# Joint frailty models for recurring events and death using maximum penalized likelihood estimation: application on cancer events.

**ABSTRACT** The observation of repeated events for subjects in cohort studies could be terminated by loss to follow-up, end of study, or a major failure event such as death. In this context, the major failure event could be correlated with recurrent events, and the usual assumption of noninformative censoring of the recurrent event process by death, required by most statistical analyses, can be violated. Recently, joint modeling for 2 survival processes has received considerable attention because it makes it possible to study the joint evolution over time of 2 processes and gives unbiased and efficient parameters. The most commonly used estimation procedure in the joint models for survival events is the expectation maximization algorithm. We show how maximum penalized likelihood estimation can be applied to nonparametric estimation of the continuous hazard functions in a general joint frailty model with right censoring and delayed entry. The simulation study demonstrates that this semiparametric approach yields satisfactory results in this complex setting. As an illustration, such an approach is applied to a prospective cohort with recurrent events of follicular lymphomas, jointly modeled with death.

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- Esther Vorovich, Benjamin French, Bonnie Ky, Lee Goldberg, James C Fang, Nancy K Sweitzer, Thomas P Cappola[Show abstract] [Hide abstract]

**ABSTRACT:**Identification of heart failure (HF) patients at risk for hospitalization may improve care and reduce costs. We evaluated nine biomarkers as predictors of cardiac hospitalization in chronic HF.Journal of Cardiac Failure 06/2014; · 3.07 Impact Factor - SourceAvailable from: Virginie Rondeau[Show abstract] [Hide abstract]

**ABSTRACT:**Frailty models are extensions of the Cox proportional hazards model which is the most pop-ular model in survival analysis. In many clinical applications, the study population needs to be considered as a heterogeneous sample or as a cluster of homogeneous groups of individuals such as families or geographical areas. Sometimes, due to lack of knowledge or for economical reasons, some covariates related to the event of interest are not measured. The frailty approach is a statistical modelling method which aims to account for the heterogeneity caused by unmea-sured covariates. It does so by adding random effects which act multiplicatively on the hazard function. FrailtyPack is an R package 1 which allows to fit four types of frailty models, for left-truncated and right-censored data, adapted to most survival analysis issues. The aim of this talk is to present the new version of the R package FrailtyPack, which is available from the Comprehen-sive R Archive Network at http://CRAN.R-project.org/, and the various new models proposed. It depends on the R survival package. 2 The initial version of this package 3 was proposed for a simple shared frailty model, and was developed for more general frailty models. 4 The shared frailty model 5 can be used, when observations are supposed to be clustered into groups. The nested frailty model 6 is most appropriate, when there are two levels of hierarchical clustering. However, several relapses (recurrent events) are likely to increase the risk of death, thus the terminal event is considered as an informative censoring. Using a joint frailty model, it is possi-ble to fit jointly the two hazard functions associated with recurrent and terminal events, 7 when these events are supposed to be correlated. The additive frailty model 8 is more adapted to study both heterogeneity across trial and treatment-by-trial heterogeneity (for instance meta-analysis or multicentric datasets study). We show how a simple multi-state frailty model can be used to study semi-competing risks while fully taking into account the clustering (in ICU) of the data and the longitudinal aspects of the data, including left truncation and right censoring. 9 We included recently parametric hazard functions and prediction methods. Depending on the models, stratification and time-dependent covariates are allowed or not. The frailty models discussed in recent literature present several drawbacks. Their convergence is too slow, they do not provide standard errors for the variance estimate of the random ef-fects and they can not estimate smooth hazard function. FrailtyPack use a non-parametric penalized likelihood estimation, and the smooth estimation of the baseline hazard functions is provided by using an approximation by splines. FrailtyPack was first written in Fortran 77 and was implemented for the statistical software R. We will present the models that FrailtyPack can fit and the estimation method, then we will describe all the functions and the arguments of FrailtyPack. Finally epidemiological il-lustrations will be provided using FrailtyPack functions. FrailtyPack is improved regularly in order to add new developments around frailty models. - [Show abstract] [Hide abstract]

**ABSTRACT:**In a typical randomized clinical study to compare a new treatment with a control, oftentimes each study subject may experience any of several distinct outcomes during the study period, which collectively define the "risk-benefit" profile. To assess the effect of treatment, it is desirable to utilize the entirety of such outcome information. The times to these events, however, may not be observed completely due to, for example, competing risks or administrative censoring. The standard analyses based on the time to the first event, or individual component analyses with respect to each event time, are not ideal. In this paper, we classify each patient's risk-benefit profile, by considering all event times during follow-up, into several clinically meaningful ordinal categories. We first show how to make inferences for the treatment difference in a two-sample setting where categorical data are incomplete due to censoring. We then present a systematic procedure to identify patients who would benefit from a specific treatment using baseline covariate information. To obtain a valid and efficient system for personalized medicine, we utilize a cross-validation method for model building and evaluation and then make inferences using the final selected prediction procedure with an independent data set. The proposal is illustrated with the data from a clinical trial to evaluate a beta-blocker for treating chronic heart failure patients.Biostatistics 08/2014; · 2.24 Impact Factor

Page 1

Biostatistics (2007), 8, 4, pp. 708–721

doi:10.1093/biostatistics/kxl043

Advance Access publication on January 30, 2007

Joint frailty models for recurring events and death

using maximum penalized likelihood estimation:

application on cancer events

VIRGINIE RONDEAU∗

Institut National de la Sant´ e et de la Recherche M´ edicale, U875 (Biostatistique),

Bordeaux, F-33076, France and Universit´ e Victor Segalen Bordeaux 2, Bordeaux, F-33076, France

virginie.rondeau@isped.u-bordeaux2.fr

SIMONE MATHOULIN-PELISSIER

Institut Bergoni´ e—Centre R´ egional de Lutte Contre le Cancer du Sud-Ouest,

Bordeaux F-33076, France

H´EL`ENE JACQMIN-GADDA

Institut National de la Sant´ e et de la Recherche M´ edicale, U875 (Biostatistique),

Bordeaux, F-33076, France and Universit´ e Victor Segalen Bordeaux 2, Bordeaux, F-33076, France

V´ERONIQUE BROUSTE, PIERRE SOUBEYRAN

Institut Bergoni´ e—Centre R´ egional de Lutte Contre le Cancer du Sud-Ouest,

Bordeaux F-33076, France

SUMMARY

The observation of repeated events for subjects in cohort studies could be terminated by loss to follow-

up, end of study, or a major failure event such as death. In this context, the major failure event could be

correlated with recurrent events, and the usual assumption of noninformative censoring of the recurrent

event process by death, required by most statistical analyses, can be violated. Recently, joint modeling

for 2 survival processes has received considerable attention because it makes it possible to study the joint

evolution over time of 2 processes and gives unbiased and efficient parameters. The most commonly used

estimation procedure in the joint models for survival events is the expectation maximization algorithm.

We show how maximum penalized likelihood estimation can be applied to nonparametric estimation of

the continuous hazard functions in a general joint frailty model with right censoring and delayed entry.

The simulation study demonstrates that this semiparametric approach yields satisfactory results in this

complex setting. As an illustration, such an approach is applied to a prospective cohort with recurrent

events of follicular lymphomas, jointly modeled with death.

Keywords: Cancer; Joint frailty models; Penalized likelihood; Recurrent events.

∗To whom correspondence should be addressed.

c ? The Author 2007. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

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Joint frailty models for recurring events and death

709

1. INTRODUCTION

In many clinical or epidemiological studies, subjects can potentially experience recurrent or repeated

events.Forinstance,patientsmayexperiencerepeatedepilepticseizuresorcancerpatientsmayexperience

recurrent superficial tumors or repeated episodes of hospitalization. Statistical models have been proposed

to analyze these recurrent event data (Cook and Lawless, 2002).

Furthermore, the time frame for an individual’s repeated event process may depend on other “termi-

nating” events, such as death. Often the recurrence of serious events, such as tumors and opportunistic

infections, is associated with an elevated risk of death. In this context, the usual assumption of nonin-

formative censoring of the recurrent event process by death, required by most statistical analyses, can be

violated. This dependence should be accounted for in the joint modeling of recurrent events and deaths.

The approach we develop in this paper is motivated by a study of patients with follicular lymphoma

(FL) undergoing episodic relapses of FL. FLs account for one-third of non-Hodgkin lymphomas in adults.

The prognosis of FL is heterogenous, and numerous treatments may be proposed (Solal-Celigny and

others, 2004). The course of this disease is usually characterized by a response to initial treatment, fol-

lowed by relapses, sometimes associated with high-grade non-Hodgkin lymphomas. After the initial treat-

ment, each patient was monitored regularly for routine visits, and the presence of FL relapses was notified

at each visit. Estimation of the risk of recurrence allows for better planning of follow-up schedules after

diagnosis or first treatment and permits clinicians to determine therapeutic approaches based on the pa-

tient’s risk of relapse. Furthermore, FL relapses may increase the risk of death. As a result, there is an

association between the FL relapses process and the survival process, which precludes the use of standard

analyses of recurrent events. Specifically, those subjects experiencing FL relapses at the highest rate are

typically observed for shorter periods of observation due to mortality. In this work, we will thus consider

the FL relapses and the terminal event process jointly, in a joint frailty model setting.

Li and Lagakos (1997) considered the marginal approach of Wei–Lin–Weissfeld (1989). They as-

sumed the terminating event as a censoring event for each recurrent event, or they treated the failure time

for each recurrence as the first occurrence of the recurring event or terminating event, whichever came

first. However, these marginal models do not specify the dependence between recurrent events and death.

Ghosh and Lin (2003) proposed a joint marginal formulation for the distributions of the recurrent event

process and the dependent censoring time.

Some methods based on counts have been proposed. Lancaster and Intrator (1998) considered joint

parametric modeling of repeated inpatient episodes (via a Poisson model) and survival time of a panel of

patients over 15 months. The model induced a correlation between hospitalization and death via a person-

specific frailty term. Sinha and Maiti (2004) used a more general joint model for panel count data and a

dependent termination, using a Bayesian approach.

Huang and Wolfe (2002) proposed to take into account the informative censoring in clustered data.

Liu and others (2004) proposed a joint semiparametric model for the intensity functions of both recurrent

events and death by a shared gamma frailty model. In these models, the frailty effect on recurrent events

and death rates is not the same. In these approaches, estimation is carried out through a Monte Carlo

expectation maximization (EM) algorithm, which could be time consuming. Furthermore, these methods

cannot be used to correctly estimate hazard functions, which often have a meaningful interpretation in

epidemiological studies. Most of the time, the baseline intensity estimate is based on Breslow’s estimate

leading to a piecewise-constant baseline hazard function or unspecified baseline hazard function.

In this paper, we propose a nonparametric penalized likelihood method for estimating hazard functions

in a general joint frailty model for recurrent events and terminal events, with both right-censored survival

data and delayed entries. This approach is of interest for several reasons. First, it makes it possible to

deal with informative censoring for recurrent event data; in addition, it also allows joint treatment of 2

processes which evolve with time leading to more accurate estimates. This work extends the previous

work by giving smoothed estimates of the 2 hazard functions which represent incidence and mortality

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710V. RONDEAU AND OTHERS

rates in epidemiology. It is natural in epidemiology to impose a continuous hazard function with small

local variations.

To analyze recurrent event data, the focus can be placed on time-between-events (i.e. gap times) or

time-to-events (i.e. calendar times) models (see Duchateau and others, 2003). These 2 timescales which

are 2 important aspects of the data can be linked to the semi-Markov models in which the transition

probability between 2 states depends only on the waiting times, whereas in the nonhomogenous Markov

models, this transition depends only on the time since inclusion in the study. The proposed approach can

deal with both situations and is illustrated in the article.

The paper is organized as follows: In Section 2, we describe the joint frailty model. The construction

of the full penalized log-likelihood is explained in Section 3. Results from a detailed simulation study are

reported in Section 4. The model is applied to the analysis of episodic relapses of FL and death in Section

5. Finally, Section 6 presents a concluding discussion.

2. JOINT MODEL FOR RECURRENT EVENTS AND A TERMINATING EVENT

2.1

The model

We denote for subject i (i = 1,..., N), Xij the jth recurrent times (j = 1,...,ni), Ci the cen-

soring times (not by death), and Di the death times. We first consider Xij as a time to event. Tij =

min(Xij,Ci, Di) corresponds to each follow-up time, and δij is a binary indicator for recurrent events

which is 0 if the observation is censored or if the subject died and 1 if Xijis observed?δij = I(Tij=Xij),

which is either a time of censoring or a time of death (T∗

we actually observe is (Tij,δij,δ∗

and others, 1993; Liu and others, 2004). Let NR∗

i(t) define the actual number of recurrent events in (0,t]

for the ith individual. Because of censoring, it is impossible to observe NR∗

process NR

i

less than NR∗

ND∗

i

cess which indicates whether the subject is still under observation at time t or not. The number of recurrent

events that occur for subject i over the small interval [t,t +dt) is dNR∗

and we have dNR

i(t).

We consider Ft the σ-algebra generated by the whole observed data and the unobserved frailty ω

(defined later), Ft = σ{Yi(u), NR

process history of subject i up to time t, the filtration is the family (Ft)t?0and with Zi(t) the covariate

process.

The following assumptions are made:

where I(·)denotes the indicator function?. Similarly, we denote T∗

i). We also use the theory of multivariate counting processes (Andersen

ithe last follow-up time for subject i,

= min(Ci, Di)), and δ∗

ii= I(T∗

i=Di). What

i(·). Rather, we observe the

i(t) = NR∗

i(t). Similarly, denote by ND

(t) = I(Di? t) the actual death indicator. Furthermore, define Yi(t) = I(T∗

(min(T∗

i,t)) which counts the observed number of recurrent events, which may be

i(t) = I (T∗

i

? t,δ∗

i= 1) the observed death indicator and

i? t), the “at-risk” pro-

i(t) = NR∗

i((t +dt)−)− NR∗

i(t−),

i(t) = Yi(t)dNR∗

i(u), ND

i(u), Zi(u),0 ? u ? t,ωi,i = 1,...,n} which represents

1) We assume continuous recurrent, terminating, and censoring processes so that recurrent events and

death cannot happen at the same time. We adopt the convention that death happens first in the small

interval [t,t + dt). For 2 subjects in the application study who died on the same day as their FL

relapsed, they only count for terminal events, not for recurrent event.

2) NR∗

i(t) is constant after time Dibut can increase after Ci. That means death precludes the observa-

tion of new FL relapses, but on the contrary censoring (as lost of follow-up) does not interrupt the

occurrence of new relapses, they are simply not observed.

3) We define Yi(t)ri(t) the intensity of the recurrent event process at time t in the filtration (Ft)t?0,

given the covariate process, the frailty, and the condition Di ? t (being alive just before time t),

Page 4

Joint frailty models for recurring events and death

711

using

ri(t)dt = dRi(t) = P?dNR∗

i(t) = 1|Zi(t),ωi, Di? t?.

i(t) = 1|Ft−) = Yi(t)dRi(t) = Yi(t)

We then wish to describe the FL relapse rate among the patients currently alive. We assume

as a characterization of the independent censoring P(dNR

ri(t)dt.

4) Similarly, we define the death intensity process Yi(t)λi(t) at time t, given the covariates, the frailty,

using

λi(t)dt = d?i(t) = P?dND∗

Independent censoring for death then requires P(dND

i (t) = 1|Zi(t),ωi, Di? t?.

i(t) = 1|Ft−) = Yi(t)d?i(t) = Yi(t)λi(t)dt.

Following the model of Liu and Wolfe (2004), the joint model for the hazard functions for recurrent

event (ri(·)) and death (λi(·)) is

?ri(t|ωi) = ωir0(t)exp(β?

λi(t|ωi) = ωα

The effect of the explanatory variables is assumed to be different for recurrent and death times. The

parameters β1and β2are interpretable in terms of the instantaneous probability of occurrence of the

recurrent events and the terminal event, respectively, conditional on the subject’s past event history and

on the subject being alive. The model and the estimation can deal with external time-dependent covariates

in the sense of Kalbfleisch and Prentice (2002, p. 197). The previous number of recurrent events can also

be considered as an internal time-dependent covariate that requires the survival of the individual for its

existence, and its path thus carries direct information on the time to failure.

The random effects ωi (frailties) are assumed independent. The gamma frailty density is adopted

here with unit mean and variance θ. The dependence between T∗

due to the fact that the unobserved ωi affects both the recurrent times and the death times. The com-

mon frailty parameter ωiwill take into account the heterogeneity in the data, associated with unobserved

covariates.

In the traditional model, the assumption is that α = 0 in (2.1), that is λi(t) does not depend on ωi, and

thus death (or the terminal event process) is not informative for the recurrent event rate ri(t), that is the

2 rates λi(t) and ri(t) are not associated, conditional on covariates. When α = 1, the effect of the frailty

is identical for the recurrent events and the terminating event. When α > 1, the recurrent rate and the

death rate are positively associated; higher frailty will result in higher risk of recurrence and higher risk

of death.

In the gap timescale formulation, Tijis replaced by Sij= Tij− Tij−1with Ti0= 0 for the recurrent

hazard functions, and the corresponding joint model is

1Zi(t)) = ωiri(t),

2Zi(t)) = ωα

iλ0(t)exp(β?

iλi(t).

(2.1)

iand Tijconditional on Zi(t) is solely

?ri(s|ωi) = ωir0(s)exp(β?

λi(t|ωi) = ωα

1Zi(t)) = ωiri(s),

2Zi(t)) = ωα

iλ0(t)exp(β?

iλi(t).

2.2

Inference in the joint frailty model

We show the expression of the full log-likelihood for calendar times (or time-to-events) and explain how

to deduce it for gap times (or time-between-events). Using the time-to-events timescale, it is easy to

incorporate time-varying covariates, and the likelihood must incorporate delayed entries. The length of

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712V. RONDEAU AND OTHERS

the time-at-risk period is the same for the 2 timescales; however, in the calendar-time formulation, the

start of the at-risk period is not reset to 0 but to the actual time since entry to the study.

Contrary to the shared gamma frailty models (Rondeau and others, 2003), the full log-likelihood of the

joint frailty model does not take a simple form because the integrals do not have a close form. Thus, using

other distributions for the frailty, such as log normal or positive stable, will not induce more difficulties.

Moreover, Pickles and Crouchley (1995) suggest that the results should not be sensitive to the choice of

the frailty distribution.

We denote φ = (r0(·),λ0(·),β β β,α,θ). The construction of the log-likelihood is detailed in Appendix

A.1. We obtain the following expression of the full marginal log-likelihood in the calendar timescale:

⎧

⎩

+ log

0

l(φ) =

?

i

⎨

?

?∞

j

δijlogri(Tij) + δ∗

ilogλi(T∗

i) − log?(1/θ) −1

θlogθ

ω(NR

i(T∗

i)+αδ∗

i+1/θ−1)exp

?

−ω

?T∗

i

0

dRi(t) − ωα

?T∗

i

0

d?i(t) −ω

θ

?

dω

⎫

⎬

⎭(2.2)

with Ti0= 0 and Tini= T∗

or a death time and not a relapse time), ?i(t) =?t

with Ri(·|ω) = ωRi(·).

In the gap timescale formulation, the likelihood expression is the same except that Tijis replaced by

Sij= Tij− Tij−1giving the expression

⎧

⎩

+ log

0

i(for each subject, we assume that the last observation time is a censoring time

0λi(u)∂ du the cumulative hazard function for death,

with ?i(·|ω) = ωα?i(·), and Ri(t) =

?t

0ri(u)du the cumulative hazard function for recurrent events,

l(φ) =

?

i

⎨

?

?∞

j

δijlogri(Sij) + δ∗

ilogλi(T∗

i) − log?(1/θ) −1

θlogθ

ω(NR

i(T∗

i)+αδ∗

i+1/θ−1)exp

⎛

⎝−ω

ni

?

j=1

?Sij

0

dRij(s) − ωα

?T∗

i

0

d?i(t) −ω

θ

⎞

⎠dω

⎫

⎬

⎭.

3. THE SEMIPARAMETRIC PENALIZED LIKELIHOOD APPROACH

We introduced a semiparametric penalized likelihood approach to estimate the different parameters β β β, α,

θ, and the baseline hazard function r0(t) for recurrent events or λ0(t) for death times.

In most situations, it is reasonable to expect smooth baseline hazard functions, piecewise constant

modeling for the hazard functions being often unrealistic. To introduce such a priori knowledge, we

penalize the likelihood by a term which has large values for rough functions (O’Sullivan, 1988; Joly and

others, 1998). The roughness penalty function is represented by the sum of 2 squared norms of the second

derivative of the hazard functions (O’Sullivan, 1988). The penalized log-likelihood is thus defined as

pl(r0(·),λ0(·),β β β,α,θ) = l(φ) − κ1

?∞

0

r??2

0(t)dt − κ2

?∞

0

λ??2

0(t)dt,

(3.1)

where l(λ0(·),β β β,α,η) is the full log-likelihood defined in (2.2) and κ ? 0 is a positive smoothing para-

meter which controls the trade-off between the data fit and the smoothness of the functions. Maximization

of (3.1) defines the maximum penalized likelihood estimators (MPnLE) ˆ r0(t),ˆλ0(t),ˆβ β β, ˆ α, andˆθ. We

Page 6

Joint frailty models for recurring events and death

713

directly use ˆ H−1as a variance estimator, where H is minus the converged Hessian of the penalized log-

likelihood. Furthermore, to deal with the constraint on the variance component (θ > 0), we used a squared

transformation and the standard error of θ was computed by the ?-method (Knight and Xekalaki, 2000).

The estimators ˆ r0(t) andˆλ0(·) cannot be calculated explicitly but can be approximated on the basis of

splines. Splines are piecewise polynomial functions that are combined linearly to approximate a function

on an interval. We use cubic M-splines, which are a variant of cubic B-splines (for more details, see

Ramsay, 1988). M-splines are nonnegative and easy to integrate or differentiate. As we use cubic spline

(or of order 4), the second derivative of r or λ is approximated by a linear combination of piecewise

polynomial of order 2. This approximation allows flexible shapes of the hazard functions while reducing

the number of parameters. If we denote ˜ r(·) an approximation to the MPnLE ˆ r(·), the approximation error

can be made as small as desired by increasing the number of knots. In our approach, although there are

2 different hazard functions (for recurrent events and death), we use the same basis of splines for each

function, but the spline coefficients are different for the distinct functions.

We have previously shown that to obtain a good estimation of the theoretical hazard function, the

more knots we used, the closer the MPnLE was to the true hazard function (Rondeau and others, 2003).

The smoothing parameters can be chosen by maximizing a likelihood cross-validation criterion as in Joly

and others (1998). Another approach consists in fixing the number of degrees of freedom to estimate the

hazard function, as has been previously described (Rondeau and others, 2003; Gray, 1992). We thus use

the relation linking the model degrees of freedom (mdf) and the smoothing parameter κ to evaluate the

smoothing parameter: mdf = trace([ˆ H]−1ˆI) (with I the Hessian matrix of the log-likelihood computed

at the MPnLE). Indeed, it is easier to specify a number of degrees of freedom to estimate a given curve,

rather than to specify a smoothing parameter.

We proposed to directly maximize the observed log-likelihood (3.1) using a modified robust Mar-

quardt (1963) optimization algorithm, which is a combination between the Newton–Raphson algorithm

and the steepest descent algorithm. This algorithm is more stable than the Newton–Raphson algorithm

(Fletcher, 2000) but preserves its fast convergence property near the maximum. The integrations in the full

log-likelihood expression in (2.2) were evaluated using Gaussian quadrature. Laguerre polynomials with

20 points were used to treat the integration [0,∞).

4. SIMULATIONS

A simulation study of the joint frailty model was performed to evaluate the performance of the estimators

and to compare a joint frailty model with a single/reduced frailty model. In order to investigate the effect

of increased sample size on estimator performance, we considered 2 sample sizes with a variable number

of subjects and a variable number of recurrent events by subject. There were 200 or 500 subjects and

1000 simulated data sets for each case. For each simulation run, the joint frailty model (2.1) was used. We

treated the right-censored case only and used a calendar timescale representation.

For each subject i:

1) we generated the random variables ωi,i = 1,..., N, i.i.d. ?(1/θ; 1/θ) with θ = 0.5, the variance

of the random effect.

2) a fixed right-censoring variable was used, Ci= 0.8 (i = 1,..., N).

3) we generated an exponential death time Diusing λi(t|ωi) = ωα

and δ∗

4) we generated the gap times Xikusing ri(t|ωi) = ωir0(t)exp(β1Z1ij+β2Z2ij) with an exponential

r0(t) = 1.0; the corresponding observed calendar times are Tij= min?Ci, Di,?j

iλ0(t)exp(β∗

1Z1i) with λ0(t) = 2.0

i= 1 if Di< Ci.

k=1Xik

?, δi= 1

if Tij=?j

k=1Xikwith Ti0= 0. This simulation scheme is valid since r0(t) is constant.

Page 7

714V. RONDEAU AND OTHERS

To summarize, if the observed time is a recurrent event time Tij =?j

if the observed time is a death time Tij= Di, δij= 0 and T∗

Death times and recurrent event times have in common only one explanatory variable Z1ij. The binary

explanatory variables Z1ijand Z2ijwere generated from a Bernoulli distribution with P(Z = 1) = 0.5.

We set β1= 1.0, β2= −0.5, and β∗

setting II α = −0.5, and setting III α = 0.

We used cubic splines to approximate each hazard function. The number of equidistant knots was

5 for all simulations. For the first replicate of each simulation (i.e. for a given θ and sample size), we

estimated κ using the cross-validation method, the same κ was used for the other 999 generated data sets.

We eliminated the rare cases (less than 5%) when convergence or numerical problems occurred in the

estimation of the parameters.

k=1Xikand δij = 1, the data

i= Ci, δ∗

i= 1, the data generation stops.

generation continues; if the observed time is a censoring time Tij= Ci, δij= 0 and T∗

i= 0; or

i= Di, δ∗

1= 0.7. We consider 3 settings for α, setting I corresponds to α = 0.5,

4.1

Results

The death rate ranges from 27.1% to 49.4%. The average number of observed recurrent events by subject

ranges from 0.60 to 1.52 in the conducted simulation studies with a maximum fixed at 24. Between

40.5% and 70.0% of the subjects did not have a recurrent event. The results of simulation studies using

a penalized likelihood estimation are summarized in Tables 1–5 of the supplementary material (available

at Biostatistics online, http://www.biostatistics.oxfordjournals.org). The regression coefficients from the

joint model were very well estimated in the 3 settings. We observe in the first setting (α = 0.5), a bias

on the regression coefficients using the simple, shared frailty model instead of the joint model. The bias

on the estimates of the variance of the random effects (θ) was very small in the joint model. In setting

II (α = −0.5), that is with a negative association between recurrent events and death, we observed a

significant bias using the simple frailty model (ˆθ = 0.365 with N = 500 andˆθ = 0.359 with N = 200).

This demonstrates that ignoring the dependence between the terminal and the recurrent events can lead to

erroneous results. It can be seen that in the 3 settings for the joint model, ˆ α was unbiased. As expected,

in all simulations the estimates for the standard errors were smaller for N = 500 than for N = 200. In

setting III (α = 0), the 2 models (joint and reduced) are valid and give similar results.

We also increased the degree of dependency between the recurrent events and death with α = 1.0

(results not shown). We observed larger differences between the joint and the reduced models. The re-

sults increasing the number of recurrent events by subject are summarized in the supplementary material

available at Biostatistics online. We obtained a clear improvement in the estimations.

We evaluated the estimation of the survival functions and the hazard functions for the recurrent events

using the mean integrated squared error (H¨ ardle, 1990). More details and results are described in Table 5

of the supplementary material available at Biostatistics online. We observed that our penalized likelihood

estimation gives good estimates for the survival and hazard functions. This also illustrates that better

estimations are obtained using the joint model instead of the reduced model.

5. FL, RELAPSES AND DEATH

The scope of our investigation was to estimate a joint model to describe the risk factors associated with the

recurrences of FLs and death, taking into account the informative censoring by death. If the death times

depend on the recurrent event times, it is necessary to use a joint model to make valid inferences. Another

important point was to study whether the subjects who are at higher risk of FL recurrences tend to be at

an elevated risk of death or inversely at a lower risk of death. This approach allows us to quantify the

association between the recurrent events of FL and death. From 1965 to 2000, 409 patients with FL (190

Page 8

Joint frailty models for recurring events and death

715

males, 46.5%) were monitored at Institut Bergoni, a regional comprehensive cancer center in south west

France. All the patients were prospectively included by one research assistant in a clinical, histological,

and biologic database. A FL recurrence was defined as the first clinical sign of FL. Patients came to the

hospital for a routine visit every 4 months for 3 years, every 6 months for 2 years, once a year for 5 years,

and then every 2–3 years. Some other spontaneous inter-visits could take place. The FL recurrence was or

was not detected at each visit.

Information on patient gender, age, and the number of recurrent events or deaths is given in Table 1.

A total of 249 (60.9%) patients died during the follow-up and 49.1% of subjects did not have a recurrent

event. For 2 subjects who died on the same day as their recurrence, they only count for terminal events,

not for recurrent event. The follow-up period thus varied between 11 days and 30 years. The median

follow-up of surviving patients was 9.8 years. Table 1 shows that the older subjects (?60 years) have

fewer recurrences but more deaths. This would suggest that older subjects could die before developing a

recurrence.

The number of recurrences ranges from 0 to 4, averaging 0.71 per patient. Episodes were categorized

into 1, 2, 3, 4, or 5 corresponding to the number of observation times for each subject. The fifth episode

number corresponds to a censoring time or death.

Figure 1 presents the survival functions following successive recurrences of FL. This figure does not

illustrate clear trends in the evolution of the risk of recurrence.

We modeled the joint distribution of the inpatient recurrences and the survival times (model (2.1))

using the fact that we wished to describe the relapse’s rate among patients currently alive. The person-

specific frailty term represents the effect of unmeasured factors on the chances of both recurrence and

death. The time variable (“gap time”) was the time since the latest episode. We expected that the hazard

rate would not change substantially over time but as a function of the time since the last event. The

covariates included in the analyses were the number of prior episodes (as an internal time-dependent

covariate), gender, age at diagnosis (60 years or older versus younger than 60 years), the tumor burden

with the Ann Arbor stage (III–IV versus I–II), the number of nodal areas involved (?4 versus <4), and

the initial treatment at diagnosis classified as “any type of radiotherapy” versus “chemotherapy alone,

another treatment, or no treatment.” We did not adjust for the serum lactate dehydrogenase levels (for

tumor aggressiveness) or for the hemoglobin levels (consequences of the lymphoma on the host) even if

Table 1. Number of FL recurrences and death according to age and gender

No. of

patients

No. of

deaths

116

61.1

No. of recurrences since diagnosis

1

9476

49.5 40.0

023

5

2.6

4

Male

(%)

190

100

141

0.57.4

Female

(%)

219

100

133

60.7

104

47.5

74

33.8

30

13.7

6

2.7

5

2.2

Diagnosis age < 60 years

(%)

208

100

96

46.2

96

46.2

79

37.9

25

12.0

6

2.9

2

0.9

Diagnosis age ? 60 years

(%)

201

100

153

76.1

102

50.7

71

35.3

19

9.5

5

2.5

4

2.0

Total

(%)

409

100

249

60.8

198

48.4

150

36.7

44

10.8

11

2.7

6

1.5

Page 9

716V. RONDEAU AND OTHERS

Fig. 1. Survival functions for successive FL recurrences.

they are also involved in the FL International Prognostic Index (Solal-Celigny and others, 2004), because

there were too many missing data.

The statistical softwares used were R and the library “frailtypack” (version 2.0-0) with the function

“frailtyPenal” for the shared frailty models (Rondeau and Gonzalez, 2005), and a fortran program was

developed for the joint modeling and will be inserted in the frailtypack. Penalized likelihood maximization

was used. In the reduced models, κ1and κ2were evaluated using the cross-validation method, thereafter

this value was used in the joint model.

Table 2 presents the results using adjusted joint models and reduced shared frailty models. The rate

of recurrence increased with age (age ? 60 years, RR = 1.93, 95% confidence interval [CI] (1.39–2.67))

was higher for women and was associated with the stage of the tumor. These 3 effects were underestimated

using the reduced shared frailty model instead of the joint model. It is clear that ignoring the dependence

between the terminal event and recurrent events resulted in significant biases in the independent shared

frailty model compared to the joint model. For instance, the effect of gender was greater using the joint

frailty model compared to the reduced shared frailty model (1.53 versus 1.16). As a result, some covariates

can be incorrectly observed as nonsignificant variables using a simple, reduced shared frailty model which

does not take into account the informative censoring by death. Age, gender, and the stage of the tumor

were also identified as significant prognostic factors.

The positive value of α = 2.17 in the joint model indicates that the incidence of recurrences is pos-

itively associated with death after controlling for the number of past events. Patients with a large frailty

value tend to have a high rate of recurrence after any episode, whatever the number of past relapses is.

The same positive association was also obtained without the adjustment for the number of past relapses

(α = 1.63).

The number of previous episodes influenced the risk of recurrence or death given the frailty, however,

it was significantly associated with a decreased risk of recurrence (RR = 0.60, 95% CI (0.49–0.73))

and a decreased risk of death (RR = 0.22, 95% CI (0.15–0.31)). This protective effect of the number of

recurrences on the risk of death (RR = 0.22) could be explained by the probable existence of at least

2 different types of FL based on clinical observation: FL with large tumor mass and FL with several

small and disseminated nodes. Patients of the first group often behave more aggressively with higher risk

Page 10

Joint frailty models for recurring events and death

717

Table 2. Analysis of the recurrences and death for FLs using gap times

Covariate Joint modelReduced model

RRRR 95% CI95% CI

For recurrences

Sex

Men

Women

Age

Younger than 60 years

60 years or older

Original Ann Arbor stage

I–II

III–IV

Number of prior episodes

1

1.53

1

1.16 (1.13–2.06) (0.89–1.50)

1

1.93

1

1.45(1.39–2.67) (1.09–1.94)

1

1.43

0.60

1

1.22

0.78

(1.05–1.97)

(0.49–0.73)

(0.93–1.61)

(0.55–1.09)

For survival

Sex

Men

Women

Age

Younger than 60 years

60 years or older

Original Ann Arbor stage

I–II

III–IV

Number of prior episodes

1

2.92

—

(1.69–5.02)

1

8.79

—

(4.85–15.94)

1

3.68

0.22

—

(2.12–6.41)

(0.15–0.31)—

θ(SE)

α(SE)

1.19 (0.09)

2.17 (0.22)

0.41 (0.34)

RR, relative risk; CI, confidence interval; SE, standard error of the mean.

of treatment failure and death in the short term, while the others generally have a slow progression, a

good response to treatment, and are more often in remission (partial or complete). These last patients

will correspond to patients with a higher risk of recurrences but with a longer survival. However, this

assumption remains to be formerly validated.

These models attempt to capture the effect of process history through a single covariate, which indi-

cates the number of previous recurrences of FL occurred by time t. An alternative would be to consider

a model without this variable but with r0(t) replaced by r0j(t). This stratified analysis can be easily con-

ducted with j = 2 or 3 using the MPnLE but can become less tractable with more recurrences by subject

simply because the number of parameters to estimate will increase with the number of different baseline

hazard functions. We did not perform a stratified analysis. Models including the number of prior episodes

as category variables (3 binary variables for 5 classes) confirmed the above estimations.

The variance of the frailties is a measure of the heterogeneity of the observations. The recurrence

rate varied greatly among patients (ˆθ = 1.19 in the joint model), even after adjustment for the individual

variables. We observed a greater heterogeneity using the joint frailty model.

Figure 2 illustrates the hazard of recurrence using the joint or the shared frailty model. We did not

present the hazard function after 15 years because of the lack of information in the data set after this

Page 11

718V. RONDEAU AND OTHERS

Fig. 2. Joint modeling and reduced shared frailty modeling for FL recurrence hazards.

period. We observe that the hazard function was underestimated when using the shared frailty model

because this model does not correctly model death. Indeed, when α > 0, the frail subjects with higher

failure risk are also frail subjects for death and are more likely to die before we observe their failures. The

recurrence risk is then underestimated. In contrast, when α < 0 the recurrence risk is overestimated.

The analyses were solely based on the gap timescales, and we studied how the hazard rate evolves

after an event has taken place; time is reset to zero after a recurrence. Hence, one neglects the recurrence

history when describing the inter-recurrences time. If it is expected that the recurrence rate changes as a

function of time since inclusion in the cohort, the analysis can be based on the calendar timescales, that

is the time since inclusion in the cohort. As the model can be set up in a counting process framework,

it is easy to incorporate time-varying covariates and delayed entry. The joint model using the calendar

timescale led to equivalent results but slightly smaller gender effect (RR = 1.34, 95% CI (0.98–1.84)) or

stage effect (RR = 1.37, 95% CI (0.99–1.90)), and these covariates were no longer significant.

6. CONCLUSION

This paper proposed a method of estimation in joint modeling for 2 survival processes which enables

us to study the joint evolution over time of recurrent events and death and gives unbiased and efficient

parameters. The most commonly used estimation procedure in the joint models for survival events is the

EM algorithm. The strength of this article is that it shows how MPnLE can be applied to nonparametric

estimation of the continuous baseline hazard functions in a joint frailty model with right-censored data

and delayed entry. The method of estimation proposed and the program used also have the advantage of

not being time consuming even for large applications. For instance, the joint model presented in Table 2

used 50 s of CPU time. Valid and rapid inferences under minor assumptions are then obtained.

A major advantage of joint frailty models is their ability to analyze simultaneously the recurrent events

data and a terminating event that can be associated and to assess their degree of dependence. We have

shown by simulation that using a reduced shared frailty model instead of a joint frailty model when

Page 12

Joint frailty models for recurring events and death

719

there is a significant dependence between the 2 processes leads to unreliable estimates, with regression

factors falsely nonsignificant or with an underestimation of the recurrence risk. This implies that the

noninformative censoring of the recurrent event process by death needs to be taken into account in survival

analysis to obtain accurate inferences. In general, omission of important features of dependence in the data

fromthemodelsweestimateresultsinbiasedandinefficientestimates.Ontheotherhand,ifnoassociation

exists between the 2 processes, a more restricted model might be acceptable, such as a reduced shared

frailty model.

The marginal model has already been proposed and compared to the frailty model to deal with the

dependence between recurring events and death (Schaubel and Cai, 2005). This joint frailty approach

compared to the marginal approach has the advantage to quantify this dependence. The frailty model is

implicitly conditional on the previously described filtration (Ft)t?0and the frailty term; marginal models

are in this sense marginal as opposed to conditional and can be seen as having averaged over all possible

filtrations. Furthermore, the regression coefficients of the frailty model are interpreted conditionally, given

the unobserved frailty, and do not have a clear interpretation marginally since the marginal RR does not

equal exp(β).

We applied our approach to the joint modeling of FL recurrences and death, and we found a positive

association between these 2 processes. The censoring by death was informative for the risk of recurrences,

and this was taken into account in the joint modeling. There are cases when a history of higher rates of

recurrent events implies an expected delay in the favorable termination event such as cure or discharge

from hospital. The flexible model that we used, introduced by Liu and others (2004), can accommodate

this kind of negative (α < 0) relationship between recurrent event history and risk of termination.

Other approaches allowing for additional correlation structures on the random effects may provide

valuable insight for future research.

ACKNOWLEDGMENTS

Conflict of Interest: None declared.

APPENDIX

A.1

Construction of the full log-likelihood for the joint frailty model (2.1) with calendar timescale

We denote Tijthe jth follow-up time for subject i and δijthe failure indicator for the recurrent events.

Similarly, we define T∗

i

= min(Di,Ci) the last follow-up time for subject i and the death indicator

δ∗

The marginal contribution to the likelihood Li(r0(·),λ0(·),β β β,α,θ) = Li(φ) for subject i and for

j = 1,...,niis Li(φ) =?

1) The conditional distribution of the survival times given ωiis the product of the individual contribu-

tions:

⎡

ij= I(Di< Ci).

ωLi(φ|ω) f (ω)dω

Li(φ|ω ω ωi) =

ni ?

j=1

⎣dRi(Tij|ωi)δij× exp

× d?i(T∗

⎛

⎝−ω

−ωα

ni

?

j=1

?Tij

Tij−1

dRi(t)

⎞

⎠

.

⎤

⎦

i|ωi)δ∗

i × exp

?

?∞

0

Yi(t)d?i(t)

?

2) The probability density function for the random effects ω is f (ω) =ω(1/θ−1)exp(−ω/θ)

?(1/θ)θ1/θ

.

Page 13

720V. RONDEAU AND OTHERS

3) Using the previous expressions, the ith marginal contribution to the likelihood is obtained by inte-

grating out the random effects:

Li(φ) =

?ni

j=1dRi(Tij)δij× d?i(T∗

?(1/θ)θ1/θ

⎛

j=1

i)δ∗

i

×

?∞

?∞

0

ω

?

NR

i(T∗

i)+αδ∗

i+1

θ−1?

⎞

×exp

⎝−ω

ni

?

?Tij

Tij−1

dRi(t) − ωα

0

Yi(t)d?i(t) −ω

θ

⎠dω.

We then obtain the expression (2.2) of the full log-likelihood by using

l(φ) = log

N

?

i=1

Li(φ).

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[Received April 11, 2006; first revision September 18, 2006; second revision December 11, 2006;

accepted for publication December 20, 2006]