Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 131Likelihoods for censored datapiece together from various partsfor uncensored data, same as always: f(t) (in likelihood,formally )for right-censored data, probability of surviving beyondCr: pr(T > Cr) = S(Cr)for left-censored data, probability of having failure by Cl:pr(T # Cl) = 1 - pr(T > Cl) = 1 - S(Cl)for interval censored data, probability of failing betweenCR and CL:pr(CR < T # CL) = pr(T > CR) ! pr(T > CL) = S(CR) ! S(CL)2for left-truncated data, conditional probability ofsurviving to or beyond given time given that observed oralive at earlier truncation time Yif failure observed: f(t)/S(Y)if right censored, S(Cr)/S(Y)interval censored: {S(CR) ! S(CL)}/S(Y)calculations assume that left truncation is independent offailure-timeright-truncation: f(Y)/{1-S(Y)}3So, for fixed/type I censoring, i.i.d. observations, getlikelihood multiplying likelihood contributions togethercan be generalized for where each individual has adifferent failure-time distribution, as when doingregressionwhere fi(T) / f(T|X), etc.4Likelihood construction for type I censoring: justifyabove formulaswhere both are considered:T* / true failure timeT / min(T*,C) = length of individual’s follow-upconsider as observable random variables (T,*)Can derive from (T, C) (which has more information)5then probability / likelihood pr(T,*)combine into single expression:likelihood is product over individuals6random censoringfollow book:g(t), G(t) density, survival function of random censoringtime Crif censoring independent of failure-time:likelihood factorizes into parts for censoring, survival:if parameters for censoring, survival distinct, can ignorepart for censoring, because factorizesso7example:exponential likelihoodwrite downmaximize8f(x;8) = 8exp(-8x); S(x;8) = exp(-8x)sufficient statistic from grouped data is {3*,3t); i.e., number of failures r / 3*, amount of follow-up time ST / 3texponential distribution (without censoring) is member ofexponential family9maximum likelihood estimate of rate:maximize w.r.t. 8:This is the usual estimate of rates from grouped failuredata; in epidemiology, may have information onaggregate person-time in study, number of failures;information on individuals not available in data. Still, canget same estimates.10Further, consider Poisson likelihoodrate: 8; amount of aggregate follow-up time ST;expected count: 8ST (poisson parameter)Poisson likelihood: the same as the exponential likelihood (can ignoreconstant R!) in likelihood calculationsupshot: can use Poisson likelihood, regression forgrouped failure-time, rate data, to get same findings aswith exponential distributionbetter yet, can make Poisson likelihoods mimic piecewiseexponential likelihoodsa homework problem (3.11) asks you to use the countingprocess formulation for Poisson processesvalue of censored-data likelihood construction:understanding; parametric models, semiparametric methods (proportional hazards model)11For regression models, make argument conditional onfixed covariatesexample: exponential likelihoodassume hazard in group defined by baseline covariates Xis constant across time, may depend on covariates:f(t|X;8) = 8Z exp(-8Zt); S(t|X;8) = exp(-8Zt)so, one observation’s contribution to exponentiallikelihood for right-censored data isHave written baseline covariates X as random variables;conventionally treated as fixedWhat are consequences of treating baseline covariates(including fixed censoring times) as random?12If performing likelihood-based inference, it makes nodifference if consider baseline covariates as fixed orrandomEven if random, should treat as fixed in inferenceFactor joint densityif parameters distinct, then X is ancillarymaximum likelihood estimate of $2 does not depend ondensity f(X) except through observed Xsame true of partial likelihood13Importance of this (fixed versus random baselinecovariates): theory of semiparametric inference (e.g.,Newey, 1990; Bickel et al., 1992)There, theory treats all observed data as (if appropriate)i.i.d. random vector