis•Lengthoftimeanindividual“survives”•Packages–SAS:ProcLIFEREG∗Fixedeffectsmodels–SurvivalKit∗Mixedmodels•Requireabasicunderstanding•Breeder’stestthelimitsofgeneralpackagesA“Survivable”IntroductiontoSurvivalAnalysis3Objective•Quickintroductiontotheanalysisofsurvivaldata–Survivalfunction–Hazardfunctionasafunctionforbuildingsurvivalfunctions–InterpretationofriskfactorsunderaWeibullmodel–Estimatingequationscomparedtothemixedmodelequations–CensoringA“Survivable”IntroductiontoSurvivalAnalysis4Model•Ti:Failuretimeofanimali•Influencedby•RiskFactorη=Xβ+Zu–Riskfactorforeachanimal–Increasedriskleadstoshortersurvivaltimes•FixedEffectsβ–Breed,seasonofcalving,heterosis•RandomEffectsu∼N(0,G)–GeneticmeritG=Aσ2aA“Survivable”IntroductiontoSurvivalAnalysis5SurvivalFunctionS(t;ηi)=Pr(Ti≥t)•Probabilitythatanindividualwithagivenriskfactorηiwillsurvivetilltimet•Lengthoftime,impliesthatSurvivalis100%attime0•Decreasingfunction–Nobody“unfails”•Features–Shapeofthesurvivalfunction–Location“Stretchingofthetimescale”A“Survivable”IntroductiontoSurvivalAnalysis6SimmentalDataSet•7,429cows–Lengthofproductivelife–Censor1=uncensored,0=censored–∼25%censored–Sireandmaternalgrandsire–Herd–Season–PercentSimmentalA“Survivable”IntroductiontoSurvivalAnalysis7•SimpleModel–ContemporaryGroup:Herd*Year*Season–PercentSimmental–Sire–1,019equationsA“Survivable”IntroductiontoSurvivalAnalysis8SimmentalsurvivalfunctionChallenge:FindareasonablemodelforthesurvivalfunctionA“Survivable”IntroductiontoSurvivalAnalysis9SurvivalFunction:PercentSimmental•Similarshape–Mediansurvivaltime50%5.0yearsor∼150%ofthe94%Simmental75%3.7yearsor∼110%ofthe94%Simmental94%3.3years–Between10-20%ofcowsareculledeachyear–About28%ofthe“50%Simmental”cowswhichentertheir4thyearareculledinthenextyearofproductionA“Survivable”IntroductiontoSurvivalAnalysis10HazardFunctionShorttermriskoffailureforanimalaliveattimetλ(t;ηi)Overshortperiodsoftime,theprobabilitythatananimalfailsisapproximatelyequaltothehazardratetimestheperiodoftime.•Thehigherthehazardratetheshorterthetimeperiodthatthisapproximationisreasonable.•Dramaticshiftswillalsomaketheapproximaterelationshippoorer.A“Survivable”IntroductiontoSurvivalAnalysis11Simmentalhazardfunction(monthly)•Sharppeaksat“Weaning”withsmallerpeaksat“Calving”•Generalriseastimeincreases•Focusonshorttermeffects•Hazardratesarenonnegative•CanbegreaterthanoneA“Survivable”IntroductiontoSurvivalAnalysis12Simmentalhazardfunction(yearly)•Smoothesovershorttermeffects•LongtermeffectsaremoreevidentA“Survivable”IntroductiontoSurvivalAnalysis13HazardandSurvivalfunctions•Giveneitherthesurvivalfunction,thedensityfunction,orthehazardfunctiontheothertwocanbefoundS(t)=e−Λ(t)f(t)=λ(t)S(t)λ(t)=f(t)/S(t)Λ(t)=Zt0λ(w)dwA“Survivable”IntroductiontoSurvivalAnalysis14ExponetialmodelConstanthazardλ(t;ηi)=λS(t;ηi)=e−λt•Animal’schanceofsurvivinganadditional5yearsisthesamewhentheanimal–entersproduction–5yearsafterenteringproduction–10yearsafterenteringproductionA“Survivable”IntroductiontoSurvivalAnalysis15Simmentalexponentialsurvivalfunction•Underestimatethetail•Approximately2/3ofthecowsareculledwithin1/λ=4.63yearsofenteringproductionA“Survivable”IntroductiontoSurvivalAnalysis16Simmentalexponentialhazardfunction•Itisclearthattheexponentialhazardfunctionunderestimatesthecullingrateforoldcows.A“Survivable”IntroductiontoSurvivalAnalysis17Weibull•Populargeneralizationoftheexponentialmodel•Hazardfunctionisnonnegative•Lookatthelogofthehazardfunctionln(λ(t,ηi))•Linearinlogtime•ParameterizesotheSurvivalfunctionlooksniceln(λ(t;ηi))=[ln(ρ)+(ρ)ln(λ)]+(ρ−1)ln(t)S(t;ηi)=e−(λt)ρ•Watchoutfortimezero!A“Survivable”IntroductiontoSurvivalAnalysis18Weibullhazardfunction•ρ1–IncreasingHazardfunction–λ(0)→0•ρ1–DecreasingHazardfunction–λ(0)→∞A“Survivable”IntroductiontoSurvivalAnalysis19Weibullsurvivalfunction•–ρ1startsoutflatandspeedsup–ρ1startsoutsteepandflattensout–1/λisapproximatelyequaltothetimewhen2/3oftheanimalshavebeenculledA“Survivable”IntroductiontoSurvivalAnalysis20SimmentalWeibullhazardfunction•Startsat0–Failuresat“Zero”willrestricthowlargetherateparametercanget.•Asaresultitalsohastroublepickingupthetail.A“Survivable”IntroductiontoSurvivalAnalysis21SimmentalWeibullsurvivalfunction•IntheWeibullλplaystheroleofaninterceptandρisarateparameter•ToemphasizethiswecanreparameterizeasS(t;ηi)=e−exp[ρln(t)+ρln(λ)]=e−tρeη•whereη=ρln(λ)A“Survivable”IntroductiontoSurvivalAnalysis22ProportionalHazardModelsλ(t;ηi)=ρtρ−1eη•Baselinehazardλ0(t)•Scalingfactoreηλ(t;ηi)=λ0(t)eηΛ(t;ηi)=Λ0(t)eηA“Survivable”IntroductiontoSurvivalAnalysis