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Joint Models of Longitudinal and Time-to-Event Data with More Than One Event Time Outcome: A Review

Graeme L. Hickey
  • Department of Biostatistics,University of Liverpool, Waterhouse Building, 1-5 Brownlow Street, Liverpool, L69 3GL, UK
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/ Pete Philipson
  • Department of Mathematics,Physics and Electrical Engineering, Northumbria University, Ellison Place, Newcastle upon Tyne, NE1 8ST, UK
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/ Andrea Jorgensen
  • Department of Biostatistics,University of Liverpool, Waterhouse Building, 1-5 Brownlow Street, Liverpool, L69 3GL, UK
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/ Ruwanthi Kolamunnage-Dona
  • Corresponding author
  • Department of Biostatistics,University of Liverpool, Waterhouse Building, 1-5 Brownlow Street, Liverpool, L69 3GL, UK
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Published Online: 2018-01-31 | DOI: https://doi.org/10.1515/ijb-2017-0047

Abstract

Methodological development and clinical application of joint models of longitudinal and time-to-event outcomes have grown substantially over the past two decades. However, much of this research has concentrated on a single longitudinal outcome and a single event time outcome. In clinical and public health research, patients who are followed up over time may often experience multiple, recurrent, or a succession of clinical events. Models that utilise such multivariate event time outcomes are quite valuable in clinical decision-making. We comprehensively review the literature for implementation of joint models involving more than a single event time per subject. We consider the distributional and modelling assumptions, including the association structure, estimation approaches, software implementations, and clinical applications. Research into this area is proving highly promising, but to-date remains in its infancy.

Keywords: Joint models; multivariate data; longitudinal data; time-to-event data; recurrent events

1 Introduction

In clinical studies, measurements are often recorded about subjects at each follow-up visit; these response data give rise to longitudinal data. Subsequently, times to one or more clinically significant events are also recorded. The longitudinal data might be censored by one of these clinical events; for example, if the event was death or treatment failure. A growing field of research has emerged that seeks to jointly model these two outcomes —so-called joint modelling. When the outcome processes are correlated, joint modelling has been empirically demonstrated to lead to improved efficiency and reduced bias [1, 2, 3], improved prediction [4], and be applicable to outcome surrogacy [5]. The literature is extensive, with comprehensive reviews given by Hogan and Laird [6], Tsiatis and Davidian [7], Diggle et al. [8], Sousa [9], Proust-Lima et al. [10], and Gould et al. [11].

The classical joint model, from which most research has spawned, involves a single continuous longitudinal outcome and a single right-censored event time. Notwithstanding this simplicity, the joint modelling methodology has been recently extended to generalize both submodels. For the longitudinal submodel, developments include the incorporation of multiple outcomes [12], binary [13], count [14], and ordinal [15] outcomes, and extensions of the classical error and random effects distribution assumptions [16]. For the time-to-event submodel, extensions have involved the modelling of interval- [17] and left-censored [18] data, discrete event times [19], competing risks [20], parametric models [21], spline models [22], and subject- and institutional-level frailty effects [23]. Commensurate with this methodological research, there has been an increase in use of joint models in a wide –range of clinical settings [23, 24, 25, 26] and development of several mainstream statistical software packages [27, 28, 29, 30, 31, 32, 33, 34].

Due to current trends towards personalized medicine, models that utilise all available information more efficiently are of considerable value. In health research, patients may often experience multiple, recurrent, or a succession of clinical events, thus potentially admitting more than one event time. In this article, we comprehensively review the methodological literature for joint models involving multivariate event time data. Although the primary focus is on the ubiquitous shared random effects models, we also describe the growing framework of joint latent class models. Our review encapsulates multiple events, recurrent events (either in the presence of a terminal event, or not) and succession of events data. Although competing risks data can also be considered as multivariate time-to-event data, we do not review these models here as each subject still only admits a single event time. Furthermore, competing risks joint models have been extensively reviewed elsewhere in the joint model literature [35].

2 Longitudinal data submodels

Let Yik(tijk) denote the j-th observed value of the k-th longitudinal outcome for subject i, measured at time tijk, for i=1,,N,k=1,,K, and j=1,,nik. In some cases, only a single longitudinal outcome (i. e. K=1) is considered, which greatly simplifies the model. We will consider both univariate (K=1) and multivariate (K>1) longitudinal data in this review, depending on the methodology presented in each article, but exclusively reserve the subscript k to denote multivariate cases.

In the framework of joint models involving more than one event time, the corresponding longitudinal measurements have predominantly been continuous. However, some models have considered binary and count data (Table 1). As noted earlier, some models have also considered multiple longitudinal outcomes. For a full review of joint models involving multivariate longitudinal outcomes, see Hickey et al. [12]. Król et al. [36] also considered left-censored longitudinal measurements, which is pertinent to biomarker measurements that involved minimum detection thresholds. There are a plethora of modelling approaches for multivariate longitudinal data [37]. In most cases, a generalized linear mixed model (GLMM) [38] is specified. Namely,

hk{E[Yik(tijk)]}=μik(tijk),(1)

Table 1:

Summary of longitudinal submodels.

where hk() denotes a known one-to-one link function for the k-th outcome, E is the expectation operator, and μik() is the linear predictor:

μik(tijk)=Xik(1)(tijk)βk(1)+W1i(k)(tijk),(2)

where

E{W1i(k)(tijk)}=Zik(tijk)bik,(3)

and Xik(1)(tijk) and Zik(tijk) are vectors of (possibly time-varying) covariates for subject iassociated with fixed and random effects respectively, which can vary by outcome, βk(1) is a vector of fixed effects parameters for the k-th outcome, and bik is a vector of subject-specific random effects for the k-th outcome. We denote the stacked vector of subject-specific random effects for all K outcomes by bi=(bi1,bi2,,biK). Some authors have considered including spline terms in Xik(1)(tijk) to capture complex functional forms between the outcome and measurement time [25, 39]. On the other hand, Dantan et al. [40] specified a segmented GLMM with a random change-point, which was intrinsically linked to the time-to-event submodel through one of the transition hazard functions. Random change-points were shown to be particularly useful for capturing changes in the longitudinal trajectory of the outcome following a clinical (pre-)diagnosis.

Generally, for continuous longitudinal outcomes, independent and identically distributed normal errors are assumed. However, extensions to robust skew-normal distributed errors have also been proposed [39]. Subject-specific random effects are generally modelled as being multivariate normally distributed, reducing to a normal distribution in the case of a random-intercepts only model. Different modelling approaches have also been considered. Notably, Huang et al. [41] adopted discrete independent probability distributions. Njagi et al. [14] considered over-dispersed data, and proposed conjugate Beta and Gamma random effects for binary and count outcomes respectively. Several authors who considered multivariate longitudinal outcomes have proposed capturing the cross-sectional association between repeated measures through a correlated errors structure rather than a correlated random effects structure, i. e. Yik(tij)=μik(tij)+εijk, with εij.NK(0,Σ) and bikNvk(0,Ψk) [39, 42, 43, 44]. This allows for separate estimation of correlation between repeated measures and between different longitudinal outcomes.

In some cases, a semiparametric paradigm has been adopted. Within the Bayesian framework, Tang et al. [44] and Tang and Tang [39] assumed a Dirichlet process prior for the random effects, removing the need to assume a fixed parametric form, which is therefore robust to potential misspecification. Li et al. [45] suggested a time-dependent vector of random effects, which are independently and identically distributed according to an unknown multivariate distribution. The longitudinal submodels are also specified as marginal proportional rates models; namely, as eq. (1) with hk(.) given by the exponential link function, and linear link functions are also suggested [46]; the time-dependent fixed effects are absorbed into an unspecified smooth baseline function.

Following Henderson et al. [47], additional autocorrelation structure can be incorporated into the model by augmenting eq. (3) to include a zero-mean stationary Gaussian process term. However, such models are met with a substantially increased computational burden, so it is not unexpected that very few methodological articles have considered this extension [48, 49]. Zhang et al. [49], as well as considering correlation for W1i(k)(t) in eq. (3), also allowed for correlation of errors within an outcome over time by letting εik=(εi1k,,εink) have zero-mean multivariate normal distribution with a u-lag correlation function given by

ρ1k(α1k,u)=exp{α1k|u|δ},0<δ2.

A summary of the longitudinal data submodels used in joint models involving multivariate time-to-event data is given in Table 1.

3 Time-to-event data submodels

Let Tig denote the g-th event time for the i-th subject (i=1,,n). Also, let Ci be a censoring time for the subject such that we actually observe Tig=min(Tig,Ci). Typically, continuous event times are observed. Two exceptions were Huang et al. [41], who considered discrete event times, and Rouanet et al. [50] who allowed one of the semi-competing event times to be interval-censored. For each subject i, let the vector Xi(2)(t), which may be time-varying, denote the observed covariate data, and βg(2) denote the coefficient parameters associated with these covariates for the g-th event time. Similarly, for models involving a third submodel (e. g. a joint model of longitudinal data, recurrent events, and a terminal event), we will use the notation Xi(3)(t) and β(3), as appropriate. However, in practice, there will be an overlap between baseline measurements in Xi(1)(t),Xi(2)(t), and Xi(3)(t). Specification of the time-to-event model depends on the type of multivariate event time data and the association structure that gives rise to the joint model. These are described below and succinctly summarized in Table 2 and Table 3. We will denote the association parameters by γg, and any extra random effects terms by θi.

Table 2:

Summary of time-to-event submodels.

Table 3:

Summary of association structure, estimation method, and software implementation.

3.1 Multiple events

Multiple (unordered) events occur when more than one event is observed, and interest lies with all of them. A joint model can be specified to capture the association between a longitudinal process and multiple failure times; for example, the time to cancer relapse in two separate organs.

Chi and Ibrahim [42] derived a novel yet complex bivariate survival model from first principles of latent precursor events modelled by a Poisson process. The model accommodates both zero and non-zero cure fractions, and the survival distribution is given by

S(ti1,ti2|θi)=exp{θi[0ti1λi1(u)F1(ti1u)du+0ti2λi2(u)F2(ti2u)du]},

where θi is a subject-specific frailty term that follows a positive stable law distribution indexed by the parameter ρ, which accounts for the correlation between the pair of event times, and F1(t) and F2(t) are distribution functions for the latent precursors, and later specified as exponential distributions. A current values parameterization was assumed to link the longitudinal and time-to-event submodels through

λig(t)=exp{k=1Kγgkμik(t)+Xi(2)βg(2)}.

It was noted that both the conditional and marginal survival function satisfies the proportional hazards property so long as the baseline covariates are modelled as per above, and Xi(2) is independent of time.

Zhu et al. [43], Tang et al. [44], and Tang and Tang [39] used the more ubiquitous piecewise constant proportional hazards model for the baseline hazard function, with knots placed at times vgq {q=1,Q} for the g-th time-to-event outcome, such that 0=vg0<vg1<<vgQ, with vgQ being greater than max(T1g,,Tng); namely

λ0g(t)=q=1QξqgI(vg,q1<tvgq),

where I() denotes the indicator function, and ξqg denotes the value of the event-specific hazard function in the interval (vg,q1,vgq] for event g. The separate event time and longitudinal submodels are subsequently linked through a current values parameterisation:

λig(t)=λ0g(t)exp{k=1Kγgkμik(t)+Xi(2)βg(2)}.

Huang et al. [41] adopted a discrete time hazard model of the form

log(fijgSijg)=Xi(2)(tj)βg(2)+γg(1)ηij+γg(2)θi+γg(3)ηijxi(3),

where fijg=P[Tig=j], Sgij=1j=1jfijg for discrete times tj (j=1,,J), and {γg(1),γg(2),γg(3)} are a set of association parameters. The first discrete random effect, ηij, links the longitudinal submodel to the event process by a random effects parameterisation, which includes an interaction with one of the baseline covariates, xi(3). The second discrete random effect, θi, captures additional association between the multivariate event times, beyond what is predicted by ηij. An additional discrete multivariate distributed random effect was included in the multivariate longitudinal outcome submodel only.

3.2 Recurrent events

Recurrent (ordered) events occur when the same non-terminal event can be observed multiple times over a follow-up period. Henderson et al. [47] first presented a joint model compatible with recurrent events data, but this was ultimately simplified to the case of a single event time (i. e. a time to a single terminal event).

3.2.1 Without a terminal event

The simplest situation is when the recurrent events process is observed without a terminating process. For example, an epileptic patient can undergo multiple seizures in a day, and targeted treatments for epilepsy may be dependent on biomarker values [51]. A joint model of the recurrent events process and longitudinal outcomes data can capture this dependence.

Han et al. [51] adopted the general recurrent event model of Peña and Hollander [52] within a latent class framework, similar to that of Lin et al. [53], with the intensity function defined according to

ri(t)=θir0r(Ei(t))ρ(Ni(t),ar)ψ(Xi(2)(t)β(2)),

where θi is a mean-one Gamma distributed frailty term, r0r(t) denotes the latent class-specific baseline intensity function (with r=1,,R), Ei(t) is the “effective age” of subject i at time t, Ni(t) is the effective number of accumulated events just prior to time t, ρ(,ar) is an event accumulation function parameterized by ar, and ψ(Xi(2)(t)β(2)) is a function of the covariate linear predictor term, for example ψ(x)=exp(x), as in the aforementioned models. The “effective age” is a predictable process that reflects the effect of interventions after each failure. In the simplest case, Ei(t)=t, corresponding to a “minimal repair”. At the other extreme, the “effective age” may be reset to zero. The effective number of accumulated events is zero if a successful intervention is applied just prior to time t, else it equals the cumulative number of failures. The function ρ(,ar) captures the effect of recurrent events on the subject, which might be non-linear; for example, ρ(n,ar)=arn. The model specification is complete once a parametric distribution for r0r(t) is specified, which can be generalized to multiple families. The association between the longitudinal and event time processes is captured entirely through the latent class, with the class membership probabilities modelled according to a multinomial distribution. Although latent class models are distinct from shared random effects models, they can be considered as semiparametric analogues.

Njagi et al. [14] considered the Weibull-gamma-normal model for recurrent events. In short, this is a Weibull regression model conditional on independent random effects biN(0,D), as per the longitudinal submodel, and θigΓ(a,b), a frailty term such that the intensity function can be written as

ri(tig)=λgρgtigρg1θigexp{Ligλgtigρgθigexp{Lig}},

where Lig=Xig(2)β(2)+γigbi, and γig is a vector of scale factors. The association between the event time and longitudinal submodel is captured through the shared random effects bi, and the correlation between the recurrent events is captured by the θig. It was noted by the authors that this model encompasses shared and correlated random effects parameterisations. In the example, the authors impose further conditions; namely, ρgρ, γigγ, and θigθiΓ(a,a1) for identifiability purposes. Efendi et al. [54] also adopted a version of this model.

Shen et al. [48] proposed modelling the recurrent events as per the model formulation in Henderson et al. [47], namely through the intensity function

ri(t)=r0(t)exp{Xi(2)(t)β(2)+W2i(t)},

where r0(t) is a baseline intensity function at time t, and W2i(t) is a zero-mean latent process term. In general, W2i(t)=Zi(2)(t)bi+V2i(t), where V2i(t) is a stationary Gaussian process. The model was simplified by specifying W2i(t)=γ1bi+γ2V1i(t), assuming μi(t)=Xi(1)(t)β(1)+bi+V1i(t) for the longitudinal submodel, with V1i(t) a second stationary Gaussian process. However, the model was ultimately reframed as a conditional rates function, namely E[ri(t)|Yi], in order to exploit and extend an estimating equations methodology approach.

Zhang et al. [49] proposed a recurrent events model with two non-absorbing states, each with separate intensity functions. Essentially, this model is a special case of the multi-state model (discussed below), known as the illness-recovery model. For states g=1,2, the intensity functions were defined as

ri(t)=r0gexp{Xi(2)(t)βg(2)+W2ig(t)},

where the baseline intensity is constant, r0g, and W2ig(t)=γ0gθi+γgWi1(t) a zero-mean Gaussian process with u-lag correlation function

ρ2(α2,u)=exp{α2|u|δ},0<δ2,

with θi a normally distributed subject-specific random effect, and Wi1(t)Wi1(k)(t) for all k.

Li [55] proposed a joint model that assumed the same intensity model as per Liu et al. [56] (with γ1=0; described below). However, the repeated binary measure was modelled using a discrete-time Markov model. A joint model was formed by factorizing the likelihood into a selection model [9], which lies outside the scope of this review.

3.2.2 With a terminal event

A natural extension to the joint model of longitudinal outcome data and a recurrent events process is to consider the situation of a terminating event process; for example, time to death. In this scenario, a third type of submodel is required to capture this additional event time, which may also be associated with the longitudinal outcomes and the recurrent events process.

Liu and Huang [57] and Liu et al. [56] considered a recurrent events submodel with a separate terminal event submodel. A random effects parameterization was used in both the recurrent events intensity function, ri(t), and the terminal event hazard function, λi(t):

ri(t)=r0(t)exp{Xi(2)(t)β(2)+γ1bi0+θi},

λi(t)=λ0(t)exp{Xi(3)(t)β(3)+γ2bi0+γ3θi}.

The standard model assumption of piecewise constant baseline hazards for r0(t) and λ0(t) was assumed. In addition, the terminal event submodel has a random effect parameterization linking it to the recurrent events submodel, where random effect term, θi, captures the correlation between recurrent events independent of bi. Rizopoulos [38] described a similar model, but only briefly described the estimation procedure, and furthermore a clinical application was not provided to illustrate the model. Król et al. [36] also adopted this model, with some slight modifications. Firstly, the baseline intensity and hazard functions were approximated by cubic M-splines on Q-knots; namely

r0(t)=q=1Q+2ξrqMq(t)andλ0(t)=q=1Q+2ξλqMq(t),

where {ξrq;q=1,,Q+2} and {ξλq;q=1,,Q+2} are the spline coefficients for the baseline intensity and hazard functions, respectively, corresponding to M-spline basis functions, Mq(t). Secondly, the association terms with the event time submodels and the longitudinal submodel were specified more flexibly as γ1fr(bi,β(1),Zi(t),Xi(1)(t)) and γ2fλ(bi,β(1),Zi(t),Xi(1)(t)). For example, fr() and fλ() might admit the current values or random effects parameterization.

Kim et al. [58] also proposed a joint model for a longitudinal outcome and a recurrent events process with a terminal event process. The recurrent events process, modelled using a broad class of transformation models, was linked by extra random effect terms θi, that are correlated with bi,

ri(t)=ddtFR(0tr0(s)exp{Xi(2)(s)β(2)+Zi(2)(s)θi}ds),

with ηi=(bi,θi) jointly distributed, and FR() a specified transformation function. The terminal event submodel—again modelled using a transformation model—was associated with the longitudinal and recurrent events submodels through a random effects parameterization with interaction with (possibly time-varying) subject-specific covariates:

λi(t)=ddtFT(0tλ0(s)exp{Xi(3)(s)β(3)+Zi(3)(s)(γηi)}ds),

with FT() a separate specified transformation function. The authors explicitly used the logarithmic and Box-Cox transformation models for analysis in their data application. The baseline functions r0(t) and λ0(t) were modelled semiparametrically, with mass at each unique observed event time.

3.2.3 As a device for informative observation times

Joint models are usually based on the assumption of non-informative observation times for the repeated measurement process. This is generally reasonable for randomized control trials, but perhaps not so for observational data studies, where sicker patients (possibly indicated through their longitudinal measurement data) present more frequently to their physician, and whom are more likely to experience an event. Several models have been proposed to account for this potentially informative observational times protocol, which fall under the umbrella of joint models of longitudinal data and recurrent events, either with or without a separate terminal event process. In fact, the model by Liu et al. [56] was motivated by this situation, but the subject-specific shared random effects model is widely applicable to other data. This emerging field of joint modelling has its own substantive and rapidly growing literature, but clearly warrants a discussion here. In the interests of brevity, we do not review the entire literature on this particular joint model, and instead illustrate the ideas through the model proposed by Li et al. [45], which is representative of the model specification and estimation methodology in the literature. Readers should consult Li et al. [59], Han et al. [60], and references therein for more details on this model framework.

Working within a semiparametric framework, a flexible proportional rates marginal model for the observation (recurrent events) process was specified by Li et al. [45]; namely

E[dNi(t)|Xi(3),bi(t)]=exp{Xi(3)β(3)+bi3(t)}dr0(t),

where dr0(t) is an unknown baseline rate function, and bi(t)=(bi1(t),bi2(t),bi3(t)) is a vector of possibly correlated subject-specific time-dependent random effects with 3 components corresponding to the longitudinal measurements, terminal event and recurrent events, respectively. The terminal event was modelled as a semiparametric additive hazards model [45], namely,

λi(t)=λ0(t)+Xi(2)β(2)+bi2(t),

with the baseline hazard λ0(t) left unspecified; however, parametric and semiparametric proportional hazards regression models could also be integrated into this framework [46, 61]. Association between the submodels is induced through the joint distribution of bi(t).

3.2.4 Multiple recurrent events

Musoro et al. [25] were motivated to unify both multiple and recurrent event types (Sections 3.1 and 3.2) into a single joint model. For G multiple event outcomes, which can be recurrent, they specified an intensity model

λig(t)=λ0g(t)exp{k=1Kγgkμik(t)+Xi(2)βg(2)+θig+ψi},

where θig and ψi are zero-mean independent Gaussian random effect terms that account for within and between event types, respectively. As above, λ0g(t) was modelled semiparametrically.

3.3 Succession of events

A succession of events occurs when non-fatal events can precede an absorbing state event, e. g. death. The intermediate events provide information on the disease progression, and can be viewed as transitions from one state to another. Multistate models provide a framework for analysing this data [62]. Longitudinal measurements that are collected over time may have different associations with progression between separate health states. We also note that multistate models can also be viewed as an extension of the competing risks model framework, where interest continues after the first event. Joint models of longitudinal data and standard competing risks data are described elsewhere [35].

Multistate models have also been applied in what is essentially the univariate event time joint modelling framework. For example, Deslandes and Chevret [63] discretized the longitudinal outcome space to form states that were combined with the event. However, clinical events of interest—disease progression or death—were combined into a single composite event. Hu et al. [64] also considered a multistate model where the longitudinal outcome was discretized according to quartiles to form transition states, augmented with additional states defined by competing risks data. Neither of these two articles considered an actual succession of event times, and therefore are not discussed further. Le Cessie et al. [65] adopted a simple model where hazard functions for disease state transitions were estimated using separate Cox proportional hazards regression models. However, the joint model was effectively constructed through a type of pattern mixture model, in which the conditional responses per disease state were estimated using a generalized estimating equations framework, and the disease state probabilities were combined to estimate the marginal mean response over time. Pattern mixture models (and similarly, selection models) have their own dedicated literature in the model-based literature [9].

Ferrer et al. [66] proposed a Markovian multi-state transition submodel with proportional hazards, such that the transition intensity at time t from state g to h is

λigh(t)=λ0gh(t)exp{Xghi(2)βgh(2)+γghfgh(bi,β(1),Zi(t),Xi(1)(t))},(4)

where the baseline intensity function λ0gh(t) can be specified as a Weibull, piecewise constant, or B-splines function, and γgh are transition-specific parameters corresponding to fgh()—a flexible association function that links the multistate submodel to the longitudinal data submodel by any function of the random effects. Special cases include the current values parameterization, the random slopes parameterization, and a linear combination of both aforementioned parameterizations.

Dantan et al. [40] proposed a multi-state model with transition between states specified as per eq. (4), subject to the association structures f01()=0, f12() a random effects parameterization, and fg3() a current values parameterization, for g=0,1,2, and other transitions were discounted. In addition, the baseline hazards were defined by Weibull distributions for the non-absorbing transitions, and a piecewise constant function for all transitions to the absorbing (death) state. Dantan and colleagues also extended the model to incorporate left-truncation to account for subjects already in the disease state entering the study late.

As noted earlier, competing risks data can be viewed as a special case of multistate models. In the context of multiple event times data, semi-competing risks model is of most interest. In this situation, a terminal event censors a non-terminal event, but not vice versa; hence, it is possible to observe more than one event time. Rouanet et al. [50] proposed two joint models for this data within a latent class framework. The first was a Markovian multi-state (or illness-death) model, as per above, with

λghri(t)=λghr0(t)exp{Xghi(2)βghr(2)},

where λghr0(t) is a baseline intensity function for the transition from states g to h in latent class r (modelled as either a Weibull function or using M-splines), and βghr(2) are class and transition-specific parameters corresponding to baseline covariates Xghi(2). The second was a semi-Markovian model, where one specific transition (from illness to death) depends on the time spent in the illness state, i. e. λ12ri(tTi1), as opposed to just the time elapsed. As per other latent class models, the association between the submodels is captured entirely through the latent classes, with class membership modelled separately.

4 Model estimation

Several different estimation approaches have been utilized to fit the models described above (Table 3). Loosely, these methods can be separated as either likelihood maximisation or Bayesian model fitting.

Extending the original joint model developments of Wulfsohn and Tsiatis [67], the expectation-maximization (EM) algorithm has been used in some cases. In the case of Han et al. [51], the latent class membership, longitudinal data submodel random effects, and the time-to-event submodel frailty terms were treated as missing data. In the case of Kim et al. [58], only the random effects were treated as missing data, and recursive formulae used to reduce the number of model parameters required for estimation. Król et al. [36] used penalized maximum likelihood estimation using the Marquardt algorithm, with the penalization performed to obtain smooth estimates of the baseline hazard and intensity functions. Rouanet et al. [50] also utilized the Marquardt algorithm, with the number of latent classes selected according to the Bayesian Information Criterion. Dantan et al. [40] reported using a Newton-Raphson-like algorithm. Huang et al. [41] used automatic differentiation—a numerical technique for simultaneously evaluating a function and its derivatives—with a Newton-Raphson algorithm, which was purportedly faster than the EM algorithm. Njagi et al. [14] and Efendi et al. [54] used a partial marginalisation approach [68] whereby the conjugate random effects are analytically integrated out, and the normal random effects are numerically integrated using standard software. Efendi et al. [54] then exploited the ideas of Heagerty and Zeger [69] to establish marginal effects. Liu et al. [56] and Liu and Huang [57] reported using numerical likelihood maximisation via standard software. Standard errors of all these aforementioned model fits can be estimated from the inverse of the observed information matrix; however, Han et al. [51] reported using the bootstrap method.

Zhang et al. [49] proposed a two-stage estimation strategy. In the first stage, the covariance parameters were estimated from the repeated measures marginal likelihood function, with the mean function estimated by a weighted moving average. In the second stage, the expected likelihood function for the time-to-event data were maximized by an EM algorithm, with Gibbs sampling implemented for the high-dimension numerical integration, and a Newton-Raphson step used for the M-step. Shen et al. [48] developed a two-stage conditional estimating equations approach for model fitting, followed by a bootstrap approach for standard error estimating. As a precursory step, the authors reframed the time-to-event submodel from an intensity function to a conditional rate function. For models that accounted for informative observation times, generalized estimating equations in a semiparametric framework was the standard approach, which yielded consistent estimators [45, 46, 61]. In these cases, theoretical results have been derived on the asymptotic normality, which is subsequently used to make inference on the estimated parameters.

Bayesian estimation of standard univariate joint models has seen increased attention over recent years [28, 30], especially as it is a natural tool for dynamic prediction and model averaging [4]. Moreover, there are multiple disadvantages to the ubiquitous frequentist estimation approach, including but not limited to, computational challenges—something one would expect to be particularly burdensome in a multivariate framework, the dependence on asymptotic approximations, and the complexity of model assessment and comparison. In joint models involving multivariate longitudinal data, Liu and Li [70] compared the performance of Bayesian approaches to maximum likelihood approaches under different strengths of association, and demonstrated superiority of the Bayesian methods with respect to bias, root-mean square error, and coverage. Of the joint models involving multivariate event time data that were estimated using Bayesian statistics [25, 39, 42, 43, 44], Markov chain Monte Carlo (MCMC) methods were employed in all cases with default non-informative prior distributions chosen for the parameters. As noted earlier, Tang et al. [44] and Tang and Tang [39] also assumed a Dirichlet process prior for the random effects, removing the need to assume a fixed parametric form, which is therefore robust to potential misspecification. Tang and Tang [39] explored the sensitivity of results to prior distribution inputs, showing that good prior knowledge led to marginally improved estimation. The Gibbs sampling algorithm was used in all cases, with non-standard conditionals sampled using adaptive rejection or Metropolis-Hasting algorithms. Chi and Ibrahim [42] specifically noted that hierarchical centring [71], as well as some parameter transformations were used to facilitate convergence of the MCMC algorithms. The posterior conditional distributions for each parameter were derived analytically by all authors, except Musoro et al. [25], who exploited the automation provided by the OpenBUGS software. In all cases, assessment of convergence was made using general diagnostic methods; for example, examination of trace plots, autocorrelation plots, and the Gelman-Rubin statistics [72].

5 Software

The ability to fit the models discussed is severely limited by the availability of software packages or modifiable code. Several authors have made code available either in an appendix, an online supplement, or via an online code repository system (Table 3). However, many authors do not report what software was used, or make said code available. Only one article released their code in the form of a software package, namely Król et al. [36], which fits a joint model for a single longitudinal outcome, a recurrent events process, and a single terminal event, and which is available through the trivPenal() function in the R package frailtypack [73].

6 Clinical applications

Development of novel methodology of joint models of longitudinal data and multivariate event times data have predominantly been motivated by real-world clinical datasets. Here, we summarize the applications that have led to the models discussed in this review.

6.1 Multiple events

Chi and Ibrahim [42] were interested in assessing whether four different quality of life measures (appetite, mood, coping, and physical wellbeing) were prognostic and predictive of breast cancer progression in a drug randomized controlled trial (RCT). The study monitored patients concerning two different failure times: death and cancer recurrence. A joint model was constructed to model these 4 longitudinal outcomes and 2 event time outcomes. Tang et al. [44], Tang and Tang [39], and Zhu et al. [43] each proposed multiple event joint models as per above, motivated by the same objectives and breast cancer dataset described above, but with novel model innovations including semiparametric Bayesian random effects modelling, robust errors, different association structures, and event-time submodels. Musoro et al. [25] considered a case of multiple recurrent events, where each patient could become repeatedly infected with one of 9 different infections (including upper respiratory, fungal, and parasitic infections) following kidney transplantation surgery. The objective of the study was to evaluate the effect of 4 repeatedly measured immune system biomarkers (CD4 + T cells, CD8 + T cells, natural killer cells, and B cells) on the risk of each infection type in a single joint model of multiple recurrent events and multivariate longitudinal data. This particular clinical application also falls under the umbrella of multiple events and recurrent events (6.2). Huang et al. [41] analyzed data from a complex prevention trial, with an interest on whether different interventions were associated with times to initiation of alcohol use and tobacco use. It was hypothesized that a psychiatric distress latent variable, which is reflected in multiple repeatedly measured mental health items, affects substance initiation; hence, a joint model was constructed.

6.2 Recurrent events

Njagi et al. [14] and Efendi et al. [54] were interested in jointly modelling the recurrent time to re-hospitalization and a repeated measure of heart rate from the same dataset of patients with chronic heart failure who were discharged from hospital. Efendi et al. [54] modelled heart rate as a continuous outcome, whereas Njagi et al. [14] modelled it as a count response based on the number of times the heart rate was classified as “abnormal”. Han et al. [51] considered repeated times to seizure in an epilepsy cohort study. Serial blood measures were also recorded for 3 blood plasma lipids; however, based on clinical knowledge, a single longitudinal outcome was constructed from 2 of the biomarkers by taking a ratio at each measurement time; —the lecithin–cholesterol ratio, with the third biomarker discounted, as this ratio was believed to be elevated during periods of the day when seizures occurred. Shen et al. [48] jointly modelled time to cocaine-use relapse, a recurrent events outcome, and a repeated measure of psychiatric symptoms used to assess stress and cocaine craving levels in patients enrolled in a clinical intervention study. The primary objective was to understand whether the randomly assigned intervention (contingency management or not) treatment affects either stress or drug relapse after adjustment for demographic variables. Zhang et al. [49] were interested in investigating the health effects of air quality on respiratory symptoms. Four measures of air quality were recorded daily, as were three symptoms recorded per subject (runny nose, cough, sore throat/general sickness). Each day, subjects could be in either a symptomatic or asymptomatic state, which they transition between (i. e. an illness-recovery model). For each symptom in turn, a recurrent events joint model with the 4 longitudinal measures was fitted.

Liu and Huang [57] hypothesized that repeatedly high CD4 cell counts in HIV positive patients are associated with low risk of opportunistic disease, which is a potentially recurring event. They further hypothesized that a higher CD4 cell count and lower rate of opportunistic disease are associated with better survival, which is a terminal event. The interplay between these three processes might, however, be motivated by different application-specific reasons. Similarly, Kim et al. [58] modelled the recurrent time to a coronary heart disease event and time to death with repeated measurements on systolic blood pressure in patients previously diagnosed with hypertension. Within the context of a clinical trial for metastatic colorectal cancer, Król et  [36] were interested in the predictive ability of tumour size (a possibly left-censored repeated measurement), and the recurrent appearance of new lesions and the terminal outcome death.

Recurrent events are a particularly attractive modelling component for observational studies. Namely, when the follow-up protocol is not pre-specified or random, one might expect that the sickest subjects are those both more likely to experience the event of interest, as well as visit their physician more regularly where they will have biomarker measurements recorded. A recurrent events process can therefore be used to account for the correlation between observation times and repeated measures process. This was the case in Liu et al. [56], who considered recurrent times to hospital visits for diagnosis or treatment of heart failure alongside time to death, with repeated measurements on medical costs. Data from a skin cancer clinical trial was analyzed in a similar fashion by Li et al. [45], with the number of observed tumours at each observation time modelled as the longitudinal outcome.

6.3 Succession of events

Ferrer et al. [66] analyzed data from a multi-centre clinical trial treated with external beam radiotherapy for localized prostate cancer. Prostate-specific antigen (PSA) was repeatedly measured during follow-up. In addition, times of transitions between different disease states were recorded: radiotherapy cessation, local recurrence, distant recurrence, initiation of hormonal therapy, and death. The association between PSA and clinical relapse is well-known from univariate joint models; however, it is also of value to clinicians and patients to be able to distinguish between the different phases of disease progression as PSA may be differently correlated at each stage.

Rouanet et al. [50] analyzed a cohort study of patients to model pre-dementia cognitive decline, as measured by a psychometric test score to assess verbal fluency, in the presence of semi-competing risks of dementia onset and death. That is, the risk of dementia is null after death has occurred, but death can occur after dementia. As the diagnosis of dementia cannot be precisely recorded due to intermittent assessment, it is interval-censored, thus known to have occurred between two follow-up appointments. It is important to account for that this interval, as it is known as the risk of dementia may be underestimated otherwise. Using data from the same cohort study, Dantan et al. [40] also analyzed the dependency of cognitive ageing—repeatedly measured using a psychometric test used to assess cognitive ability—on the progression from healthy, pre-diagnosis, illness, and death states. A fundamental difference of the latter model compared to the former is that an interim “pre-diagnosis” state was included, which was modelled by a segmented linear mixed model with a random change point.

7 Discussion

The case for use of joint models has been made already [1, 74, 75]. Namely, when the longitudinal and event time processes are correlated they reduce the bias obtained from simpler methods, including separate models (e. g. separate LMMs, survival models, recurrent event models, and multistate models), or even the two-stage approach. There has been a myriad of extensions in the joint modelling framework over the past few years, including extensions to multivariate longitudinal data [12] and competing risks data [35]. Relatively fewer developments have been made pertaining joint models involving more than a single event time, which includes multiple events, recurrent events, and a succession of events. Yet, as shown, there are wide-ranging clinical applications for these models. In particular, motivation has stemmed from disease areas representing cancer, infection, cardiovascular disease, neurological disease, mental health, and respiratory disease. Moreover, data were derived from both randomized controlled trials and cohort studies.

The review presented here contributes to this narrowly focused but important topic in joint models by bringing together in a single place and juxtaposing the models and distributional assumptions, outcome types, estimation and software implementations alongside clinical applications. This is a research area of growing interest and clinical importance, and the extensions developed are necessary to appropriately analyze this complex data. However, we found that availability of mainstream statistical software to fit these models is severely limited, and this will ultimately pose problems, since the complexity of the models means that ad hoc programming is required. This is not unexpected as joint models are computationally difficult to fit; a problem that is exacerbated by the extension to joint models involving more than a single event time. In fact, Musoro and colleagues [25] noted that their ambitious attempt to fit a model to 4 longitudinal outcomes and 9 recurrent event outcome types was precluded by computational time; development of approaches that reduce this computational burden are therefore of paramount importance.

The extension of joint models to more than a single event time offers not only improved inference, but also opportunity for dynamic prediction. This has received growing interest in the classical joint model framework [4], but less so in the extension of multivariate event time data. Król et al. [36] developed dynamic prediction and predictive assessment tools for their recurrent events joint model. Others have also discussed prediction in the context of joint models involving multivariate event time data [14, 50, 66]. Dynamic prediction is easily encompassed in a Bayesian joint model framework. Despite this, the use of Bayesian methods for model fitting has been rather limited in the methodological developments of joint models involving multivariate event time data. Moreover, there is also limited research on the role of prior distribution selection. Research to-date has been predominantly technical, and more attention is required on the interpretability of these models in clinical applications. Moreover, the complexity of these models requires further development on diagnostics that will facilitate model selection, including the choice of association structure.

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Footnotes

    About the article

    Received: 2017-06-30

    Accepted: 2018-01-17

    Revised: 2017-12-07

    Published Online: 2018-01-31


    This research was funded by Medical Research Council (MRC) grant MR/M013227/1.


    Citation Information: The International Journal of Biostatistics, Volume 14, Issue 1, 20170047, ISSN (Online) 1557-4679, DOI: https://doi.org/10.1515/ijb-2017-0047.

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    © 2018 Hickey et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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