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 , and be applicable to outcome surrogacy . The literature is extensive, with comprehensive reviews given by Hogan and Laird , Tsiatis and Davidian , Diggle et al. , Sousa , Proust-Lima et al. , and Gould et al. .
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 , binary , count , and ordinal  outcomes, and extensions of the classical error and random effects distribution assumptions . For the time-to-event submodel, extensions have involved the modelling of interval-  and left-censored  data, discrete event times , competing risks , parametric models , spline models , and subject- and institutional-level frailty effects . 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 .
2 Longitudinal data submodels
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. . Król et al.  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 . In most cases, a generalized linear mixed model (GLMM)  is specified. Namely,
Summary of longitudinal submodels.
|Article||Ref.||Multivariate||Outcome types||Model||Error distribution||Random effects distribution|
|Huang et al. (2001)||||Yes||Binary||– Logistic regression model given the latent variable||N/A||Discrete independent probability distributions|
|– Marginal log-odds model for the longitudinal latent process|
|Chi & Ibrahim (2006)||||Yes||Continuous||LMM||MVN||MVN|
|Zhang et al. (2008)||||Yes||Continuous||LMM||MVN – stationary Gaussian process with exponential correlation||MVN – stationary Gaussian process with exponential correlation|
|Zhu et al. (2012)||||Yes||Continuous and/or discrete||GLMM||MVN for continuous outcomes||MVN|
|Tang et al. (2014)||||Yes||Continuous and/or discrete||GLMM||MVN for continuous outcomes||Unspecified distribution modelled with a Dirichlet process prior (with MVN base distribution)|
|Tang & Tang (2015)||||Yes||Continuous||LMM + P-splines||Multivariate skew-normal||Unspecified distribution modelled with a Dirichlet process prior (with MVN base distribution)|
|Multiple events + recurrent events|
|Musoro et al. (2015)||||Yes||Continuous||LMM + thin-plate splines||Normal||MVN + normal for thin-plate spline effects|
|Han et al. (2007)||||No||Continuous||LMM||Normal||MVN|
|Liu et al. (2008)||||No||Continuous||LMM||Normal||Normal|
|Liu & Huang (2009)||||No||Continuous||LMM||Normal||Normal|
|Kim et al. (2012)||||No||Continuous||LMM||Normal||MVN|
|Efendi et al. (2013)||||No||Continuous||LMM||Normal||MVN|
|Njagi et al. (2013)||||No||Continuous, binary or count||– LMM for continuous outcomes||Normal for continuous outcomes||MVN Separate Beta or Gamma random effects for binary or count outcomes, respectively|
|– Probit for binary outcomes|
|– Poisson for count outcomes|
|Król et al. (2014)||||No||Continuous||LMM||Normal||MVN|
|Li et al. (2016)||||No||Continuous||Marginal proportional means model||N/A||Multivariate – left unspecified|
|Shen et al. (2016)||||No||Continuous||LMM||Normal||MVN + stationary Gaussian process|
|Succession of events|
|Dantan et al. (2012)||||No||Continuous||Segmented LMM with random change-point||Normal||MVN|
|Ferrer et al. (2016)||||No||Continuous||LMM||Normal||MVN|
|Rouanet et al. (2016)||||Yes*||Continuous (normal and non-normal)||LMM*||Normal||MVN|
Abbreviations: LMM = linear mixed model, GLMM = generalized linear mixed model, MVN = multivariate normal, N/A = not applicable
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 . 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.  adopted discrete independent probability distributions. Njagi et al.  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.
In some cases, a semiparametric paradigm has been adopted. Within the Bayesian framework, Tang et al.  and Tang and Tang  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.  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
Following Henderson et al. , 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. , as well as considering correlation for
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
Summary of time-to-event submodels.
|Article||Ref.||Multiple events||Recurrent events||Succession of events||Model||Random effects distribution&|
|Huang et al. (2001)||||✓||X||X||Discrete-time hazard log-linear models||Discrete probability|
|Chi & Ibrahim (2006)||||✓||X||X||Novel time-to-event joint model with conditional and marginal proportional hazards structure, and capable of accommodating zero- and non-zero cure rate fractions||Positive stable law§|
|Han et al. (2007)||||X||✓||X||General recurrent events model of Peña and Hollander ||Gamma§|
|Liu et al. (2008)||||X||✓||X||Proportional hazards with piecewise constant baseline hazard and intensity functions||Normal|
|Zhang et al. (2008)||||✓||X||X||Constant baseline intensities||Normal|
|Liu & Huang (2009)||||X||✓||X||Proportional hazards with piecewise constant baseline hazard and intensity functions||Normal|
|Dantan et al. (2012)||||X||X||✓||Proportional transition intensity model with Weibull and piecewise constant baseline functions||N/A|
|Kim et al. (2012)||||X||✓||X||Transformation models||Normal|
|Zhu et al. (2012)||||✓||X||X||Proportional hazards with piecewise constant baseline hazard functions||N/A|
|Efendi et al. (2013)||||X||✓||X||Weibull-gamma-normal model||Gamma§|
|Njagi et al. (2013)||||X||✓||X||Weibull-gamma-normal model||Gamma§|
|Tang et al. (2014)||||✓||X||X||Proportional hazards with piecewise constant baseline hazard functions||N/A|
|Musoro et al. (2015)||||✓||✓||X||Proportional semiparametric intensity model||Independent normal (two random effects present for within and between event types)|
|Tang & Tang (2015)||||✓||X||X||Proportional hazards with piecewise constant baseline hazard functions||N/A|
|Ferrer et al. (2016)||||X||X||✓||A proportional hazards Markovian intensity model (with Weibull, piecewise constant, or B-spline baseline intensity function)||N/A|
|Król et al. (2016)||||X||✓||X||Proportional hazards with cubic M-spline baseline hazard and intensity functions||Normal|
|Li et al. (2016)$||||X||✓||X||Terminal event: additive hazards with unspecified baseline hazard function||Left unspecified|
|Recurrent events: marginal proportional rates model|
|Rouanet et al. (2016)||||X||X||✓#||Two models proposed:||N/A|
|1.A proportional hazards Markovian intensity model (with Weibull or M-spline baseline intensity function)|
|2. A semi-Markovian model where transition intensity to death from disease state depends on time with illness|
|Shen et al. (2016)||||X||✓||X||Proportional semiparametric intensity model, which was reframed as a conditional rate function for the purpose of estimation||N/A*|
& Random effects in the time-to-event submodels other than those shared with the longitudinal data submodel.
* In principle, separate normal frailty terms can be included, as per Henderson et al. .
§ Denotes distributions of frailties that act multiplicatively on the hazard. All other distributions correspond to random effects that act additively on the log-hazard scale.
Summary of association structure, estimation method, and software implementation.
|Article||Ref.||Association structure*||Estimation method||Software implementation & availability|
|Huang et al. (2001)||||Current value of true latent variable + interaction terms with external covariates||MLE: Newton-Raphson algorithm with automatic differentiation and iterative proportional fitting||S-Plus: AD09 module available online to implement automatic differentiation and Newton-Raphson algorithm1|
|Chi & Ibrahim (2006)||||Current value parameterization||Bayesian MCMC: Gibbs sampling algorithm (with adaptive rejection algorithm and Metropolis algorithm)||N/S|
|Han et al. (2007)||||Latent class membership||MLE: EM algorithm||N/S|
|Liu et al. (2008)||||Random effects parameterization||MLE: Gaussian quadrature tools in standard statistical packages||SAS: code provided online|
|Both recurrent and terminal time-to-event models additionally correlated through common frailty, which is independent of longitudinal process|
|Zhang et al. (2008)||||Random effects parameterization||MLE: two-stage approach with one component estimated using the EM algorithm||N/S|
|Liu & Huang (2009)||||Random effects parameterization||MLE: Gaussian quadrature tools in standard statistical packages||SAS: code provided online2|
|Dantan et al. (2012)|||
|Kim et al. (2012)||||Correlated random effects between longitudinal and recurrent events submodels, with time-dependent covariate vector interactions||MLE: EM algorithm with a recursive formula proposed to reduce the number of parameters to be maximised||R: code provided online|
|Zhu et al. (2012)||||Current value parameterization||Bayesian MCMC: Gibbs sampling algorithm (with Metropolis-Hastings algorithm)||N/S|
|Efendi et al. (2013)||||Random effects parametrization||MLE: via partial marginalization ; i. e. where the conjugate random effects are analytically integrated out, followed by numerical integration of shared normal random effects||SAS: code provided in the Appendix|
|Njagi et al. (2013)||||Random effects parameterization||MLE: via partial marginalization ; i. e. where the conjugate random effects are analytically integrated out, followed by numerical integration of shared normal random effects||SAS: code provided in the Appendix|
|Tang et al. (2014)||||Current value parameterization||Bayesian MCMC: Block Gibbs sampling algorithm (with Metropolis-Hastings algorithm)||R and Matlab: code available on request from the authors|
|Musoro et al. (2015)||||Current value parameterization||Bayesian MCMC: Gibbs sampling algorithm||OpenBUGS: code not provided|
|Tang & Tang (2015)||||Current value parameterization||Bayesian MCMC: Block Gibbs sampling algorithm (with Metropolis-Hastings algorithm)||N/S|
|Ferrer et al. (2016)||||Current value parameterization, Time-dependent slopes parameterization, both, or any other function of the random effects||MLE: hybrid algorithm that begins with an EM algorithm and switches to a quasi-Newton algorithm if the convergence is not achieved||R: code provided online and in Appendix|
|Król et al. (2016)||||Current value parameterization, Time-dependent slopes parameterization, both, or any other function of the random effects||MLE: penalized maximum likelihood estimation using the Marquardt algorithm||R: implemented in the frailtypack package (v2.8) and code provided in the Appendix|
|Li et al. (2016)||||Correlated random effects||Estimating equations||N/S|
|Rouanet et al. (2016)||||Latent class membership||MLE: Marquardt algorithm||R: code provided online|
|Shen et al. (2016)||||Random effects parameterization, with separate coefficients for the time-independent and -dependent random effects||Two-stage conditional estimating equation approach||N/S|
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  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
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
Zhu et al. , Tang et al. , and Tang and Tang  used the more ubiquitous piecewise constant proportional hazards model for the baseline hazard function, with knots placed at times
Huang et al.  adopted a discrete time hazard model of the form
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.  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 . A joint model of the recurrent events process and longitudinal outcomes data can capture this dependence.
Han et al.  adopted the general recurrent event model of Peña and Hollander  within a latent class framework, similar to that of Lin et al. , with the intensity function defined according to
Njagi et al.  considered the Weibull-gamma-normal model for recurrent events. In short, this is a Weibull regression model conditional on independent random effects
Zhang et al.  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
where the baseline intensity is constant,
Li  proposed a joint model that assumed the same intensity model as per Liu et al.  (with
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  and Liu et al.  considered a recurrent events submodel with a separate terminal event submodel. A random effects parameterization was used in both the recurrent events intensity function,
The standard model assumption of piecewise constant baseline hazards for
Kim et al.  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
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.  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. , which is representative of the model specification and estimation methodology in the literature. Readers should consult Li et al. , Han et al. , 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. ; namely
with the baseline hazard
3.2.4 Multiple recurrent events
Musoro et al.  were motivated to unify both multiple and recurrent event types (Sections 3.1 and 3.2) into a single joint model. For
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 . 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 .
Multistate models have also been applied in what is essentially the univariate event time joint modelling framework. For example, Deslandes and Chevret  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.  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.  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 .
Ferrer et al.  proposed a Markovian multi-state transition submodel with proportional hazards, such that the transition intensity at time
where the baseline intensity function
Dantan et al.  proposed a multi-state model with transition between states specified as per eq. (4), subject to the association structures
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.  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
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 , the expectation-maximization (EM) algorithm has been used in some cases. In the case of Han et al. , 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. , 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.  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.  also utilized the Marquardt algorithm, with the number of latent classes selected according to the Bayesian Information Criterion. Dantan et al.  reported using a Newton-Raphson-like algorithm. Huang et al.  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.  and Efendi et al.  used a partial marginalisation approach  whereby the conjugate random effects are analytically integrated out, and the normal random effects are numerically integrated using standard software. Efendi et al.  then exploited the ideas of Heagerty and Zeger  to establish marginal effects. Liu et al.  and Liu and Huang  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.  reported using the bootstrap method.
Zhang et al.  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.  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 . 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  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.  and Tang and Tang  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  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  specifically noted that hierarchical centring , 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. , 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 .
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. , 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 .
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  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. , Tang and Tang , and Zhu et al.  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.  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.  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.  and Efendi et al.  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.  modelled heart rate as a continuous outcome, whereas Njagi et al.  modelled it as a count response based on the number of times the heart rate was classified as “abnormal”. Han et al.  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.  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.  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  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.  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  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. , 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. , with the number of observed tumours at each observation time modelled as the longitudinal outcome.
6.3 Succession of events
Ferrer et al.  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.  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.  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.
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  and competing risks data . 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  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 , but less so in the extension of multivariate event time data. Król et al.  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.
Ibrahim JG, Chu H, Chen LM. Basic concepts and methods for joint models of longitudinal and survival data. J Clin Oncol. 2010;28:2796–801.
Chen LM, Ibrahim JG, Chu H. Sample size and power determination in joint modeling of longitudinal and survival data. Stat Med. 2011;30:2295–309.
Hogan JW, Laird NM. Increasing efficiency from censored survival data by using random effects to model longitudinal covariates. Stat Methods Med Res. 1998;7:28–48.
Rizopoulos D, Hatfield LA, Carlin BP, Takkenberg JJM. Combining dynamic predictions from joint models for longitudinal and time-to-event data using Bayesian model averaging. J Am Stat Assoc. 2014;109:1385–97.
Tsiatis AA, DeGruttola V, Wulfsohn MS. Modeling the relationship of survival to longitudinal data measured with error - applications to survival and CD4 counts in patients with AIDS. J Am Stat Assoc. 1995;90:27–37.
Hogan JW, Laird NM. Model-based approaches to analysing incomplete longitudinal and failure time data. Stat Med. 1997;16:259–72.
Tsiatis AA, Davidian M. Joint modeling of longitudinal and time-to-event data: an overview. Stat Sin. 2004;14:809–34.
Diggle PJ, Sousa I, Chetwynd AG. Joint modelling of repeated measurements and time-to-event outcomes: the fourth Armitage lecture. Stat Med. 2008;27:2981–98.
Sousa I. A review on joint modelling of longitudinal measurements and time-to-event. Revstat Stat J. 2011;9:57–81.
Proust-Lima C, Sene M, Taylor JMG, Jacqmin-Gadda H. Joint latent class models for longitudinal and time-to-event data: a review. Stat Methods Med Res. 2012;23:74–90.
Gould AL, Boye ME, Crowther MJ, Ibrahim JG, Quartey G, Micallef S, et . Joint modeling of survival and longitudinal non-survival data: current methods and issues. Report of the DIA Bayesian joint modeling working group. Stat Med. 2015;34:2181–95.
Hickey GL, Philipson P, Jorgensen A, Kolamunnage-Dona R. Joint modelling of time-to-event and multivariate longitudinal outcomes: recent developments and issues. BMC Med Res Methodol. 2016;16:1–15.
Rizopoulos D, Ghosh P. A Bayesian semiparametric multivariate joint model for multiple longitudinal outcomes and a time-to-event. Stat Med. 2011;30:1366–80.
Njagi EN, Molenberghs G, Rizopoulos D, Verbeke G, Kenward MG, Dendale P, et . A flexible joint modeling framework for longitudinal and time-to-event data with overdispersion. Stat Methods Med Res. 2013;0:1–16.
Li N, Elashoff RM, Li G, Saver J. Joint modeling of longitudinal ordinal data and competing risks survival times and analysis of the NINDS rt-PA stroke trial. Stat Med. 2010;29:546–57.
Huang X, Li G, Elashoff RM, Pan J. A general joint model for longitudinal measurements and competing risks survival data with heterogeneous random effects. Lifetime Data Anal. 2011;17:80–100.
Gueorguieva R, Rosenheck R, Lin H. Joint modelling of longitudinal outcome and interval-censored competing risk dropout in a schizophrenia clinical trial. J R Stat Soc Ser A Stat Soc. 2012;175:417–33.
Thiébaut R, Jacqmin-Gadda H, Babiker AG, Commenges D. Joint modelling of bivariate longitudinal data with informative dropout and left-censoring, with application to the evolution of CD4+ cell count and HIV RNA viral load in response to treatment of HIV infection. Stat Med. 2005;24:65–82.
- Export Citation
Thiébaut R, Jacqmin-Gadda H, Babiker AG, Commenges D. Joint modelling of bivariate longitudinal data with informative dropout and left-censoring, with application to the evolution of CD4+ cell count and HIV RNA viral load in response to treatment of HIV infection. Stat Med. 2005;24:65–82.)| false 10.1002/sim.1923 15523706
Albert PS, Shih JH. An approach for jointly modeling multivariate longitudinal measurements and discrete time-to-event data. Ann Appl Stat. 2010;4:1517–32.
Williamson PR, Kolamunnage-Dona R, Philipson P, Marson AG. Joint modelling of longitudinal and competing risks data. Stat Med. 2008;27:6426–38.
Crowther MJ, Abrams KR, Lambert PC. Flexible parametric joint modelling of longitudinal and survival data. Stat Med. 2012;31:4456–71.
Brown ER, Ibrahim JG, DeGruttola V. A flexible B-spline model for multiple longitudinal biomarkers and survival. Biometrics. 2005;61:64–73.
Luo S, Wang J. Bayesian hierarchical model for multiple repeated measures and survival data: an application to Parkinson’s disease. Stat Med. 2014;33:4279–91.
Touloumi G, Pantazis N, Babiker AG, Walker SA, Katsarou O, Karafoulidou A, et . Differences in HIV RNA levels before the initiation of antiretroviral therapy among 1864 individuals with known HIV-1 seroconversion dates. Aids. 2004;18:1697–705.
- Export Citation
Touloumi G, Pantazis N, Babiker AG, Walker SA, Katsarou O, Karafoulidou A, et . Differences in HIV RNA levels before the initiation of antiretroviral therapy among 1864 individuals with known HIV-1 seroconversion dates. Aids. 2004;18:1697–705.)| false 10.1097/01.aids.0000131395.14339.f5 15280781
Musoro JZ, Geskus RB, Zwinderman AH. A joint model for repeated events of different types and multiple longitudinal outcomes with application to a follow-up study of patients after kidney transplant. Biometrical J. 2015;57:185–200.
Ibrahim JG, M-H C, Sinha D. Bayesian methods for joint modeling of longitudinal and survival data with applications to cancer vaccine trials. Stat Sin. 2004;14:863–83.
Guo X, Carlin BP. Separate and joint modeling of longitudinal and event time data using standard computer packages. Am Stat. 2004;58:16–24.
Philipson P, Sousa I, Diggle PJ, Williamson PR, Kolamunnage-Dona R, Henderson R, Hickey GL. Package “joineR”. R Foundation for Statistical Computing. Version 1.2.2 2017.
Rizopoulos D. The R package JMbayes for fitting joint models for longitudinal and time-to-event data using MCMC. J Stat Softw. 2016;72:1–45.
Zhang D, Chen M-H, Ibrahim JG, Boye ME, Shen W. JMFit: a SAS macro for joint models of longitudinal and survival data. J Stat Softw. 2016;71.
Rizopoulos D. JM: an R package for the joint modelling of longitudinal and time-to-event data. J Stat Softw. 2010;35:1–33.
Proust-Lima C, Philipps V, Liquet B. Estimation of latent class linear mixed models: the new package lcmm. arXiv Prepr. 2015.
Hickey GL, Philipson P, Jorgensen A, Kolamunnage-Dona R. joineRML: joint modelling of multivariate longitudinal data and time-to-event outcomes. Comprehensive R Archive Network; 2017.
Hickey GL, Philipson P, Jorgensen A, Kolamunnage-Dona R. A comparison of different joint models for longitudinal and competing risks data: with application to an epilepsy drug randomised control trial. J R Stat Soc Ser A Stat Soc. DOI: .
Król A, Ferrer L, Pignon JP, Proust-Lima C, Ducreux M, Bouché O, et . Joint model for left-censored longitudinal data, recurrent events and terminal event: predictive abilities of tumor burden for cancer evolution with application to the FFCD 2000-05 trial. Biometrics. 2016;72:907–16.
- Export Citation
Król A, Ferrer L, Pignon JP, Proust-Lima C, Ducreux M, Bouché O, et . Joint model for left-censored longitudinal data, recurrent events and terminal event: predictive abilities of tumor burden for cancer evolution with application to the FFCD 2000-05 trial. Biometrics. 2016;72:907–16.)| false 26890381 10.1111/biom.12490
Verbeke G, Fieuws S, Molenberghs G, Davidian M. The analysis of multivariate longitudinal data: A review. Stat Methods Med Res. 2012;23:42–59.
Rizopoulos D. Joint models for longitudinal and time-to-event data, with applications in R. Boca Raton, FL: Chapman & Hall/CRC; 2012.
Tang AM, Tang NS. Semiparametric Bayesian inference on skew–normal joint modeling of multivariate longitudinal and survival data. Stat Med. 2015;34:824–43.
Dantan E, Joly P, Dartigues J-F, Jacqmin-Gadda H. Joint model with latent state for longitudinal and multistate data. Biostatistics. 2012;12:723–36.
Huang W, Zeger SL, Anthony JC, Garrett E. Latent variable model for joint analysis of multiple repeated measures and bivariate event times. J Am Stat Assoc. 2001;96:906–14.
Chi YY, Ibrahim JG. Joint models for multivariate longitudinal and multivariate survival data. Biometrics. 2006;62:432–45.
Zhu H, Ibrahim JG, Chi YY, Tang NS. Bayesian influence measures for joint models for longitudinal and survival data. Biometrics. 2012;68:954–64.
Tang NS, Tang AM, Pan DD. Semiparametric Bayesian joint models of multivariate longitudinal and survival data. Comput Stat Data Anal. 2014;77:113–29.
Li Y, He X, Wang H, Sun J. Regression analysis of longitudinal data with correlated censoring and observation times. Lifetime Data Anal. 2016;22:343–62.
Sun L, Song X, Zhou J, Liu L. Joint analysis of longitudinal data with informative observation times and a dependent terminal event. J Am Stat Assoc. 2012;107:688–700.
Henderson R, Diggle PJ, Dobson A. Joint modelling of longitudinal measurements and event time data. Biostatistics. 2000;1:465–80.
Shen Y, Huang H, Guan Y. A conditional estimating equation approach for recurrent event data with additional longitudinal information. Stat Med. 2016.
Zhang H, Ye Y, Diggle PJ, Shi J. Joint modeling of time series measures and recurrent events and analysis of the effects of air quality on respiratory symptoms. J Am Stat Assoc. 2008;103:48–60.
Rouanet A, Joly P, Dartigues J-F, Proust-Lima C, Jacqmin-Gadda H. Joint latent class model for longitudinal data and interval-censored semi-competing events: application to dementia. Biometrics. 2016;72(4):1123–1135.
Han J, Slate EH, Pena EA. Parametric latent class joint model for a longitudinal biomarker and recurrent events. Stat Med. 2007;26:5285–302.
Pena EA, Hollander M. Models for recurrent events in reliability and survival analysis. In: Soyer R, Mazzuchi T, Singpurwalla N, editors. Mathematical reliability: an expository perspective. Dordrecht: Kluwer Academic Publishers; 2004.
Lin H, Turnbull BW, McCulloch CE, Slate EH. Latent class models for joint analysis of longitudinal biomarker and event process data: application to longitudinal prostate-specific antigen readings and prostate cancer. J Am Stat Assoc. 2002;97:53–65.
- Export Citation
Lin H, Turnbull BW, McCulloch CE, Slate EH. Latent class models for joint analysis of longitudinal biomarker and event process data: application to longitudinal prostate-specific antigen readings and prostate cancer. J Am Stat Assoc. 2002;97:53–65.)| false 10.1198/016214502753479220
Efendi A, Molenberghs G, Njagi EN, Dendale P. A joint model for longitudinal continuous and time-to-event outcomes with direct marginal interpretation. Biometrical J. 2013;55:572–88.
Li S. Joint modeling of recurrent event processes and intermittently observed time-varying binary covariate processes. Lifetime Data Anal. Springer US. 2016;22:145–60.
Liu L, Huang X, O’Quigley J. Analysis of longitudinal data in the presence of informative observational times and a dependent terminal event, with application to medical cost data. Biometrics. 2008;64:950–58.
Liu L, Huang X. Joint analysis of correlated repeated measures and recurrent events processes in the presence of death, with application to a study on acquired immune deficiency syndrome. J R Stat Soc Ser C Appl Stat. 2009;58:65–81.
Kim S, Zeng D, Chambless L, Li Y. Joint models of longitudinal data and recurrent events with informative terminal event. Stat Biosci. 2012;4:262–81.
Li Y, He X, Wang H, Sun J. Joint analysis of longitudinal data and informative observation times with time-dependent random effects. In: Jin Z, Liu M, Luo X, editors. New developments in statistical modeling, inference and application. Switzerland: Springer International Publishing; 2016.
Han M, Song X, Sun L. Joint modeling of longitudinal data with informative observation times and dropouts. Stat Sin. 2014;24:1487–504.
Miao R, Chen X, Sun L. Analyzing longitudinal data with informative observation and terminal event times. Acta Math Appl Sin. 2016;32:1035–52.
Putter H, Fiocco M, Geskus RB. Tutorial in biostatistics: competing risks and multi-state models. Stat Med. 2007;26:2389–430.
Deslandes E, Chevret S. Assessing surrogacy from the joint modelling of multivariate longitudinal data and survival: application to clinical trial data on chronic lymphocytic leukaemia. Stat Med. 2007;26:5411–21.
Hu B, Li L, Wang X, Greene T. Nonparametric multistate representations of survival and longitudinal data with measurement error. Stat Med. 2012;31:2303–17.
Le Cessie S, De Vries EGE, Buijs C, Post WJ. Analyzing longitudinal data with patients in different disease states during follow-up and death as final state. Stat Med. 2009;28:3829–43.
Ferrer L, Rondeau V, Dignam J, Pickles T, Jacqmin-Gadda H, Proust-Lima C. Joint modelling of longitudinal and multi-state processes: application to clinical progressions in prostate cancer. Stat Med. 2016.
Wulfsohn MS, Tsiatis AA. A joint model for survival and longitudinal data measured with error. Biometrics. 1997;53:330–39.
Molenberghs G, Verbeke G, Demétrio CGB. An extended random-effects approach to modeling repeated, overdispersed count data. Lifetime Data Anal. 2007;13:513–31.
Liu F, Li Q. A Bayesian model for joint analysis of multivariate repeated measures and time to event data in crossover trials. Stat Methods Med Res. 2014;0:1–13.
Gelfand AE, Sahu SK, Carlin BP. Efficient parameterizations for normal linear mixed models.
Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian data analysis, 3rd ed. Boca Raton, FL: CRC Press; 2013.
Król A, Mauguen A, Mazroui Y, Laurent A, Michiels S and Rondeau V. Tutorial in Joint Modeling and Prediction: A Statistical Software for Correlated Longitudinal Outcomes, Recurrent Events and a Terminal Event. Journal of Statistical Software; 2017: 81(3), 1–52.
Sweeting MJ, Thompson SG. Joint modelling of longitudinal and time-to-event data with application to predicting abdominal aortic aneurysm growth and rupture. Biometrical J. 2011;53: 750–763. doi:
Wang P, Shen W, Boye ME. Joint modeling of longitudinal outcomes and survival using latent growth modeling approach in a mesothelioma trial. Heal Serv Outcomes Res Methodol. 2012;12:182–99.
Molenberghs G, Verbeke G, Demétrio CGB, Vieira AMC. A family of generalized linear models for repeated measures with normal and conjugate random effects. Stat Sci. 2011;25:325–47.