Bayesian Cure Rate Frailty Models with Application to a Root Canal Therapy Study

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1 Bometrcs 61, June 2005 DOI: /j x Bayesan Cure Rate Fralty Models wth Applcaton to a Root Canal Therapy Study Guosheng Yn Department of Bostatstcs and Appled Mathematcs, M. D. Anderson Cancer Center, The Unversty of Texas, Houston, Texas 77030, U.S.A. emal: gyn@odn.mdacc.tmc.edu Summary. Due to natural or artfcal clusterng, multvarate survval data often arse n bomedcal studes, for example, a dental study nvolvng multple teeth from each subject. A certan proporton of subjects n the populaton who are not expected to experence the event of nterest are consdered to be cured or nsusceptble. To model correlated or clustered falure tme data ncorporatng a survvng fracton, we propose two forms of cure rate fralty models. One model naturally ntroduces fralty based on bologcal consderatons whle the other s motvated from the Cox proportonal hazards fralty model. We formulate the lkelhood functons based on pecewse constant hazards and derve the full condtonal dstrbutons for Gbbs samplng n the Bayesan paradgm. As opposed to the Cox fralty model, the proposed methods demonstrate great potental n modelng multvarate survval data wth a cure fracton. We llustrate the cure rate fralty models wth a root canal therapy data set. Key words: Bayesan nference; Cure fracton; Fralty model; Gbbs samplng; Multvarate falure tme data; Proportonal hazards. 1. Introducton As possbly the last endeavor to save a natural tooth before extracton, root canal therapy (RCT) corrects the dsorder and dysfuncton of the dental pulp when the cares or restoratons are deep. RCT usually nvolves removng the tooth crown and the affected pulpal tssue, followed by cleanng the surroundng nfected area to provde a healthy and bondable surface for a permanent fller. After fllng, a crown s fabrcated to complete the procedure. Many root-canal-flled (RCF) teeth last for a lfetme whle some may be lost shortly after completon of endodontc therapy. Non-RCF teeth can be lost due to nonrestorable cares, advanced alveolar bone loss, or catastrophc fracture. Besdes the above reasons, RCF teeth can be lost due to endodontc mshaps (e.g., perforaton) or postendodontc restoratons (e.g., vertcal root fracture from ntracanal posts). To quantfy the degree to whch endodontc nvolvement affects tooth survval, a retrospectve RCT study was conducted n the School of Dentstry at the Unversty of North Carolna at Chapel Hll (Caplan et al., 2005). Usng databases of the Kaser Permanente Northwest Dvson Dental Care Program, and followng the crtera of the study desgn, 202 elgble patents were dentfed. Each patent contrbuted data from one RCF tooth and one smlar non-rcf tooth. If the contralateral tooth was present, t was selected as the matchng non-rcf tooth. If that tooth was mssng or was already endodontcally treated, a tooth of the same type (.e., anteror, premolar, or molar) adjacent to the contralateral tooth was selected. Follow-up for both the RCF and non-rcf teeth started on the ndex date and contnued through the date of extracton or the end of the study, whchever came frst. Tme to extracton was the outcome of nterest whch could be censored by the study termnaton or by the RCT. From a dental scentfc perspectve, many RCF teeth are consdered sound and wll last for a lfetme after successful endodontcal treatment, and thus can be vewed as cured. If a sgnfcant number of patents are cured and thus rsk-free of the dsease of nterest, the populaton s then a mxture of susceptble and nsusceptble subjects. In these cases, the Cox (1972) proportonal hazards model may not be approprate, because t nherently assumes that all the subjects have the same susceptblty to the dsease and wll eventually experence the event over a suffcently long perod of follow-up. Cure rate models are ntended to model falure tme data wth a survvng fracton, whch becomes ncreasngly mportant and popular n clncal trals and medcal research, especally n varous types of oncology studes, such as breast cancer, leukema, and melanoma. et S (t Z )beaproper survval functon (.e., lm t S (t Z )=0), where Z and Z may share common components and the frst component of Z s 1. The mxture cure model proposed by Berkson and Gage (1952) assumes that a certan probablty θ(z )ofbeng cured s mxed wth the remanng 1 θ(z )ofnot beng cured, 552 S pop ( t Z, Z ) = θ(z )+{1 θ(z )}S ( t Z ), (1) where S pop (t Z, Z ) s the populaton survval functon and S (t Z ) s that for the uncured subjects. The cure fracton n model (1) s lm t S pop (t Z, Z ) = θ(z ). A logstc regresson structure s usually assumed so that

2 Bayesan Cure Rate Fralty Models 553 θ(z )=exp(β Z )/{1 +exp(β Z )}, where β s the parameter vector of nterest ncludng an ntercept. The mxture cure model (1) has been extensvely studed n the lterature, ncludng Gray and Tsats (1989), Kuk and Chen (1992), Taylor (1995), Maller and Zhou (1996), Sy and Taylor (2000), Peng and Dear (2000), and Betensky and Schoenfeld (2001), among others. Although (1) s ntutvely attractve and wdely used, t does not have a proportonal hazards structure n the presence of covarates, whch s an undesrable feature when dong covarate analyss. In the Bayesan computaton, f β takes an mproper unform pror,.e., π(β) 1, the posteror dstrbuton based on (1) s mproper (Chen, Ibrahm, and Snha, 1999). In partcular, when modelng the heterogenety n the populaton usng fralty models, t would be more appealng and convenent to employ the proportonal hazards modelng scheme. An alternatve defnton of the unvarate cure rate model, whch has been nvestgated by Yakovlev and Tsodkov (1996), Tsodkov (1998), Chen et al. (1999), and Ibrahm, Chen, and Snha (2001a) among others, s gven by S pop (t Z )=exp{ θ(z )F (t)}, (2) where F(t) saproper cumulatve dstrbuton functon. The correspondng cure rate n model (2) s lm t S pop (t Z )= exp{ θ(z )}. For subject, θ(z )slnked to the covarates va the exponental relaton θ(z )=exp(β Z ). For a more detaled dscusson of (2), see Chapter 5 of the book by Ibrahm, Chen, and Snha (2001b). The aforementoned methods are based on a crtcal assumpton that the survval tmes are ndependent. However, we often encounter multvarate falure tme data n bomedcal research where the correlaton may be nduced by natural or artfcal clusterng effects. In famly studes of genetc dseases, ltter-matched mce experments, or ophthalmologc research, observatons n the same cluster or group may be correlated. The underlyng correlaton needs to be carefully adjusted to ensure vald estmaton and nference. In the RCT example, each subject contrbuted a par of observatons from RCF and non-rcf teeth. Clearly, data collected on two teeth from the same ndvdual cannot be assumed to be ndependent. Extensve research has been carred out for multvarate falure tme data. The fralty model and the margnal model are the most popular approaches. Focusng on subject-specfc effects, the fralty model explctly formulates the nature of the underlyng dependence structure. The margnal model takes a populaton-average approach to model the margnal mean whle treatng the correlaton as a nusance. In ths artcle, we concentrate on the fralty model whch accommodates the ntraclass correlaton through an unobservable random effect, or a fralty. For the lth subject n the th cluster, wth a covarate vector Z l ( =1,..., n; l =1,..., ), the usual Cox shared fralty model s gven by λ(t Z l,w )=λ(t)w exp(β Z l ), (3) where λ(t) sthe unknown and unspecfed baselne hazard functon and W s the unobservable fralty nduced by the th cluster. Condtonal on W, the falure tmes n cluster are assumed to be ndependent. The most studed parametrc assumpton s that the W s are ndependent and dentcally dstrbuted (..d.) from a gamma densty wth mean 1 (Clayton, 1978). Another popular dstrbuton for W s the postve stable dstrbuton (Hougaard, 2000), whch preserves the proportonal hazards structure uncondtonally (after ntegratng W out). Semparametrc Bayesan methods for multvarate falure tme data have been proposed n varous contexts (Clayton, 1991; Snha, 1993; Sargent, 1998; Qou, Ravshanker, and Dey, 1999; among others). However, lmted attenton has been pad to the research n multvarate cure rate models. In the frequentst framework, Chatterjee and Shh (2001) proposed a margnal approach usng bvarate Copula models. Prce and Manatunga (2001) mposed fralty to account for correlaton and conducted the maxmum lkelhood estmaton under a parametrc model assumpton. Both methods were based on the mxture cure model (1). In the Bayesan paradgm, Chen, Ibrahm, and Snha (2002) generalzed the work of Chen et al. (1999) to bvarate falure tme data by ntroducng a postve stable fralty, where an llustratve example was gven for smultaneously modelng two dstnct events,.e., tme to cancer relapse and tme to death. A recent revew paper by Tsodkov, Ibrahm, and Yakovlev (2003) gves a comprehensve treatment and dscusson of the development of the cure rate model (2). Motvated by the RCT study, n whch cure s apparently a possblty and correlaton naturally arses from the pared teeth of the same subject, we propose two new cure rate fralty models for multvarate falure tme data wth a survvng fracton. The proposed methods are closely related to the unvarate cure rate model (2), because t mght not be ntutvely convenent to ncorporate fralty to model (1), and the nterpretaton could be potentally cumbersome. We formulate the model n a Bayesan framework, so that the pror nformaton can be easly ncorporated through hstorcal data. The rest of ths artcle s organzed as follows. In Secton 2, we motvate one form of a cure rate fralty model from a clonogenc tumor cell example, and propose the other by mmckng the Cox fralty model. In Secton 3, we derve the lkelhood functons for the proposed cure fralty models wthn the Bayesan paradgm, and obtan the full condtonal dstrbutons based on sutable pror dstrbutons. In Secton 4, we propose a model selecton technque usng the devance nformaton crteron (DIC) proposed by Spegelhalter et al. (2002). We llustrate the proposed methods wth the RCT example n Secton 5, and provde concludng remarks n Secton Cure Rate Fralty Models We ntroduce a cure rate fralty model that s motvated by the followng clonogenc tumor cell example. For the th ndvdual n the populaton, let N be the number of tumor cells that have the potental of metastaszng,.e., N s the number of metastass-competent tumor cells. Assume that N has a Posson dstrbuton wth mean θ(z ). Gven N = K, let (X 1,..., X K )bethe promoton tmes for all the K tumor cells n the th subject. That s X k (k =1,..., K) sthe tme for the kth metastass-competent tumor cell n subject to produce a detectable tumor mass. Because the N cells belong to the same subject, we assume a random effect W for subject to account for the wthn-subject correlaton among

3 554 Bometrcs, June 2005 the X k s. Condtonal on N and W,weassume that the X k s are..d. from F(t). Here, we emphasze that both N and (X 1,...,X N ) are unobservable random varables. The tme to cancer relapse for the th subject, whch s observed, s defned as T = mn(x 1,...,X N ). Therefore, the populaton survval functon for the th subject s gven by S pop (t Z,W ) =Pr(N =0)+ =Pr(N =0)+ = exp{ θ(z )} + Pr(T >t N = K, W )Pr(N = K) K=1 Pr(X 1 >t,...,x K >t N K=1 = K, W )Pr(N = K) exp{ W Λ(t)K} K=1 θ(z ) K exp{ θ(z )} K! = exp θ(z )+θ(z ) exp{ W Λ(t)}], (4) where Λ(t) s the common cumulatve hazard functon of the X k s. Note that λ(t) =dλ(t)/dt, then the populaton hazard functon of (4) s λ pop (t Z,W )=θ(z )W λ(t) exp{ W Λ(t)}. (5) Model (5), referred to as the promoton tme cure rate fralty model, s a generalzaton of the work of Chen et al. (1999), whch dd not consder the heterogenety of metastaszng tumor cells from dfferent subjects. The formulaton of model (5) s substantally dfferent from that of Chen et al. (2002) where they ncorporated the fralty through the Posson means to model parallel dstnct types of falures. In contrast, we focus on the tme to the same type of event whle correlaton arses from the clusterng effects. We assume W n (5) to be a random varable from a gamma dstrbuton wth mean 1 and varance η 1,.e., W Ga(η, η). In the lmt η,model (5) reduces to the unvarate cure rate model (2). Asde from the bologcal motvaton, model (5) s sutable for correlated falure tme data wth a cure fracton n a wde varety of contexts. Thus, clustered survval data wth a cure fracton, whch may not ft the defnton of a metastaszng tumor cell gven above, can stll be modeled by (5). For ease of exposton, we formulate the cure rate fralty models n the followng setup. Suppose that there are n clusters, and wthn cluster, there are subjects. For = 1,..., n, and l =1,...,, let T l be the falure tme for the lth member n the th cluster, C l be the censorng varable, and Y l = mn(t l, C l )bethe observed tme. Defne the censorng ndcator ν l = I(T l C l ), where I( ) sthe ndcator functon. et Z l be the (p +1) 1vector of bounded covarates, where the frst component of Z l s 1 correspondng to the ntercept. Falure tmes are assumed to be ndependent of censorng tmes condtonal on Z l. Wthn cluster, {(T l, C l, Z l ), l =1,..., } may be dependent but exchangeable. We frst propose a promoton tme cure rate fralty model, of whch the populaton hazard s λ (1) pop(t Z l,w )=λ(t)w exp{ Λ(t)W } exp(β Z l ). (6) As an alternatve, we then ntroduce a dfferent form of cure rate fralty model that s analogous to the Cox fralty model (3). Model (2) can be rewrtten as λ pop (t Z ) = f(t) exp(β Z ), where f(t) s an unknown baselne densty functon. Hence, we propose the followng cure gamma fralty model, λ (2) pop(t Z l,w )=f(t)w exp(β Z l ). (7) Both cure rate fralty models (6) and (7) are constructed to take the wthn-cluster correlaton nto consderaton, whch apparently reduces to the unvarate case f W kelhoods and Full Condtonals We assume a pecewse exponental dstrbuton for the baselne hazard functon λ(t). The pecewse exponental model s useful and smple for modelng survval data, whch serves as a benchmark for comparsons wth other semparametrc and fully parametrc models. The lkelhood functon s constructed as follows. et J be the fnte number of parttons of the tme axs,.e., 0 <s 1 < <s J, wth s J >y l for =1,..., n; l =1,...,.Thus, we have J ntervals, (0, s 1 ], (s 1, s 2 ],...,(s J 1, s J ], where each nterval contans at least one falure and a reasonable way to allocate the data s to balance the number of events among ntervals. The pecewse exponental model assumes that λ(y) =λ j for y (s, s j ], j =1,..., J. Defne δ lj =1fthe lth subject n the th cluster fals or s censored n the jth nterval, and 0 otherwse. When J = 1, namely wth no partton, the baselne hazard reduces to that of an exponental dstrbuton wth λ(t) λ 1.Byncreasng J, wewould obtan fner parttons of the tme scale such that a more flexble structure of the underlyng baselne hazard can be captured. et D denote the observed data, W = (W 1,..., W n ) and λ = (λ 1,..., λ J ). The random effects W ( =1,..., n) are usually assumed to follow a gamma dstrbuton, W Ga(η, η), wth mean 1 and varance η 1.Thus, the condtonal lkelhood functon concernng model (6) s gven by (1) (β, λ W,D)= n (β, λ W,D), where =1 (1) (1) (β, λ W,D) = λ(yl )W exp{ Λ(y l )W } exp(β Z l ) ] ν l { 1 exp{ Λ(y l )W }] exp(β Z l )} { { J = λ j W exp λ j (y l s ) j=1 } ] + λ q (s q s q 1 ) W exp(β Z l ) ( { δlj 1 exp { λ j (y l s ) } νl δ lj } ]) } + λ q (s q s q 1 ) W exp(β Z l ).

4 Bayesan Cure Rate Fralty Models 555 Smlarly, for model (7), (2) (β, λ W,D)= n λ W,D), where (2) (β, λ W,D) = {f(y l )W exp(β Z l )} ν l { F (y l )W exp(β Z l )} { J = λ j exp λ j (y l s ) j=1 } λ q (s q s q 1 ) W exp(β Z l ) { δlj 1 exp { λ j (y l s ) =1 (2) ] νl δ lj (β, }] } λ q (s q s q 1 ) W exp(β Z l ). We take nonnformatve prors for all the parameters such that the lkelhood functons domnate the posteror dstrbutons. Wthout loss of generalty, we assume that β and λ are ndependent, and ther components are ndependent, a pror. Specfcally, we take β k N(µ, σ 2 ) for k =0,1,..., p, and λ j Ga(α, γ) for j =1,..., J. Furthermore, we take W Ga(η, η) and assume that η Ga(a, b), where the hyperparameters a and b are chosen to yeld a large pror varance for W. et U V ] denote the posteror dstrbuton of U gven V. For m =1,2;k =0,1,..., p; j =1,..., J; and =1,..., n, the full condtonal dstrbutons of the parameters are gven as follows: βk β ( k), λ, W,D ] (m) (β, λ W,D)π(β k ), λj β, λ ( j), W,D ] (m) (β, λ W,D)π(λ j ), W β, λ, W ( ),η,d ] (m) (β, λ W,D)W η 1 exp( ηw ), ( n ) η 1 { ( n )} η nη+a 1 W exp η W + b η W,D] =1 =1 {Γ(η)} n, where β ( k) s the rest of β after deletng the kth component, λ ( j) and W ( ) are defned smlarly, and π(β k ) and π(λ j ) are the pror denstes. In partcular for model (7),.e., m =2, due to the conjugate property, the complete condtonal dstrbuton of W has the closed form of ( { J Ga η + ν l,η+ δ lj 1 exp λ j (y l s ) j=1 }] ) λ q (s q s q 1 ) exp(β Z l ). 4. Model Adequacy Evaluaton An mportant part of selectng regresson models s evaluatng the adequacy of the model ft. It s crtcal to compare several competng models for a gven data set and select the one that best fts the data. The DIC, recently proposed by Spegelhalter et al. (2002), s a Bayesan model selecton crteron, whch s gven by DIC = Dev(β, λ)+p Dev. The term p Dev s a penalty term for model complexty, whch s reflected by the effectve number of parameters n the model. The devance s obtaned from the condtonal lkelhood,.e., Dev(β, λ) = 2 log (β, λ W,D), Dev(β, λ) sthe posteror mean of Dev(β, λ),p Dev = Dev(β, λ) Dev( β, λ) and thus DIC = 2Dev(β, λ) Dev( β, λ), where β and λ are the posteror means of β and λ, respectvely. Wth nonnformatve prors, DIC s approxmately equvalent to the Akake nformaton crteron (AIC) proposed by Akake (1973). Specfcally, for the proposed cure fralty models, m =1and 2, DIC m = 4 G G log (m)( β g], λ g] Wg],D ) g=1 +2log (m) ( β, λ W,D), where β g], λ g], and W g] are the correspondng posteror samples of the gth Gbbs teraton, W s the posteror mean, and G s the number of Gbbs teratons after burn-n. The smaller the DIC value, the better the model fts. 5. Example As an llustraton, we appled the proposed methods to the RCT data. In ths analyss, we had three covarates: the RCF tooth ndcator, tooth type, and pocket varables. We combned the anteror and premolar teeth together (nonmolar), because there were relatvely fewer anteror teeth. Among 404 teeth n the data set, there were 176 molars and 228 nonmolars (64 anterors and 164 premolars). Pocket depths had been recorded at sx stes for each tooth. If at least one of the sx perodontal pockets was 5 mm, a bnary varable took a value of 1 (31%), and was otherwse 0 (69%). Fgure 1 shows the Kaplan Meer survval curves for four groups stratfed by the root canal treatment and tooth type. The approprateness of the applcaton of cure rate models needs to be examned cautously. The survval curves level off and show plateaus at the tal parts, whch suggests a possblty of cure. The length of the follow-up was 2916 days, and the last event occurred at the 2678th day for the RCF group and the 2662th day for the non-rcf group. There were 56 teeth censored between the last event and the end of the study for each group. A general gudelne for the proper usage of cure rate models s to have a suffcent perod of follow-up, and a strong bologcal justfcaton for the cure or mmune possblty, as n the RCT study. We ncorporated a gamma fralty W to account for the correlated observatons from the pared teeth of the same patent. We took β =(β 0, β 1, β 2, β 3 ) and λ =(λ 1,..., λ J ) to

5 556 Bometrcs, June 2005 Survval Probablty Non-RCF Non-Molar Non-RCF Molar RCF Non-Molar RCF Molar Tme to Extracton (days) Fgure 1. Kaplan Meer curves stratfed by the root canal treatment and tooth type. be ndependent, a pror, and gave them nonnformatve pror dstrbutons, for example, β k N(0, 100) for k =0,1,2,3, and λ j Ga(α, γ) wth α =2and γ = 0.1, and ndependent for j =1,..., J. For the fralty, we took W Ga(η, η), where η Ga(a, b) and we specfed a =2and b = 100. We chose prors n such a way that the lkelhood functons clearly domnated the posteror dstrbutons. We took the parttons of the tme scale, J =1upto J =5.Alarger J gves more flexblty to model the baselne hazard, whereas at the same tme t brngs n more unknown parameters (the λ j s) that need to be estmated. For comparson, we also appled the Cox shared gamma fralty model (3) to the data, for whch the correspondng condtonal lkelhood based on the pecewse constant hazards s Cox (β, λ W,D)= n (β, λ W,D), where =1 Cox Cox (β, λ W,D) = {λ(y l )W exp(β Z l )} ν l yl j=1 { δlj 0 {λ(t)w exp(β Z l )} dt J = {λ j W exp(β Z l )} δ lj ν l λ j (y l s ) } ] + λ q (s q s q 1 ) W exp(β Z l ). ] We ran 30,000 Gbbs samples for each Markov Chan Monte Carlo (MCMC) chan and recorded a sample every fve teratons, after 3000 burn-ns. The chans appeared to mx well and the convergence could usually be acheved after 500 teratons. We used the dagnostc methods recommended by Cowles and Carln (1996) to montor the chans. Table 1 shows the DICs wth respect to three competng models and fve dfferent J s. The DIC statstcs clearly ndcate that the promoton tme fralty model wth J =3s the best fttng one, wth the smallest DIC = Table 2 summarzes the analyss of the RCT data under the Cox gamma fralty model and the proposed cure rate fralty models, usng J = 1 and 3, respectvely. We present the posteror mean, standard devaton, and 95% hghest posteror densty (HPD) nterval for each parameter. The three dfferent models consstently show that the root canal treatment sgnfcantly reduced tooth survval, whereas the tooth type and pocket varables were not mportant factors. We carred out senstvty analyses on the pror dstrbutons by varyng the hyperparameters (σ and γ). The results n Table 3 demonstrate that the posteror estmaton s very robust wth respect to a wde range of prors. Table 1 Model selecton crteron based on DIC wth respect to J, for the RCT data Fralty model Cox gamma Promoton tme Cure gamma J

6 Bayesan Cure Rate Fralty Models 557 Table 2 The posteror mean, standard devaton, and 95% HPD nterval for the RCT data J Fralty model Covarate Mean Std. Dev. 95% HPD nterval 1 Cox gamma Root (1.7106, ) Molar ( , ) Pocket ( , ) Fralty (η) (0.2192, ) Promoton tme Intercept ( , ) Root (1.7108, ) Molar ( , ) Pocket ( , ) Fralty (η) (0.2068, ) Cure gamma Intercept ( , ) Root (1.7791, ) Molar ( , ) Pocket ( , ) Fralty (η) (0.2137, ) 3 Cox gamma Root (1.8133, ) Molar ( , ) Pocket ( , ) Fralty (η) (0.2034, ) Promoton tme Intercept ( , ) Root (1.5256, ) Molar ( , ) Pocket ( , ) Fralty (η) (0.1952, ) Cure gamma Intercept ( , ) Root (1.9098, ) Molar ( , ) Pocket ( , ) Fralty (η) (0.2154, ) Table 3 Senstvty analyss wth dfferent prors, based on the promoton tme fralty model wth J =3 Std. 95% HPD σ γ Covarate Mean Dev. nterval Intercept ( , ) Root (1.4973, ) Molar ( , ) Pocket ( , ) Fralty (η) (0.1890, ) Intercept ( , ) Root (1.5857, ) Molar ( , ) Pocket ( , ) Fralty (η) (0.1947, ) Intercept ( , ) Root (1.5463, ) Molar ( , ) Pocket ( , ) Fralty (η) (0.1809, ) Intercept ( , ) Root (1.4965, ) Molar ( , ) Pocket ( , ) Fralty (η) (0.2064, ) 6. Remarks We have proposed cure rate fralty models for multvarate falure tme data ncorporatng a survval fracton n the Bayesan paradgm. The proposed methods have attractve features n model formulaton and Bayesan computaton. Model (6) has a strong bologcal motvaton whle model (7) s statstcally and computatonally desrable. It s partcularly appealng that the full condtonal dstrbuton of W under model (7) has a closed form due to conjugacy. Both cure fralty models reduce to the same unvarate case (2) when all the observatons are ndependent. The exstence of nsusceptble or mmune ndvduals n the populaton s the key condton for the applcablty of cure rate models. In such stuatons, censored data are a mxture of cured subjects and uncured subjects who are censored due to ncomplete follow-up. For..d. survval data, Maller and Zhou (1995) have nvestgated the Kaplan Meer estmator of the cumulatve rsk functon for testng for suffcent follow-up and a heterogeneous populaton. The development of a correspondng formal test wth correlated falure tme data s mportant and requres further nvestgaton. Acknowledgements We would lke to thank the edtor, an assocate edtor, and two referees for ther constructve comments and suggestons, whch led to substantal mprovement of the artcle. We thank Dongln Zeng and Joseph Ibrahm for helpful dscussons, and Danel Caplan n the School of Dentstry at the Unversty of North Carolna at Chapel Hll, for provdng the RCT data. References Akake, H. (1973). Informaton theory and an extenson of the maxmum lkelhood prncple. In Internatonal

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