MODELING LEFT-TRUNCATED AND RIGHT …

1 [ ] 24 Sep 2012 The Annals of Statistics2012, Vol. 40, No. 3, 1465 1488 Institute of Mathematical Statistics, 2012 MODELING LEFT-TRUNCATED AND RIGHT -CENSOREDSURVIVAL DATA WITH LONGITUDINAL COVARIATES1By Yu-Ru Su and Jane-Ling WangUniversity of California, Davis and National Cheng Kung University, andUniversity of California, DavisThere is a surge in medical follow-up studies that include longi-tudinal covariates in the MODELING of survival data. So far,the focushas been largely on RIGHT -censored survival data. We consider sur-vival data that are subject to both left truncation and rightcensor-ing. Left truncation is well known to produce biased bias issue has been resolved in the literature for the casewhich involves baseline or time-varying covariates that are observ-able. The problem remains open, however, for the important casewhere longitudinal covariates are present in survival models. A jointlikelihood approach has been shown in the literature to provide an ef-fective way to overcome those difficulties for RIGHT -censored data, butthis approach faces substantial additional challenges in the presenceof left truncation.

2 Here we thus propose an alternative likelihood toovercome these difficulties and show that the regression coefficient inthe survival component can be estimated unbiasedly and about the bias for the longitudinal component are new approach is illustrated numerically through simulations anddata from a multi-center AIDS cohort the seminal paper by Wulfsohn and Tsiatis (1997),longitudinal covariates have played an increasingly important role in themodeling of survival data. One major challenge to incorporate longitudinalcovariates is that simple approaches, such as the partial likelihood methodfor the Cox proportional hazards model [Cox (1972)], often require knowl-edge of the entire longitudinal process. This is often not feasible in realityfor follow-up checks at discrete and intermittent time points. A commonpractice is to impute the values of the missing longitudinalprocesses andReceived March 2011; revised March in part by NIH Grant 2000 subject 62N02; secondary words and approach, semiparametric efficiency, biased sample,EM algorithm, Monte Carlo is an electronic reprint of the original article published by theInstitute of Mathematical StatisticsinThe Annals of Statistics, 2012 , Vol.

3 40, No. 3, 1465 1488. This reprint differs from the original inpagination and typographic SU AND WANG then apply the partial likelihood approach to the imputed data. This iscalled a two-stage approach, where the longitudinal process is imputed atthe first stage before the partial likelihood approach is employed to estimateparameters in the survival model at the second stage. The most commonimputation method is to use the last and most recent value of the patientto impute a missing value, the so-called last-value-carry-forward method,which has been adopted in standard software such as SAS and procedures were developed by Tsiatis, DeGruttola and Wulfsohn(1995) and Dafni and Tsiatis (1998).It is easy to foresee serious biases with such an imputation method if thefollow-up schedule is infrequent over time and also when thelongitudinalcovariates are contaminated by noises or measurement errors. Both scenar-ios provide strong motivation to find alternative approaches. The approachdeveloped by Wulfsohn and Tsiatis (1997) to model the survival and longi-tudinal data simultaneously through their joint likelihood is attractive ontwo counts: (i) the resulting parametric estimators are semiparametricallyefficient when the baseline hazard function is unknown, and (ii) the jointlikelihood procedure is often insensitive to the normalityassumption on thelongitudinal data, if there is a reasonable number of repeated measurementsavailable for the longitudinal processes; see Zeng and Cai (2005) and Dupuy,Grama and Mesbah (2006) for (i) and Song, Davidian and Tsiatis (2002),Tsiatis and Davidian (2004) and Hsieh, Tseng and Wang (2006) for (ii).

4 The above joint likelihood approach not only successfully removes the bi-ases on the survival component but also leads to efficient estimation. A his-torical example for the joint likelihood approach is the investigation of CD4T-cell counts as a biomarker of time-to-death or time-to-AIDS [DeGruttolaand Tu (1994), Wulfsohn and Tsiatis (1997), Henderson, Diggle and Dob-son (2000)]. In these and other works, the survival time is subject to theusual RIGHT censoring. However, left truncation is common for studies withdelayed entry. Specifically, if the recruitment of patientscontinues after theonset time of a study, those that have already experienced the event areoften excluded from the study, which then results in left truncation of theevent-time. Patients who remain in the study are further subject to the usualright censoring, so the sample consists of LEFT-TRUNCATED and RIGHT -censored(LTRC) survival times. It is well known that left truncationis a biased sam-pling plan as subjects with shorter survival times tend to beexcluded fromthe sample.

5 As a result, the longitudinal measurements are also sampledwith example of LEFT-TRUNCATED and RIGHT -censored longitudinal study is theItalian multi-center HIV (human immunodeficiency virus) study [Rezza et al.(1989), The-Italian-Seroconversion-Study (1992)], where the primary end-point is the time from HIV positive to AIDS onset, that is, theincubationperiod of AIDS. In this study, patients who have developed AIDS at the timeof recruitment were excluded from the study, resulting in left truncation ofJT MODELING SUBJECT TO LTRC3the survival data, and CD4 counts for those who were HIV positive butADIS free were measured at each follow-up visit. As there areno proceduresavailable to handle such data properly, we develop in this paper a semipara-metric joint likelihood approach to accommodate LTRC survival data withlongitudinal covariates that are measured there is a sizable literature to jointly model RIGHT -censored sur-vival and longitudinal data [see Wulfsohn and Tsiatis (1997), Henderson,Diggle and Dobson (2000), Song, Davidian and Tsiatis (2002) and the reviewpapers by Tsiatis and Davidian (2004)], the extension to LTRC survival dataturns out nontrivial due to the left-truncation feature of the data.

6 To see this,consider first the simpler case of LEFT-TRUNCATED data with time-independentcovariates or no covariates at all. Lynden-Bell (1971), Woodroofe (1985) andWang (1987) investigated estimation of the survival function when subjectscome from the same population, that is, there are no covariates , one only needs to adjust the risk set for truncated datato reacha suitable extension of the Kaplan Meier estimator. For time-independentcovariates Andersen et al. (1993) considered estimation under the Cox modeland showed that the partial likelihood approach for RIGHT -censored data stillworks for LTRC survival data when one conditions on the values of thecovariates and truncation time-dependent covariate, the Andersen et. al. (1993) partial likeli-hood approach can still be employed if the entire covariate history is avail-able for all subjects. This is not the case for longitudinal covariates thatare observed intermittently at discrete time points. Sinceimputation meth-ods lead to biases of the estimates, bias corrected approaches have beenemployed in the literature for RIGHT -censored data with longitudinal covari-ates.

7 In particular, Wang (2006) proposed a method to correct the biasthrough the partial score equation. Such an approach is termed correctedscore methods, which originates from studies of measurement errors. Whilecorrected score methods typically lead to n-consistent estimators for theregression parameters in the Cox model, they are not efficientand easy toderive. Extension of the corrected score methods to LTRC (left-truncatedand RIGHT -censored) data might be feasible but have not beenexplored. Inthis paper, we adopt the full and joint likelihood approach of the survivaland longitudinal data due to its aforementioned efficiency and robustnessfeatures. Unfortunately, direct maximization of the full joint likelihood ismuch more complicated than the cases with no left truncation. We discov-ered a modified likelihood that is simpler, yet retains the efficiency of thefull likelihood approach, as described in rest of the paper is organized as follows. In Section2, we introducea joint model setting for both the survival time and longitudinal processesand propose a modified likelihood approach for statistical inference.

8 An EMalgorithm to maximize the modified likelihood is derived in Section3, alongwith the large sample properties of the nonparametric maximum SU AND WANG likelihood estimator (NPMMLE), including consistency, asymptotic normal-ity and efficiency. Numerical performance of the proposed estimating pro-cedure is validated through simulation studies in Section4and illustratedthrough the Italian HIV study in Section5. Section6contains some Joint MODELING under consider the setting that the sur-vival timeY of a subject is subject to random left truncation byT , soa subject is enrolled in a study only ifY T . Letnbe the total numberof subjects enrolled in the study. With such a biased sampling plan, to avoidconfusion of notation, we denote the survival and truncation time of theithenrolled subjects as (Yi, Ti), which are sampled from the joint subpopulationof (Y i, T i), whereY i T i. Upon entering the study, thesensubjects aresubject to the usual RIGHT censorship, so the final observed survival data fortheith subject is a triplet (Ti, Zi, i), whereZi= min(Yi, Ci) is the time ofthe endpoint event or drop-out (censoring) timeCi, whichever occurs first,and i=I(Yi Ci) is the censoring reality, drop-out or censoring only occurs when a subjectis enrolled intothe study.

9 This fact implies that the RIGHT -censoring timeCiis greater thanthe truncation timeTi, fori= 1, .. , n. Therefore, we introduce a positiverandom variableUito represent the time from entry into the study to drop-out from the study, that is,Ui=Ci addition to the survival data, baseline and longitudinalcovariates arecollected intermittently for theith subject from the time the subject en-ters the study until the observational limitZi. This results innirepeatedmeasurements, denoted by~Wi= (Wi1, Wi2, .. , Wini), where the measure-ments are taken at time points~si= (si1, si2, .. , sini). It is important tomake a note here that the observed~Wiare also subject to the same biasedsampling plan as the survival data, so there is a background longitudinalvector, which we will denote as~W ifor theith subject enrolled in the ,~Wiis sampled from the subpopulation of~W , whereY i T i,and values beyondZiare not observed. For simplicity of notation, we assumein this section that there is only one longitudinal covariates, but additionallongitudinal or baseline covariates can be handled easily and the AIDS datadiscussed in Section5contain two longitudinal covariates, one observed in-termittently but the complete history of the other one, the time-dependenttreatment indicator, is joint repeated measurements from the same sub-jects are likely to be correlated, we introduce a latentq 1 random vectorA ito account for their dependency and assume a common parametric densityfunctionf A( | ) with an unknown parameter forA i.

10 A linear mixed effectsmodel will be considered for the longitudinal covariate~W i=X(~si) + i=g(~si)A i+ i,( )JT MODELING SUBJECT TO LTRC5whereg( ) is a knownq-dimensional function and theni 1 vector iplaysthe role of measurement errors, sampled from a multivariatenormal distri-bution with independent marginal distributionN(0, 2), and independentof all other aforementioned random the survival timeY i, a proportional hazards model is employed, andthe hazard rate ofY iat timetgivenA iis Y i(t|A i) = 0(t)exp( Xi(t)),( )where 0is the baseline hazard rate and is the regression coefficient. Thetruncation timeT iand the timeUi, from entry to drop-out, are assumedto have distribution functionFT ( ) andFU( ), respectively. We adopt thestandard assumption in survival analysis, thatY i,T iandUiare condi-tionally independent given the covariates. This is equivalent to assumingconditional independence ofY i,T iandUigiven the value ofA i. We alsoassume thatT iandUiare independent ofA i, and the parameters in themodels for either the survival or longitudinal parts are modified likelihood the model described in the pre-vious subsection, the parameters of interest are ( , , 2and 0( )), wherethe first three components are in the Euclidean space whereas 0(t) = t0 0(u)du, the cumulative hazard function, is in a functional space, hencethe model is semiparametric.