Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 13Page 14Page 15Page 16Page 171Survival analysisAlso known as failure-time analysis, event historyanalysisOutcome: time until the occurrence of an eventWhat is (almost) a defining characteristic of survivalanalysis?censoring: outcome not observed, but known to occurin particular intervalright censoring, left censoring, interval censoringleft truncation, right truncation2examples:many given in bookexamples from my earlier or current work/discussthroughout course:1. Health Insurance Plan Study:Randomized trial of screening for breast cancerTwo arms:Control arm: no screeningTreatment arm: screening by mammography,physician-performed clinical breast examscreens performed at outset, annually for 3 yearsInterested in effect of screening on mortality, mortalityfrom breast cancerFollowed subjects up to 18 yearsMost women still alive after study endsWomen still alive said to be censored32. Multicenter AIDS Cohort Study (MACS)Observational study of subjects infected with HIVMultiple endpoints: DeathDeath from AIDS, specific causesDiagnosis of AIDSDiagnosis of Kaposi’s Sarcoma, other AIDS-definingdiseaseMany predictors: drugs, levels of CD4 cells, age,demographic variables, markers of diseaseprogressionSome fixed, some vary with time; similar in someways to issues in longitudinal data analysisIn course, will deal with fixed predictors first, thentime-varying onesMany subjects still alive, alive without specific disease43. Observational studies of survival of kidneytransplantsMany outcomes:Failure of kidney graft (i.e., stops functioning)Acute rejection (immunological reaction): can havemultiple events4. Study of intravenous iron exposure in subjects onhemodialysisSubjects on hemodialysis lose iron, become anemicReplace iron lost intravenouslyLook at effect of iron on mortalityMany subjects alive at end of planned follow-upIron may cause increased susceptibility to infection,increased mortality55. Randomized trial of new asthma medication/protocolOutcome: return visit to emergency department (ED)3 arms for ED phase; essentially 2 arms for post-EDphase6Will cover Introductory/conceptual units:basic quantities of interestincludes distributions/modelscensoring and truncationdiscussion of likelihoodsbrief discussion of counting processesDefinitions of what we are interested in, then considernuisancesUnits on most commonly used procedures:nonparametric estimationnonparametric testingCox proportional hazards model7Additional topics:Accelerated failure-time modelMultiple failuresCompeting risksCausal inferencecomputing:will illustrate with SAS and Stata:in homeworks: may use any package with necessarycapabilitieswill be some requirements to do things not “canned” invarious proceduresprogramming desirable8Basic quantitiessurvival functionhazard functioncumulative hazard functionprobability density functionmean residual life functionLet T denote time until specified event;T is non-negative random variablewill use capital letters, e.g., T, to denote randomvariables; small letters (t) denote number9Survival functionprobability of individual surviving beyond t; proportionof population surviving beyond S(t) / pr(T>t)monotone nonincreasing function; takes value 1 at origin,may take 0 at infinitycomplement of cumulative distribution function; i.e., S(t)= 1!F(t), where F(t) = pr(T#t)10probability density function for continuous survival times:f(t)= - dS(t)/dtfor discrete survival timessurvival function samereplace probability density function by probability massfunctionp(t) = pr(T=t)probability density/mass function not used often inpractical work11hazard function/hazard ratefor continuous random variables, h(t) = f(t)/S(t) = -dln{S(t)}/dtrepresents “force of mortality”infinitesimal rate of events, scaled by thenumber/proportion of subjects at risk12cumulative hazardthus, for continuous outcomescumulative hazard will be useful in derivations, ingraphical checks of models; less interest as quantity of interest in itself13for discrete failure-time variablesh(t) = pr(T=t|T$t) = p(t)/S(t!)Survival function is product of conditional survivalprobabilities:Note that conditional survivalThus,14cumulative hazard function; alternate definitions; recommended by KM; no longer have ; suggested by Cox and Oakes(1984)when are definitions close?15for small discrete hazards h(tj), the different definitionsare close1st order Taylor series expansion around y=0: ln(1-y) . ywill consider small discrete hazards (“rare diseaseassumption” in epidemiology) later in course to see whenother approximations/estimators converge16Residual life: T-t (for subjects who have not yet failed att)Mean residual lifemrl(t) / E(T!t|T>t) median lifetime t0.5 (for continuous failure-time variable)S(t0.5) = 0.5Book discusses mathematical relations among thesequantities17Main topics of course:Estimating survival and hazard functionsComparing survival and hazard functionsDescriptive, predictive, and causal/prescriptive aspects toboth topicsDescriptive: what is survivor function for subjects with certaincharacteristics (e.g., e.g., time until death forpopulation of 70-year olds)?Causal: Is treatment harmful or beneficial? If all subjectsreceived given treatment, what would survivalfunction be? How does this compare to what wouldhappen if received other level of treatment? In practical examples, will try to distinguish betweendescriptive/predictive and causal questions; can bear onthe correct analysis to