Joint modelling of longitudinal response and time-to-event data using conditional distributions: a Bayesian perspective

Abstract

Over the last 20 or more years a lot of clinical applications and methodological development in the area of joint models of longitudinal and time-to-event outcomes have come up. In these studies, patients are followed until an event, such as death, occurs. In most of the work, using subject-specific random-effects as frailty, the dependency of these two processes has been established. In this article, we propose a new joint model that consists of a linear mixed-effects model for longitudinal data and an accelerated failure time model for the time-to-event data. These two sub-models are linked via a latent random process. This model will capture the dependency of the time-to-event on the longitudinal measurements more directly. Using standard priors, a Bayesian method has been developed for estimation. All computations are implemented using OpenBUGS. Our proposed method is evaluated by a simulation study, which compares the conditional model with a joint model with local independence by way of calibration. Data on Duchenne muscular dystrophy (DMD) syndrome and a set of data in AIDS patients have been analysed.

1. Introduction

In many clinical studies, information is collected on subjects at regular time intervals. Sometimes an event time, for example, death or dropout, also becomes a point of interest. If these two responses are correlated then it may be of interest to examine the association structure between longitudinal measurements and event times. To address this issue, a class of statistical models has been developed, referred to as joint models for longitudinal and time-to-event data. This area of research has received considerable attention in recent years. A standard joint model was developed by Faucett and Thomas [16] and Wulfsohn and Tsiatis [53]. Prior to their work, joint modelling approaches with application in AIDS research have been carried out by Self and Pawitan [49] and De Gruttola and Tu [15]. An overview of this can be found in [33,34,39].

This work is motivated by a study on DMD syndrome data. This genetic disease cannot be cured and affected individuals die after some time. Some steroids are given to improve their quality of life as measured by their muscle scores based on six important muscles. Composite muscle scores may vary from 0 to 10. Severity of the disease may be assessed from these scores where low score indicates bad health condition. Two highly correlated time-to-event indicators (time to walk four steps and time to get up from lying state) were also noted. Furthermore, these indicators were subject to right censoring. It may be noted that these time-to-event indicators are highly dependent on muscle scores. In this situation joint modelling of these two correlated processes is highly desirable. Another data set, which has been used by a number of researchers, is also analysed. This is known as ddI/ ddC study and is available in JM package in R. In this data set, square root of CD4 cell count which is a biomarker for AIDS is used as longitudinal response. Death due to AIDS may depend on the CD4 cell count and hence a joint model approach may be considered.

A well known technique for joint modelling of two simultaneous processes is based on introducing a partly observed random variable following a bivariate normal distribution, where one component defines the longitudinal outcome and the second component defines time-to-event. This approach has been used by many authors (see, e.g. [10,13,17,18,37,46]). Extensions of bivariate models to incorporate a multivariate model generalization have been implemented by Molenberghs and Lesaffre [32]. Further, Fieuws et al. [17] proposed an approach for modelling multivariate longitudinal data where all the possible pairs of longitudinal data were separately modelled and were later combined in the final step. Again, Rizopoulos et al. [45] proposed a new method for calculating residuals and producing diagnostic plots in joint models, based on the idea of multiplying and imputing the missing longitudinal responses under the fitted joint model, thus creating random versions of the completed data set, which can then be used to extract conclusions regarding model assumptions.

In the case of joint models for longitudinal and time-to-event data, the Bayesian perspective enhances and improves estimation and prediction, incorporating prior information into the study [22,25,27,48,54]. However, our main objective is to estimate the survival function, the prediction of survival times, or the computation of other posterior distributions for relevant probabilities or rates. The frequentist approach is not adequate to answer all these questions because the probabilistic characteristics of the relevant estimators are practically impossible to compute [26].

Considerable attention has been given to the area of Bayesian approaches to joint models. The primary source of such work comes from medical science (see [3–7,24,28,52,56]). Conceptual simplicity and dynamic inference, based on posterior distributions, may be considered as main factors contributing to this approach, forming a natural framework for joint models. Again, Rizopoulos et al. [42] used dynamic predictions with time-dependent covariates where the main interest was in optimally utilizing the recorded information and providing the medically relevant summary measures, such as survival probabilities, which eventually aided in decision making. Two measures were presented by Rizopoulos et al. [43] based on information theory that can be used to dynamically select models in time and optimally schedule longitudinal biomarker measurements. Further, Andrinopoulou et al. [2] proposed a Bayesian joint model that allows for a time-varying coefficient to link the longitudinal and survival processes, using P-splines to improve the dynamic predictions.

In our present work, we extend the bivariate approach to a data set with n + 1 components where the first n components define the longitudinal process, and the later components describe the time-to-event. In this work, we are using fully parametric accelerated failure time (AFT) model for the time-to-event response. The main concern is to see how the time-to-event is dependent on the longitudinal process. We have used a Bartlett decomposition of the covariance matrix to define the association between these two associated processes. This modelling approach may be considered as an alternative to the existing joint models where a strong assumption regarding the independence of multiple processes is made. In traditional joint models, it is assumed that if the random effects, present in both the models, is given, the processes become independent. Assuming a structural dependency is more general; it does reduce to the standard approach whenever the correlation between two processes becomes zero. In many situations, the longitudinal response may directly influence the time-to-event process. In previous research, Chakraborty and Das [12] used the longitudinal response as a covariate in the exponent part of the Cox proportional hazard model. In case of fully parametric models, like the model we are using here, this dependency can easily be obtained from the conditional mean of the AFT variable.

Most of the packages like JM [39] for joint models have been implemented from a frequentist perspective. In the R language, the only package that uses a Bayesian perspective is JMbayes. The R package JMbayes was introduced by Rizopoulos [40] that fits joint models under a Bayesian approach, which can fit a wide range of joint models, including joint models for continuous and categorical longitudinal responses. It provided several options for modelling the association structure between two outcomes. Though it does not draw fully Bayesian dynamic inferences, it does propose a mechanism for dynamic prediction [38]. The BUGS language Win/OpenBUGS [30] and JAGS [35], Stan [9], and INLA [47] can also be used to fit joint models using Bayesian inference.

This paper is organized as follows. In Section 2, we propose our joint model and discuss Bayesian inferences. We also mention different model selection criteria. Section 3 deals with analyses of two real data sets, one obtained from a study in muscular dystrophy syndrome and the other drawing from an AIDS study. In Section 4, the performance of the proposed methodology is described in some detail and we conclude with some discussions in Section 5.

2. Joint models and estimation

2.1. Joint models

Joint models for the analysis of longitudinal and time-to-event data have been developed by several authors. The most common method is to consider a latent variable model (or a shared random-effects model [23,51]). Again, Rizopoulos and Ghosh [41] developed a spline-based semi-parametric multivariate joint model for multiple longitudinal outcomes and a time-to-event, and the Dirichlet process prior formulation for the latent terms that allow for general shapes of their distribution.

Let $y_{ij}$ denote the longitudinal response for subject $i=1,\dots ,m$ at time j for $j=1,\dots ,n_i$. Let $T_i$ denotes the event time for individual i, with ${\delta }_i$ being the censoring indicator. Then the joint distribution of longitudinal and time-to-event data, given subject-specific random-effects ${\mathit{b}}_i$, can be modelled by using the multivariate normal specification:

$${\mathit{V}}_i=\left(\begin{array}{c}{\mathit{y}}_i\\ \log T_i\end{array}\right)={\mathit{x}}_i\mathit{\beta }+{\mathit{z}}_i{\mathit{b}}_i+{\mathit{ϵ}}_i$$

where ${\mathit{x}}_i$ and ${\mathit{z}}_i$ are design matrices for fixed effects and for random-effects, respectively, $\mathit{\beta }$ is the vector of regression parameters and

$${\mathit{ϵ}}_i\sim N(\mathbf{0},{\mathbf{\Sigma }}_i)\text{with}{\mathbf{\Sigma }}_i=\left(\begin{array}{cc}{\mathbf{\Sigma }}_{y_i}& {\mathit{\sigma }}_{1i}\\ {\mathit{\sigma }}_{1i}^{\mathrm{\prime }}& {\sigma }_T^2\end{array}\right)$$

The vector ${\mathit{\sigma }}_{1i}$ denotes the association between the longitudinal outcome vector and the time-to-event observation for the $i\mathrm{th}$ individual, i.e. $Cov({\mathit{y}}_i,\log T_i)={\mathit{\sigma }}_{1i}$. In the existing literature, the association between these two simultaneous processes has been captured through the subject-specific random-effects ${\mathit{b}}_i$. However, when random-effects are present, the estimation involves intractable integrals thus necessitating numerical integration. A new computational approach for fitting such models was proposed by Rizopoulos et al. [44] which was based on the Laplace method for integrals, facilitating the inclusion of high-dimensional random-effects structures. In our model, mainly the within-subject longitudinal correlation is captured by ${\mathit{b}}_i$. So, even if the two processes do not share a common ${\mathit{b}}_i$, dependency can still be captured using the conditional distribution.

It is difficult to model the entire covariance matrix ${\mathbf{\Sigma }}_i$ because of two reasons: positive-definiteness constraints [14,36] and high-dimensionality [29] for each subject. Factorization of the joint distribution of $({\mathit{y}}_i,\log T_i)$ may be used to address this problem. In our setting, we factor the joint distribution of ${\mathit{y}}_i$ and $\log T_i$ into two components: a marginal model for ${\mathit{y}}_i$ and a correlated regression model for $\log T_i$ given ${\mathit{y}}_i$. In the context of joint modelling of longitudinal discrete and continuous processes, a similar type of modelling has been used [18,21,29]. In the presence of subject-specific random-effects ${\mathit{b}}_i$, let

$${\mathit{x}}_i=\left(\begin{array}{cc}{\mathit{x}}_{i1}& \mathbf{0}\\ \mathbf{0}& {\mathit{x}}_{i2}\end{array}\right),{\mathit{z}}_i=\left(\begin{array}{cc}{\mathit{z}}_{i1}& \mathbf{0}\\ \mathbf{0}& {\mathit{z}}_{i2}\end{array}\right)\text{and}\mathit{\beta }=\left(\begin{array}{c}{\mathit{\beta }}_1\\ {\mathit{\beta }}_2\end{array}\right).$$

Then, using a Bartlett decomposition of a covariance matrix, the new models can be expressed as:

$${\mathit{y}}_i|{\mathit{b}}_i={\mathit{x}}_{i1}{\mathit{\beta }}_1+{\mathit{z}}_{i1}{\mathit{b}}_i+{\mathit{ϵ}}_{i1}$$

and

$$\log T_i|{\mathit{y}}_i,{\mathit{b}}_i={\mathit{x}}_{i2}{\mathit{\beta }}_2+{\mathit{z}}_{i2}{\mathit{b}}_i+{\mathit{B}}_i({\mathit{y}}_i-{\mathit{x}}_{i1}{\mathit{\beta }}_1-{\mathit{z}}_{i1}{\mathit{b}}_i)+{ϵ}_{i2},$$ (1)

where ${\mathit{B}}_i={\mathit{\sigma }}_{1i}^{\mathrm{\prime }}{\mathbf{\Sigma }}_{y_i}^{-1}$ is the vector reflecting structural association between these two processes and ${\mathit{b}}_i$ is used to capture within-subject longitudinal correlation. Again, we have assumed ${\mathit{\sigma }}_{1i}^{\mathrm{\prime }}={\sigma }_{\mathrm{cov}}^2{\mathbf{1}}^{\mathrm{\prime }}$ . We have also considered the local dependency through non-zero ${\mathit{z}}_{i1}$ and ${\mathit{z}}_{i2}$. Further let us assume, ${\mathit{ϵ}}_{i1}\sim N(\mathbf{0},{\mathbf{\Sigma }}_{y_i})$ and ${ϵ}_{i2}\sim N(0,{\sigma }_T^2)$. Thus it can be clearly seen that the two submodels are linked via the structural association as well as the by the shared random-effects association.

To capture longitudinal correlation, we assume ${\mathit{b}}_i=(b_{i1},b_{i2}{)}^{\mathrm{\prime }}$ to be a bivariate normal vector following $N(\mathbf{0},{\mathbf{\Sigma }}_b)$ where ${\mathbf{\Sigma }}_b$ is structured as $((1,\rho {\sigma }_b),(\rho {\sigma }_b,{\sigma }_b^2))$ with the variance of $b_{i1}$ being set to 1 for identifiability issues also addressed by Luo [31]. Again, Zhu et al. [55] also considered similar structure for capturing the dispersion of the random-effects.

We incorporate the censoring information for the $i\mathrm{th}$ individual through ${\delta }_i$ where ${\delta }_i$ takes value 1 in case the $i\mathrm{th}$ event time is censored and 0 in the case of non-censored. Thus the time-to-event observation can be expressed as $T_i^{\ast }=min(\log T_i,C_i)$ where $C_i$ represents the censored time point for the $i\mathrm{th}$ individual and $T_i$ the time to event observations obtained from the $i\mathrm{th}$ individual.

The contribution to the likelihood can be expressed as:

$$f(T_i^{\ast }|{\mathit{y}}_i,{\mathit{b}}_i)=h(T_i^{\ast }|{\mathit{b}}_i{)}^{{\delta }_i}S(T_i^{\ast }|{\mathit{b}}_i),$$

where $h(T_i^{\ast }|{\mathit{b}}_i)$ and $S(T_i^{\ast }|{\mathit{b}}_i)$ signifies the conditional hazard and conditional survival function. It is easy to note that,

$$T_i^{\ast }|{\mathit{y}}_i,{\mathit{b}}_i\sim N(\lambda ({\mathit{y}}_i,{\mathit{x}}_i,{\mathit{b}}_i),{\sigma }_A^2),$$

Under the log-normal assumption, we have

$$S(T_i^{\ast }|{\mathit{y}}_i,{\mathit{b}}_i)=1-\mathrm{\Phi }\left(\frac{T_i^{\ast }-\lambda ({\mathit{y}}_i,{\mathit{x}}_i,{\mathit{b}}_i)}{{\sigma }_A}\right).$$

where

$$\lambda ({\mathit{y}}_i,{\mathit{x}}_i,{\mathit{b}}_i)={\mathit{x}}_{i2}{\mathit{\beta }}_2+{\mathit{z}}_{i2}{\mathit{b}}_i+{\mathit{B}}_i({\mathit{y}}_i-{\mathit{x}}_{i1}{\mathit{\beta }}_1-{\mathit{z}}_{i1}{\mathit{b}}_i)$$

and

$${\sigma }_A^2={\sigma }_T^2-{\mathit{\sigma }}_{1i}^{\mathrm{\prime }}{\mathbf{\Sigma }}_{y_i}^{-1}{\mathit{\sigma }}_{1i}$$

signifies the conditional mean and variability of the time-to-event observations of the $i\mathrm{th}$ individual when the longitudinal and subject-specific random-effects are given.

It is more difficult to handle the AFT structure [11] in the joint modelling setting than for the Cox model, since $f(T_i^{\ast }|{\mathit{y}}_i,{\mathit{b}}_i)$ is more complicated and the baseline function involves unknown quantities, which is completely different from the case in the Cox model. Moreover, for the baseline hazard function, it is not possible to use the point mass function with masses assigned to all uncensored survival times $T_i$. Now, the complete data likelihood for the $i\mathrm{th}$ individual:

$$L_i=(\prod _{j=1}^{n_i}f(y_{ij}|{\mathit{b}}_i))f(T_i^{\ast }|{\mathit{y}}_i,{\mathit{b}}_i)f({\mathit{b}}_i|{\sigma }_b^2,\rho ).$$

We take ${\mathbf{\Sigma }}_{y_i}={\sigma }_y^2{\mathit{I}}_{n_i}$ with the assumption of independence among subjects.

We let the combined observed data from all the submodels to be $\mathbf{{\rm Y}}=\{y_{ij}\}\cup \{T_i^{\ast }\}\cup \{{\delta }_i\}$ with $\mathbf{\Psi }=({\mathit{\beta }}_1^{\mathrm{\prime }},{\mathit{\beta }}_2^{\mathrm{\prime }},{\sigma }_y^2,{\sigma }_b^2,{\sigma }_{\mathrm{cov}}^2,\rho ,{\sigma }_T^2,{\mathit{\nu }}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$ be the unknown parameter vector. Here, $\mathit{\nu }=({\nu }_1,{\nu }_2{)}^{\mathrm{\prime }}$ is a coefficient row vector reflecting the association strength between the longitudinal and time-to-event submodels captured via subject-specific random-effects. It can be noted that, if ${\mathit{z}}_{i2}=({\nu }_1,{\nu }_2{)}^{\mathrm{\prime }}=0$, the two submodels are linked only via the structural association. This fact has been explored by us in Section 4 where we consider two parallel setups for demonstrating the efficiency of the structural association in the context of linking the two submodels. The complete data likelihood for the $i\mathrm{th}$ individual:

$$\begin{array}{rl}& L(\mathbf{\Psi }|\mathbf{{\rm Y}})=L(\mathbf{\Psi }|{\mathit{y}}_i,T_i^{\ast },{\mathit{b}}_i)\\ & =g_1\times g_2\times g_3\\ & =g_1\times g_2^{\ast }(W_i{)}^{{\delta }_i}\times (1-G(W_i){)}^{1-{\delta }_i}\times g_3\\ & =\exp [-{({\mathit{y}}_{\mathit{i}}-{\mathit{x}}_{i1}{\mathit{\beta }}_1-{\mathit{z}}_{i1}{\mathit{b}}_i)}^{\mathrm{\prime }}{\mathbf{\Sigma }}_{y_i}^{-1}({\mathit{y}}_{\mathit{i}}-{\mathit{x}}_{i1}{\mathit{\beta }}_1-{\mathit{z}}_{i1}{\mathit{b}}_i)]\\ & \times {[\exp \left(-\frac{(T_i^{\ast }-\lambda ({\mathit{y}}_i,{\mathit{x}}_i,{\mathit{b}}_i){)}^2}{2{\sigma }_A^2}\right)]}^{{\delta }_i}\times {[1-\int \exp \left(-\frac{(T_i^{\ast }-\lambda ({\mathit{y}}_i,{\mathit{x}}_i,{\mathit{b}}_i){)}^2}{2{\sigma }_A^2}\right)]}^{1-{\delta }_i}\\ & \times \exp (-{\mathit{b}}_i^{\mathrm{\prime }}{\mathbf{\Sigma }}_b^{-1}{\mathit{b}}_i)\end{array}$$

where, $g_1(.)$ denotes the longitudinal contribution for the $i\mathrm{th}$ individual; $g_2(.)$ denotes the time-to-event contribution for the $i\mathrm{th}$ individual given longitudinal response; $g_2^{\ast }(.)$ denotes the probability density function of the standard normal variate $W_i$; $G(.)$ cumulative distribution function of the standard normal variate $W_i$; $g_3(.)$ signifies the contribution of the subject-specific random-effects for the $i\mathrm{th}$ individual; and $W_i=\frac{T_i^{\ast }-\lambda ({\mathit{y}}_i,{\mathit{x}}_i,{\mathit{b}}_i)}{{\sigma }_A}$.

2.2. Bayesian inference

In this paper, to estimate the unknown parameters we have adopted a fully Bayesian approach via the Markov Chain Monte Carlo (MCMC) method. We need to consider prior distributions for all unknown parameters in the parameter space $\mathbf{\Psi }$. Due to the absence of prior information on the parameters of interest, non-informative priors are assigned. Non-informative priors for all the elements of $\mathbf{\Psi }$ were chosen.

The prior distribution of all the elements of ${\mathit{\beta }}_1$, ${\mathit{\beta }}_2$ and $\mathit{\nu }$ are assumed to be normal with high degree of precision. We use prior distribution of $\rho \sim Uniform[-1,1]$ and ${\sigma }_{\mathrm{cov}}^2\sim Uniform[0,0.6]$ . The priors of variability parameters ${\sigma }_y^2,{\sigma }_b^2$ and ${\sigma }_T^2$ are assumed to be Inverse Gamma with high degree of precision.

The model fitting is implemented with the help of OpenBUGS (OpenBUGS version 3.2.2). The joint posterior density finally takes the form:

$$\begin{array}{rl}& L(\mathbf{\Psi }|\mathbf{{\rm Y}})\propto \exp [-{({\mathit{y}}_i-{\mathit{x}}_{i1}{\mathit{\beta }}_1-{\mathit{z}}_{i1}{\mathit{b}}_i)}^{\mathrm{\prime }}{\mathbf{\Sigma }}_{y_i}^{-1}({\mathit{y}}_i-{\mathit{x}}_{i1}{\mathit{\beta }}_1-{\mathit{z}}_{i1}{\mathit{b}}_i)]\\ & \times {[\exp \left(-\frac{(T_i^{\ast }-\lambda ({\mathit{y}}_i,{\mathit{x}}_i,{\mathit{b}}_i){)}^2}{2{\sigma }_A^2}\right)]}^{{\delta }_i}\times {[1-\int \exp \left(-\frac{(T_i^{\ast }-\lambda ({\mathit{y}}_i,{\mathit{x}}_i,{\mathit{b}}_i){)}^2}{2{\sigma }_A^2}\right)]}^{1-{\delta }_i}\\ & \times \exp (-{\mathit{b}}_i^{\mathrm{\prime }}{\mathbf{\Sigma }}_b^{-1}{\mathit{b}}_i)\times \pi ({\mathit{\beta }}_1)\times \pi ({\mathit{\beta }}_2)\times \pi ({\sigma }_y^2)\times \pi ({\sigma }_b^2)\times \pi ({\sigma }_T^2)\times \pi ({\sigma }_{\mathrm{cov}}^2)\\ & \times \pi (\rho )\times \pi (\mathit{\nu }).\end{array}$$

The posterior distributions of the parameters do not yield standard forms and hence are sampled by a Metropolis-Hastings Acceptance-Rejection algorithm to obtain suitable estimates. The Gelman–Rubin diagnostic [19] is used to ensure the scale reduction $\hat{R}$ of all parameters are smaller than 1.1. We have run multiple chains with over-dispersed initial values to analyse data. Moreover, to ensure the chain convergence we have used the trace plots and auto-correlation functions as mentioned by Gelman et al. [19]. Other choices of priors such as half-Cauchy priors for variability parameters and normal priors varying in mean and degree of precision for regression parameters are explored. The results thus studied show no significant difference indicating robustness when it comes to prior specifications.

2.3. Model selection

A wide variety of model selection criteria in Bayesian inference are available. Among these, we have adopted the deviance information criterion (DIC) [50], the expected Akaike information criterion (EAIC), the expected Bayesian (or Schwarz) information criterion [8]. The deviance information criterion (DIC) may be considered a hierarchical modelling generalization of the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). When the posterior distributions of the models have been obtained by Markov chain Monte Carlo (MCMC) simulation, DIC may be used effectively in Bayesian model selection problems. This quantity is based on a deviance, defined by:

$$D(\mathbf{\Psi })=-2\log L(\mathbf{{\rm Y}}|\mathbf{\Psi })+2\log \zeta (\mathbf{{\rm Y}}),$$

where $L(\mathbf{{\rm Y}}|\mathbf{\Psi })$ is the likelihood function for the observed data and where $\zeta (\mathrm{{\rm Y}})$ is some fully specified standardizing term, a function of the data alone. Then, the effective dimension $p_D$ is defined as

$$p_D=\overline{D}-D(\overline{\mathbf{\Psi }}),$$

where

$$\overline{D}=E_{\mathbf{\Psi }}[-2\log L(\mathbf{{\rm Y}}|\mathbf{\Psi })|\mathbf{{\rm Y}}]+2\log \zeta (\mathbf{{\rm Y}})$$

is the posterior mean deviance, and $\overline{\mathbf{\Psi }}$ is an estimate of $\mathbf{\Psi }$ depending on $\mathbf{{\rm Y}}$. The Deviance Information Criterion (DIC) for model comparison is defined as

$$DIC=\overline{D}+p_D=2\overline{D}-D(\overline{\mathbf{\Psi }}).$$

A small DIC value indicates a better fit of the model. Alternatively one may use expected AIC and expected BIC as suggested by Carlin and Louis [8]. Both criteria are of the form $-2\overline{D}+\text{penalty}\cdot \text{(Number of~parameters in~the model)}$ , where the mean is averaged over all MCMC samples. EAIC applies a penalty of 2, while EBIC applies a penalty of $\log (N)$. Smaller EAIC and EBIC indicate that the model has a better predictive ability.

3. Data analysis

3.1. Duchenne muscular dystrophy data

The motivating data set is from a study on muscular dystrophy conducted in children and collected by the National Neurosciences Centre (NNC), India. Duchenne and Becker muscular dystrophy (DMD/BMD) is a rapidly progressive form of genetic neuromuscular disease of childhood, caused by the alteration of the dystrophin gene, called the DMD gene located at the short arm $(p)$ of the X chromosome ( $X_{p}21.2$). The dystrophin plays a crucial role in maintaining the membrane of muscle cells. Inherited as X-linked recessive diseases, it also often occurs in people from families without a known history of this condition. This basically signifies that it manifests in males whereas females carrying two X chromosomes can only be carriers of this genetic condition (where any one X chromosome carries the defective or mutated gene). Males having one X and one Y chromosome become affected if the X chromosome carrying the defective gene is inherited from their carrier mother. Symptoms can be perceived before the child reaches age six or even in the early infancy characterized by a steady decline in muscle strength (biomarker) leading to a condition where braces will be required by the child for walking before the age of 10. By the time of 12 years of age most of the patients are compelled to be dependent on wheelchairs. Other clinical conditions include abnormal development of bones leading to skeletal deformities in the body, wasting of thigh and pectoral muscles, breathing troubles, cardiomyopathy (enlarged heart condition) and even intellectual or cognitive impairment. Patients usually die around age 25, the main contributing causes of death being respiratory failure and cardiomyopathy.

The data set of our study consists of composite scores based on longitudinal observations on 6 different muscles, i.e. Neck, Deltoid, Biceps, Iliopsas, Quadriceps, and Hamstrings, which are mainly responsible for all movements of an individual. Time-to-event indicators constitute: (a) time taken by the patient to walk four steps; (b) time to get up from lying state.

A censoring mechanism is incorporated in such a way that censored observations occur when the patient is unable to complete four steps in within 90 seconds or fails to stand up within 60 seconds. The two time-to-event indicators are highly correlated as can be seen form Figure 1. It can be concluded from the figure that it might be sufficient to consider any one of the time-to-event indicators for our analysis. We have considered indicator (a), which is regarded as severer condition by the clinicians. The basic objective of the study was to observe how quality of life, as indicated by time taken to walk four steps, depends on composite muscle scores.

Figure 1.

DMD study. Comparing survival curves.

The profile plot in Figure 2(a) exhibits the highly unbalanced nature of the data. The number of observations for an individual ranges over 3–10. There are no covariates in the data set. Figure 2(b) reveals the relationship of the composite muscle score with Visit obtained using lowess smoothing. There exists an inverse relationship between the composite score and months from baseline for a patient; it is quite evident in the case of a progressive genetic neuromuscular disease. Hence we have included the effect of time (i.e. Obstime) in the longitudinal profile of each individual expressed in the form of a linear mixed-effects model:

$$y_{ij}=\beta +b_{i1}+b_{i2}{\text{Obstime}}_{ij}+{ϵ}_{ij1},$$

where $y_{ij}$ denotes the composite muscle score vector (obtained by taking the mean of six major muscles responsible for movement) for the $i\mathrm{th}$ patient at the $j\mathrm{th}$ time point, β the intercept for that $i\mathrm{th}$ patient for the $j\mathrm{th}$ time point, whereas $b_{i1}$ and $b_{i2}$ signify the intercept and slope components of the subject-specific random-effects for the $i\mathrm{th}$ patient. Here, ${ϵ}_{ij1}\sim N(0,{\tau }_z^{-1})$ and Obstime signifies the inclusion of the effect of time in the longitudinal profile, as suggested by the lowess smoothing curve. For the time-to-event submodel, we have the AFT model:

$$\log T_i={\beta }_2+{\nu }_1{b}_{i1}+{\nu }_2{b}_{i2}+{\sigma }_{\mathrm{cov}}^2\frac{\sum _{j=1}^{n_i}(y_{ij}-\beta -b_{i1}-b_{i2}{\text{Obstime}}_{ij})}{{\sigma }_y^2}+{ϵ}_{ij2},$$

where $T_i$ is the time to climb four steps within 90 seconds. The data exhibited varying values of $T_i$ over different time points for a particular patient, hence, we have used the maximum of them as the final time-to-event observation in the sense of capturing the worst possible condition. Here, ${\beta }_2$ denotes the intercept for $i\mathrm{th}$ individual time-to-event observation, ${\nu }_1$ and ${\nu }_2$ the regression coefficients corresponding to $b_{i1}$ and $b_{i2}$ for the particular subject and ${ϵ}_{ij2}\sim N(0,{\tau }_t^{-1})$. Again, ${\sigma }_{\mathrm{cov}}^2$ captures the structural bond between the two sub-processes.

Figure 2.

Exploratory data analysis on 26 patients suffering from DMD. (a) DMD study. Profile plot and (b) DMD study. Smoothing.

We explore this data set by fitting our proposed model with the help of two setups as described in Section 4. The results under joint model ${\text{JM}}_1$ (with both forms of associations between the two processes) and joint model with local independence JM ${}_2$ (with only parametric structural association) are displayed in Table 1 displaying the posterior mean, standard deviation (SD) of the estimated posterior means, and the 95% credible intervals (CI). Both the models display low bias and standard deviation (SD) and SD is pretty close to SE.

Table 1.

DMD study.

		JM ${}_1$				JM ${}_2$
Parameter	Name of the effect	Mean	SD	95% CI		Mean	SD	95% CI
For longitudinal
$\mathit{\beta }$	Intercept	1.377	0.097	1.189	1.566	1.377	0.098	1.188	1.572
${\mathit{\tau }}_z$	Precision	0.947	0.091	0.779	1.133	0.950	0.090	0.780	1.132
Both processes
${\mathit{\sigma }}_{\mathrm{cov}}^2$	Covariance between two processes	0.301	0.172	0.017	0.586	0.302	0.173	0.014	0.585
${\mathit{\sigma }}_b^2$	Variance component	0.342	0.178	0.122	0.737	0.339	0.163	0.127	0.731
$\mathit{\rho }$	Correlation	0.634	0.148	0.345	0.900	0.634	0.141	0.346	0.878
For time-to-event
${\mathit{\beta }}_2$	Intercept	1.000	0.101	0.801	1.196	0.991	0.100	0.803	1.195
${\mathit{\tau }}_t$	Presicion	1.000	0.100	0.813	1.206	0.999	0.099	0.813	1.199
${\mathit{\nu }}_1$	Random intercept	1.000	0.102	0.803	1.201
${\mathit{\nu }}_2$	Random slope	0.999	0.100	0.802	1.194

Note: Results of model fitting. Bold implies significant covariates.

It can be clearly seen that both the joint models report comparable estimates for all the parameters thus establishing the effectiveness of the parametric structural association. Again, parameters such as ${\tau }_z$, ${\sigma }_b^2$, ρ in JM ${}_2$ are seen to be performing better in terms of lower SD. Same can be exhibited in the time-to-event submodel for the intercept ${\beta }_2$ and ${\tau }_t$, the latter signifying the unconditional variability of the time-to-event observations. The parameters ${\nu }_1$ and ${\nu }_2$ display positive mean values with low standard deviations signifying the role of subject-specific random-effects in binding the two sub models in JM ${}_1$. All the parameters for both the joint models are seen to be significant.

Table 1 also shows $\rho =0.634$ for both JM ${}_1$ and JM ${}_2$ indicating the fact that longitudinal correlation can also be captured by the subject-specific random-effects. The structural dependence parameter ${\sigma }_{\mathrm{cov}}^2$ is non-zero and significant for both the joint models. This implies the fact that in situations where $b_i^{\mathrm{\prime }}s$ are not considered, ${\sigma }_{\mathrm{cov}}^2$ can efficiently capture the assocation between the two processes. This idea is also confirmed by Table 2 where the Dbar, EAIC, EBIC and DIC values for both the joint models are reported and JM ${}_2$ reveals more efficiency in terms of model fitting

Table 2.

Model comparison.

	DMD study				Aids study
	Dbar	EAIC	EBIC	DIC	Dbar	EAIC	EBIC	DIC
JM ${}_1$	203.000	221.000	215.735	232.700	1165.00	1193.00	1202.37	1741.00
JM ${}_2$	201.800	215.800	211.705	231.900	1158.00	1182.00	1190.03	1738.00

The trace plots for each model parameter display evidence of good mixing properties of the chains. This claim is also supported by $\hat{R}$ (Gelman–Rubin Diagnostic), which displays values around 1.001 in the case of both the joint models.

3.2. AIDS data

Four hundred and sixty-seven patients were enrolled for a ddI/ddC study where 230 patients were randomized, each of them to receive didanosine (ddI) at a dose of 500 mg/day and each of 237 patients were to receive zalcitabine (ddC) at a dose of 2.25 mg/day. The study conducted by the Terry Beirn Community Programs for Clinical Research on AIDS was a randomized open-label clinical trial carried out in order to compare the effectiveness of two alternative antiretroviral drugs, ddI and ddC, in patients affected with HIV who were intolerant to ZDV (zidovudine) or had a symptom of disease progression (a CD4 cell count of 300 or fewer) during the period of ZDV intake.

The study revealed that about 66% (309) patients had experienced progression of disease or death, and about 40% (188) had died after an average follow-up of of 15.6 months. A segment of patients of about 35% (164) had become study drug intolerant, and about 57% (143) of the still living and followed-up had been discontinued permanently from the original study drug. For only 1% (4) patients the vital status was not known and about 7% (31) patients were no longer participating in the study. The CD4 cells are a type of white blood cells made in the spleen, lymph nodes, and thymus gland and are part of the infection fighting system. The absolute number of CD4 lymphocyte cells per cubic millimeter of peripheral blood serves as a reasonable biomarker for progression of AIDS and so it has been measured repeatedly over follow-up periods of 2, 6, 12, and 18 month visits. A decrease in the CD4 cell count over time is an indication of the worsened condition of the immune system of the patient and thus enhancing the risk to infection. We intended to study about how the CD4 count in the patient's blood contribute to the disease progression in the patient. More detail about this study can be found in Abrams et al. [1] and Goldman et al. [20].

The CD4 trajectories for each individual in the study acting as indicator of progression of AIDS can be expressed in terms of a mixed-effects model with covariate Observation time (Obstime) and binary explanatory variables Gender ( $1=\mathrm{male}$, $-1=\mathrm{female}$), AZT-Status ( $-1=\mathrm{intolerance}$, $1=\mathrm{failed}$), Drug ( $0=\mathrm{ddC}$, $1=\mathrm{ddI}$), Previous Opportunistic Infection, i.e. PrevOI ( $1=\mathrm{AIDS}$ diagnosis at baseline, 0=no AIDS diagnosis at baseline) acting as influencing factors on the CD4 cell count in blood and can be expressed as:

$$\begin{array}{rl}{y}_{ij}& ={\beta }_{11}+{\beta }_{12}{\text{Obstime}}_{ij}+{\beta }_{13}{\text{Drug}}_i+{\beta }_{14}{\text{Gender}}_i+{\beta }_{15}{\text{PrevOI}}_i\\ & +{\beta }_{16}{\text{AZT-Status}}_i+b_{i1}+b_{i2}{\text{Obstime}}_{ij}+{ϵ}_{1ij},\end{array}$$

where $y_{ij}$ denotes the square root of the CD4 cell count for the $i\mathrm{th}$ patient at the $j\mathrm{th}$ time point with ${ϵ}_{1ij}\sim N(0,{\tau }_z^{-1})$. Here, ${\mathit{\beta }}_1=({\beta }_{11},{\beta }_{12},{\beta }_{13},{\beta }_{14},{\beta }_{15},{\beta }_{16}{)}^T$ is the vector of regression coefficients for the longitudinal trajectory. The square root transformation has been adopted as the CD4 cell count data exhibited right skewness. Again, according to our model, the time-to-event outcome variable, i.e. death due to acquired immunodeficiency syndrome can be expressed as:

$$\begin{array}{rl}\log T_i& ={\beta }_2+{\nu }_1{b}_{i1}+{\nu }_2{b}_{i2}+{\sigma }_{\mathrm{cov}}^2({\sigma }_y^2{)}^{-1}\sum _{j=1}^{n_i}(y_{ij}-{\beta }_{11}-{\beta }_{12}{\text{Obstime}}_{ij}-{\beta }_{13}{\text{Drug}}_i\\ & -{\beta }_{14}{\text{Gender}}_i-{\beta }_{15}{\text{PrevOI}}_i-{\beta }_{16}{\text{AZT-Status}}_i-b_{i1}-b_{i2}{\text{Obstime}}_{ij})+{ϵ}_{2ij},\end{array}$$

where $\log T_i$ denotes the time-to-event observation indicating the time upto which $i\mathrm{th}$ patient has survived (if death occurs) or time at which the observation was censored (if the patient was alive at the end of the study), ${\beta }_2$ being the intercept of the time-to-event trajectory, ${\nu }_1$ and ${\nu }_2$ the coefficients of the intercept and slope components of the random-effects and ${ϵ}_{2ij}\sim N(0,{\tau }_t^{-1})$. Here, ${\sigma }_{\mathrm{cov}}^2$ captures the structural covariance that is present between the CD4 cell count observations of a patient and the time to death or progression in disease severity.

As our proposed model relies on conditional dependence of the longitudinal biomarker directly on the event time process, it is clear that all the variables affecting the CD4 biomarker in the blood cell are directly affecting the time-to-event outcome, i.e. the event of death or deteriorated condition, even if the covariates were not included in the time-to-event submodel separately. We have also checked this fact by including covariates such as Drug and Previous Opportunistic Infection (i.e. AIDS or no AIDS at the baseline) in the time-to-event submodel (as these two can be considered capable of influencing the time-to-event directly) and obtained comparable results.

We have adopted a fully parametric accelerated failure time model for modelling the time-to-event part of the process where censoring occurs if the person is alive till the end of the study period. The analysis was carried out under two setups as described in Section 4. The models were compared and the findings displayed in Table 3, which shows that the proposed joint model (JM ${}_1$) and the joint model with local independence (JM ${}_2$) are equally competent with comparable estimates and low SD for all the parameters.

Table 3.

AIDS study.

		JM ${}_1$				JM ${}_2$
Parameter	Name of the effect	Mean	SD	95% CI		Mean	SD	95% CI
For longitudinal
${\mathit{\beta }}_{11}$	Intercept	2.599	0.102	2.403	2.798	2.604	0.094	2.425	2.793
${\mathit{\beta }}_{12}$	Obs. time	−0.398	0.039	−0.475	−0.322	−0.401	0.039	−0.479	−0.323
${\beta }_{13}$	Drug	0.088	0.095	−0.094	0.278	0.075	0.095	−0.106	0.259
${\beta }_{14}$	Gender	−0.003	0.081	−0.159	0.156	−0.010	0.082	−0.173	0.148
${\mathit{\beta }}_{15}$	Prev opp infection	−0.442	0.059	−0.553	−0.326	−0.433	0.059	−0.548	−0.324
${\beta }_{16}$	Intol to ZDV	−0.028	0.059	−0.143	0.089	−0.042	0.057	−0.156	0.065
${\mathit{\tau }}_z$	Precision	3.239	0.134	2.985	3.506	3.232	0.134	2.976	3.902
For both processes
${\mathit{\sigma }}_{\mathrm{cov}}^2$	covariance between two processes	0.501	0.289	0.024	0.974	0.497	0.289	0.023	0.975
${\mathit{\sigma }}_b^2$	variance component	0.203	0.046	0.123	0.294	0.211	0.046	0.126	0.308
$\mathit{\rho }$	correlation	−0.216	0.099	−0.395	−0.005	−0.209	0.097	−0.392	−0.021
For time-to-event
${\beta }_2$	Intercept	0.904	1.118	−1.277	3.069	0.904	1.110	−1.272	3.064
${\mathit{\tau }}_t$	precision	1.002	0.101	0.812	1.208	0.999	0.099	0.813	1.204
${\mathit{\nu }}_1$	random intercept	1.000	0.100	0.806	1.195
${\mathit{\nu }}_2$	random slope	1.000	0.099	0.805	1.193

Note: Results of model fitting. Bold implies significant covariates

Further, Table 2 displaying the values of Dbar, EAIC, EBIC and DIC suggests that JM ${}_2$ is a preferable model fit than JM ${}_1$. This asserts the fact that structural association between the two submodels can solely serve the role of association structure between the two submodels in the joint model framework.

The estimated regression coefficient of the follow up time was found to be significant with ${\hat{\beta }}_{12}=-0.398$ for JM ${}_1$ and ${\hat{\beta }}_{12}=-0.401$ for JM ${}_2$ implying detoriating CD4 cell counts of the patients under study. The covariate Drug held no significance thus conveying the ineffectiveness of one of the drug (ddI/ ddC) over the other in improving the clinical condition. The diagnosis of AIDS at study entry level is found to be significant with estimated coefficients ${\hat{\beta }}_{15}=-0.442$ for JM ${}_1$ and ${\hat{\beta }}_{15}=-0.433$ for JM ${}_2$. Therefore, the patients who were diagnosed with the disease at the start of the study were showing detoriated CD4 cell count as the study progressed. Other factors such as gender and the the patient's intolerance to zidovudine therapy recorded at the entry of the study were found to be insignificant.

It may be noted that, in the time-to-event model, we have not considered any covariate directly. Effects of covariates are incorporated through the conditional mean which involves all the covariates used in longitudinal submodel. This implies that the presence of structural association is very much needed. The structural association parameter was significant with an estimate of ${\hat{\sigma }}_{\mathrm{cov}}^2=0.501$ for JM ${}_1$ and ${\hat{\sigma }}_{\mathrm{cov}}^2=0.497$ for JM ${}_2$. This reflects the fact that structural association between the observed CD4 cell counts and the time-to-event, i.e. death or deteriorated condition of the patient under this modelling framework is alone sufficient to arrest the correlation between two processes. The precision parameter for the unconditional variability of the time-to-event observations is found to be significant with low standard deviation. The estimated regression coefficients for the intercept and slope for the subject-specific random-effects show significance with positive values and low standard deviation thus conveying the importance of random-effects in binding the two submodels in JM ${}_1$.

4. Simulation study

We conduct extensive simulation studies to evaluate the performance of the proposed joint model and to examine the effectiveness of the structural association based on conditional distribution. Thus we consider two parallel setups and design the simulation study as the proposed joint model (JM ${}_1$) and the joint model with local independence (JM ${}_2$). It is to be noted that, JM ${}_1$ has the two submodels linked by the structural association captured by ${\sigma }_{\mathrm{cov}}^2$ as well as by the shared parameter in the form of random-effects. JM ${}_2$ links the two submodels only through the structural association as $z_{i2}=({\nu }_1,{\nu }_2{)}^{\mathrm{\prime }}=0$ is considered in this setup.

For both the set up we consider 100 datasets with m = 200 individuals each having varying number $(n_i)$ of longitudinal outcomes. This is generated through $Uniform(4,10)$). A single continuous covariate is used for simulation. We simulate covariate values for the $i\mathrm{th}$ patient at the $j\mathrm{th}$ time point, i.e. $x_{ij}^{\mathrm{\prime }}s$ from a $N(3,0.5)$ distribution. The random-effects ${\mathit{b}}_i^{\mathrm{\prime }}s$ are simulated from $N_2(\mathbf{0},{\mathbf{\Sigma }}_b)$, where ${\mathbf{\Sigma }}_b$ is of the form $((1,\rho {\sigma }_b),(\rho {\sigma }_b,{\sigma }_b^2))$.

Longitudinal observations ${\mathit{y}}_i^{\mathrm{\prime }}s$ are generated from $N_{n_i}({\mathit{\mu }}_{1i},{\mathbf{\Sigma }}_{y_i})$ where the linear trajectory ${\mathit{\mu }}_{1i}=\alpha \mathbf{1}+\beta {\mathit{x}}_i+b_{i1}\mathbf{1}+b_{i2}{\mathit{t}}_i$ and ${\mathbf{\Sigma }}_{y_i}={\sigma }_y^2{\mathit{I}}_{n_i}$, where ${\mathit{I}}_{n_i}$ denotes an identity matrix of order $n_i$ and ${\mathit{t}}_i$ denotes vector representing the time points. Time-to-event observations, i.e. $\log T_i^{\mathrm{\prime }}s$ are generated from $N({\beta }_2+{\nu }_1{b}_{i1}+{\nu }_2{b}_{i2}+{\sigma }_{\mathrm{cov}}^2({\sigma }_y^2{)}^{-1}\sum _{j=1}^{n_i}(y_{ij}-\alpha -\beta x_{ij}-b_{i1}-t_{ij}{b}_{i2}),({\sigma }_T^2-n_i{\sigma }_{\mathrm{cov}}^4)).$

We draw the censoring times $C_i$ from an exponential distribution with mean 0.5 and $T_i^{\ast }=min(\log T_i,C_i)$. The censoring percentage was observed to be around 40%. The whole setup was also simulated with a censoring percentage of around 20% and comparable results were obtained. The parameter values were chosen as $\alpha =$1.85, $\beta =$1.3, ${\beta }_2=$1, ${\sigma }_{\mathrm{cov}}^2=$ 0.3, ${\sigma }_T^2=$0.99, ${\sigma }_b^2=$0.8, $\rho =$0.3, ${\nu }_1=$0.9, ${\nu }_2=$0.9 and ${\sigma }_y^2=$1.11. The data ( ${\mathit{x}}_{\mathit{i}},{\mathit{y}}_{\mathit{i}},T_i^{\ast },{\delta }_i)$ is generated from these parametric assumptions where ${\delta }_i$ is the censoring indicator.

The Bayesian framework in Section 2.2 is implemented for obtaining the inference. We examined the Bayesian trace plots and the autocorrelation plots from the summary to ensure the desired number of burn-in iterations to be 5000 and have examined the MCMC convergence and the mixing of the chains. We also examined three parallel MCMC chains with overdispersed initial values and the inference has been based on 5000 iteration values for each chain. The chains exhibit rapid convergence, all achieving convergence within 5000 iterations with all the parameters displaying $\hat{R}$ (the between-chain and within-chain standard deviation ratio) to be around 1.01.

The estimation results are reported in Table 4, displaying the bias (the deviation of the estimated value from the true parameter value), standard deviation (SD) of the estimated posterior means) standard error (SE), i.e. the square root of the average of the posterior variance, and the coverage probabilities (CP) of $95\mathrm{\%}$ equal-tailed credible intervals (CI). Table 4 supports the conclusion that the proposed joint model framework provides estimates of all the model parameters with negligible bias where SE is pretty close to SD and the credible interval coverage probabilities exceed the nominal level (95%) for both the joint models. We can see that the Joint model with local independence (JM ${}_2$) slightly outperforms the proposed joint model (JM ${}_1$) in terms of smaller bias and standard deviation for most of the parameters. This highlights the fact that even if the two processes do not share common subject-specific random-effects, association can be successfully captured by the structural dependency.

Table 4.

Simulation study.

		Joint Model-JM ${}_1$				JM with local independence-JM ${}_2$
Parameter	True	BIAS	SD	SE	CP	BIAS	SD	SE	CP
For longitudinal outcomes
α	1.850	−0.072	0.183	0.175	1.000	−0.003	0.194	0.187	1.000
β	1.300	0.004	0.062	0.053	0.990	0.002	0.055	0.057	0.990
${\tau }_z$	0.900	−0.036	0.038	0.042	0.990	−0.039	0.038	0.077	0.990
For both processes
${\sigma }_{\mathrm{cov}}^2$	0.300	−0.006	0.174	0.143	1.000	0.003	0.173	0.145	1.000
${\sigma }_b^2$	0.800	0.023	0.088	0.073	0.990	0.019	0.085	0.077	0.990
ρ	0.300	0.025	0.090	0.085	1.000	0.004	0.093	0.086	1.000
For time-to-event
${\beta }_2$	1.000	0.001	0.099	0.103	1.000	0.003	0.102	0.097	1.000
${\tau }_t$	1.010	−0.012	0.098	0.091	0.990	−0.008	0.099	0.093	0.990
${\nu }_1$	0.900	0.097	0.101	0.097	0.990
${\nu }_2$	0.900	0.096	0.101	0.096	0.990

Note: Operational characteristics for simulation studies.

Here, in our theoretical framework, we have modelled the time-to-event process by a parametric AFT model, so it is essential that we verify whether our proposed joint model is sufficiently robust under model misspecification. Table 5 displays the results of fitting our proposed joint model and the joint model with local independence when time-to-event data are being simulated from a log-logistic distribution. Here, in most of the cases bias increased to some extent as apprehended, but still within acceptance limit. For example, in longitudinal submodel, it may be seen that bias for β has increased to 0.089 from 0.004. SD and SE also increased as expected. The MCMC convergence and the mixing of the chains has been examined. All chains exhibit rapid convergence with convergence being achieved within 5000 iterations with good mixing of the chains with $\hat{R}$ value within 1.02 for all the parameters.

Table 5.

Simulation study.

		Joint Model-JM ${}_1$				JM with local independence-JM ${}_2$
Parameter	True	BIAS	SD	SE	CP	BIAS	SD	SE	CP
For longitudinal
α	1.850	−0.149	0.259	0.242	1.000	−0.144	0.254	0.239	1.000
β	1.300	0.089	0.078	0.080	0.990	0.085	0.077	0.079	0.990
${\tau }_z$	0.900	−0.087	0.037	0.038	0.970	0.089	0.039	0.084	0.970
For both processes
${\sigma }_{\mathrm{cov}}^2$	0.300	0.005	0.174	0.292	1.000	−0.008	0.172	0.141	1.000
${\sigma }_b^2$	0.800	−0.024	0.087	0.083	0.990	−0.044	0.085	0.075	0.990
ρ	0.300	−0.038	0.124	0.114	1.000	−0.030	0.113	0.115	1.000
For time-to-event
${\beta }_2$	1.000	−0.004	0.102	0.102	1.000	−0.005	0.103	0.106	1.000
${\tau }_t$	1.010	−0.010	0.103	0.196	0.990	−0.008	0.100	0.094	0.990
${\nu }_1$	0.900	0.101	0.095	0.185	0.990
${\nu }_2$	0.900	0.104	0.098	0.098	0.990

Note: Robustness check under model misspecification.

5. Discussion

In this manuscript, efforts have been undertaken towards introducing a new class of models based on a conditional distribution. This approach may further be extended to other cases where the conditional distribution of one process carries information about the other process. This proposed model can further be generalized. In the longitudinal sub-model, instead of using the linear mixed effects model, one may use some nonlinear or partial linear mixed effects model (PLMM) to capture nonlinear disease progression rate. In the accelerated failure time model (AFT) one can introduce some smoothing terms for covariate modelling. A joint model having submodels like PLMM and AFT model with splines will pose computational challenges.

Bayesian framework gives an additional advantage in terms of computational complexity. Frequentist approaches are mostly based on EM algorithm which is highly dependent on the choice of the parameter values at the initial stage. Using OpenBUGS the posterior computation becomes easier. A blend of OpenBUGS and R has been used for computation. The OpenBUGS code used for this model is provided as Supplementary information. Sensitivity of prior selection has also been studied with different standard priors. Two different data sets have been analysed in this work. The AIDS data set is available in R and the DMD data set is also available subject to the approval of National Neurosciences Centre.

It may also be noted that under the proposed set up, it may not be possible to test the dependency of the simultaneous processes which may be done for traditional set up. This shortcoming may be considered as a future research problem. Another challenge will be to incorporate the missing data (non-ignorable) in this model. We are currently working in this direction.

Disclosure statement

No potential conflict of interest was reported by the author(s).