A joint latent class changepoint model to improve the prediction of time to graft failure

A joint latent class changepoint model to improve the prediction of time to graft failure
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  © 2008 Royal Statistical Society 0964–1998/08/171299 J.R.Statist.Soc. A (2008) 171 ,  Part   1 ,  pp. 299–308 A joint latent class changepoint model to improvethe prediction of time to graft failure Francisca Galindo Garre, TNO Quality of Life, Leiden, The Netherlands  Aeilko H.Zwinderman and Ronald B.Geskus University of Amsterdam, The Netherlands  andYvo W.J.Sijpkens Leiden University Medical Center, The Netherlands  [Received October 2006.Revised July 2007] Summary.  The reciprocal of serum creatinine concentration, RC, is often used as a biomarkerto monitor renal function. It has been observed that RC trajectories remain relatively stableafter transplantation until a certain moment, when an irreversible decrease in the RC levelsoccurs.This decreasing trend commonly precedes failure of a graft.Two subsets of individualscan be distinguished according to their RC trajectories: a subset of individuals having stableRC levels and a subset of individuals who present an irrevocable decrease in their RC levels.To describe such data, the paper proposes a joint latent class model for longitudinal and sur-vival data with two latent classes. RC trajectories within latent class one are modelled by anintercept-only random-effects model and RC trajectories within latent class two are modelledby a segmented random changepoint model.A Bayesian approach is used to fit this joint modelto data from patients who had their first kidney transplantation in the Leiden University Medi-cal Center between 1983 and 2002.The resulting model describes the kidney transplantationdata very well and provides better predictions of the time to failure than other joint and survivalmodels. Keywords  : Changepoint; Joint models; Kidney failure; Latent class; Predictions 1. Introduction Many clinical studies have evaluated long-term success of renal transplantation (Paul, 1999;de Bruijne  et al. , 2003). These studies generally involve a collection of repeatedly measuredmarker data and an observation on a possibly censored time to graft failure. Since graft failureis usually preceded by chronic transplant dysfunction, repeated measurements of the reciprocalof serum creatinine concentration, RC, are commonly monitored to evaluate renal function. Asan illustration, Fig. 1 presents longitudinal RC trajectories for four patients who were treatedat Leiden University Medical Center. The horizontal axis represents time measured in months,with time starting 6 months after kidney transplantation, and the vertical axis represents 1000times the reciprocal of serum creatinine concentrations measured in micromoles per litre. A fullvertical line in Fig. 1 indicates that the patient has suffered a graft failure, and a broken verticalline indicates that a patient has been censored at this point. Note that for the two patients with Address for correspondence : Francisca Galindo Garre, TNO Quality of Life, PO Box 2215, 2301 CE Leiden,The Netherlands. E-mail:  300  F.Galindo Garre, A.H.Zwinderman, R.B.Geskus andY.W.J.Sijpkens  05101520051015200 (a)(c) (d) 50 100 1500 50 100 150month        d      a       t      a (b) Fig. 1.  RC trajectories for four selected cases: (a) patient 1; (b) patient 2; (c) patient 3; (d) patient 4 graftfailureasuddenchangeintheRCtrajectoryoccurs.Detectionofchangesinrenalfunctionmay provide essential information to predict the start of dialysis.Several models have been proposed for jointly modelling both longitudinal and survivaldata (e.g. Faucett and Thomas (1996) and Wulfsohn and Tsiatis (1997)). These models com-monly assume a linear mixed effects model for the longitudinal process, and a Cox regressionmodel for the hazard rate. Although such joint models provide better estimates of the hazardrate than the Cox model with measured time-dependent covariates, simple linear mixedeffects models are not always appropriate to describe the longitudinal process. In acquiredimmune deficiency syndrome clinical trials, for example, CD4 cell counts are usually monitoredto assess immunological health of human immunodeficiency virus patients. This biomarker hashigh variability within patients, and more flexible mixed effects models are needed to describeits trajectory. Wang and Taylor (2001) proposed a mixed effects model with an integratedOrnstein–Uhlenbeck process that allows for interrelationships between consecutive measure-ments of CD4 cell counts to predict time to event in acquired immune deficiency syndromeclinical trials, and Brown  et al.  (2005) proposed a mixed effects model that includes cubic B  -splines to describe CD4 cell counts and viral load trajectories. In the latter model, boththe number of knots and the location of the knots must be chosen in advance. Linear mixedeffects models are also not appropriate to describe RC trajectories. A better approach consistsofestimatingasegmentedrandom-effectsmodelwithonlytwosegmentsseparatedbyachange-point to be estimated for each person. Random-effects models with a changepoint were firstproposed by Carlin  et al.  (1992) and were first used in the context of joint models by Pauler andFinkelstein (2002) to describe prostate-specific antigen trajectories in prostate cancer patients.  Prediction of Time to Graft Failure   301 Aparticularfeatureofourapplicationisthatonlyasubsetofthepatientsexperienceachange-point whereas RC trajectories remain stable for the rest of the patients. Pauler and Finkelstein(2002) also noted the presence of these two subsets within the prostate cancer patients, butthey did not take it into account explicitly in their model. Instead they assumed that accurateestimates of the changepoint would only be reached if the data clearly indicate the existenceof a changepoint. The presence of patient subsets was explicitly distinguished, however, in themodel that was proposed by Pauler and Laird (2000) to identify subjects who switch regimesduring a clinical trial, and in the latent class joint model that was proposed by Lin  et al.  (2002)to describe heterogeneity in prostate-specific antigen trajectories in subpopulations of prostatecancer patients.The approach that is presented in this paper can be viewed as a combination of the modelsthatwereproposedbyPaulerandLaird(2000)andbyLin etal. (2002).WesuggestajointmodelinwhichRCtrajectoriesaredescribedbymeansofalatentclassmodelwithtwolatentclasses.Inthe first latent class, RC trajectories are modelled with an intercept-only random-effects model,and, in the second latent class, RC trajectories are modelled with a segmented random-effectsmodel. A description of this joint model for longitudinal and time-to-event data can be foundin Section 3. A Bayesian approach is used to estimate the posterior distribution of the modelparameters. Results from the resulting model applied to the kidney transplantation data aredescribed in Section 4. The paper ends with a discussion. 2. Longitudinal kidney data The model will be illustrated with data from 698 patients who had their first kidney transplan-tation in the Leiden University Medical Center between January 1st, 1983, and January 1st,2002, and who had a functioning kidney for at least 6 months after transplantation. A detaileddescription of these data can be found in de Bruijne  et al.  (2003). Since we were interested in thestudy of long-term success of kidney transplantion, serum creatinine levels were collected foreach patient at unspecified time points beyond 6 months after transplantation. As is commonlyused in practice to assess renal function, 1000 times the reciprocal of the serum creatinine con-centration, RC, was used in the analyses. The mean number of recordings per patient was 76(range 2–294) values. Late graft failure is defined as a return to dialysis. Patients who died withafunctioninggraftarecountedasnon-failures.InadditiontoRC,severaltimeinvariantcovari-ates were evaluated as risk factors of graft failure. These covariates were recipient age, recipientpanelreactiveantibodies,PRA,thelevelofcross-reactivegroupsthataresharedbetweendonorand recipient, CREG, and the number of treated acute rejection episodes. The covariates werechosenbecausetheyreachedsignificanteffectsintheanalysesthatwereperformedbydeBruijne et al.  (2003). 3. The joint latent class changepoint model In this section the joint model for the RC trajectories and graft failure outcomes is presented.For the RC trajectories, a two-latent-class model is proposed, where each latent class has itsown random-effects model. For the survival outcomes, a Cox regression model is used, with adifferent baseline hazard for each latent class. The joint model will be estimated under a Bayes-ian approach. Marginal posterior distributions of the parameters of interest will be obtainedwith the statistical package WinBUGS 1.4.1 (Spiegelhalter  et al. , 2003). Next, we describe thelog-likelihood function and the prior functions that are used in this paper.  302  F.Galindo Garre, A.H.Zwinderman, R.B.Geskus andY.W.J.Sijpkens  Suppose that a sample of   N   subjects is drawn with observations at time  t  ij  , with  j  = 1,..., m i and  m i  denoting the number of observations for subject  i  . Let  Y  i .t  ij  /  denote 1000 times thereciprocal of the serum creatinine concentration at time  t  ij   and  Y i  be the complete vector of observations for subject  i  . Each subject is assumed to belong to one of   k = 1,2 latent classesand  L ik  is an unobserved variable that indicates to which latent class  k   subject  i   belongs.  L ik is 1 if subject  i   belongs to latent class  k   and 0 otherwise. It is assumed that  . Y i | L ik = 1 /  hasthe  k  th component distribution  f  k . Y i | θ ik / , which is a normal distribution that depends on asubject and class-specific parameter vector  θ ki . Denote by  π k  the proportion of the populationbelonging to latent class  k  , which satisfies  Σ 2 k = 1 π k = 1. The joint distribution of the observeddata and the unobserved indicators conditional on the model parameters (Gelman  et al. , 2003)can be written as p. Y , L | θ , π / = p.L | π /p. Y | L , θ / = N   i = 12  k = 1 { π k f  k . Y i | θ ik / } L ik : . 1 / 3.1. Submodels for kidney transplant recipients  Inthissubsectionthetwocompetingindividuallevelmodelsthatareusedwithineachlatentclassare described. The first model is an intercept-only random-effects model for patients whose kid-ney function remains stable over time. The second model is a segmented random-effects modelwith a constant level before a random changepoint and a linearly declining trend thereafter, forpatients who suffer an irreversible kidney dysfunction. The value for subject  i   at time t  ij   is givenby Y  i .t  ij  / ∼ N  { µ i .t  ij  / , τ  2 "  } ,with . µ i .t  ij  / | L i 1 / = a 1 i if individual  i   belongs to subgroup 1, and . µ i .t  ij  / | L i 2 / = a 2 i − b + i  .t  ij  − c i / + if individual  i   belongs to subgroup 2. Here,  a ki ,  b i  and  c i  are random effects and  τ  2 "  denotes therandom-error variance.  a ki  denotes the subject-specific intercept in latent class  k  . For the sec-ondmodel, b i  denotesthesubject-specificslopeaftertherandomchangepoint,and c i  representsthe time at which the irreversible kidney dysfunction starts. The superscript ‘+’ is an indicatorfunction with  z + = z  for  z> 0 and  z + = 0 otherwise.It is assumed that  a 1 i  follows a normal distribution  N. α , σ 2 α / , the vector  .a 2 i , b i /  follows abivariate normal distribution  N. µ , σ 2 µ /  and  c i  follows a normal distribution  N. γ  , σ 2 γ  / , which istruncated below at 0. Minimum informative priors are used for the hyperparameters. A normalprior distribution  N. 0,500 /  is chosen for  α  and  µ 1 , a normal prior distribution  N. 1,200 /  ischosen for  µ 2  and an  N. 50,200 /  distribution is chosen for  γ  , with 50 being the average numberof observed months per patient. An inverse Wishart prior with location matrix  1 0 : 0050 : 005 1  and degrees of freedom  p = 4 is chosen for  σ 2 µ , and inverse gamma(0.01,0.01) distributions arechosen for  τ  2 "  ,  σ 2 α  and  σ 2 γ  . Finally, each unobserved vector  L i = .L i 1 , L i 2 /  is regarded as a bino-mial random variable with parameters  π , whose natural conjugate prior distribution is a flatbeta distribution with parameters equal to  . 1,1 / .  Prediction of Time to Graft Failure   303 3.2. Submodel for time to chronic rejection  For the survival time data, a Cox proportional hazards model is assumed with a different base-linehazard λ 0 k .t/ foreachlatentclass.deBruijne etal. (2003)showedthattheCoxproportionalhazards model is a suitable model to describe the kidney transplantation data. Let  F  i  denotethe failure time for subject  i  . Since failure time can be right censored, let  C i  be the censoringtime for subject  i  . Data from .T  i , D i / were observed, where T  i = min .F  i , C i / , and D i  is the failureindicator,whichtakesthevalueof1ifthefailureisobservedandof0otherwise.TheCoxmodelspecifies that the hazard of failing at time  t  for a subject  i   who belongs to subpopulation  k   isequal to λ i .t  | L ik = 1 / = λ 0 k .t/ exp[ υ k { µ i .t/ | L ik = 1 } + ζ  w i ],where  υ k  is the parameter linking the trajectories in latent class  k   to the hazard function and ζ   is a parameter vector linking a vector  w i  of baseline covariates for subject  i   to the haz-ard function. All these parameter vectors were assumed to have minimum informative normalpriors with zero mean and large variance  . σ 2 = 1000 / . Finally, we adopted independent inversegamma(0.01,0.01)priorsforthebaselinehazards λ 0 k .t/ .Theapplicationofthismodeltokidneytransplantation data will be illustrated in the next section. 3.3. Predicting time to event  Suppose that a new sequence of serum creatinine measurements is available for patient  i   witha kidney transplant. On the basis of this information we can then obtain the probability of agraft failure at a certain moment during the follow-up. A method for doing this is based oncalculating the posterior predictive survival function S  i  to t  + ∆ t  , given the value of  Y  i .t/ at time t  and the fact that the patient has survived to time  t : S  i { t  + ∆ t  | T   t  , Y  i .t/ } = 2  k = 1 P.L ik = 1 | π /P  i { t  + ∆ t  | T   t  , Y  i .t/ , L i = k } ,  . 2 / which is a mixture of survival distributions for patient  i   conditional on group  k  . The survivalfunction for patient  i   in group  k   is given by P  i { t  + ∆ t  | T   t  , Y  i .t/ , L i = 1 } =    exp  − u = t  + ∆ t   u = 0 λ 0 k .u/ exp[ υ k { µ i .t/ | θ ik , L ik = 1 } + ζ  w i ]  f  { θ ik | Y  i .t/ , T   t  } d θ ik    exp  − u = t   u = 0 λ 0 k .u/ exp[ υ k { µ i .t/ | θ ki , L ik = 1 } + ζ  w i ]  f  { θ ik | Y  i .t/ , T   t  } d θ ik ,where  θ ik  denotes a subject- and class-specific parameter vector. This integral is approximatedby using Monte Carlo simulation.Equation (2) is similar to the survival function of the cure rate models (see Ibrahim  et al. (2001), chapter 5). In the latter models, the survival function for cured patients is 1 becausethese patients will never experience the event. We cannot talk about a cured group in our case,however, because observations of RC decline are not available when patients suffer an acute RCdecline which occurs so shortly before graft rejection that it is not observed. This is why ourmodel includes a survival function for the patients with a stable RC trajectory as well. 4. Application to kidney transplant data 4.1. Estimates for the joint model  We now describe the analysis of the kidney transplant data with the joint model that was pro-posedinSection3.UsingWinBUGS1.4.1(Spiegelhalter etal. ,2003),twoMarkovchainMonte
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