## Jirss.irstat.ir

**JIRSS (2013)**
**Vol. 12, No. 1, pp 71-112**
**Modeling and Inferential Thoughts for**
**Consecutive Gap Times Observed with Death**
e de Strasbourg, Strasbourg, France.

**Abstract. **In the perspective of biomedical applications, consider a re-

current event situation with a relatively low degree of recurrence. In this

setting, the focus is placed on successive inter-event gap times which are

observed in the presence of both a terminal event like death and inde-

pendent censoring. The terminal event is potentially related to recurrent

events while the censoring process is an independent nuisance that bears

on the total observation time i.e. on the sum of the successive gap times.

We review diﬀerent modeling and inferential strategies. We also present

a nonparametric estimation method of joint distribution functions and

outline the need for future developments.

**Keywords. **Consecutive gap times, independent censoring, joint mod-

eling, recurrent events, terminal event.

**MSC: **62G05, 62N01, 62N02, 62P10.

Many clinical and epidemiological cohort studies involve health out-comes that a participant may experience a few times during the follow-up period. Interest is particularly centered on life-threatening recurrentevents that each patient may experience at most a very few times. Clas-
Received: June 23, 2011; Accepted: July 17, 2012
sical examples of such recurrent events from longitudinal studies includesolid tumor recurrences in cancer patients, transient ischaemic attacksin atherosclerotic patients and recurrent leukaemia in patients undergo-ing allogeneic haemopoietic stem cell transplantation. The occurrenceof recurrent events as serious as tumor recurrence or ischaemic attack isassociated with a high risk of death so that the subject may die duringthe study. Obviously, the death of a patient stops any subsequent occur-rence of recurrent events, hence the name "terminal event". In addition,during a trial, right-censoring phenomenon is common and should alsobe accounted for. Studies such as CARE (Pfeﬀer et al., 1996), ASPIRE(Bowker et al., 1996), CAPRIE (Gent et al., 1996), LIPID (The LIPIDStudy Group, 1995, 1998 and Marschner et al., 2001) fall within thisframework.

For a given patient, the occurrence of a recurrent event often im-
pacts the risk of novel recurrent event and even of death. Therefore theassumption of independence among the gap times of an individual isoften violated for recurrent event data and the death time is also likelyto be dependent on the recurrent event history. This dependence shouldbe taken care of in the inferential procedures and accounted for in thejoint modeling of recurrent and terminal events. On the opposite, whenno covariates are available, the censoring process is often assumed to beindependent of both the recurrent and terminal processes. Possible rea-sons for such an independent censoring are loss to follow-up (non-relatedto side-eﬀects of the treatment under study) or end of study. When co-variates are available, it is often assumed that the censoring process isindependent of both the recurrent and terminal processes conditionallyon covariates.

Throughout, the recurrent and the terminal processes are all assumed
to have distribution such that recurrent event and death cannot happenat the same time.

The kind of data collected during such a trial is illustrated on Fig-
ure 1 for six subjects labeled

*S*1 to

*S*6. The follow-up time for a givenpatient is represented by a straight line along which the diﬀerent eventsare indicated. Recurrence time data can be regarded as multivariatedata that have speciﬁc characteristics:

*• *the diﬀerent recurrence times of a given subject are stochastically

*• *the diﬀerent patients do not experience the same number of events
and the number of observed events for each subject is not knownin advance,

*Modeling and Inferential Thoughts for Consecutive Gap Times*
*• *the number of patients still alive and still under study decreases
as the events occur,

*• *the terminal event stops the further occurrence of recurrent events,

*• *the last recurrence time of a subject is either a censoring time or

*• *censoring occurs at most once for a given patient and prevents any
further event from being observed.

Figure 1: Example of data (RE = recurrent event).

Traditional statistical methods for the analysis of cohort study have
been focused either on the survival time or on the ﬁrst occurrence ofa composite outcome due to lack of appropriate methodology. For in-stance, the primary pre-speciﬁed endpoint in the LIPID (1995) studywas coronary heart disease related death and for secondary analyzes acomposite of coronary heart disease related death or non-fatal myocar-dial infarction.

Focusing only on the most serious issue i.e. on the fatal one and
comparing treatments only with respect to total lifetime would lead toeﬃcacy problems. Coping with these eﬃcacy problems would requirelonger trial duration and larger sample size. It would also result in aconsiderable loss of information that would obscure the issue of seriousnon-fatal event recurrence which is also a major concern even if the re-currence degree is relatively low. This was a serious matter in Jokhadar
et al. (2004) who outlined that, as myocardial infarction hospital fatal-ities decline, survivors are candidates for recurrent events. This paperalso states that the question of morbidity after non-fatal myocardialinfarction and how it may have changed over time with the arrival ofcontemporary treatments is still of interest. Assessment of prognostic forfurther recurrence is a critical step in evaluating the need for treatmentand lifestyle modiﬁcations to manage the risk of future events.

On the other hand, the most common analysis consists of restrict-
ing attention to ﬁrst event occurrence and of focusing on a predeﬁnedcomposite endpoint that combine fatal and non-fatal events to demon-strate treatment eﬃcacy. The ﬁrst consequence of this is bias problemstoward shorter lifetimes. The second consequence is more subtle andhas been outlined in a number of paper including Pocock (1997), Mah´
and Chevret (1999), Ferreira-Gonz
ales et al. (2007) and Kleist (2007)
among many others. All outlined the fact that composite endpointsshould be clinically meaningful and that the expected eﬀects on eachendpoint component should be similar, based on biological plausibility.

All components of the composite endpoint need to be analyzed sepa-rately. Diﬃculties in interpretation then arise when the results on singlecomponents of the composite endpoint go in opposite directions andwhen hard clinical outcomes are combined with soft endpoints, particu-larly if the latter occur more frequently but are of inferior relevance.

As a conclusion to this discussion, patients need to be followed up on
assigned treatment until death or end of planned follow-up in the absenceof events and must not be regarded as 'a trial completer' after occurrenceof the ﬁrst component event. More speciﬁc regulatory guidelines, betterreporting standards and appropriate statistical methodology are neededto this aim. In this set-up, some progress are still to be made to supportclinical decision making.

Various modeling approaches have been considered with recurrent
event data to address diﬀerent types of questions. The appropriatenessof a selected model depends on the nature of the recurrent event dataas well as on the interest of the study. In the analysis of recurrent eventdata, the focus can be placed either on the times to recurrent event oron the gap times between successive events or on the recurrent eventprocess

*N ∗*(

*.*) where, for

*t ≥ *0, the process

*N ∗*(

*t*) records the number ofrecurrent events occurring in the time interval [0

*, t*].

Note that a connection can be made with multi-state models by
considering the multi-state model depicted in Figure 2 with boxes rep-resenting the states and arrows the possible transitions. In this model,

*Modeling and Inferential Thoughts for Consecutive Gap Times*
the ﬁrst (

*k*0 + 1) states represent the cumulative number of events expe-rienced, the last state is absorbing and represents death. Only forwardtransitions are possible. The "gap time" timescale can be linked to time-homogeneous semi-Markov models in which the transition probabilitybetween two states only depends on the gap time whereas the "time-to-event" scale can be linked to time-inhomogeneous Markov models inwhich the transition probability between two states only depends on thetime since inclusion in the study. Dealing in full details with multistatemodels is beyond the scope of this paper. Nevertheless, we refer tothe recent papers by Andersen and Pohar Perme (2008) and by Meira-Machado et al. (2009).

Figure 2: Multistate model view (RE = recurrent event).

From now on, interest is speciﬁcally focused on successive gap times
since this approach is rather suited to studying recurrent events dynam-ics. Modeling and analyzing the waiting times between successive eventsis attractive in speciﬁc settings. First of all, analyzes based on waitingtimes are often useful when the recurrence degree is relatively low. More-over, when evaluating the eﬃcacy of a treatment on a life-threateningillness where only a few recurrences are expected, it is important to as-sess whether or not the treatment delays the time from treatment startto the ﬁrst episode, the time from the ﬁrst episode to the second episodeand so on. Indeed, several phenomenons may aﬀect the understandingof the illness mechanism. On the one hand, a treatment which delays theﬁrst episode will inevitably lengthen the total time to the second episodeeven if it becomes ineﬀective after the ﬁrst episode. On the other hand,in some cases, a compensating phenomenon between the diﬀerent stagesof a disease or of a treatment may exist. For example, a treatment maydelay the occurrence of the ﬁrst recurrent event but have the reverseeﬀect on the occurrence of the subsequent recurrent events. It is impor-tant to detect such a phenomenon. The distribution of successive gaptimes between recurrent events is then a valuable information.

At last, joint modeling of successive gap times including a possible
fatal event and independent censoring is needed for practical purposesas stated in the paper of Cui et al. (2010). This problem is currentlynot fully addressed, up to our knowledge.

In Section 2, diﬀerent modeling strategies frequently used in the
literature for gap times inference are exposed. In Section 3, we inves-tigate relevant cause-speciﬁc distribution functions. Some perspectivesare given in Section 4.

**A Review of Modeling Strategies for Gap**

Times
In the literature, various modeling approaches have been adopted for theanalysis of recurrent event data. Amongst these are renewal processes,frailty models or multi-state models. Regression models that incorpo-rate either the past event history as covariates or explanatory covariatesor both are also of much use. At last, a purely non-parametric approachwith as few assumptions as possible regarding either the gap times dis-tribution or association structure is also of much interest.

The literature about gap times distribution function or hazard func-
tion inference may be broadly classiﬁed into three categories accordingto whether authors focused on univariate functions, joint functions orconditional-on-past-event-history functions. These three analyzes canalso be carried out conditional on explanatory covariates if some areavailable. As we will see in the sequel, the two dominant methods to in-corporate explanatory covariates are Cox's proportional hazards modeland the accelerated failure time model. We now review some importantcontributions.

In the discussion to come, let us denote by

*Y *[

*k*] for

*k *= 1

*, *2

*, . *the
successive gap times between successive recurrent events and by

*D *the

death time if considered. We adopt the convention that

*Y *[0] = 0 and that

**Z **is a vector of explanatory covariates when available. Time-dependent

vectors of covariates are written as

**Z**(

*.*). Throughout, the use of

*y *is

dedicated to the gap timescale while the use of

*t *is dedicated to the

calendar timescale.

**Models for Univariate Functions**
Models for univariate functions are useful when interest lies in under-standing the evolution through time of separate gap times even though

*Modeling and Inferential Thoughts for Consecutive Gap Times*
possible association is accounted for by some authors. In the absence of(within-subject) gap time independence, two major statistical issues inthe marginal analysis of gap times are identiﬁability and induced depen-dent censoring. On the one hand, since the study duration is typicallyless than the support of the ﬁrst failure time, the marginal distributionof, say, the second gap time is not identiﬁable unless (within-subject)successive gap times are independent as discussed for instance by Wangand Wells (1998), Wang (1999), Lin et al. (1999). This explains theneed for more or less strong modeling assumptions even though marginalmethods are often said to be robust to the subject-speciﬁc correlationstructure between gap times. On the other hand, even if censoring actsindependently on the ﬁrst gap time and on the total observation time,the second and subsequent gap times are subject to induced depen-dent censoring. For example, a greater ﬁrst event time implies highercensoring probabilities for the second and subsequent gap times sinceindependent censoring bears on the times-to-event ie on the sums of thesuccessive gap times. Failures to account for this association may leadto substantial bias in dealing with gap times after the ﬁrst.

We now review diﬀerent approaches to univariate modeling.

Renewal processes have the property that the gaps between succes-
sive events are independent and identically distributed. Even thoughmuch less suited to biomedical applications, one can nonetheless men-tion the work of Pe˜
na et al. (2001) under this renewal process assump-
tion. The authors established Nelson-Aalen-like and Kaplan-Meier-likeestimators of the gap times marginal cumulative hazard function and dis-tribution function using the method of moments and then argued thatthey are NPMLE respectively for the gap times marginal cumulativehazard function and the marginal distribution function. The authorsshowed that any deviation from the independence assumption provokesbias problems that logically increase as the level of association betweengap times increases.

The assumption that gap times are independent and identically dis-
tributed is very strong when no covariates are present. Therefore it isimportant to consider diagnostic checks. An important way of modelchecking is by ﬁtting models that include renewal processes as specialcase. The independence assumption can also be checked informally whenno covariates are present by, for example, looking at scatter plots of suc-cessive gap times within individuals. There should be an absence oftrend if the renewal assumption is valid. If the gap times are indepen-dent, then informal checks on the assumption of a common distribution
can also be made by comparing separate empirical distributions for thediﬀerent gaps. Anyway, the independence assumption is a very strongcondition and renewal processes are mainly useful in reliability when asubject is repaired in some sense after each event. The renewal assump-tion is clearly untenable in most biomedical applications.

Speciﬁc extensions of renewal processes can be found in Cook and Law-less (2008) and in Gill and Keiding (2010).

A classical generalization of renewal models that allows association
between gap times consists of considering a frailty model in which a la-tent variable is used to take into consideration a subject-speciﬁc randomeﬀect. Speciﬁcally, it is supposed that, for any given subject, there existsan unobserved random variable

*U *, called the frailty, with distribution

*FU *such that given

*U *=

*u *the successive gap times of the individual arei.i.d. with some distribution

*F *(

*. u*). This unmeasured eﬀect is assumedto follow a distribution with mean equal to one and unknown ﬁnite vari-ance. A distribution such as the Gamma (most popular frailty model)or Inverse Gaussian or positive stable or log-normal can be assumed forthe frailty. The goal of the frailty model analysis is generally to esti-
mate the distribution function

*F *(

*.*) =

*F *(

*. u*)

*dFU *(

*u*) unconditional on
the frailty or the hazard function also unconditional on frailty. Tech-nically speaking, frailty models can be ﬁtted either with a frequentistapproach by maximizing the marginal likelihood or with a Bayesian ap-proach by computing parameter posteriors densities. Hougaard (2000)and Duchateau and Janssen (2008) provided a comprehensive coverageof this area.

Wang and Chang (1999) focused on a marginal approach for the
gap times between successive events using this less restrictive frailtyapproach. They derived a weighted moment estimator for the marginalgap times distribution under a nonparametric frailty model assumingthat, given the frailty, the successive duration times are independentand identically distributed. Their correlation structure is quite generaland contains both the i.i.d. and multiplicative (hence Gamma) frailtymodel as special cases.

na et al. (2001) also proposed an estimator when the gap times
follow a Gamma frailty model and compared the performance of theirestimator to Wang and Chang's (1999) estimator by simulations. Theauthors found out that, when applied to i.i.d. gap times, their estimatoris expected to be more eﬃcient than that of Wang and Chang (1999).

From a practical viewopint, it is interesting to note that both Pe˜
et al. (2001) and Wang and Chang (1999) estimators are implemented

*Modeling and Inferential Thoughts for Consecutive Gap Times*
in the R package survrec.

One limitation of many types of random eﬀect models is that only
one or two parameters are used to model association among a numberof successive gap times. This can be inadequate when association struc-tures are complex or changing over time. Moreover these approaches donot readily deal with negative associations. At last, it should be notedthat misspeciﬁcation of the frailty distribution can cause severe bias inestimation procedures with recurrent event data, see e.g. Kessing et al.

(1998). This remark entails that the possibility of testing model ade-quacy is an important issue that should be dealt with. Kvist et al. (2007)developed a procedure for checking the adequacy of Gamma frailty torecurrent events. To apply their model checking procedure, a consis-tent non-parametric estimator for the marginal gap time distributionsis needed. The performance of the model checking procedure dependsheavily on this estimator. Moreover, the authors concluded that theirprocedure in its current state only works when the within-subject asso-ciation between gap times is weak. They suggested possible future im-provement of their methods consisting of checking of the Gamma frailtymodel for recurrent events from a comparison of conditional distribu-tions instead of marginal distributions. Up to now, there is still a roomfor improvements on that issue.

When covariates are available, marginal methods are also helpful to
understand how either population-level characteristics (e.g. treatmentgroup) or subject-speciﬁc (e.g. sex) or gap-time speciﬁc characteristics(e.g. some biological marker) inﬂuence the marginal gap time distribu-tion. Explanatory covariates are often incorporated through Cox-like oraccelerated-failure-time-like assumptions for their ease of interpretation.

Assuming that the successive gap times of each individual are i.i.d.

(unconditional on covariates), Huang and Chen (2003) considered a pro-portional hazards assumption of the form
Λ(

*y ***Z**) = Λ0(

*y*) exp(

*β′.***Z**)

to assess the eﬀect of a vector

**Z **of time-independent covariates on the

common baseline cumulative hazard function Λ0(

*.*) of the successive gap

times. The authors developed an inferential procedure that improves the

functional formulation of Cox regression by Huang and Wang (2000)

with respect to eﬃciency. To this aim, the authors noticed that the

uncensored gap times are exchangeable provided the model assumptions

are valid, then constructed speciﬁc clustered data. For each cluster, the

ﬁrst gap time is chosen if the subject has only one censored gap time,

otherwise all the uncensored gap times are selected. Then they got theirnew estimates of

*β *and Λ0(

*.*) through a modiﬁed estimating equationobtained from the clustered data. This procedure is shown to performwell for practical sample sizes. The authors also noted that their modeland inferential procedure still apply for covariates that depend on timefrom earlier episode and have uniform eﬀects across all gap times butthat diﬃculties arise if time-varying covariates are episode-speciﬁc.

Obviously, the validity of statistical inference depends on the ad-
equacy of the model.

Recent progress have been made in Cox-type
model checking for gap times in Huang et al. (2010) who proposed bothgraphical techniques and formal tests for checking the Cox model withrecurrent gap time data to assess diﬀerent aspects of goodness-of-ﬁt forthis model.

In the same setting, ie also assuming that the successive gap times of
each individual are i.i.d. (unconditional on covariates), Sun et al. (2006)considered an alternative model under the form of an additive hazardsmodel deﬁned as

*λ*(

*y ***Z**) =

*λ*0(

*y*) +

*β′.***Z**
where

*λ*0(

*.*) is the common baseline instantaneous hazard function of thesuccessive gap times. The authors used the same inferential procedureas in Huang and Chen (2003) to ensure satisfying eﬃciency properties.

Strawderman (2005) also proposed a marginal regression model for
consecutive gap times of the accelerated failure type but alleviated the

i.i.d. property of gap times. Speciﬁcally, he assumed that, conditional

on a vector of covariates

**Z**, the variables

*Y *[

*k*] exp(

*β′.***Z**) for

*k *= 1

*, *2

*, .*

are i.i.d. or equivalently that the common hazard function of

*Y *[

*k*] con-

ditional on

**Z **is of the form

where

*λ*0(

*.*) is an unspeciﬁed baseline function. Similarly to the accel-erated failure time model, explanatory population-level (ie not episode-speciﬁc) covariates serve to accelerate or decelerate a baseline gap timehazard function. The problem of obtaining an eﬃcient estimation of

*β*is investigated.

However, the same restrictions and pitfalls as previously also apply
to regression models in which the gap times are independent conditionalon covariates. Here again, the conditional independence is questionable.

We also mention that questions exist concerning the interpretation ofbaseline functions when there is association between the successive gaptimes.

*Modeling and Inferential Thoughts for Consecutive Gap Times*
A possible extension consists of incorporating episode-speciﬁc covari-
Chen, Wang and Huang (2004) considered a situation in which episode-
speciﬁc vectors of covariates, say

**Z**[

*k*] for

*k *= 1

*, *2

*, . *are available and

assumed that the gap times are i.i.d. conditional on covariates. Account-

ing for right-truncation phenomenon in the observation of successive gap

times, they worked with a subject-speciﬁc reverse-time hazards function

deﬁned for subject

*i *by

(

*y ***Z **) =

P

*y − h ≤ Y*
*≤ y, ***Z**
*h→*0+

*h*[

*d *log P

*Y*
*≤ y ***Z**
Their approach relies on modeling proportional reverse-time hazardsfunctions so that, for individual

*i*, we have
(

*y ***Z **) =

*κ*
i.e. each individual is assumed to have its own baseline reverse-timehazard function

*κi,*0(

*.*). Thus the model copes with high heterogeneityacross the patient population. The prize to pay for this generality is po-tential identiﬁability problems. The authors suggested as a special casethat the baseline reverse-time hazard function could be modeled using afrailty

*Ui *for subject

*i *as

*κi,*0(

*y*) =

*Uiκ*0(

*y*). Note that the eﬀect of theepisode-speciﬁc covariates on the baseline reverse-time hazard functionis constrained to be identical across recurrences. The interpretation ofsuch a constraint may be tricky but has the advantage of allowing moreeﬃcient estimation of

*β*.

Similarly, Du (2009) assumed that each gap time

*Y *[

*k*] depends only
on an episode-speciﬁc covariate

**Z**[

*k*] such that the

*Y *[

*k*] conditionally on

the

**Z**[

*k*] are i.i.d. Stating that the history of a subject before each recur-

rence conveys information for that recurrence, Du (2009) suggested to

include the number of past recurrences in the episode-speciﬁc covariates.

The author investigated a nonparametric estimator for the marginal gap

time hazard function of

*Y *[

*k*] conditional on

**Z**[

*k*] =

**z**, denoted by

*λ*(

*. *),

using a functional ANOVA decomposition of log

*λ *of the form

log

*λ*(

*y ***z**) =

*η*0 +

*η*gap(

*y*) +

*η*cov(

**z**) +

*η*gap,cov(

*y, ***z**)

where

*η*0 represents the grand mean,

*η*gap(

*.*) represents the main gap

time eﬀect,

*η*cov(

**z**) represents the main covariates eﬀect and

*η*gap,cov(

*y, ***z**)

represents the interaction eﬀect between gap time and covariates, pro-vided some identiﬁability conditions are ensured. As a consequence, thismodel can be applied to assess the validity of the proportional hazardsassumption by examining the interaction between gap times and covari-ates. The inferential procedure is based on non-parametric penalizedlikelihood with a cross-validation step to select smoothing parameter.

This model has the advantage of allowing greater ﬂexibility for the func-tional form of the diﬀerent eﬀects and of generalizing multiplicative formof the hazard at the expense of loosing the usual interpretation in termsof risk ratio.

An alternative approach to these models consists of accounting for as-
sociation between within subject gap times via random eﬀects to lightenthe i.i.d. assumption on successive gap times, conditionally on covariatesin the regression setting.

Therneau and Grambsch (2000) considered that the successive gap
times are i.i.d. conditionally on both an (unobservable) frailty

*U *and

an (observed) episode-speciﬁc vector of covariates

**Z**[

*k*]. The authors

discussed the ﬁtting of a model for the conditional hazard function of

*Y *[

*k*] of the form

*λ*[

*k*](

*y ***Z**[

*k*]

*, U *) =

*U λ *(

*y*) exp(

*β′*
(

*.*) is an episode-speciﬁc baseline hazard function and where

*βk *is the episode-speciﬁc eﬀect on the episode-speciﬁc baseline hazard

function of the episode-speciﬁc vector of covariates

**Z**[

*k*]. Note that here

the

*Y *[

*k*] are conditionally independent but not identically distributed.

This enlarged ﬂexibility may lead to inconsistency problems as

*k *grows

if too few patients experience

*k *events. Except for the case where

*U *has

a positive stable distribution, these models do not give unconditional

(on

*U *) distributions for

*Y *[

*k*] given

**Z**[

*k*] of proportional hazards form.

Chang (2004) considered a marginal accelerated failure time frailty
model of the form
log

*Y *[

*k*] =

*U *+

*β′.***Z **+

*ε*[

*k*]

*, k *= 1

*, *2

*, .*
where the variables

*ε*[

*k*] for

*k *= 1

*, *2

*, . *are i.i.d. This model assumes thatthe covariates eﬀect and the subject-speciﬁc frailty

*U *are additive on thegap time logarithm and that the covariates eﬀect remains the same overdistinct episodes. The distributions of the frailty and the random errorin the model are left unspeciﬁed which decreases adequacy issues. Theauthor developed two estimation methods, the second of which beingrobust to deviation from the hypothesis that the

*ε*[

*k*] for

*k *= 1

*, *2

*, . *are

*Modeling and Inferential Thoughts for Consecutive Gap Times*
identically distributed. The authors mentioned the possibility to extendtheir model to allow the incorporation of an episode-speciﬁc covariateeﬀect
log

*Y *[

*k*] =

*U *+

*β*[

*k*]

*′ .***Z **+

*ε*[

*k*]

*, k *= 1

*, *2

*, .*
even though consistent estimation of

*β*[

*k*] may not be possible if thenumber of subjects experiencing

*k *events is not large enough.

All these methods, however, assume that recurrent events are not
terminated by death during the study.

Rondeau et al. (2007) accounted for death in their analysis. Specif-
ically, they jointly modeled the association between survival time and

within subject gap times through a Gamma frailty

*U *. Conditional on

*U *and on an external time-dependent vector of covariates

**Z**(

*.*), they as-

sumed that the

*Y *[

*k*] for

*k *= 1

*, *2

*, . *are i.i.d. with conditional hazard

function given by

*λ*(

*y U, ***Z**(

*.*)) =

*U λ*0(

*y*) exp(

*β′.***Z**(

*t*))

and are independent of the death time with conditional hazard functiongiven by

*λD*(

*t U, ***Z**(

*.*)) =

*U αλ*0

*,D*(

*t*) exp(

*γ′.***Z**(

*t*))

*.*
The frailty eﬀect on recurrent events and death is diﬀerent unless

*α *= 1.

When

*α > *1, the recurrent rate ad the death rate are positively asso-ciated since higher frailty results in both higher risk of recurrence andhigher risk of death. The authors proposed a semiparametric penalizedlikelihood estimation method in which the model degree of freedom isused to specify the smoothing parameter. Their method yields unbiasedand eﬃcient estimates. It is noteworthy to say that the work of Rondeauet al. is implemented in a very complete R package named frailtypack.

Huang and Liu (2007) considered a similar situation. Conditional on
a Gamma frailty

*U *, on a baseline vector of covariates associated with

survival

**Z***D *and on an episode-speciﬁc vector of covariates

**Z**[

*k*], they

assumed that the

*Y *[

*k*] for

*k *= 1

*, *2

*, . *are independent (but not identically

distributed) with

*Y *[

*k*] having conditional hazard function given by

*λk*(

*y U, ***Z**[

*k*]) =

*U λ *(

*y*) exp(

*β*[

*k*]

*′.***Z**[

*k*])

and are independent of the death time with conditional hazard functiongiven by

*λD*(

*t U, ***Z***D*) =

*U αλ*0

*,D*(

*t*) exp(

*γ′.***Z***D*)

*.*
The authors mentioned the fact that if covariate eﬀects are believed to behomogeneous across gap times in some appropriate practical situation, a
common

*β *may be use instead of the

*β*[

*k*] for

*k *= 1

*, *2

*, . *in order to gaineﬃciency. However, the additional ﬂexibility of this model with respectto that of Rondeau et al. (2007) induced by episode-speciﬁc baselinehazard functions and episode-speciﬁc covariate eﬀects may be of limitedpractical use if data are sparse as

*k *grows.

The standard assumption that the frailty

*U *is ﬁxed over time and
independent of observed covariates is still strong. With this respect,

Du et al. (2011) proposed a more general model even though they do

not account for the possibility of associated death. Conditional on an

unobserved vector of random eﬀects

**U **and on two vectors of observed

covariates

**Z **and e

**Z**, Du et al. (2011) assumed that the

*Y *[

*k*] for

*k *= 1

*, *2

*, .*
are independent with conditional hazard function satisfying
log

*λ*(

*y ***U***, ***Z***, *e

**Z**) =

*η*(

*y, ***Z**) + e

The vector of covariates

**Z **is expected to impact the gap time dis-

tribution while the vector of covariates e

**Z **is expected to impact the

random eﬀects. The usual frailty model correspond to e

**Z***′.***U **=

*U *for

a scalar-valued random variable

*U *that is both time-independent and

covariate-independent. The random eﬀects multivariate distribution is

left completely unspeciﬁed which allows to incorporate time-varying

frailty. Moreover, the very general form of the hazard function gives

the possibility to investigate a general shape of the conditional hazard

function and extract useful information that might be missed by para-

metric or semiparametric models. Inference is carried out by iteratively

minimizing a penalized likelihood in which the smoothing parameter

selection is reported as potentially challenging. Extension of episode-

speciﬁc covariates

**Z**[

*k*] is claimed to be straightforward.

As as summary of this subsection, in the current state of the lit-
erature, a balance has to be made between models relying on strongassumptions that are more or less hard to check and, on the other hand,more general ﬂexible models in which one may have to face identiﬁabilityand eﬃciency problems.

As a second modeling strategy, it is possible to estimate meaningfuland identiﬁable conditional distributions related to the gap times in thepresence of non-informative censoring. Such models usually specify howthe probability (or hazard) function of subsequent recurrence depends

*Modeling and Inferential Thoughts for Consecutive Gap Times*
on the past event history which may not be a trivial task. Typically lessrobust to bad speciﬁcation of subject-level correlation structure betweenevents, these models are useful for studying local process dynamics andpredicting recurrence experience at the subject level.

Amongst the primary attempt to estimate P[

*Y *[2]

*≤ y*2

* Y *[1]

*≤ y*1] is
the proposal by Lin et al. (1999) who investigated the estimate deﬁnedas the ratio of the estimate of the joint distribution function of (

*Y*1

*, Y*2)(see also next subsection) over the estimate of the marginal distributionof

*Y *[1]. Their inferential procedure is based on the inverse probability ofcensoring weights which is a well-known and useful tool for adjusting theinduced dependent censoring when analyzing multiple gap times betweenrecurrent events. Adjusting for induced dependent censoring consistsof weighting risk set contributions by the inverse of the probability ofremaining uncensored. The estimate is obtained without any modelingassumptions regarding the dependence structure of the successive gaptimes. The standard errors are also derived.

Quite similarly, Schaubel and Cai (2004) proposed an estimator of
the conditional survival function for the

*k*-th gap time conditional onthe (

*k − *1)-th event occurring prior to some ﬁxed time point. Theirwork shares with Lin et al. (1999) the fact that the estimate is obtainedwithout any modeling assumptions regarding the dependence structureof the successive gap times. However, instead of being based on a ratio ofestimate, Schaubel and Cai (2004) proposed an estimator that is deriveddirectly from a cumulative hazard function. From a technical viewpoint,their estimator is not subject to negative mass which is a problem thatmay arise with an estimate that depends on the joint distribution ofthe successive gap times. Another advantage of the proposed techniquesis the ease of computing standard errors which may be important topractitioners. A method for computing simultaneous conﬁdence bandsis also provided.

Regression methods are also available to incorporate covariates into
the analysis of conditional distributions.

Chang and Wang (1999) focused on semi-parametric regression for
conditional gap times analysis using a Cox model incorporating time-dependent covariates and in which the number of past episodes servesas a stratiﬁcation variable. Two types of time-dependent covariates areincluded. The ﬁrst type of covariates has an eﬀect which is expected toremain constant through the distinct episodes while the second kind of
covariates eﬀect is episode-speciﬁc. Setting

*k *(

*y*) =

**Z**(

*u*) =

: 0

*≤ u ≤*
*Y *[

*j*] +

*y*
*λ*[

*k*](

*y Y *[1]

*, ., Y *[

*k−*1]

*, Zk*(

*y*)) =
P

*y ≤ Y *[

*k*]

*≤ y *+

*h**Y *[

*k*]

*≥ y, Y *[1]

*, ., Y *[

*k−*1]

*, Zk*(

*y*)

*,*
*h→*0+

*h*
their model can be written as follows for

*y ≥ *0

*λ*[

*k*](

*y Y *[1]

*, ., Y *[

*k−*1]

*, Zk*(

*y*)) =
(

*y*) exp

*β′.***Z **

*Y *[

*j*] +

*y* +

*γ′*
*Y *[

*j*] +

*y*

*.*
*k .***Z**2

Implicitly, the prior event history is summed up by the time-dependentcovariates. To estimate the parameter

*β*, a proﬁle likelihood approachbased on all of the data is adopted to handle the nuisance parameters

*γk*.

In the data, because the number of subjects who experience at least

*k*recurrent events decreases as

*k *increases, a limitation in the estimationof the

*γk *is the lack of suﬃcient data for consistent estimation when

*k*has large values, as already mentioned elsewhere. However, the authorspoint out that, with appropriate conditions, the regression coeﬃcient

*β*can be consistently estimated regardless of whether the parameters

*γk*can or cannot be.

Lawless et al. (2001) reviewed conditional regression models applied
to shunt failure data. The following model

*λ*[

*k*](

*y Y *[1]

*, ., Y *[

*k−*1]

*, ***Z**[

*k*]) =

*λ *(

*y*) exp

*β′*
*k .*(

*Y *[1]

*, ., Y *[

*k−*1])

*′ *+

*α′k .***Z**[

*k*]

which gives a symmetric role to episode-speciﬁc covariates

**Z**[

*k*] and past

gap times was considered with an emphasis on the condition

*βk *=

(0

*, ., *0

*, bk*)

*′ *so that their model incorporates ﬁrst-order dependence.

The main diﬀerence with the model of Chang and Wang (1999) is

that the conditional hazard function now explicitly depends on previ-

ous event. Besides the fact that the functional relationship between the

gap times should be adequate, a potential drawback to the conditional

approach is that the parameters have to be interpreted conditionally to

previous event times.

*Modeling and Inferential Thoughts for Consecutive Gap Times*
Schaubel and Cai (2004b) considered estimation via semi-parametric
Cox regression models for conditional gap times hazard functions. Re-specting the identiﬁability issues, the authors focused on the followinggap-time-speciﬁc hazard functions
P

*y ≤ Y *[

*k*]

*≤ y *+

*h**Y *[

*k*]

*≥ y,*
*Y *[

*j*]

*≤ t*
*h→*0+

*h*
for some pre-speciﬁed

*tk−*1 chosen in the support of the total observa-

tion time distribution and for some external time-dependent covariates

**Z**[

*k*](

*.*). They assumed the proportional hazards formulation

*λ*[

*k*](

*y*;

*tk−*1

* ***Z**[

*k*](

*y*)) =

*λ *(

*y*;

*t*
(

*.*) is an unspeciﬁed continuous function. A sensibility analysis
is recommended for an appropriate choice of ﬁxed time point

*tk−*1. Infer-ence can be carried out without making assumptions about associationamong individual's gap times.

Clement and Strawderman (2009) proposed a method for estimating
the parameters indexing the conditional means and variances of the gaptime distributions conditional on all the available explanatory covariateshistory as well as on past gap times. Precisely, their work deal with
E

*Y *[

*k*]

* Y *[1]

*, ., Y *[

*k−*1]

*,*

**Z**(

*u*) : 0

*≤ u ≤*
*Y *[

*j*] =

*µk*(

*θ*)
Var

*Y *[

*k*]

* Y *[1]

*, ., Y *[

*k−*1]

*,*

**Z**(

*u*) : 0

*≤ u ≤*
*Y *[

*j*] =

*σ*2

*Vk*(

*θ*)2 (2)
where

*µk*(

*.*) and

*Vk*(

*.*) are known scalar functions of the unknown pa-rameter

*θ*. The scalar parameter

*σ*2

*> *0 is also to be estimated. Theproposed methodology is an adaptation of generalized estimating equa-tions for longitudinal data and permits the use of both time-ﬁxed andtime-varying covariates, as well as transformations of the gap times.

Censoring is dealt with by imposing a parametric assumption on thecensored gap times. Simulations report the relative robustness to devia-tions from this assumption although this supposed adequacy is identiﬁedas a potential issue. It shall be emphasized that the parametric assump-tions in (1) and (2) bear on the two ﬁrst moments of the conditional
distribution and not on the conditional hazard function itself. The Rpackage condGEE implements this conditional GEE for recurrent eventgap times.

Note also that all these methods do not account the possibility of
The main issue with conditional models lies in the more or less ques-
tionable assumptions made to incorporate past events.

**Models for Multivariate Functions**
Several non-parametric statistical analysis have been proposed for jointinference on consecutive gap times through multivariate functions inthe absence of death but accounting for non-informative censoring. Thenonparametric approach is quite classical to this aim. Nonparametricstatistics have the beneﬁt of avoiding too restrictive assumptions espe-cially regarding would-be independence or memoryless-type conditions.

It is a good method to understand basics and to produce descriptiveresults. It also allows a ﬁrst investigation of eﬀects of covariates shownby stratifying data into groups.

Such an approach was originally developed in Visser (1996). The
author considered joint nonparametric estimation for two successive du-ration times in the presence of independent right-censoring restricted tothe setting where the gap times and the censoring variable are discrete.

His method can deal with situations where censoring may depend uponprevious gap times but relies on estimating the cumulative conditionalhazard of the second gap time given the ﬁrst one and therefore discretecensoring time and gap times are mandatory.

Wang and Wells (1998) studied the same problem but for any arbi-
trary distributions of the gap times and the censoring variable. Theyalso considered joint nonparametric estimation for two successive du-ration times.

They proposed an estimator for the bivariate survival
function of (

*Y *[1]

*, Y *[2]) by estimating the cumulative conditional hazardof

*Y *[2] given

*Y *[1]

*> y*1. The estimator was shown to be consistent andasymptotically normal, but do not guarantee a non-negative weightingof the data. Moreover, no analytical variance expression is given due tothe complicated expression of the estimator.

Lin et al. (1999) proposed a nonparametric estimator for the joint
distribution function of the gap times. Their estimator is based on theinverse probability censoring weighted method used with the Kaplan-Meier estimator.

To enable comparison with other proposals, let us

*Modeling and Inferential Thoughts for Consecutive Gap Times*
introduce the observable gap times

*T *[1] = min(

*Y *[1]

*, C*) and

*T *[2] =min(

*Y *[2]

*, *(

*C − Y *[1])

*I*(

*Y *[1]

*≤ C*)), the observable total duration time

*T *= min(

*Y *[1] +

*Y *[2]

*, C*) and the observable indicator variable

*δ *=

*I*(

*Y *[1] +

*Y *[2]

*≤ C*) when

*C *is the censoring variable independent of
(

*Y *[1]

*, Y *[2]). Let (

*T*
*i, δi*) for

*i *= 1

*, ., n *be i.i.d. replications of
(

*T *[1]

*, T *[2]

*, T, δ*). Lin et al. (1999)'s estimator of P[

*Y *[1)

*≤ y*1

*, Y *[2]

*≤ y*2]can be written as
1 ∑

*I*(

*T*
∑

*I*(

*T ≤ y*
*i*=1 1

*− *b

*Gn *is a suitable Kaplan-Meier estimate of the censoring distri-
bution function

*G*. However, their estimator is not always a properdistribution function in that it may have negative mass points thoughit converges to a proper distribution function as

*n *goes to

*∞*. Meira-Machado and Moreira (2010) found out in simulation studies that thisestimator is almost unbiased but may have important variance.

Van der Laan et al. (2002) considered more general problems of
estimation that can be exploited for successive gap times. They alsoused inverse probability of censoring weights techniques. The statisticalnovelty of their approach lies in the derivation of locally eﬃcient one-stepestimator.

In a more general situation of dependent censoring including the
present setting as special case, Van Keilegom (2004) derived a nonpara-metric estimator for the bivariate and marginal distribution functionsof two gap times. The proposal by Van Keilegom (2004) consists ofwriting the joint distribution function of (

*Y *[1]

*, Y *[2]) as an average of theconditional distribution

*F*2

* *1(

*y*2

* y*) = P[

*Y *[2]

*≤ y*2

* Y *[1) =

*y*] ie as
P[

*Y *[1]

*≤ y*1

*, Y *[2]

*≤ y*2] =
where

*F*1(

*y*) = P[

*Y *[1]

*≤ y*1]. The conditional Kaplan-Meier estimator ofBeran (1981) is used to estimate

*F*2

* *1. This relies on a kernel smoothingaround

*Y *[1] =

*y *with the modiﬁcation that only uncensored observa-tions of

*T *[1] are allowed in the window. The practical choice of theaforementioned window may be a limiting factor to a more frequent useof this estimator even though this choice is reported as non crucial andif a bootstrap procedure is advocated for.

Alvarez and Meira-Machado (2008) proposed another non-
parametric estimator of bivariate distribution function of two consecu-tive gap times. The estimator of de U˜
Alvarez and Meira-Machado
(2008) is a weighted bivariate distribution function of the form

*Wi I*(

*T*
where the weight

*Wi *is the Kaplan-Meier weight attached to

*Ti *whenestimating the marginal distribution of

*Y *[1] +

*Y *[2] from the observablerandom variables (

*Ti, δi*). Their estimator is a proper distribution func-tion contrarily to the proposal of both Wang and Wells (1998) and Linet al. (1999). Simulations revealed that this new estimator is reasonablyunbiased and may achieve eﬃciency levels clearly above the previous pro-posals, which is promising. However, theoretical investigation is neededto get general conclusions. It is also noted that the method can easilybe extended to cope with more than two successive gap times. Meira-Machado and Moreira (2010) found out in simulation studies that thisestimator is almost unbiased but may still have important variance.

In general, the prize to pay for the absence of restrictive assump-
tions is a lack of eﬃciency. Some methods however exists to deal withthis issue. Presmoothing techniques may be useful to gain eﬃciency.

The idea of presmoothing goes back at least to Dikta (1998), see alsoDikta (2000), (2001) and Dikta et al. (2005). Presmoothing consists ofreplacing the censoring indicator by a smooth ﬁt of a binary regressionof the indicator on observable gap times. This replacement usually re-sults in estimators with improved variance. That is why de U˜
and Amorim (2011) applied the idea of presmoothing to the estimationof the bivariate distribution function of censored gap times. As in thepaper by de U˜
Alvarez and Meira-Machado (2008), the estimator of
Alvarez and Amorim (2011) is a weighted bivariate distribution
function of the form

*Wi I*(

*T *[1]

*≤ y*1

*, T *[2]

*≤ y*2)

*Wi *now uses a presmoothed version of the preceding
Kaplan-Meier weight

*Wi*. The consequence of this is that the estimatorde U˜
Alvarez and Amorim (2011) can attach positive mass to pair of
gap times with censored second gap times which is not the case with theestimator of de U˜
Alvarez and Meira-Machado (2008). Note that in
the limiting case of no presmoothing, the estimator de U˜
Amorim (2011) reduces to that of de U˜
Alvarez and Meira-Machado
(2008). A simulation study by Meira-Machado and Moreira (2010) logi-cally concluded that the presmoothed estimator improves eﬃciency with

*Modeling and Inferential Thoughts for Consecutive Gap Times*
respect to the estimator of de U˜
Alvarez and Meira-Machado (2008)
but its bias is greater.

Van Keilegom et al. (2011) considered a non-parametric location-
scale model for the two ﬁrst gap times assuming that the vector of gaptimes (

*Y *[1]

*, Y *[2]) satisﬁes

*Y *[2] =

*m*(

*Y *[1]) +

*σ*(

*Y *[1])

*ε*
where the functions

*m *and

*σ *are smooth and

*ε *is independent of

*Y *[1].

This allows the transfer of tail information from lightly censored areasto heavily ones. Under this model, the authors proposed estimators ofP[

*Y *[1]

*≤ y*1

*, Y *[2]

*≤ y*2], P[

*Y *[2]

*≤ y*2

* Y *[1] =

*y*1] and other related quan-tities. In a related paper, Meira-Machado et al. (2011) discussed thepractical implementation and performance of the aforementioned esti-mators and proposed some modiﬁcations. In an extensive simulationstudy, the good performance of the method is shown. The main limita-tion of their work lies in the fact that the adequacy of the model to thedata needs to be tested. However, the authors mentioned that derivingsuch a test in the present setting is far from being straightforward.

The R package survivalBIV is much helpful to calculate the diﬀerent
estimates for the bivariate distribution function.

We already mentioned the possible use of presmoothing to improve
eﬃciency. General and testable assumptions such as Koziol-Green modelalso termed as informative censoring (Koziol and Green (1976), Chengand Lin (1987)) or proportionality constraints (Dauxois and Kirmani(2003), Geﬀray and Guilloux (2011)) can also be used leading to moreeﬃcient semi-parametric inference under not so much restrictive assump-tions. Adekpedjou et al. (2010) adopted this strategy to tackle theeﬃciency problem.

Two-sample tests have been brieﬂy considered in the literature. Lin
and Ying (2001) proposed several classes of two-sample nonparametricstatistics for comparing the gap time distributions based on the nonpara-metric estimator of the gap time distribution given by Lin et al. (1999).

These statistics are analogous to familiar censored data statistics, suchas weighted log-rank statistics.

Huang (2000) proposed a semi-parametric accelerated failure time
model to compare two treatment groups in terms of their successivegap times. Let ∆ be the indicator function that takes value 1 whenthe subject is in the ﬁrst treatment group and 0 when the subject is inthe second treatment group. Speciﬁcally, the author parametrized thegroup eﬀect on gap times and survival time by a scale transformation
assuming that the random variables exp(

*β*1∆)

*Y *[1]

*, ., *exp(

*βk *∆)

*Y *[

*k*0]
follow an unspeciﬁed multivariate continuous distribution function thatis independent of ∆. A log-rank type statistic is then derived.

Joint regression models have also been considered.

Huang (2002) considered multivariate accelerated failure time models
for which the variables log

*Y *[

*k*] for

*k *= 1

*, ., k*0 follow a multivariatelocation-scale distribution of the form:
log

*Y *[

*k*] =

*β′k.***Z**[

*k*] +

*ε*[

*k*]

where (

*ε*[1]

*, ., ε*[

*k*0]) have an unspeciﬁed joint distribution that does not

depend on the vector of episode-speciﬁc covariates

**Z**[

*k*]. Note that infer-

ence is robust to misspeciﬁcation of the gap time association structure

at the expense of strong assumptions on the censoring mechanism. It

turns out that, in this paper, censoring is assumed to be independent of

both covariates and the recurrent event process. Moreover, it is implic-

itly considered that each subject can experience at most

*k*0 events. This

model may consequently appear less suited when the numbers of events

vary substantially across subjects.

He and Lawless (2003) presented multivariate parametric regression
models for proportional hazards speciﬁed either within a copula model orwithin a frailty model. The method employs ﬂexible piecewise constantor spline speciﬁcations as baseline hazard functions in either models.

Because all the models considered are parametric, ordinary maximumlikelihood can be applied. The adequacy to the parametric assumptionsis crucial to get unbiased estimates which may be a drawback.

All these methods, however, assume that recurrent events are not
terminated by death during the study. Some eﬀorts have been madeto account for death in a joint analysis for two-sample comparison pur-poses. Chang (2000) proposed a semi-parametric accelerated failure timemodel to compare two treatment groups jointly in terms of their suc-cessive gap times and survival time. This model is similar to that ofHuang (2000) but accounts for death. Let ∆ be the indicator func-tion that takes value 1 when the subject is in the ﬁrst treatment groupand 0 when the subject is in the second treatment group.

cally, the author parametrized the group eﬀect on gap times and sur-vival time by a scale transformation assuming that the random variablesexp(

*α*∆)

*D, *exp(

*β*1∆)

*Y *[1]

*, ., *exp(

*βk *∆)

*Y *[

*k*0] follow an unspeciﬁed mul-
tivariate continuous distribution function that is independent of ∆. Alog-rank type statistic is then derived.

As a brief summary, models for multivariate functions mostly belong
to the realm of non-parametric statistics in the absence of covariates

*Modeling and Inferential Thoughts for Consecutive Gap Times*
information. Fewer papers are available for multivariate functions inthe regression framework or in the presence of death.

**Nonparametric Estimation of Cause-Specific**

Distributions
In the recurrent events with death framework, functions describing thestochastic dynamics in the tree of Figure 3 can be much useful. Theapproach of Li and Lagakos (1997) and Derzko and Leconte (2004)who treated death as a competing risk acting at each recurrence canbe adopted for that purpose. They modeled the terminal event as a de-pendent competing event for each recurrent event i.e. they treated thefailure time for each recurrence as the ﬁrst occurrence of the recurringevent or terminating event whichever came ﬁrst. Thus, for each recur-rence, the patient is submitted to two dependent competing risks (REand death) in the presence of independent right-censoring provided heor she survived the previous occurrences. These step-by-step competingrisks models do not specify the association structure between recurrentevents and death. The work is centered on crude functions since theseare the only identiﬁable quantities without any assumptions regardingthe association structure among the competing risks. Non-parametricinference under minimal assumption is investigated.

At risk subjects
Figure 3: Competing risks at each recurrence in the presence of independentcensoring (RE = recurrent event).

We assume that the observed data consist of i.i.d.

(

*Y *[0]

*, ., Y *[

*K*]

*, *(

*D ∧ C*)

*−*
*Y *[

*k*]

*, I*(

*D ≤ C*)) where

*D *is the death
time,

*C *is the independent right-censoring. The number

*K ∈ *N is ran-dom as in Wang and Chang (1999) and Pe˜
na et al. (2001),

*Y *[0] is set
as 0, if

*K ≥ *1, the

*Y *[

*k*] for

*k *= 1

*, ., K *are the observed gap times untila recurrent event while the last gap time ends either with a death or acensoring event.

With these remarks in view, the functions that can serve as useful
descriptive devices are the following. We consider for

*y*1

*, y*2

*≥ *0:

*F *[1(2)](

*y*1) = P

*D ≤ y*1

*, Y *[1]

*> D ,*
*F *[1(1)

*,*2(1)](

*y*1

*, y*2) = P

*Y *[1]

*≤ y*1

*, Y *[1]

*≤ D, Y *[2]

*≤ y*2

*, Y *[2]

*≤ D − Y *[1]

*,*
*F *[1(1)

*,*2(2)](

*y*1

*, y*2) = P

*Y *[1]

*≤ y*1

*, Y *[1]

*≤ D, D − Y *[1]

*≤ y*2

*, Y *[2]

*> D − Y *[1]

*.*
This can be straightforwardly extended to further recurrences provided
the data are not too sparse.

Let

*FD *be the distribution function of

*D*. Denote by

*C *the non-
negative random variable that stands for the independent right-censoringwith distribution function

*G*. Let

*H *be the distribution function deﬁnedby 1

*− H *= (1

*− FD*)(1

*− G*) and let

*τH *= sup

*{x *:

*H*(

*x*)

*< *1

*} *be theright-endpoint of the distribution function

*H*. The functions (3) to (5)can be consistently estimated and it can be shown that the correspond-ing estimators have an asymptotic Gaussian behavior on compact setssuch that the corresponding total observation time is inferior to

*τH *. No-tice that

*F *[1](

*y*1) is estimable only if

*y*1

*< τH *, that

*F *[1(1)

*,*2](

*y*1

*, y*2) isestimable only if

*y*1 +

*y*2

*< τH *and so on. The objective of this section isto justify nonparametric estimation for the functions displayed in Equa-tions (3) to (5) without any assumption regarding either the dependencestructure among the multiple endpoints.

For ease of exposition, note that the observable random variables
can be coded as follows.

*• *Let

*K *+1 (with

*K ∈ *N) be the total number of observed events for
a given individual (including recurrent events, death and censoringevents).

For

*k *= 1

*, . . , K *+ 1, let

*T*
be the random variable that stands
for the gap time between the (

*k − *1)-th and the

*k*-th event and set

*• *For

*k *= 1

*, . . , K *+ 1, the random variable

*Modeling and Inferential Thoughts for Consecutive Gap Times*
0 if the

*k*-th event is censored

*J *[

*k*] = 1 if the

*k*-th event is a recurrent event
2 if the

*k*-th event is a death
indicates the nature of the

*k*-th observed event.

We suppose that observations are taken on an i.i.d. sample of

*n*
individuals. For

*i *= 1

*, . . , n*, the data for the

*i*-th individual consists of

*Ki *+ 1 couples where

*Ki *is the number of observed (non-fatal) recurrent
events. For

*k *= 1

*, . . , Ki *+ 1, the

*k*-th couple is given by (

*T*
which is distributed as (

*T*
*, J *[

*k*]).

**Estimation of the Censoring Distribution Function**
An estimate of the censoring distribution function

*G *will be used. Thissubsection deals with this preliminary step.

As noted in Section 1, the last observation for a given patient is either
a censoring time or a death time. For a given patient, we do not observeboth the censoring event and the death but only the ﬁrst event thatoccurs. Since the death and the censoring processes are independent,the censoring distribution function may be estimated by the Kaplan-Meier estimator based on the total observation time for each patient i.e.

on the data (

*T*
) for

*i *= 1

*, . . , n*.

The Kaplan-Meier estimator of the censoring distribution function

*G *is given for

*t ≥ *0 by:

*i ≤ t, Ji*
*n*(

*t*) = 1

*−*
*I *(

*Tℓ ≥ Ti*)
If there are ties between recurrent event times and censoring times, theKaplan-Meier estimator of

*G *cannot be obtained by using the indicatorstatus equal to zero. In such cases, the R package prodlim provides auseful alternative to estimate the censoring distribution.

**"Plug-in" Estimation of the Functions of Interest**
To derive an estimator for the functions

*F *[1(2)],

*F *[1(1)

*,*2(1)] and

*F *[1(1)

*,*2(2)],we introduce the following distribution functions for

*y*1

*, y*2

*≥ *0:

*H*[1(1

*,*2)](

*y*1) = P

*T*
1

*, J *[1] = 2

*H*[1(1

*,*1)

*,*2(1

*,j*)](

*y*1

*, y*2) = P

*T*
*, j *= 1

*, *2

*.*
2

*, J *[1] = 1

*, J *[2] =

*j*
For

*y ≥ *0, we obtain the following relation:

*H*[1(1

*,*2)](

*y*) = P

*Y *[1]

*≤ y, Y *[1]

*≤ C, *C[1] = 2

*I *(

*u ≤ y, u ≤ c*)

*G*(

*dc*)

*F *[1(2)](

*du*)
1

*− G−*(

*u*)

*F *[1(2)](

*du*)

*.*
with

*G− *being the left-continuous modiﬁcation of

*G*.

*F *[1(2)](

*y*) can be written in terms of the estimable functions

*G *and

*H*[1(1

*,*2)]:

*F *[1(2)](

*y*) =

*u≤y *1

*− G−*(

*u*)
We can obtain in the same way for

*j *= 1

*, *2 and

*y*1

*, y*2

*≥ *0 that

*I *(

*u ≤ y*1

*, v ≤ y*2

*, c ≥ u *+

*v*)

*× G*(

*dc*)

*F *[1(1)

*,*2(

*j*)](

*du, dv*)
1

*− G−*(

*u *+

*v*)

*F *[1(1)

*,*2(

*j*)](

*du, dv*)

*F *[1(1)

*,*2(

*j*)](

*y*1

*, y*2) =
1

*− G−*(

*u *+

*v*)
Consequently, we propose "plug-in" estimates of the functions

*F *[1(2)]and

*F *[1(1)

*,*2(

*j*)] for

*j *= 1

*, *2 by means of "plug-in" estimators denoted
respectively by b
for

*j *= 1

*, *2. These estimators are
obtained by replacing

*G *by its Kaplan-Meier estimator deﬁned in Sub-section 3.1 and

*H*[1(1

*,*2)] and

*H*[1(1

*,*1)

*,*2(1

*,j*)] by their empirical counterpartswhich are deﬁned respectively for

*y*1

*, y*2

*≥ *0 by:

*y*1

*, J*
(

*y*1

*, y*2) =

*y*1

*, T*
*y*2

*, J*
*Modeling and Inferential Thoughts for Consecutive Gap Times*
Consequently, we let for

*y*1

*, y*2

*≥ *0

*n *(

*u*)
(

*y*1

*, y*2) =

*j *= 1

*, *2

*n *(

*u *+

*v*)

*n *is the left-continuous modiﬁcation of b

*1. For any σ < τH , the estimator *b

*is strongly consistent on *[0

*, σ*]

*for F *[1(2)]

*.*
*2. For any σ < τH , the estimators *b

*are strongly consistent for*
*F *[1(1)

*,*2(

*j*)]

*, for j *= 1

*, *2

*, on the set Tσ *=

*{*(

*y*1

*, y*2) :

*y*1 +

*y*2

*< σ}.*
**Remark 3.1.**
If

*y*1 is taken equal to

*∞ *in the deﬁnition of the esti-
, one would have b
(

*∞*) equal to b
is the last order statistic of the sample (

*T*
)

*→ F *[1(2)](

*τ*
1(2)

*∧ τG*) holds in probability where

*τ*1(2) is the right-endpoint of

*F *[1(2)] and where

*τG *is the right-endpointof

*G*. But

*F *[1(2)](

*τ*1(2)

*∧ τG*) may be strictly inferior to

*F *[1(2)](

*∞*). Thisis fulﬁlled in particular if

*τG < τ*1(2) which is the case in a clinical trialfor example where this is to hope that some patients won't experience arecurrence by the end of study. This situation would lead to a biased es-timation of

*F *[1(2)](

*∞*). The same kind of restriction holds for the otherestimators mentioned here.

*Assume that G is continuous. For any σ < τH , the*
*− F *[1(2)]

*and n F *[1(1)

*,*2(

*j*)]

*− F *[1(1)

*,*2(

*j*)]

*for j *= 1

*, *2

*converge jointly in distribution to zero-mean Gaussian pro-cesses in the Skorohod space of c *
*ag functions on Tσ.*
**Remark 3.2.**
The condition that

*G *is continuous is restrictive since
it does not allow for a ﬁxed time to follow-up. Further work would beneeded for such an extension.

To save place, the large sample arguments are purposefully sketchy.

**Proof of Proposition 3.1. **We decompose

*Fn*
*F *[1(1)

*,*2(

*j*)] for

*j *= 1

*, *2 into
(

*y*1

*, y*2)

*− *b

*F *[1(1)

*,*2(

*j*)](

*y*1

*, y*2) =
1

*− G−*(

*u *+

*v*)

*n *(

*u *+

*v*)
(

*du, dv*)

*− H*[1(1

*,*1)

*,*2(1

*,j*)](

*du, dv*))
1

*− G−*(

*u *+

*v*)
We carry out integration by parts on the second term in the aboveequality and get straightforwardly
(

*y*1

*, y*2)

*− *b

*t≤σ * 1

*− G*(

*t*)
1

*− G*(

*σ*)
(

*y*1

*, y*2)
The fact that

*G*(

*σ*)

*< *1 together with Glivenko-Cantelli's theorem validwith and without independent right-censoring give the required almost
sure convergence on

*Tσ*. The proof is identical for b

**Proof of Proposition 3.1. **First, we endow the space of cadlag func-

tions on

*Tσ *with the appropriate topology. This can be obtained by

transporting the Skorohod topology of the space of cadlag functions on

[0

*, *1]2 build in Neuhaus (1971) since the spaces

*Tσ *and [0

*, *1]2 are home-

omorphic.

The weak convergence result is then obtained by empirical processes
techniques. It relies on appropriate decomposition of the processes

*− F *[1(2)] and

*n Fn*
*− F *[1(1)

*,*2(

*j*)] for

*j *= 1

*, *2 that
permits to apply the joint convergence of univariate and multivariateempirical processes based on the observed data. The functional delta-method as in Andersen et al.

(1993) is also of much use.

ingredient is the use of the existing results for the Kaplan-Meier processwhich makes the assumption that

*G *is continuous necessary. Let us
begin with the decomposition of

*− F *[1(1)

*,*2(

*j*)] for

*j *=

*Modeling and Inferential Thoughts for Consecutive Gap Times*
1

*, *2. The decomposition of

*− F *[1(2)] is left to the reader.

(

*y*1

*, y*2)

*− F *[1(1)

*,*2(

*j*)](

*y*1

*, y*2)

*y*2

*Hn*
(

*du, dv*)

*− H*[1(1

*,*1)

*,*2(1

*,j*)](

*du, duv*)
1

*− G*(

*u *+

*v*)

*n *(

*u *+

*v*)

*− G*(

*u *+

*v*)

*H*[1(1

*,*1)

*,*2(1

*,j*)](

*du, dv*)
(1

*− G*(

*u *+

*v*))2

*Gn*(

*u *+

*v*)

*− G*(

*u *+

*v*)
(1

*− G*(

*u *+

*v*))2

*n *(

*u *+

*v*)

*− G*(

*u *+

*v*)
1

*− G*(

*u *+

*v*)

*n *(

*u *+

*v*)
=

*I*1(

*y*1

*, y*2) +

*I*2(

*y*1

*, y*2) +

*I*3(

*y*1

*, y*2) +

*I*4(

*y*1

*, y*2)

*.*
Terms

*I*3 and

*I*4 are negligible uniformly on

*Tσ *thanks to the functionaldelta-method and to the weak convergence of both the Kaplan-Meier
process and the empirical process

*I*2 needs to be further decomposed. Let

*H*(0) be the censoring subdis-tribution function deﬁned by

*H*(0)(

*y*) = P[

*T ≤ y, JK*+1 = 0]

*,*
let its empirical counterpart be deﬁned by

*n *(

*y*) =

*I*(

*Ti ≤ y, JKi*+1 = 0)
and let

*Hn *be the empirical counterpart of

*H *deﬁned by

*Hn*(

*y*) =

*I*(

*Ti ≤ y*)

*.*
Applying the methods and results of Cs¨
o (1996), we can state since

*G *is assumed continuous that

*n*(

*t*)

*− G*(

*t*)

*d*(

*Hn *(

*s*)

*− H*(0))(

*s*)
1

*− G*(

*t*)
1

*− H−*(

*s*)

*n *(

*s*)

*− H −*(

*s*)

*dH*(0)(

*s*) =

*o*P

*√*
(1

*− H−*(

*s*))2
Consequently, the process

*− F *[1(1)

*,*2(

*j*)] is asymptotically

*y*2

*Hn*
(

*du, dv*)

*− H*[1(1

*,*1)

*,*2(1

*,j*)](

*du, dv*)
1

*− G*(

*u *+

*v*)
(

*u*+

*v*)

*− d*(

*Hn *(

*s*)

*− H*(0))(

*s*)
1

*− H−*(

*s*)

*n *(

*s*)

*− H −*(

*s*)

*dH*(0)(

*s*)

*H*[1(1

*,*1)

*,*2(1

*,j*)](

*du, dv*)

*.*
(1

*− H−*(

*s*))2
It remains to get the joint convergence of

*n*(

*Hn − H*(0)),

*n*(

*Hn −*
*−H*[1(1

*,*1)

*,*2(1

*,j*)]) in order to apply once again
the functional delta-method and get the result. To see this, set

*T R *=
and

*J R *=

*J Ki*+1 for

*i *= 1

*, ., n *such that

*K*
*i *+ 1

*≥ *3. Set
also

*T R *=

*K*+1

*T*
with the sum being null if the summation index
set is void and

*J R *=

*J K*+1. Then, decompose (

*Hn − H*(0)) into

*n *(

*y*)

*− H *(0)(

*y*)

*≤ y, J *= 0)

*− *P[

*T ≤ y, J*[1] = 0]

*≤ y, J *= 1

*, J *= 0)

*≤ y, J*[1] = 1

*, J*[2] = 0]

*≤ y, J *=

*J *= 1

*, J*
*≤ y, J*[1] =

*J*[2] = 1

*, J*[

*R*] = 0
and decompose (

*Hn − H*) into

*Hn*(

*y*)

*− H*(

*y*) =

*≤ y, J ̸*= 1)

*− *P[

*T ≤ y, J*[1]

*̸*= 1]

*Modeling and Inferential Thoughts for Consecutive Gap Times*
*≤ y, J *= 1

*, J ̸*= 1)

*≤ y, J*[1] = 1

*, J*[2]

*̸*= 1]

*≤ y, J *=

*J *= 1

*, J*
*≤ y, J*[1] =

*J*[2] = 1

*, J*[

*R*]

*̸*= 1
Pollard's (1982) theorem valid for the empirical processes indexed by
the VC-class of sets

*{ *2 [0

*, y*
*i*] :

*y*1 +

*y*2

*≤ σ} *concludes the argument.

Integration by parts permits to have the empirical processes in the in-tegrand rather than in the measure of integration. This technicality isleft to the reader.

*2*
The variance function of the limiting process is not mentioned. Ob-
taining it through empirical processes techniques is quite cumbersome.

Moreover, a classical problem with competing risks is the variance ex-plosion which entails far too wide conﬁdence bands, see e.g. Geﬀray(2009), making this calculus less interesting.

Martingale methods have been successfully used for survival analysispurposes from the mid 1970's. These developments go back to Aalen'swork, see e.g. Aalen (1978a), (1978b), Aalen et al. (1978) then moved totwo-sample tests, Cox's proportional hazards regression model, Markovtransition probabilities estimation among many other situations, see e.g.

Gill (1980), (1983), (1994), Andersen et al. (1993). It emerges that thecounting process and stochastic integral approach provides relativelysimple methods of inference in some situations where standard methodsof inference are too cumbersome or require too restrictive assumptions.

The martingale inference methods provide systematic methods for mo-ment calculation, establish asymptotic normality of empirical processesand gave rise to a variety of results in settings where a single time runs,in particular, it enables to extend asymptotic results up to the last or-der statistic of the observation instead of restricting their validity tocompact time-intervals.

The martingale methods of use in survival analysis could be gen-
eralized to our setting where multiple times run. This would requiremulti-parameter counting processes and martingales.

time-continuous multi-parameter martingale and stochastic integral hasdeveloped extensively from the 1970's, see e.g.

Bickel and Wichura
(1971), Zakai (1981), Zakai et al.

(1974), (1976), Merzbach (1988),
Ivanoﬀ (1996), with a special emphasis to set-indexed martingales inthe nineties, see e.g.

Ivanoﬀ and Merzbach (2000) for an extensive
study of this subject. These probability results opened a new perspec-tive for statisticians involved with multi-time periods inference prob-lems. The use of martingale methods for multi-parameter problems insurvival analysis was initiated by Pons (1986) in the setting of bivari-ate survival function estimation. In a tremendous paper, Ivanoﬀ andMerzbach (2002) developed a general model for survival analysis wherecensored data are parametrized by sets instead of time points. Dis-appointingly, their work is of little help for our purpose. Our speciﬁccensoring scheme complicate technical matters considerably and, in par-ticular, invalidate the direct use of their methods. However we anticipatethat some progress could be made in this direction.

We pointed out earlier the fact that the pure nonparametric approach
may suﬀer from a lack of eﬃciency. Considering a presmoothed versionof the estimate of functions (3) to (5) as in Cao et al. (2004), (2005),Amorim et al. (2011), de U˜
na-Alvarez and Amorim (2011) could be an
interesting remedy.

In the framework of Section 3, conditional analysis could be interest-
ing since it allows dynamical prediction while incorporating a patient'shistory. The estimation of conditional probabilities such as
P

*Y *[2]

*≤ y*2

*, Y *[2]

*≤ D − Y *[1]

* Y *[1] =

*y*1

*, Y *[1]

*≤ D*
P

*D ≤ y*2

*, Y *[2]

*> D − Y *[1]

* Y *[1] =

*y*1

*, Y *[1]

*≤ D*
is currently under investigation via projection methods without any as-sumptions regarding dependence structure of successive gap times.

It is worth mentioning that investigators increasingly encounter data-
sets in which some patients are expected to be cured. This is a seri-ous matter because a patient surviving the trial is considered censoredwhereas the patient is cured if he or she will never experiment the eventunder study. The diﬃculty comes from the fact that a cure can neverbe observed due to a ﬁnite monitoring time. To address this problem,cure rate models have been proposed and have received intensive atten-tion for their ability to account for the probability of a patient being

*Modeling and Inferential Thoughts for Consecutive Gap Times*
cured, see e.g. Yin and Ibrahim (2005). Recently, some progress havebeen made to incorporate the possibility of cure into recurrent eventsmodeling. Rondeau (2010) developed a cure frailty model to evaluatetime-dependent medical treatment eﬀects on the times to recurrenceamong the uncured patients and on the cure probability. The probabil-ity of cure here may evolve with time and is deﬁned as the probabilityof not developing further event after each event. Rondeau et al. (2011)compared several forms of cure rate model within a frailty model for therecurrent event part. To analyze recurrent events, it is ﬁrst necessaryto deﬁne the cured proportion to be modeled. The ﬁrst model considersthat immune patients are those who are not expected to experience theevents of interest over a suﬃciently long time period. The other investi-gated models account for the possibility of cure after each event i.e. theprobability of cure may evolve with time. The focus is placed on timesto recurrence and death is accounted for.

Account for the possibility of cure should be dealt with in the frame-
work of joint multivariate approach as well as th possibility of incorpo-rating covariates acting on the uncured population survival. Investiga-tion of both their practical and theoretical properties should be carefulreported for applied purposes.

Another point that is worth mentioning is the issue due to non-
reliable cause of death. In this situation, relative survival models canbe of use, see e.g. Lambert et al. (2010). Subjects may die of thedisease they are diagnosed with but they may also die of something else.

Deaths due to another cause than strictly the disease under study canbe broadly classiﬁed into "totally independent death" i.e. death froma cause related neither to the disease under study nor to the treatmentand "possibly related death". The "totally independent death" usuallyconstitutes part of the independent right-censoring process and is notan issue.

But the class "possibly dependent" leads to diﬃculties in
interpreting the results. Interest mostly lies in mortality strictly due tothe disease of interest and not to related causes. How to classify, forexample, deaths due to treatment complications? Consider a patientdiagnosed with lung cancer who dies following a myocardial infarction.

Do we classify this death as ‘due entirely to lung cancer' or ‘due entirelyto other causes' ? There may also exist problems with cause-speciﬁcdeath distribution due to inaccuracy of death certiﬁcates. An alternativeto cause-speciﬁc distribution estimation is then to model relative survivalor its converse which is termed as excess mortality. Suppose that,

*S∗*(

*t*)is the expected survival. Then the total survival

*S*(

*t*) can be written as
the product of the relative survival

*R*(

*t*) and the expected survival

*S∗*(

*t*)i.e.

*S*(

*t*) =

*S∗*(

*t*)

*R*(

*t*). Relative survival is often preferred over cause-speciﬁc survival for the study of cancer patient survival. This issue isparticularly relevant here where possible applications are infarction orcancer recurrence and related death and is worth investigating.

A last interesting point to note is that in our setting some events
may be rare. For instance, Cui et al. (2010) noted that the chance ofhaving two myocardial infarction events within 5 years was low amongall participants in the LIPID study. As a consequence, when analyzingthe pre-speciﬁed set, say (

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*, Y *[2]), the second gap times

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Source: http://jirss.irstat.ir/article-1-210-en.pdf

Cheryl Lopate, MS, DVMDiplomate, American College of Theriogenologists Pyometra in the Bitch Pyometra is a condition that affects intact bitches, causing a variety of clinical signs and symptoms. Pyometra is typically pre-empted by pathologic changes in the uterus. The Greekderivation of pyometra is: pyo = pus and metra = uterus, so pyometra = an accumulation of pus inthe uterus.

esclaves au XXIe siècle la couleur des jours 1 · automne 2011 La traite d'êtres humains,une réalité invisibleen Suisse romande Chaque année, des centaines d'hommes et de femmes sont victimes de traite des êtres humains en Suisse, pays de transit et de destination de ce commerce d'un autre temps. Les autorités helvétiques commencent à prendre la mesure du phénomène et plusieurs outils de lutte ont été développés cette dernière décennie. Mais la Suisse romande est en retard. Dans nos cantons se cache un esclavagisme moderne à l'abri des regards et souvent des consciences. Témoignages et analyses.