Bayesian analysis of multi-type recurrent events and dependent termination with nonparametric covariate functions

Abstract

Multi-type recurrent event data occur frequently in longitudinal studies. Dependent termination may occur when the terminal time is correlated to recurrent event times. In this article, we simultaneously model the multi-type recurrent events and a dependent terminal event, both with nonparametric covariate functions modeled by B-splines. We develop a Bayesian multivariate frailty model to account for the correlation among the dependent termination and various types of recurrent events. Extensive simulation results suggest that misspecifying nonparametric covariate functions may introduce bias in parameter estimation. This method development has been motivated by and applied to the lipid-lowering trial (LLT) component of the Antihypertensive and Lipid-Lowering Treatment to Prevent Heart Attack Trial (ALLHAT).

1 Introduction

Recurrent events of the same type occur frequently in longitudinal clinical trials and observational studies. Examples include repeated lung infections in people with cystic fibrosis¹, recurrent shunt failures in children with hydrocephalus², and recurrent strokes in older adults³. An important feature of recurrent events is that the event times within the same subject are correlated. The correlation among event times violates the independence assumption of the Cox proportional hazards model. As a result, the cox model produces estimates that are both biased and insufficient in the context of repeated events^4–6. Many authors have investigated the analysis of recurrent event data, e.g., Vaida and Xu⁷, Ibrahim et al ⁸, Pepe and Cai⁹, and Lin et al ¹⁰. Moreover, multi-type recurrent event data arise when one subject experiences two or more different types of recurrent events during the course of the study. A scientific objective is to examine covariate effects on the risks of different types of recurrent events. Different types of recurrent events may be correlated to each other, e.g., acute myocardial infarction increases the occurrence of heart failure¹¹. Therefore, it may not be sufficient to perform separate analyses for each type of recurrent events, while ignoring the correlation among event types¹². To this end, Cai and Schaubel¹³ proposed a class of semiparametric marginal mean/rates models for multi-type recurrent event data with a general relative risk form and established a partial likelihood score function to estimate the covariate effects. Chen et al ¹² used the Gibbs sampling algorithm to fit a Bayesian model for the interval-censored recurrent event data.

Another important feature of recurrent event data is that recurrent events are often subject to termination. Such termination may be either independent (the terminal time is independent of the recurrent event times, e.g., administrative censoring, dropout due to moving) or dependent (the terminal time is correlated to the recurrent event times) on the recurrent events. Dependent termination has a non-neglectable impact on the occurrences of the recurrent events. For example, the risk of recurrent strokes is positively associated with the risk of death and no further stroke events can occur once patients die. Ignoring dependent termination leads to bias in model parameter estimation^14–16. To resolve this issue, Chen and Cook¹⁷ developed methods for treatment comparison in the presence of multi-type recurrent events and a terminal event. Zhu et al ¹⁸ treated both the distributions of the dependent termination and latent variables as nuisance parameters and developed statistical methods for estimating regression parameters of recurrent events. Mazroui et al ¹⁹ later developed a multivariate frailty model for multi-type recurrent events with dependent termination and applied a Gauss-Hermite quadrature approximation with the penalized likelihood method for statistical inference.

In the regression analysis of multi-type recurrent events with a dependent termination, it is common that the true underlying covariate effect functions are nonlinear, rather than linear. For example, age is one of the largest risk factors for cardiovascular diseases, and its functional form for risk of cardiovascular diseases may not be linear²⁰. Multiple studies have reported that the risk of cardiovascular diseases increases greatly for older adults, e.g., the risk of stroke doubles for every decade of age increase after age 55²¹. Modeling nonparametric covariate functions in the presence of single type recurrent events and a dependent termination has been discussed by Yu and Liu²², who used the penalized partial likelihood (PPL) method²³ to estimate regression parameters. However, to the best of our knowledge, there is no research conducted to simultaneously consider multi-type recurrent events, dependent termination, and nonparametric covariate functions in a Bayesian inference framework. It is not clear how the dependence structure of these three features interact and influence the statistical inference.

The goal of this article is to investigate the effects of nonparametric covariate functions in the joint modeling framework of multi-type recurrent events with dependent termination. We use multivariate frailty distributions to model the heterogeneity among subjects, the correlation among different event types, in addition to the correlation between recurrent events and dependent termination. Gaussian quadrature method and penalized partial likelihood have been developed to fit frailty models in recurrent event data. However, Gaussian quadrature method cannot incorporate the penalty term of the penalized spline function into the convenient SAS procedure PROC NLMIXED (SAS Institute Inc., Cary, NC, USA)²². And, the penalized partial likelihood method ²³ underestimates the frailty variance which is an essential parameter to quantify the association between different types of recurrent events and a terminal event. Therefore, we will use Bayesian methodology for statistical inference.

The remainder of this article is organized as follows. In Section 2, we describe the lipid-lowering trial (LLT) component of the Antihypertensive and Lipid-Lowering Treatment to Prevent Heart Attack Trial (ALLHAT) that motivates this methodological development and propose the joint models for the multi-type recurrent events, dependent terminal event, and nonparametric functions of covariates. Section 3 discusses Bayesian inference and Bayesian model selection criteria. Section 4 presents an extensive simulation study to evaluate the performance of the proposed models. In Section 5, we apply the proposed models to the LLT component of the ALLHAT study and present analysis results. In Section 6, we provide concluding remarks and discussion.

2 Data and Joint Models

2.1 A Motivating Clinical Trial

This methodological research is motivated by the lipid-lowering trial (LLT) component of the Antihypertensive and Lipid-Lowering Treatment to Prevent Heart Attack Trial (ALLHAT) study²⁴, referred to as the ALLHAT-LLT study, which was a randomized, non-blinded, large pragmatic trial conducted from February 1994 through March 2002 at 513 clinical centers in the United States, Puerto Rico, US Virgin Islands, and Canada²⁵. The principal objective of the ALLHAT-LLT study was to evaluate the impact of large sustained cholesterol reduction on all-cause mortality in a hypertensive cohort. Other outcomes included CHD (composite of fatal coronary heart disease and nonfatal myocardial infarction), stroke (fatal and nonfatal), and heart failure (hospitalized or fatal). A total of 10,355 subjects were randomly assigned to receive either pravastatin (5,170 subjects) or usual care (5,185 subjects). The detailed results of the ALLHAT-LLT study are described in the final ALLHAT-LLT article²⁵.

The follow-up time was 4.8 years in mean duration, with a maximum of 7.8 years. At the end of the trial, 84.8% of subjects were known to be alive, 12.3% were confirmed dead, 0.5% were reported dead with confirmation pending, and 2.4% were lost to follow-up or withdrawal from the study. We excluded the subjects with death confirmation pending and those with missing data in covariates, and obtained the final dataset with a sample size of 9,901. The subjects who were alive at the end of the trial, or lost to follow-up during the trial are considered as independent termination (right censored) and the time to the last follow-up visit is used as the survival time. During the follow-up of the study, subjects may experience three types of recurrent cardiovascular disease events (CHD, stroke, and heart failure) before death or censoring. To visualize the data structure, Figure 1 displays the event occurrence times (from randomization) for four selected subjects. The follow-up times are displayed on the x-axis and the subjects’ IDs are on the y-axis. For example, subject 2 experienced one CHD event at 4.4 months and one stroke event at 35.7 months before he/she died at 67.6 months from randomization. Some important characteristics of the data are: (1) each subject may experience multiple occurrences of one type of events (e.g., subject 3 had 4 recurrent heart failure events, while subject 4 had 3 recurrent CHD events); (2) subjects may experience more than one type of recurrent events prior to death or censoring (e.g., subjects 2, 3, and 4); (3) the frequencies and event times of the recurrent events vary across subjects. Table 1 lists the numbers of subjects with various types of recurrent CHD, stroke, and heart failure events. For example, in the pravastatin group, 331 subjects had one CHD occurrence and 29 subjects had at least two CHD occurrences. There were 583 and 596 all-cause mortalities in the pravastatin and usual care groups, respectively. If mortality occurred immediately after the occurrence of CHD, stroke, or heart failure, then the time to all-cause mortality and the time to the last recurrent event are identical.

Figure 1.

Time plots of four selected subjects with occurrences of CHD, stroke, heart failure, and death or censoring.

Table 1.

Numbers of subjects with CHD, stroke, and heart failure events.

	CHD		stroke		heart failure
	pravastatin	usual care	pravastatin	usual care	pravastatin	usual care
1 event	331	349	176	184	154	183
2+ events	29	48	25	30	78	58

Subjects with at least one type of cardiovascular disease events have higher risks of all-cause mortality and other types of cardiovascular disease events than subjects without any cardiovascular disease events^20,26,27. To visualize these correlations, Figure 2 displays the estimates of cumulative hazard rates of death (panel a), CHD (panel b), stroke (panel c), and heart failure (panel d). Panel a suggests that subjects with occurrences of CHD, stroke, or heart failure had higher cumulative hazard of death than those without. This phenomena manifests the strong correlation between the recurrent cardiovascular disease events and death. Similarly, panels b, c, and d suggest that subjects with occurrence of one type of cardiovascular disease events had higher cumulative hazards of other types of cardiovascular disease events than those without. Moreover, among the risk factors for cardiovascular disease events, age is the most important one; and it has been suggested that the age effect on the risk of cardiovascular disease events is nonlinear^20,28. Therefore, it is essential to develop a joint model framework for the analysis of the multi-type recurrent events (CHD, stroke, heart failure) and the dependent termination (all-cause mortality) with nonlinear covariate functions.

Figure 2.

Cumulative hazard curves. (a) Cumulative hazard of death for subjects with and without the occurrence of cardiovascular disease events (CHD, stroke, and heart failure). (b) Cumulative hazard of CHD for subjects with and without the occurrence of stroke or heart failure. (c) Cumulative hazard of stroke for subjects with and without the occurrence of CHD or heart failure. (d) Cumulative hazard of heart failure for subjects with and without the occurrence of CHD or stroke.

2.2 Joint Models of Multi-type Recurrent Events and Dependent Termination with Nonparamertric Covariate Function

Let r_ij(t) be the hazard function of type j recurrent events for subject i at time t since study onset, where i = 1, …, I, and j = 1, …, J. To analyze recurrent event data, the time scale can be gap times (time between two successive events) or calendar time (time since study onset). This article focuses on calendar time while the proposed method can be applied to both time scales. Let n_ij be the number of type j recurrent events, and t_ijr be the time to the rth (r = 1, …, n_ij) occurrence of type j recurrent event of subject i, since study onset. All types of recurrent event are stopped by T_i = min(D_i,C_i), where D_i is time to a terminal event (e.g., all-cause mortality), C_i is time to censoring (e.g., end of study or dropout). Let λ_i(t) be the hazard of death for subject i at time t and the death indicator be Δ_i = I(D_i < C_i), where I(·) is the indicator function. The observed data structure is (t_ij, T_i) with t_ij = (t_ij1, …, t_ijnij)′.

For simplicity of illustration, we consider a single nonparametric covariate function in the hazard functions of each type of recurrent events and the terminal event. Additional nonparametric covariate functions can be incorporated in a similar fashion. The joint model with nonparametric covariate functions can be defined as

$$\begin{array}{l}{r}_{ij}(t\mid b_{ij})=r_{0j}(t)exp(f_j(Z_i)+{\mathit{X}}_i^{\prime }(t){\mathit{\beta }}_j+b_{ij}),\\ {\lambda }_i(t\mid {\mathit{b}}_i)={\lambda }_0(t)exp(f_0(Z_i)+{\mathit{X}}_i^{\prime }(t){\mathit{\beta }}_0+\sum _{j=1}^J{\delta }_j{b}_{ij}),\end{array}$$ (1)

where r₀₁(t), …, r_0J(t), and λ₀(t) are baseline hazard functions for all types of recurrent events and the terminal event, respectively, β₀, β₁, …, β_J are the regression coefficient vectors associated with the covariate vector ${\mathit{X}}_i^{\prime }(t)$. The covariate vector could be different in the hazard functions of different events and could be time-dependent. The covariate Z_i has nonlinear effects on the hazard functions of the recurrent events and terminal event. And, f₀(·), f₁(·), …, f_J(·) are unspecified (nonparametric) smooth functions whose functional forms are of interest. The random effects vector b_i = (b_i1, …, b_iJ)′ is assumed to follow a J-dimensional multivariate normal distribution with mean 0 and covariance matrix Σ, whose off-diagonal entries govern the correlation among risks of multi-type recurrent events. Specifically, positive entry σ_jj′ of matrix Σ indicates positive correlation between the risks of type j and type j′ recurrent events. The vector δ = (δ₁, …, δ_j, …, δ_J)′ incorporates the correlation between time to all-cause mortality and times to multi-type recurrent events. Indeed, a positive parameter δ_j indicates that the patients with higher risk of type j recurrent events tend to have all-cause mortality earlier.

The nonparametric covariate functions in survival analysis have been widely investigated in the literature^29–31. In Eq (1), the nonparametric functions can be modeled by penalized splines. The crucial problem is how to select the number and locations of knots. In general, a large number of knots can ensure satisfactory fit and can capture the variability of the data, but can also dramatically increase the computational overhead. Thus, the idea of using fewer spline knots has been discussed, e.g., O’Sullivan³², and Kelly and Rice³³. Higher degree basis functions may need smaller number of knots than the linear spline basis functions. To this end, Ruppert and Wand³⁴ suggested that quadratic or cubic regression splines can fit the data better than the linear splines when the number of knots is small. However, the columns of the model matrix for cubic splines tend to be highly correlated because each column is a transformed version of z, which can induce considerable collinearity³⁵, resulting in a nearly singular model matrix and imprecision in the spline fit³⁴. In comparison, the cubic B-spline basis rescales the cubic spline basis, and is remarkably more stable in numerical computation. Therefore, we adopt the cubic B-spline basis and denote the spline model as $f_j(z)={\sum }_{k=1}^m{u}_{jk}{B}_k(z)$, where B(z) = (B₁(z),B₂(z), …, B_m(z)) is the vector of rescaled spline basis as (z, z²,z³, ${(z-{\kappa }_1)}_1^{+},\dots ,{(z-{\kappa }_K)}_{+}^3$), m = K +3, (z−κ_k)₊ = z−κ_k if z > κ_k and 0 otherwise. Gray³⁶ suggested to use cubic B-spline with fairly small number of basis functions (10–20 knots) and that the number of knots has little effect on the fixed effects parameter estimation.

Moreover, we approximate the baseline hazard functions using piecewise constant functions, which have been frequently used in survival and event history data analysis^37,38. Lawless and Zhan³⁹ demonstrated that models with piecewise constant baseline hazard functions using 8 to 10 intervals often yield excellent estimation for fixed and random effects parameters. More precisely, we divide the follow-up time into S intervals where each interval has 1/S quantile of the distinct terminal event times, denoted by $0=Q_0^D<Q_1^D<\dots <Q_S^D$. Then, the piecewise constant baseline hazard function for the terminal event can be expressed as ${\lambda }_0(t)={\sum }_{s=1}^S{\lambda }_{0s}I(Q_{s-1}^D<t\le Q_s^D)$. Similarly, we can define the piecewise constant baseline hazard function for each type of recurrent events.

2.3 Likelihood Formulation

For notational ease, we denote the vector of B-spline coefficients for event type j as u_j = (u_j1, …, u_jm) and denote the corresponding smoothing parameter as ζ_j, for j = 0, 1, …, J, with j = 0 denoting the terminal event. To avoid over-fitting by splines, we impose the penalty based on finite differences of the coefficients of adjacent B-splines⁴⁰. This type of penalty is a discrete approximation to the traditional integrated square of the finite derivation and it reduces the computation complexity. Define the coefficients for the first order differences as Δu_jl = u_jl−u_jl−1, the penalized log-likelihood conditional on the random effects vector b_i is

$$l(\mathit{\theta },\mathit{u};·)=\sum _{i=1}^I\sum _{j=1}^J{l}_{ij}^R(\mathit{\theta };b_{ij},{\mathit{t}}_{ij},T_i,Z_i,{\mathit{X}}_i(t))+\sum _{i=1}^I{l}_i^D(\mathit{\theta };{\mathit{b}}_i,Z_i,{\mathit{X}}_i(t),T_i,{\mathrm{\Delta }}_i)-\sum _{j=0}^J{\zeta }_j\sum _{l=2}^m{(\mathrm{\Delta }{u}_{jl})}^2,$$ (2)

where

$$\begin{array}{lll}{l}_{ij}^R(\mathit{\theta };b_{ij},{\mathit{t}}_{ij},T_i,Z_i,{\mathit{X}}_i(t))\hfill & =\hfill & \sum _{r=1}^{n_{ij}}\left\{log\left[\sum _{s=1}^S{r}_{0js}I(Q_{s-1}^{(j)}<t_{\mathit{ijr}}\le Q_s^{(j)})\right]+{\mathit{B}}_j(Z_i){\mathit{u}}_j+{\mathit{X}}_i^{\prime }(t_{\mathit{ijr}}){\mathit{\beta }}_j+b_{ij}\right\}\hfill \\ \hfill & -\hfill & \sum _{s=1}^S{\int }_{Q_{s-1}^{(j)}}^{Q_s^{(j)}\wedge T_i}{r}_{0js}exp\{{\mathit{B}}_j(Z_i){\mathit{u}}_j+{\mathit{X}}_i^{\prime }(t){\mathit{\beta }}_j+b_{ij}\}\mathrm{d}t,\hfill \\ l_i^D(\mathit{\theta };{\mathit{b}}_i,Z_i,{\mathit{X}}_i(t),T_i,{\mathrm{\Delta }}_i)\hfill & =\hfill & {\mathrm{\Delta }}_i\left\{log\left[\sum _{s=1}^S{\lambda }_{0s}I(Q_{s-1}^D<T_i\le Q_s^D)\right]+{\mathit{B}}_0(Z_i){\mathit{u}}_0+{\mathit{X}}_i^{\prime }(T_i){\mathit{\beta }}_0+\sum _{j=1}^J{\delta }_j{b}_{ij}\right\}\hfill \\ \hfill & -\hfill & \sum _{s=1}^S{\int }_{Q_{s-1}^D}^{Q_s^D\wedge T_i}{\lambda }_{0s}exp\{{\mathit{B}}_0(Z_i){\mathit{u}}_0+{\mathit{X}}_i^{\prime }(t){\mathit{\beta }}_0+\sum _{j=1}^J{\delta }_j{b}_{ij}\}\mathrm{d}t,\hfill \end{array}$$

where the parameter vector θ = (β₀, β₁, …, β_J, δ₁, …, δ_J, Σ) and l ∧ m = min(l,m).

3 Bayesian Inference

3.1 Prior Specification

In the classical approach, the selection of smoothing parameters via cross validation is difficult to implement and often fails when the number of smooth functions is large⁴¹. Lang and Brezger⁴¹ derived the Bayesian B-spline which replaced the penalties by random walk prior distributions. To do this, the prior distributions for the vector of B-spline coefficients u_j can be expressed as u_j1 ~ Uniform(−10, 10) and u_jl|u_jl−1 ~ N(u_jl−1, τ_j), where j = 0, …, J and l = 2, …, m, and the variance parameter τ_j controls the amount of smoothness of spline functions, and it corresponds to the smoothing parameter ζ_j in model (2). We augment the parameter vector as θ = (β₀, β₁, …, β_J, δ₁, …, δ_J, Σ, τ₀, τ₁, …, τ_J). We impose the prior distribution τ_j ~ Gamma(0.01, 0.01), which is a non-informative prior distribution with mean being 1 and variance being 100.

Because no prior information of the model parameters are available, we specify non-informative prior distributions for all unknown parameters. A commonly-used prior distribution for the covariance matrix Σ is a inverse-Wishart distribution because it is a conjugate prior distribution. However, the inverse-Wishart distribution uses a single parameter to control the precision of all entries in matrix Σ. Thus, using the standard “noninformative” inverse-Wishart prior distribution is not essentially noninformative⁴². To this end, we reparameterize the covariance matrix Σ using the Cholesky decomposition. This approach is widely used in survival and longitudinal data analysis, and yields satisfactory results^12,37. Specifically, we let Σ = ΨΨ′, where Ψ is a lower triangular matrix with real and positive diagonal entries. Let ϕ_lm be the (l,m)th entry of Ψ for 1 ≤ m ≤ l ≤ J and let the vector a_i = (a_i1, …, a_iJ)′ follow a standard normal distribution. Then the random effects vector b_i = Ψa_i has mean 0 and variance Σ. Moreover, the entries of matrix Σ can be expressed as ${\sigma }_{lm}={\sum }_{k=1}^{l\wedge m}{\varphi }_{lk}{\varphi }_{mk}$, where 1 ≤ l,m ≤ J and l ∧ m = min(l,m). We impose Uniform(0, 20) prior distribution on the diagonal entries ϕ_kk, k = 1, …, J, to ensure non-negativity and Uniform(−10, 10) prior distribution on the off-diagonal entries of Ψ to allow for possible negative correlation. We assign the N(0, 100) prior distribution for covariate effects β₀, β₁, …, β_J and for the parameters δ₁, …, δ_J. The baseline hazard functions for the recurrent and terminal events are divided into 10 pieces and the baseline hazard within each piece has a prior distribution Gamma(0.01, 0.01). We have investigated other selections of non-informative prior distributions and have obtained very similar results in both the simulation studies and the application to the ALLHAT-LLT study.

3.2 Model Fitting

To obtain the estimates of the parameter vector, we use Bayesian inference based on Markov Chain Monte Carlo (MCMC) to sample from the joint posterior distribution. The MCMC sampler is implemented using BUGS language⁴³ by specifying the log-likelihood function (2) and the prior distributions of all unknown parameters. The convergence of the MCMC chains is assessed via monitoring MCMC chain histories, autocorrelation plots, and density plots. In addition, we run multiple chains with disperse initial values and compute the Gelman-Rubin scale reduction statistics (R̂) to ensure R̂ of all parameters smaller than 1.1⁴⁴. From the MCMC samples, we estimate the posterior means, posterior standard deviations, and 95% credible intervals of parameters. To facilitate easy reading and implementation of the proposed methodology, we include a sample BUGS code in the Web Supplement.

3.3 Model Selection Criteria

Among the various model selection methods available in Bayesian inference, we select the deviance information criterion (DIC), expected Akaike information criterion (EAIC), expected Bayesian information criterion (EBIC)⁴⁵, and log pseudo-marginal likelihood (LPML). The deviance information criterion (DIC) assesses model fit based on the posterior mean of the deviance and a penalty on the model complexity⁴⁶. Because of the mixture framework in our model, we use the DIC₃ measurement⁴⁷. The DIC₃ is defined as ${\mathit{DIC}}_3=\overline{D(\mathit{\theta })}+{\tau }_D$, where $\overline{D(\mathit{\theta })}=-2{\mathrm{E}}_{\mathit{\theta }\mid \mathcal{D}}\{log[{\prod }_{i=1}^{I}f({\mathit{t}}_{ij},T_i\mid \mathit{\theta })]\}$ is the posterior mean deviance, ${\tau }_D=\overline{D(\mathit{\theta })}+2log\{{\mathrm{E}}_{\mathit{\theta }\mid \mathcal{D}}[{\prod }_{i=1}^{I}f({\mathit{t}}_{ij},T_i\mid \mathit{\theta })]\}$ is a measure of the effective number of parameters in the model, and E_θ|𝒟 (.) is the expectation with respect to the joint posterior distribution π(θ|𝒟). Thus, we have ${\mathit{DIC}}_3=-4E_{\mathit{\theta }\mid \mathcal{D}}\{log[{\prod }_{i=1}^{I}f({\mathit{t}}_{ij},T_i\mid \mathit{\theta })]\}+2log\{{\mathrm{E}}_{\mathit{\theta }\mid \mathcal{D}}[{\prod }_{i=1}^{I}f({\mathit{t}}_{ij},T_i\mid \mathit{\theta })]\}$. Applying Monte Carlo approximation,

$$\widehat{{\mathit{DIC}}_3}=-\frac{4}{M}\sum _{m=1}^M\sum _{i=1}^{I}log\{f({\mathit{t}}_{ij},T_i\mid {\mathit{\theta }}^{(m)})\}+2log\{\frac{1}{M}\sum _{m=1}^M\prod _{i=1}^{I}f({\mathit{t}}_{ij},T_i\mid {\mathit{\theta }}^{(m)})\}.$$

A smaller value of DIC₃ indicates a better-fitting model. The EAIC and EBIC can be estimated as $\widehat{\mathit{EAIC}}=\overline{D(\mathit{\theta })}+2\nu$ and $\widehat{\mathit{EBIC}}=\overline{D(\mathit{\theta })}+\nu logI$, where ν is the number of parameters in the model, and I is the number of subjects. Smaller values of EAIC and EBIC indicate a better-fitting model.

Conditional predictive ordinate (CPO)⁴⁵ is a cross-validation predictive method that evaluates the predictive distribution of the model conditioning on the data but with one subject deleted. The CPO for subject i is defined as ${\mathit{CPO}}_i=\int f({\mathit{t}}_{ij},T_i\mid \mathit{\theta })\pi (\mathit{\theta }\mid {\mathcal{D}}^{(-i)})d\mathit{\theta }={\{\int \frac{\pi (\mathit{\theta }\mid \mathcal{D})}{f({\mathit{t}}_{ij},T_i\mid \mathit{\theta })}d\mathit{\theta }\}}^{-1}$, where 𝒟 is the full data, 𝒟⁽⁻ⁱ⁾ is the data with subject i deleted. In the absence of a closed form, a Monte Carlo approximation of CPO_i can be obtained using the harmonic-mean approximation⁴⁸ as ${\widehat{\mathit{CPO}}}_i={\{\frac{1}{M}{\sum }_{m=1}^M\frac{1}{f({\mathit{t}}_{ij},T_i\mid {\mathit{\theta }}^{(m)})}\}}^{-1}$, where θ^(m) is the mth sample of parameter vector θ after burn-in. A larger CPO value indicates a better model fit. The summary statistic of the CPO is the log pseudo-marginal likelihood (LPML), defined as $\text{LPML}={\sum }_{i=1}^{I}log({\widehat{\mathit{CPO}}}_i)$. A larger value of LPML suggests better model fit.

4 Simulation Study

We conduct a simulation study with two settings to compare the performance of three models: joint model (Eq (1)), reduced model (the recurrent and terminal events are modeled independently with δ_j = 0 for j = 1, …, J in Eq (1)), and parametric model (model the nonparametric covariate functions in Eq (1) as linear functions). In each simulation setting, we generate 400 datasets with 500 subjects in each dataset. For each subject, we simulate two types of recurrent events and a terminal event. The hazard functions of the multi-type recurrent events and terminal event are defined as follow:

$$\begin{array}{ll}{r}_{i1}(t)\hfill & =r_{01}(t)exp(f_1(Z_i)+X_i{\beta }_1+b_{i1}),\hfill \\ r_{i2}(t)\hfill & =r_{02}(t)exp(f_2(Z_i)+W_i{\beta }_2+b_{i2}),\hfill \\ {\lambda }_{ij}(t)\hfill & ={\lambda }_0(t)exp(Z_i{\alpha }_0+X_i{\alpha }_1+W_i{\alpha }_2+{\delta }_1{b}_{i1}+{\delta }_2{b}_{i2}),\hfill \end{array}$$

where the baseline hazard functions follow Weibull distributions with r₀₁(t) = 0.2t^0.5, r₀₂(t) = 0.1t, and λ₀(t) = 0.02t. The nonlinear smooth functions are f₁(Z) = (sin(3Z) + Z²)/5, and f₂(Z) = cos(4Z)/5 + Z²/3. Note that variable Z is linear in the terminal event model to be consistent with our data application because the preliminary analysis of the ALLHAT-LLT dataset does not support nonparametric age function as detailed in Section 5. We generate variables X and W as binary variables taking values 0 and 1 with probability 0.5, and generate variable Z from the uniform distribution between −2 and 2. The regression coefficients are β₁ = 1, β₂ = 1.5, α₀ = 0.5, α₁ = −1, and α₂ = 2. We assume the random effects vector ${\mathit{b}}_i={(b_{i1},b_{i2})}^{\prime }\stackrel{\mathit{iid}}{\sim }N(0,\mathbf{\sum })$ where the covariance matrix Σ = {(0.64, 0.32), (0.32, 1.6)}. The censoring time is generated as C_i = 15 + uniform(0, 6).

We use the statistical inference procedure described in Section 3 to estimate the unknown parameters in the model. We set the number of pieces to be 10 in all baseline hazard functions and set the number of knots to be 10 in all nonparametric functions. We have used different numbers of knots and pieces and have obtained very similar results. In each simulation setting, we run two parallel MCMC chains with over-dispersed initial values. Each chain is run for 10, 000 iterations. The first 5, 000 iterations are discarded as burn-in, and the remaining 5, 000 samples are used to obtain the posterior distributions of the parameters. The computation time of all three models is approximately 25 minutes. We compute bias (the average of posterior means minus the true values), the standard deviation of the posterior means (SD), the square root of the average of the posterior variance (SE), and coverage probability (CP) of 95% equal-tail credible intervals. Table 2 displays the simulation results of both settings (upper and lower table for simulation settings I and II, respectively).

Table 2.

Results of the simulation study. Setting I: no correlation between the multi-type recurrent events and terminal event (δ₁ = δ₂ = 0); Setting II: positive correlation between the recurrent events and terminal event (δ₁ = 0.3, δ₂ = 0.5).

Setting I	Joint Model				Reduced model				Parametric Model
	Bias	SD	SE	CP(%)	Bias	SD	SE	CP(%)	Bias	SD	SE	CP(%)
For recurrent events
β1 = 1	0.018	0.071	0.072	94.3	0.009	0.084	0.081	95.5	0.017	0.102	0.096	84.4
β2 = 1.5	−0.032	0.104	0.110	92.5	−0.030	0.121	0.129	92.3	−0.019	0.138	0.128	87.8
ρ = 0.316	0.002	0.043	0.044	93.5	0.006	0.051	0.048	94.3	0.058	0.052	0.049	74.0
σ1 = 0.8	0.002	0.030	0.029	96.5	0.000	0.035	0.037	93.0	0.050	0.039	0.037	68.8
σ2 = 1.265	0.004	0.044	0.042	94.5	0.002	0.052	0.054	93.8	0.082	0.054	0.055	66.8
For terminal event
α0 = 0.5	0.001	0.040	0.041	94.8	0.003	0.045	0.046	95.5	0.005	0.046	0.048	94.4
α1 = −1	−0.010	0.090	0.092	93.8	−0.006	0.106	0.103	96.8	−0.020	0.104	0.107	96.0
α2 = 2	0.006	0.109	0.102	96.8	0.016	0.128	0.135	93.5	0.012	0.130	0.129	94.0
δ1 = 0	−0.036	0.068	0.069	92.0					−0.039	0.079	0.078	92.8
δ2 = 0	−0.013	0.040	0.042	93.5					−0.012	0.046	0.046	94.4
Setting II	Joint Model				Reduced model				Parametric Model
	Bias	SD	SE	CP(%)	Bias	SD	SE	CP(%)	Bias	SD	SE	CP(%)
For recurrent events
β1 = 1	0.012	0.090	0.092	94.0	0.035	0.091	0.092	88.8	0.011	0.105	0.091	87.5
β2 = 1.5	−0.060	0.121	0.121	92.8	−0.108	0.124	0.132	83.8	−0.067	0.140	0.126	87.3
ρ = 0.316	−0.016	0.056	0.054	94.0	−0.020	0.052	0.051	92.5	0.048	0.055	0.052	83.0
σ1 = 0.8	−0.005	0.038	0.038	94.0	0.003	0.039	0.038	93.8	0.043	0.040	0.039	78.3
σ2 = 1.265	−0.030	0.048	0.046	91.8	0.001	0.051	0.053	93.5	0.062	0.049	0.051	77.3
For terminal event
α0 = 0.5	−0.017	0.053	0.053	93.0	−0.110	0.043	0.046	30.0	−0.050	0.052	0.048	80.3
α1 = −1	0.007	0.108	0.108	93.5	0.209	0.099	0.109	44.3	0.057	0.105	0.105	91.0
α2 = 2	−0.051	0.138	0.144	91.8	−0.465	0.112	0.121	1.8	−0.147	0.137	0.131	79.0
δ1 = 0.3	−0.022	0.083	0.084	94.0					−0.097	0.081	0.079	73.5
δ2 = 0.5	−0.032	0.052	0.049	91.8					−0.117	0.048	0.048	32.3

In simulation setting I, there is no correlation between the multi-type recurrent events and terminal event (δ₁ = δ₂ = 0), and the percentage of censoring is around 20%. The reduced model is the true model in this setting. The simulation results suggest that both the joint and reduced models generate comparable results, with small bias, SE close to SD, and coverage probabilities close to the nominal value of 95%. Under model overparameterization, the estimates of parameters δ₁ and δ₂ from the joint model are correctly close to zero, suggesting that the joint model is still a reasonable model in this simulation setting. However, by misspecifying the nonparametric covariate functions, the parametric model gives biased estimates and low coverage probabilities for all parameters, except the regression coefficients in the survival sub-model. In Figure 3, we present the joint model estimates (dotdash lines) with 95% pointwise credible intervals (dashed lines) of the nonparametric covariate functions f₁(Z) (panel a) and f₂(Z) (panel b) with the true functions (solid lines), in addition to their coverage probabilities of the 95% pointwise credible intervals (panel c for f₁(Z) and panel d for f₂(Z)). Panels a and b suggest that the estimated nonparametric functions by the joint model are reasonably close to the true functions, with 95% pointwise credible intervals always covering the true functions. In panels c and d, the empirical coverage probabilities on all values of Z are close to the nominal level of 95%.

Figure 3.

The joint model’s estimates of nonparametric covariate functions in simulation setting I. Penalized spline estimates of f₁(Z) (panel a) and f₂(Z) (panel b). True functions, solid; estimated functions, dotdash; 95% pointwise credible intervals, dashed. Coverage probabilities of the 95% pointwise credible intervals for f₁(Z) (panel c) and f₂(Z) (panel d) with a reference line (dotted) at 0.95.

In simulation setting II, there is a positive correlation between the recurrent events and terminal event (δ₁ = 0.3 and δ₂ = 0.5), so that the subjects with higher risks of recurrent events tend to have a terminal event earlier. The joint model is the true model in this setting. The percentage of censoring is also around 20%. The joint model generally provides estimates with negligible bias, SE close to SD, and CP reasonably close to 95%. These results suggest that the joint model can successfully recover the true parameters in the presence of a dependent terminal event and nonparametric covariate functions. In contrast, the reduced model gives severely biased parameter estimates and low coverage probabilities, especially for the regression coefficients β and α, because it misspecifies the model for the terminal event. The parametric model provides poor estimation on all parameters including α, even though there is no nonparametric covariate function in the hazard function of the terminal event. This is because the parametric model treats the nonparametric functions as linear, and it fails to capture the true between-subject variation in the risks of recurrent events. Figure 4 displays the joint model’s estimated nonparametric functions and their coverage probabilities in the simulation setting II. Similar to setting I, the estimated nonparametric function performs well in recovering the true covariate functions.

Figure 4.

The joint model’s estimates of nonparametric covariate functions in simulation setting II. Penalized spline estimates of f₁(Z) (panel a) and f₂(Z) (panel b). True functions, solid; estimated functions, dotdash; 95% pointwise credible intervals, dashed. Coverage probabilities of the 95% pointwise credible intervals for f₁(Z) (panel c) and f₂(Z) (panel d) with a reference line (dotted) at 0.95.

To justify the use of non-parametric covariate functions, we have added a simulation study to examine whether polynomial regression models can sufficiently estimate the nonlinear covariate functions. Polynomial regression models using higher order polynomials are commonly used to fit a nonlinear relationship between the explanatory variable and response variable. In simulation settings I and II, we adopt cubic polynomial models to estimate the nonlinear smooth functions f₁(Z) and f₂(Z), respectively, using $f_1(Z)=z_i{\beta }_{11}+z_i^2{\beta }_{12}+z_i^3{\beta }_{13}$ and $f_2(Z)=z_i{\beta }_{21}+z_i^2{\beta }_{22}+z_i^3{\beta }_{23}$. The estimated covariate functions from the polynomial regression models are given in Web Figures 1–2. The simulation results suggest that the polynomial regression models are unable to capture all the variation of the true covariate functions and the 95% pointwise credible intervals do not always cover the true functions. Moreover, the empirical coverage probabilities on all values of Z deviate markedly from the nominal level of 95%. These undesirable properties from the polynomial regression models have been reported in the literature^49,50.

In conclusion, the simulation results suggest that in the scenario of independent termination, the joint model provides results comparable to the reduced model, and both outperform the parametric models with both linear functions and polynomials. In the presence of a dependent terminal event and nonparametric covariate functions, the joint model provides more accurate parameter estimates than the reduced model and the parametric models with both linear functions and polynomials.

5 Application to ALLHAT-LLT Data

In this section, we apply all three models to the motivating ALLHAT-LLT study. We consider three types of recurrent events (CHD, stroke, and heart failure) and a terminal event (all-cause mortality). We use two parallel chains with overdispersed initial values and run each chain for 20,000 iterations. The first 10,000 iterations are discarded as burn-in and the parameter estimates are based on the remaining 10,000 iterations from each chain. For baseline hazard functions, we use piecewise constant functions with 10 intervals by every 1/10th quantile. The nonparametric functions of age in the hazard functions of the three types of recurrent events are estimated by Bayesian B-splines with 10 knots. Good mixing properties of the MCMC chains are observed in the trace plots.

Previous studies suggest a number of risk factors for the cardiovascular disease, including older age, male gender, hypertension, diabetes, tobacco use, excessive alcohol consumption, excessive sugar consumption, family history of cardiovascular disease, obesity, lack of physical activity, air pollution, among others^26,51. Based on the data availability, we include the following covariates in the models for recurrent events and all-cause mortality: lipid-lowering trial randomization group (llt=0 for pravastatin, 1 for usual care), gender (gender=0 for male, 1 for female), race (race=0 for white and others, 1 for black), diabetes (diabetes=0 for yes, 1 for no), history of CHD (lchd=0 for yes, 1 for no), cigarette smoker (cursmoke=0 for current smoker, 1 for past smoker, 2 for never smoker), aspirin (aspirin=0 for yes, 1 for no), antihypertensive treatment before trial (blmeds=0 for yes, 1 for no), antihypertensive randomization group (att=0 for doxazosin, 1 for chlorthalidone, 2 for amlodipine, 3 for lisinopril), age, body mass index (bmi), systolic blood pressure (sbp), diastolic blood pressure (dbp), high-density lipoprotein cholesterol (hdl), and low-density lipoprotein cholesterol (ldl).

To investigate the linearity of the age effects on the multi-type recurrent events and all-cause mortality, we conduct a preliminary analysis for all these events separately using coxph function with option cluster (to analyze the recurrent events) and pspline (to use a penalized spline basis) in R package survival ³¹. Table 3 compares the models with either linear or nonparametric age functions using the Akaike information criterion (AIC). All recurrent event hazard models with nonparametric age functions have smaller AIC than their counterparts with linear age functions, suggesting that nonparametric age functions are preferable in modeling all three types of recurrent events. However, the mortality hazard model with linear age function has smaller AIC compared to the counterpart with nonparametric age function, suggesting that linear age function is more appropriate in modeling the hazard of mortality. Therefore, we include nonparametric functions of age in the hazard models of the multi-type recurrent events.

Table 3.

AIC for the models with either linear or nonparametric functions of age.

Event type	Linear	Nonparametric
CHD	9822.6	9819.1
Stroke	4937.8	4934.0
Heart failure	7604.4	7592.3
All-cause mortality	20195.4	20224.5

To assess the performance of all three models (the joint, reduced, and parametric models), we use the model selection criteria discussed in Section 3.3. Table 4 presents DIC₃, EAIC, and EBIC, LPML values for all three models. The results suggest that the joint model outperforms the other two models in all criteria with smaller DIC₃, EAIC, EBIC, and larger LPML, and it is selected as the final model. Table 5 compares the posterior mean, standard deviation (SD), and 95% equal-tail credible intervals from all three models. We find that there is no significant difference between usual care and pravastatin treatment in the risks of CHD (HR= exp(0.128) = 1.137; 95% CI [0.957, 1.353]), stroke (HR= 1.106; 95% CI [0.894, 1.376]), heart failure (HR= 0.953; 95% CI [0.766, 1.196]), and all-cause mortality (HR= 1.025; 95% CI [0.815, 1.303]). The results are consistent with the final ALLHAT-LLT article²⁵. The results in Table 5 also indicate the significant effects of diabetes. For example, diabetes significantly increases the rates of CHD (HR= 1.721; 95% CI [1.406, 2.081]), stroke (HR= 1.697; 95% CI [1.323, 2.162]), heart failure (HR= 2.162; 95% CI [1.718, 2.716]), and all-cause mortality (HR= 2.275; 95% CI [1.721, 3.013]), which is also consistent with the literature⁵². The complete lists of covariate effects are given in Web Tables 1–4.

Table 4.

Model selection criteria for the ALLHAT-LLT study. DIC₃: deviance information criterion. EAIC: expected Akaike information criterion; EBIC: expected Bayesian information criterion; LPML: log pseudo-marginal likelihood.

Criteria	Joint model	Reduced model	Parametric model
DIC3	28356.9	33445.8	28443.5
EAIC	26654.5	32261.6	26765.0
EBIC	27525.7	33111.2	27636.2
LPML	−16417.4	−17708.7	−16509.6

Table 5.

Parameter estimations from the joint model, reduced model, and parametric model. PM: posterior mean.

	Joint Model				Reduced model				Parametric Model
	PM	SD	95% CI		PM	SD	95% CI		PM	SD	95% CI
For CHD events
age (years)									0.078	0.007	0.065	0.091
llt (usual care)	0.128	0.088	−0.044	0.302	0.141	0.080	−0.019	0.297	0.130	0.090	−0.044	0.308
gender (female)	−0.599	0.108	−0.811	−0.387	−0.486	0.097	−0.684	−0.302	−0.613	0.107	−0.821	−0.410
diabetes (no)	−0.543	0.099	−0.733	−0.341	−0.411	0.088	−0.578	−0.236	−0.562	0.103	−0.761	−0.358
For stroke events
age (years)									0.088	0.008	0.072	0.104
llt (usual care)	0.101	0.110	−0.112	0.319	0.105	0.110	−0.109	0.318	0.094	0.111	−0.121	0.315
gender (female)	−0.215	0.129	−0.468	0.029	−0.133	0.130	−0.381	0.125	−0.230	0.126	−0.482	0.009
diabetes (no)	−0.529	0.123	−0.771	−0.280	−0.438	0.122	−0.674	−0.201	−0.551	0.119	−0.778	−0.320
For heart failure events
age (years)									0.117	0.008	0.101	0.134
llt (usual care)	−0.048	0.113	−0.267	0.179	−0.046	0.119	−0.284	0.189	−0.045	0.115	−0.264	0.174
gender (female)	−0.213	0.135	−0.474	0.052	−0.171	0.140	−0.455	0.106	−0.219	0.140	−0.506	0.046
diabetes (no)	−0.771	0.120	−0.999	−0.541	−0.699	0.125	−0.939	−0.458	−0.772	0.128	−1.022	−0.518
For all-cause mortality
age (years)	0.155	0.011	0.134	0.177	0.079	0.004	0.071	0.087	0.159	0.012	0.138	0.182
llt (usual care)	0.025	0.119	−0.205	0.265	0.035	0.059	−0.084	0.153	0.020	0.118	−0.208	0.254
gender (female)	−0.726	0.146	−1.013	−0.444	−0.295	0.068	−0.432	−0.163	−0.728	0.146	−1.012	−0.440
diabetes (no)	−0.822	0.141	−1.103	−0.543	−0.381	0.064	−0.507	−0.253	−0.817	0.138	−1.091	−0.547
σ1	2.117	0.076	1.969	2.269	1.400	0.065	1.276	1.528	2.117	0.078	1.972	2.277
σ2	2.036	0.096	1.852	2.229	1.667	0.103	1.473	1.869	2.033	0.099	1.834	2.228
σ3	2.456	0.092	2.278	2.643	2.286	0.094	2.112	2.475	2.467	0.089	2.296	2.644
ρ21	0.864	0.034	0.792	0.923	0.576	0.074	0.424	0.708	0.875	0.035	0.797	0.937
ρ31	0.921	0.018	0.881	0.952	0.986	0.014	0.949	1.000	0.929	0.018	0.893	0.961
ρ32	0.708	0.041	0.625	0.786	0.547	0.062	0.422	0.665	0.712	0.041	0.630	0.788
δ1	1.671	0.420	0.943	2.680					0.812	0.667	0.748	2.942
δ2	0.364	0.226	−0.176	0.754					0.749	0.790	−0.348	1.656
δ3	−0.135	0.218	−0.661	0.235					0.102	0.654	−1.023	1.322

Moreover, Table 5 displays the estimates of the entries in the covariance matrix of random effects b_i. Specifically, the random effects variances are σ̂₁ = 2.117 (95% CI [1.969, 2.269]), σ̂₂ = 2.036 (95% CI [1.852, 2.229]), and σ̂₃ = 2.456 (95% CI [2.278, 2.643]), for the recurrent events of CHD, stroke, and heart failure, respectively. Conditional on the observed risk factors, we observe significant positive correlation between the risks of CHD and stroke events (ρ̂₂₁ = 0.864, 95% CI [0.792, 0.923]), between the risks of CHD and heart failure events (ρ̂₃₁ = 0.921, 95% CI [0.881, 0.952]), and between the risks of stroke and heart failure events (ρ̂₃₂ = 0.708, 95% CI [0.625, 0.786]). This phenomena suggests that subjects with one type of cardiovascular disease events (CHD, stroke, or heart failure) are very likely to experience another type of cardiovascular disease events than those without. Moreover, the risk of having recurrent CHD events is positively associated with the hazard of all-cause mortality (δ̂₁ = 1.671, 95% CI [0.943, 2.680]), conditional on the observed risk factors. It suggests that subjects with CHD events tend to die earlier. However, the other two types of recurrent events (stroke and heart failure) do not show significant correlation with all-cause mortality in this study.

To visualize the nonparametric covariate functions, Figure 5 displays the estimated covariate functions of age (solid lines) in the risks of CHD (left panel), stroke (middle panel), and heart failure (right panel) from the joint model, along with the 95% pointwise credible intervals (dashed lines). The results suggest that risks of cardiovascular disease events (CHD, stroke, and heart failure) increase as age increases, but not in a linear fashion. In general, age increases the risks of cardiovascular disease events more rapidly at older ages than at younger ages. On average, the hazard ratio of CHD for one year increase in age is 1.048 (95% CI [1.029, 1.064]) when age is less than 80 years. After that, the hazard ratio increases to 1.124 (95% CI [1.043, 1.190]). For stroke, the hazard ratio for one year increase in age increases rapidly when age is larger than 65 years (HR= 1.025, 95% CI [0.993, 1.080] for age ≤ 65; HR= 1.106, 95% CI [1.046, 1.124] for age > 65). For heart failure, the hazard ratio for one year increase in age increases rapidly when age is larger than 85 years (HR= 1.076, 95% CI [1.052, 1.094] for age ≤ 85; HR= 1.315, 95% CI [1.082, 1.520] for age > 85). The phenomena of cardiovascular disease event risks increasing nonlinearly as age growth is also reported in some epidemiology studies^20,28.

Figure 5.

Estimated nonparametric covariate functions of age in the risks of CHD (left panel), stroke (middle panel), and heart failure (right panel). Estimated functions, solid; 95% pointwise credible intervals, dashed line.

6 Discussion

In this article, we propose a multivariate frailty model to jointly analyze the multi-type recurrent events and dependent terminal event with nonparametric covariate functions. The recurrent events and dependent terminal event are linked together via shared random effects, which represent the subject-specific heterogeneity in the hazard functions. To model the nonparametric covariate functions, we use the B-spline smooth functions with the penalty terms being replaced by random walk prior distributions. Through these model specification with fairly small number of parameters in piece-wise constant functions and B-spline basis function, satisfactory results can be obtained for a moderate to large data sets. The simulation study indicates that in the scenario of independent termination, the joint model provides results comparable to the reduced model, and both outperform the parametric model. In the presence of a dependent terminal event and nonparametric covariate functions, the joint model provides more accurate parameter estimates than the reduced model and the parametric model. In the analysis of the ALLHAT-LLT study, the joint model has a better fit than the reduced model and parametric model. We have identified insignificant pravastatin treatment effects, significant diabetes effects, and nonlinear age effects on the risks of CHD, stroke, and heart failure, but insignificant pravastatin treatment effect, significant diabetes effect, and linear age effect on the hazard of all-cause mortality. Moreover, we conclude that the risk of one type of cardiovascular disease event is positively correlated to other types of cardiovascular disease events, and the risk of CHD is positively correlated to the all-cause mortality. For simplicity of illustration, we include only one nonparametric covariate function for each type of recurrent events. Additional nonparametric covariate functions can be incorporated using the same approach. The Bayesian B-spline we adopt is capable of modeling a moderate to reasonably large number of nonparametric regression functions simultaneously⁴¹.

Our proposed model has some limitations that we view as future research directions. We have chosen a multivariate normal distribution for the random effects vector because it is flexible in modeling the covariance structure within and between various types of recurrent events and it has meaningful interpretation on correlation. Due to these reasons, it has been used in modeling multi-type recurrent event data in several articles^12,19. In generalized linear mixed models, misspecification of random effects distribution has little impact on the parameters that are not associated with the random effects^53–55. The impact of random effects misspecification in our joint model framework warrants further investigation. One limitation is the shared random effects assumption, under which the model of dependent terminal event shares the frailty terms with the models of multi-type recurrent events. While this shared random effects model is easy to implement and has low dimension of random effects distribution, it makes a relatively strong assumption about the association between subject-specific heterogeneity. Molenberghs and Verbeke⁵⁶ pointed out that this kind of shared parameter model is a special case of the multivariate frailty model, which in the current context assumes the multi-type recurrent events and terminal events are linked by a four-dimensional vector of correlated random effects with a multivariate normal distribution. Moreover, we have excluded the subjects with missing data in covariates. Missing covariate is a common issue in longitudinal data analysis and is an active research area. Ibrahim et al.⁵⁷ gives excellent review of common approaches for inference in generalized linear models with missing covariate data. How to address the missing covariate issue in the proposed modeling framework is an interesting future research direction.

Dynamic prediction of risk of death using the information from recurrent events has recently become a research topic of significant clinical interest⁵⁸. How to conduct the dynamic prediction in the framework of multi-type recurrent events is an interesting further research topic. As a last note, the risks of future disease events may change after some events occur. For example, the incidence of first stroke may elevate the risks of future stroke and other cardiovascular disease events. Several multi-state models have been studied in the semi-competing risk data to model the effect of nonfatal event occurrence on the risk of terminal event⁵⁹. In the context of recurrent events, each event can be considered as one state and the number of states increases as the number of events increases. We would like to assess the event effects in our proposed model as a future research endeavor.