Bayesian Hierarchical Model for Multiple Repeated Measures and Survival Data: an application to Parkinson's Disease

Abstract

Multilevel item response theory (MLIRT) models have been increasingly used to analyze the multivariate longitudinal data of mixed types (e.g., continuous and categorical) in clinical studies. To address the possible correlation between multivariate longitudinal measures and time to terminal events (e.g., death and dropout), joint models that consist of a MLIRT submodel and a survival submodel have been previously developed. However, in multisite studies, multiple patients are recruited and treated by the same clinical site. There can be a significant site correlation because of common environmental and socioeconomic status, and similar quality of care within site. In this article, we develop and study several hierarchical joint models with the hazard of terminal events dependent on shared random effects from various levels. We conduct extensive simulation study to evaluate the performance of various models under different scenarios. The proposed hierarchical joint models are applied to the motivating DATATOP study to investigate the effect of tocopherol in slowing PD progression.

1 Introduction

Parkinson's disease (PD) is a chronic progressive neurodegenerative disorder manifested clinically by tremor, rigidity, slow movement, and impaired balance [1]. In the United States alone, PD affects about 1% of people older than 60 years [2]. Morbidity and mortality associated with PD burden result in an estimated six billion dollars in healthcare cost in the United States annually. Because the risk of PD increases with age, the financial and public health burden of PD is expected to increase as the population ages. The impairment caused by PD is multidimensional and progressive. Its multidimensional nature precludes a single outcome to measure disease progression. Many clinical trials of PD often collect multiple longitudinal outcomes of mixed types (categorical and continuous) to assess the treatment effects on overall improvement.

To analyze such multivariate longitudinal data, multilevel item response theory (MLIRT) models have been increasingly used [3–9]. According to MLIRT, the observed measurements are viewed as imperfect clinical manifestations of the interaction between a univariate subject-specific latent disease severity and measurement-specific parameters (e.g., the measurements’ ability to distinguish PD patients in disease severity) in the first level measurement model. Then in the second level structural model, the latent disease severity is regressed on predictors (e.g., treatment, disease duration, and time), and subject-specific random effects (describing between-subject differences) to study the overall treatment effects via a single hypothesis test [10–14]. Advantages of the MLIRT model include that (1) it avoids adjustment of type I error rates and provides improved treatment effect estimate [15]; (2) it separates the measurement-specific parameters and subject-specific predictors so that both may be understood and studied separately; (3) it accounts for all sources of correlation within the same subject via the random effects and it utilizes the full longitudinal information of all subjects; (4) each subject's disease progression can be readily monitored using the latent disease severity. The MLIRT model has been increasingly used in studying many diseases such as PD disability [6, 16], Alzheimer's disease [17], Huntington's disease [18], and dementia [19].

During the follow-up process of PD studies, some patients may have terminal events such as death, dropout, and withdrawal of consent. The terminal events are often correlated with the multivariate longitudinal measurements. The presence of such a dependent failure time is often termed “informative censoring”, ignoring which could lead to biased estimates [20]. To address this issue, joint analysis of survival with repeated measures has been increasingly common [20–24]. In the IRT modeling framework, Wang et al [4] proposed a joint model to analyze multiple-item ordinal quality of life data in the presence of death. He and Luo [8] developed a joint model for multiple longitudinal outcomes of mixed types, subject to outcome-dependent terminal events. Luo [9] relaxed the proportional hazard (PH) assumption in He and Luo [8] and developed a joint modeling framework replacing the PH model by various parametric accelerated failure time (AFT) models.

However, most joint models developed in the literature have focused on the subject level. In many multisite PD studies such as DATATOP study [25], ELLDOPA study [26], and LS-1 study [27], multiple patients are recruited and treated by the same clinical site. There can be a significant site correlation because the patients in the same clinical site may share common environmental (e.g., medical environment related variables) and socioeconomic status and have similar quality of care. The existing MLIRT models ignore site correlation and potentially increase the variability of parameters, resulting in considerable bias in parameter estimates [28–30].

In this article, we will develop a hierarchical joint modeling framework in which a hierarchical MLIRT model is used for the multi-level multivariate longitudinal outcomes and a Cox PH model is used for the dependent terminal event. The two models are linked via multi-level random effects. The rest of the article proceeds as follows, In Section 2 we describe a motivating clinical trial, the data structure, and the dependent terminal event. Section 3 discusses the joint random effects model, Bayesian inference, and Bayesian model selection criteria. Section 4 provides a simulation study to assess the performance of the proposed joint model. In Section 5, we apply the proposed model to a motivating clinical trial dataset. Section 6 gives some concluding remarks and discussions. To facilitate easy reading and implementation of the proposed models, the codes have been posted at the web supplement.

2 A motivating clinical trial

The methodological development in this article is motivated by the DATATOP (Deprenyl And Tocopherol Antioxidative Therapy Of Parkinsonism) study, a double-blind, placebo-controlled multicenter clinical trial to determine if deprenyl and/or tocopherol administered to patients with early PD will slow the progression of PD. A total of 800 patients were randomly assigned in a 2 × 2 factorial design to receive double-placebo, active tocopherol alone, active deprenyl alone, and both active tocopherol and deprenyl. The patients who did not receive tocopherol (double-placebo and active deprenyl alone groups, 401 patients) are combined and referred to as placebo group. The patients who received tocopherol (active tocopherol alone and both active tocopherol and deprenyl groups, 399 patients) are combined and referred to as treatment group. Please refer to Shoulson et al [25] for details of the DATATOP trial.

The multiple outcomes collected include UPDRS (Unified Parkinson's Disease Rating Scale) total score, Schwab and England activities of daily living (SEADL), modified Hoehn and Yahr scale (HY), measured at 10 visits (baseline, months 1, and every 3 months starting from month 3 to month 24). UPDRS is the sum of 44 questions each measured on a 5-point scale (0-4) and it is approximated by a continuous variable with integer value from 0 (not affected) to 176 (most severely affected). SEADL is a measurement of activities of daily living [31] and it is an ordinal variable with integer value from 0 to 100 incrementing by 5, with larger value reflecting better clinical outcomes. HY is a scale describing how the symptoms of PD progress. It is an ordinal variable with possible values at 1, 1.5, 2, 2.5, 3, 4, and 5, with higher values being clinically worse outcome. We have recoded SEADL variable so that higher values in all outcomes correspond to worse clinical conditions. In addition, we have combined some categories with zero or small counts so that SEADL and HY have 8 and 5 categories, respectively.

Before the end of the study, some patients (193 and 183 patients in the placebo and treatment groups, respectively) reached the primary study endpoint, e.g., in the judgement of the blinded investigator, the patients reached a level of functional disability sufficient to initiate levodopa therapy, which can ameliorate the disease symptoms. We hence define time to levodopa therapy as the terminal event. To visualize the correlation between the terminal event and the outcome measurements, Figure 1 displays mean UPDRS values over time for DATATOP patients with follow-up time less than 6 months (solid line), 6-12 months (dotted line), and more than 12 months (dashed line). It indicates that patients with shorter follow-up had higher UPDRS values (more severe PD).

Figure 1.

Mean UPDRS values over time.

Patients were recruited and treated in 28 research centers in the United States and Canada. There can be a significant clustering effect, which may represent homogeneity of quality of care in the same research center. In addition, patients treated in the same research center may share a common environmental and socioeconomic status. To this end, we will develop a hierarchical joint modeling framework of multivariate longitudinal outcomes and survival data that accounts for four levels of nesting, i.e., multiple outcomes (level 1) are nested within visits (level 2) which are nested within patients (level 3) clustered within centers (level 4).

3 Model and estimation

3.1 The multilevel item response model

Let y_hijk (binary, ordinal, and continuous) be the observed outcome k (k = 1, . . . , K) at visit j (j = 1, . . . , J_i, where j = 1 is baseline and J_i is the number of visits of patient i) from patient i (i = 1, . . . , I_h, where I_h is the number of patients in center h) within center h (h = 1, . . . , H). Let y_hij = (y_hij1, . . . , y_hijK)′ and let y_hi = (y′_hi1, . . . , y′_hiJi)′. All outcomes are coded so that larger observation values correspond to worse clinical conditions. Let θ_hij be a univariate latent variable measuring disease severity of patient i from center h at visit j, with a higher value denoting more severe status. In the first level measurement model, we model the binary outcomes, the cumulative probabilities of ordinal outcomes, and the continuous outcomes as functions of θ_hij and outcome-specific parameters.

$$\mathrm{logit}\left\{p(y_{hijk}=1\mid {\theta }_{hij})\right\}=a_k+b_k{\theta }_{hij},$$ (1)

$$\mathrm{logit}\left\{p(y_{hijk}\le l\mid {\theta }_{hij})\right\}=a_{kl}-b_k{\theta }_{hij},\text{with}l=1,2,\dots ,n_k-1,$$ (2)

$$y_{hijk}=a_k+b_k{\theta }_{hij}+{∊}_{hijk},$$ (3)

where ${∊}_{hijk}\sim N(0,{\sigma }_k^2)$, a_k is the outcome-specific “difficulty” parameter and b_k (always positive) is the outcome-specific “discriminating” parameter representing the discrimination of outcome k, i.e., the degree to which outcome k discriminates between patients with different latent disease severity θ_hij. In model (2), the kth ordinal outcome has n_k categories and n_k − 1 thresholds a_k1, . . . , a_kl, . . . , a_knk−1 with the order constraint a_k1 < . . . < a_kl < . . . < a_knk−1. The probability that patient i is in category l on outcome k at visit j is p(Y_hijk = l|θ_hij) = p(Y_hijk ≤ l|θ_hij) − p(Y_hijk ≤ l − 1|θ_hij).

In the second level structural multilevel model, the latent disease severity θ_hij is regressed on covariates of interest, visit time, and random effects.

$${\theta }_{hij}={\mathit{X}}_{hi0}{\beta }_0+u_{hi0}+v_h+({\mathit{X}}_{hi1}{\beta }_1+u_{hi1})t_{hij},$$ (4)

where X_hi0 and X_hi1 are vectors of covariates associated with disease severity and progression rate, respectively, X_hi0 may or may not be identical to X_hi1, t_hij is visit time variable with t_hi1 = 0 for baseline, random intercept u_hi0 and random slope u_hi1 determine the subject-specific disease severity and progression rate, respectively, and the center-specific random effects v_h follows $N(0,{\sigma }_v^2)$. It is of note that IRT models are over-parameterized. To make the model identifiable, we set V ar(u_hi0) = 1 and set a_k1 = 0 in one ordinal outcome. The random effects vector u_hi = (u_hi0, u_hi1)′ follows N₂(0, Σ), with covariance matrix Σ being denoted by $\{(1,\rho {\sigma }_u),(\rho {\sigma }_u,{\sigma }_u^2)\}$ with ${\sigma }_u^2$ being the variance of u_hi1 and ρ being the correlation coefficient. The random effects vector u_hi accounts for three sources of correlation within the same patient, i.e., longitudinal (same outcome at different visits), inter-source (different outcomes at the same visit) and cross correlation (different outcomes at different visits). The random effects v_h accounts for the within-center correlation. Under the conditional independence assumption (conditional on the random effects u_hi and v_h, all measurements from each patient are independent) [32], the conditional likelihood of the multiple longitudinal outcomes for patient i is

$$L_y({\Phi }_y\mid {\mathit{y}}_{hi},{\mathit{u}}_{hi},v_h)=\prod _{j=1}^{J_i}\prod _{k=1}^{K}p(y_{hijk}\mid {\mathit{u}}_{hi},v_h),$$ (5)

where p(y_hijk|u_i) is the conditional density function of y_hijk obtained from models (1) to (4), Φ_y is the parameter vector. For notation convenience, we let the difficulty parameter vector be a = (a′₁, . . . , a′_k, . . . , a′_K)′, with a_k being numeric for binary and continuous outcomes and a_k = (a_k1, . . . , a_knk−1)′ for ordinal outcomes. Let the discrimination vector be b = (b₁, . . . , b_K)′, and let the regression vector be β = (β′₀ ,β′₁)′. So Φ_y = (a′, b′, β′, ρ, σ_u, σ_v, σ_k)′.

Let t_hi denote the time to terminal event for patient i within center h, δ_hi (1 if the terminal event is observed and 0 if not needed) denote the censoring indicator for t_hi, and X_hi denote vector of possible risk factors. Vector X_hi can share part of or all covariates in X_hi0 and X_hi1. The hazard of having a terminal event at time t_hi is

$$\lambda \left(t_{hi}\right)={\lambda }_0\left(t_{hi}\right)\exp ({\mathit{X}}_{hi}\gamma +{\nu }_0{u}_{hi0}+{\nu }_1{u}_{hi1}+{\nu }_2{v}_h),$$ (6)

where γ, ν = (ν₀, ν₁, ν₂)′ are unknown parameters, and λ₀(t_hi) is the baseline hazard function. The associated survival function is $S\left(t_{hi}\right)=\exp [-{\int }_0^{t_{hi}}\lambda \left(s\right)ds]$. The likelihood of event outcome t_hi and δ_hi for patient i is

$$L_s({\Phi }_s\mid t_{hi},{\delta }_{hi},{\mathit{u}}_{hi},v_h)=\lambda {\left(t_{hi}\right)}^{{\delta }_{hi}}S\left(t_{hi}\right),$$ (7)

where the parameter vector Φ_s = (γ′, ν′)′. Conditional on the random effects u_hi and v_h, y_hi is assumed to be independent of t_hi. The full likelihood of the joint model for patient i within center h is

$$L(\Phi \mid {\mathit{y}}_{hi},t_{hi},{\delta }_{hi},{\mathit{u}}_{hi},v_h)=L_y({\Phi }_y\mid {\mathit{y}}_{hi},{\mathit{u}}_{hi},v_h)L_s({\Phi }_s\mid t_{hi},{\delta }_{hi},{\mathit{u}}_{hi},v_h)p\left({\mathit{u}}_{hi}\right)p\left(v_h\right),$$ (8)

where p(u_hi) and p(v_h) are the density functions of u_hi and v_h, the parameter vector Φ = (Φ′_y, Φ′_s)′. The “cross-equation correlation” between models (4) and (6) is introduced by the center level random effects v_h and subject level random effects u_hi. Specifically, positive parameters ν₀ and ν₁ indicates that the patients with worse disease severity (higher u_hi0) and faster disease progression rate (higher u_hi1) tend to need levodopa therapy earlier. Similarly, positive parameter ν₂ suggests that the patients within the center with more severe PD patients tend to have higher hazard of needing levodopa therapy. We thereafter refer to the propose joint modeling framework (8) as “full model”.

Moreover, we consider two special cases of the full model. The first case (refer to as “reduced model A”) is that the hazard of survival event only depends on the subject level random effects u_hi.

$$\begin{array}{cc}\hfill & {\theta }_{hij}={\mathit{X}}_{hi0}{\beta }_0+u_{hi0}+v_h+({\mathit{X}}_{hi1}{\beta }_1+u_{hi1})t_{hij}\hfill \\ \hfill & \lambda \left(t_{hi}\right)={\lambda }_0\left(t_{hi}\right)\exp ({\mathit{X}}_{hi}\gamma +{\nu }_0{u}_{hi0}+{\nu }_1{u}_{hi1}).\hfill \end{array}$$ (9)

Another special case (refer to as “reduced model B”) is that the hazard of survival event only depends on the center-level random effects v_h.

$$\begin{array}{cc}\hfill & {\theta }_{hij}={\mathit{X}}_{hi0}{\beta }_0+u_{hi0}+v_h+({\mathit{X}}_{hi1}{\beta }_1+u_{hi1})t_{hij}\hfill \\ \hfill & \lambda \left(t_{hi}\right)={\lambda }_0\left(t_{hi}\right)\mathrm{epx}({\mathit{X}}_{hi}\gamma +{\nu }_2{v}_h).\hfill \end{array}$$ (10)

3.2 Bayesian inference

The full likelihood formulation in model (8) involves unspecified baseline hazard function λ₀(·). We adopt a piecewise constant function to approximate λ₀(·). It has been shown that survival models using a piecewise constant baseline hazard function yield good estimators for both fixed effects and frailty [33, 34], although fixed cut points need to be specified a priori. It is more flexible than the a priori choices of baseline hazard distribution (e.g., Weibull distribution) and it retains enough model structure [35]. Specifically, we divide the follow-up period into m intervals by every 1/mth quantiles denoted by τ₁, τ₂, . . . , τ_m and τ₀ = 0 or the smallest failure time. The baseline hazard vector is g = (g₁, g₂, . . . , g_m), and the piecewise constant baseline hazard function is defined as ${\lambda }_0\left(t\right)={\sum }_{l=1}^m{g}_l{I}_l\left(t\right)$, with indicator function I_l(t) = 1 if τ_l−1 ≤ t < τ_l and 0 otherwise.

To infer the unknown parameter vector Φ, we use Bayesian inference based on Markov Chain Monte Carlo (MCMC) posterior simulations. We use vague priors on all elements in the parameter vector Φ. Specifically, the prior distributions of all elements in β andγ, ν₀, ν₁, and ν₂ are N(0, 100). We use the prior distribution b_k ~ Uniform(0, 30), k = 1, . . . , K, to ensure positivity. The prior distribution for the difficulty parameter a_k of the continuous outcomes is a_k ~ N(0, 100). To obtain the prior distributions for the threshold parameters of ordinal outcome k, we let a_k1 ~ N(0, 100), and a_kl = a_k,l−1 + δ_l for l = 2, n_k − 1, with δ_l ~ N(0, 100)I(0, ), i.e., normal distribution left censored at 0. We use the prior distribution ρ ~ Uniform[−1, 1], and σ_k, σ_u, σ_v ~ Gamma(0.01, 0.01).

The model fitting is performed in OpenBUGS (OpenBUGS version 3.2.2) by specifying the full likelihood function and the prior distributions of all unknown parameters. We use the history plots available in OpenBUGS and view the absence of apparent trend in the plot as evidence of convergence. In addition, we use the Gelman-Rubin diagnostic to ensure the scale reduction $\widehat{R}$ of all parameters are smaller than 1.1 [36].

3.3 Bayesian model selection criteria

There are a wide variety of model selection criteria in Bayesian inference. We adopt a model selection approach using the deviance information criterion (DIC) proposed by Spiegelhalter et al [37]. The DIC provides an assessment of model fitting and a penalty for model complexity. The deviance statistic is defined as D(Φ) = −2 log f(y|Φ) + 2 log h(y), where f(y|Φ) is the likelihood function for the observed data y given the parameter vector and h(y) denotes a standardizing function of the data alone that has no impact on model selection [38]. The DIC is defined as $\mathrm{DIC}=2\overline{D}-D\left(\overline{\Phi }\right)=\overline{D}+p_D$, where $\overline{D}=E_{\Phi \mid y}\left[D\right]$ is the posterior mean of the deviance, $D\left(\overline{\Phi }\right)=D\left(E_{\Phi \mid y}\left[\Phi \right]\right)$ is the deviance evaluated at the posterior mean $\overline{\Phi }$ of the parameter vector, and $p_D=\overline{D}-D\left(\overline{\Phi }\right)$ is the effective number of parameters. A smaller value of DIC indicates a better-fitting model. In addition, we use the expected Akaike information criterion (EAIC) and the expected Bayesian (or Schwarz) information criterion (EBIC) as model selection tools [38]. The EAIC and EBIC can be estimated as $\mathrm{EAIC}=\overline{D}+2p$ and $\mathrm{EBIC}=\overline{D}+p\log \left(N\right)$, where p is the number of elements in the parameter vector Φ. Smaller values of EAIC and EBIC indicate better predictive ability of the model.

Moreover, Bayes factors (BF) is a Bayesian alternative to p values for testing hypotheses and for quantifying the degree to which observed data support or conflict with a hypothesis. Let two competing models be M₁ and M₂. The Bayes factor in favor of model M₁ over M₂ is defined as:

$$\mathrm{BF}(M_1;M_2)=\frac{p(M_1\mid \mathit{y})∕p(M_2\mid \mathit{y})}{p\left(M_1\right)∕p\left(M_2\right)}=\frac{p(\mathit{y}\mid M_1)}{p(\mathit{y}\mid M_2)},$$ (11)

where p(M_i) is the prior probability of model M_i, where i = 1, 2, p(M_i|y) is the posterior probability of model M_i, and p(y|M_i) is the predictive probability of observing data y under model M_i, and $p(\mathit{y}\mid M_i)=\int f(\mathit{y}\mid {\Phi }_i,M_i)p({\Phi }_i\mid M_i)d{\Phi }_i$, where p(Φ_i|M_i) is the prior distribution for parameter vector Φ_i under model M_i. When the Bayes factor is greater than 100, decisive evidence is shown in favor of model M₁. To avoid the integral involved in computation of Bayes factor, the Laplace-Metropolis estimator based on the normal distribution [39] is adopted to approximate the predictive probability. Specifically, $p(\mathit{y}\mid M_i)\approx {\left(2\pi \right)}^{d_i∕2}{\mid {\Sigma }_i\mid }^{1∕2}f(\mathit{y}\mid {\overline{\Phi }}_i,M_i)p({\overline{\Phi }}_i\mid M_i)$, where d_i is the number of the parameters in Φ_i, Σ_i is the posterior covariance matrix of Φ_i, ${\overline{\Phi }}_i$ is the posterior mean of parameters, $p({\overline{\Phi }}_i\mid M_i)$ is the prior probability of parameters evaluated at ${\overline{\Phi }}_i$, and $f(\mathit{y}\mid {\overline{\Phi }}_i,M_i)$ is the likelihood when parameters are at the posterior mean values.

4 Simulation studies

In this section, we conduct simulation study under two settings to compare the performance of the proposed hierarchical joint models (full model and two reduced models). In both settings, we generate 200 datasets with sample size H = 25 centers and I_h = 32 patients per center, totally 800 patients. The data structure similar to the motivating dataset and it has one continuous outcome and three ordinal outcomes with 7, 7, 10 categories, and 5 visits. We generate data by the following model:

$$\begin{array}{cc}\hfill & {\theta }_{hij}={\beta }_0{x}_{hi}+u_{hi0}+v_h+({\beta }_1+{\beta }_2{x}_{hi}+u_{hi1})t_{hij}\hfill \\ \hfill & \lambda \left(t_{hi}\right)={\lambda }_0\exp (\gamma x_{hi}+{\nu }_0{u}_{hi0}+{\nu }_1{u}_{hi1}+{\nu }_2{v}_h).\hfill \end{array}$$ (12)

The covariate x_hi takes value 0 or 1 each with probability 1/2 to mimic treatment assignment. The time vector t_hi = (t_hi1, . . . , t_hi5)′ = (0, 0.17, 0.5, 1.5, 2.5)′. We set the coefficients to be β = (β₀₁, β₁₀, β₁₁)′ = (−1, 0.4, −0.5)′, γ = −0.7. The parameters for the continuous outcome are set to be a₁ = 3, b₁ = 1.5, and σ₁ = 1. The parameters for the ordinal outcomes are set to be a₂ = (0, 1, 1.8, 2.6, 3.3, 4), a₃ = (−2.7, −0.6, 2, 2.8, 5, 6), a₄ = (−1, −0.1, 0.5, 1, 1.5, 2, 2.4, 2.8, 3.3), b ₂ = 2, b₃ = 0.4, b₄ = 0.7. Assume that center-specific random effects $v_h\sim N(0,{\sigma }_v^2)$ with σ_v = 1, subject-specific random effects u_hi = (u_uhi0, u_hi1)′ ~ N₂(0, ), where $\Sigma =((1,\rho {\sigma }_u),(\rho {\sigma }_u,{\sigma }_u^2))$ with ρ = 0.4 and σ_u = 1.3. The baseline hazard λ₀ is set to 0.07. The independent censoring time is sampled from uniform(10, 20).

In setting I, we assume ν = (0.4, 1.0, 1.0)′ so that the hazard of terminal event depends on both the subject-specific and center-specific random effects. Table 1 displays bias (the average of the posterior means minus the true values), standard error (SE, the square root of the average of the posterior variance), standard deviation (SD, the standard deviation of the posterior means, and coverage probabilities (CP) of 95% equal-tail credible intervals of two reduced models and full model (true model). The results suggest that full model provides parameter estimates with very small biases, SE being close to SD, and the coverage probabilities being close to the nominal level 0.95. In contrast, when all component ν in are nonzero as in setting I, incorrectly constraining some of the model association to 0 lead to in accurate results. For example, both reduced models A and B give biased estimates and poor coverage probabilities for the regression vectors β and γ, in additional to the variance-related parameters ρ, σ_u, and σ_v. The estimates of the difficulty parameter vector a, the discriminating vector b, and σ₁ from both reduced models and full model have small bias, SE close to SD, and CP close to 95% (please refer Table 1 in the web supplement).

Table 1.

Simulations results from reduced model A, reduced model B, and full model for setting I (the terminal event dependent on both the subject-specific and center-specific random effects).

	Reduced Model A				Reduced Model B				Full Model
	Bias	SE	SD	CP	Bias	SE	SD	CP	Bias	SE	SD	CP
For longitudinal outcomes
β01 = –1.0	0.037	0.078	0.082	0.920	0.007	0.079	0.082	0.935	0.009	0.079	0.084	0.925
β1o = 0.4	0.140	0.091	0.105	0.640	–0.501	0.079	0.093	0.000	–0.007	0.086	0.088	0.945
β11 = –0.5	-0.029	0.116	0.129	0.910	0.124	0.106	0.106	0.785	–0.001	0.112	0.121	0.925
ρ = 0.4	0.009	0.043	0.044	0.945	–0.092	0.052	0.050	0.550	–0.006	0.044	0.042	0.940
σu = 1.3	0.068	0.071	0.078	0.825	–0.172	0.055	0.054	0.135	0.001	0.063	0.061	0.960
σv = 1.0	–0.247	0.126	0.117	0.600	0.047	0.165	0.157	0.940	0.030	0.162	0.142	0.965
For survival
γ = –0.7	0.055	0.153	0.150	0.945	0.225	0.098	0.122	0.395	0.011	0.150	0.159	0.930
vo = 0.4	0.110	0.094	0.114	0.730					–0.000	0.079	0.076	0.940
v1 = 1.0	–0.032	0.082	0.093	0.905					0.001	0.083	0.076	0.965
V2 =1.0					–0.303	0.067	0.065	0.010	0.009	0.079	0.081	0.935

In setting II, we assume ν = (0.4, 1.0, 0)′ so that the hazard of terminal event only depends on the subject-specific, but not the center-specific random effects. Table 2 displays the simulation results of setting II from reduced model A (true model) and full model. The results suggest that both models generate comparable results, i.e., the bias is negligible, SE is close to SD, and the credible interval coverage probabilities are reasonably close to 95%. Under model overparameterization, the estimate of η₂ from full model is correctly close to zero. The estimates of the difficulty parameter vector a, the discrimination vector b, and σ₁ from both reduced model A and full model have small bias, SE close to SD, and CP close to 95% (please refer Table 2 in the web supplement).

Table 2.

Simulations results from reduced model A and full model for setting II (the terminal event only dependent on the subject-specific random effects).

	Reduced Model A				full Model
	Bias	SE	SD	CP	Bias	SE	SD	CP
For longitudinal outcomes
β01 = –1.0	0.000	0.079	0.085	0.940	0.008	0.079	0.084	0.925
β10 = 0.4	0.003	0.086	0.090	0.930	–0.009	0.086	0.092	0.930
β11 – 0.5	–0.001	0.112	0.110	0.930	0.011	0.112	0.121	0.925
ρ = 0.4	–0.002	0.044	0.037	0.980	–0.000	0.044	0.038	0.975
σu= 1.3	0.003	0.064	0.065	0.940	–0.002	0.065	0.065	0.955
σv = 1.0	0.027	0.162	0.162	0.930	0.025	0.162	0.152	0.945
For survival
γ = –0.7	–0.007	0.149	0.165	0.935	0.007	0.149	0.159	0.945
v0 = 0.4	–0.005	0.077	0.074	0.950	–0.001	0.078	0.075	0.955
v1 = 1.0	0.002	0.083	0.084	0.950	0.000	0.084	0.085	0.955
v2 = 0.0					–0.009	0.065	0.068	0.940

From the simulation study, we conclude that when the hazard of terminal event depends on both the subject-specific and center-specific random effects, full model provides satisfactory results while reduced models A and B with incorrect assumption on the dependence structure result in considerable biases and poor coverage probabilities. On the other hand, when the hazard of a terminal event only depends on the subject-specific, but not the center-specific random effects, the overparameterized full model provides results comparable to the reduced model which has correct assumption on the dependence structure.

5 Real Data Analysis

In this section, we apply the proposed hierarchical joint models and the Bayesian inference framework to the motivating DATATOP study. Specifically, we compare full model, reduced models A and B, and the model without accounting for center information (refer to as “no-center model”). For all the results in this section, we use two parallel MCMC chains with overdispersed initial values, and run each chain for 40, 000 iterations. The first 30, 000 iterations are discarded as burn-in and the inference is based on the remaining 10, 000 iterations from each chain. Good mixing properties of the chains for the model parameter are observed in the trace plots. The scale reduction $\widehat{R}$ of all parameters are smaller than 1.1. To specify the baseline hazard functions λ₀(t_hi), we divide the time to levodopa therapy into M = 3 intervals by every 1/Mth quantiles. We have also explored other selections of M and the results are very similar.

The PD patients in the DATATOP study were recruited and treated by a total of 28 research centers. The number of patients in each center varies from 10 (center 4) to 44 (center 19) with mean and standard deviation being 24.9 and 9.8, respectively. There are 104 patients with center information missing, which we collapse into one “artificial” center to have a total of 29 research centers. Table 3 displays the research center numbers and the number of patients in each center. We consider the treatment assignment variable x_hi (1 treatment, and 0 if placebo) as a single covariate so that X_hi0 = x_hi, X_hi1 = (1, x_hi)′, and γ = x_hi. Hence, model (4) is θ_hij = β₀₁x_hi + u_hi0 + v_h + (β₁₀ + β₁₁x_hi + u_hi1)t_hij, with visit times being transformed in two-year unit as t_ij = (0, 1, 3, 6, 9, 12, 15, 18, 21, 24)/24. The Cox model for the time to levodopa therapy is λ(t_hi) = λ₀(t_hi)exp(γx_hi + ν₀u_hi0 + ν₁u_hi1 + ν₂v_h). Table 4 compares all the hierarchical joint models using the model selection criteria discussed in Section 3.2. All of the proposed hierarchical joint models perform significantly better than the no-center model with smaller Dbar, EAIC, EBIC, and DIC values, suggesting that the hierarchical joint models accounting for the center information are more preferable than the no-center model. The BFs in favor of reduced model A over reduced model B or no-center model are much larger than 100, indicating decisive evidence in favor of reduced model A [40]. In comparison, the BF in favor of reduced model A over full model is 38.9, indicating very strong evidence in favor of reduced model A [40]. Reduced model A is selected as the final model because it has the best predictive ability with the smallest Dbar, EAIC, EBIC, and DIC values, and the BF.

Table 3.

Research center numbers and the number of patients in each center. Center 31 is the "artificial" center with all patients without center information.

Center #	1	2	4	5	6	7	8	9	10	11	12	14	15	16	17
Count	23	35	10	43	23	18	22	18	19	41	16	37	25	20	26
Center #	18	19	20	21	22	23	24	25	26	27	28	29	30	31
Count	34	44	31	20	19	18	30	19	14	18	11	42	20	104

Table 4.

Model comparison statistics for the DATATOP dataset. Dbar: the posterior mean of the deviance; EAIC: expected Akaike information criterion; EBIC: expected Bayesian information criterion; DIC: deviance information criterion; BF: Bayes factor. Boldface indicates the preferred model.

	Dbar	EAIC	EBIC	DIC	BF
Reduced model A	53876.9	53930.9	54057.4	55186.9	Ref
Reduced model B	54576.9	54628.9	54750.7	55846.9	>> 100
Full model	53876.9	53932.9	54064.1	55196.9	38.9
no-center model	53976.9	54028.9	54150.7	55326.9	>> 100

Table 5 compares the posterior mean, standard deviation (SD), and 95% equal-tail credible intervals from full model and reduced models A and B. It is observed that all models give different parameter estimates, although the same set of parameters are identified for significance by all models. For instance, reduced model A suggests that tocopherol is associated with 0.021 unit decrease (${\widehat{\beta }}_{01}$, 95% CI: [−0.167, 0.102]) in disease severity, while it is associated with 0.027 (95% CI: [−0.183, 0.095]) and 0.027 (95% CI: [−0.127, 0.091]) unit decrease in disease severity in reduced model B and full model, respectively. The placebo patients have significant disease progression at the rate of 2.527 units per two year (${\widehat{\beta }}_{10}$ 95% CI: [2.261, 2.822]) in reduced model A, while the rates of progression are 2.258 (95% CI: [2.007, 2.503]) and 2.520 (95% CI: [2.251, 2.782]) units per two year in reduced model B and full model, respectively. In comparison, reduced model A suggests that the tocopherol patients have disease progression rate of 2.398 units per two year (${\widehat{\beta }}_{10}+{\widehat{\beta }}_{11}$ 95% CI: [1.794, 3.012]) with insignificant tocopherol treatment effect of slowing down the disease progression rate by −0.129 per two year (${\widehat{\beta }}_{11}$ 95% CI: [−0.467, 0.190]). Similar conclusions can be made from reduced model B and full model. Moreover, tocopherol has insignificant effects on the probability of having levodopa therapy, i.e., the time to levodopa therapy for tocopherol patients is estimated to be 0.971 times ($\exp \left(\widehat{\gamma }\right)$, 95% CI: [0.718, 1.288]) that of non-tocopherol patients in reduced model A v.s. 0.957 times (95% CI: [0.781, 1.164]) in reduced model B and 0.976 times (95% CI: [0.746, 1.402]) in full model.

Table 5.

Results of fitting reduced model A, reduced model B, and full model in the DATATOP dataset.

	Reduced Model A				Reduced Model B				Full Model
	Mean	SD	95% CI		Mean	SD	95% CI		Mean	SD	95% CI
For longitudinal outcomes
β 01	–0.021	0.067	–0.167	0.102	–0.027	0.068	–0.183	0.095	–0.027	0.056	–0.127	0.091
β 10	2.527	0.145	2.261	2.822	2.258	0.127	2.007	2.503	2.520	0.138	2.251	2.782
β 11	–0.129	0.167	–0.467	0.190	–0.085	0.164	–0.423	0.224	–0.109	0.150	–0.410	0.176
ρ	0.450	0.039	0.372	0.523	0.422	0.044	0.334	0.507	0.450	0.041	0.370	0.529
σ u	2.190	0.108	1.987	2.413	1.995	0.102	1.800	2.205	2.185	0.112	1.969	2.400
σ v	1.757	0.250	1.350	2.318	1.819	0.260	1.392	2.412	1.763	0.255	1.346	2.336
For survival
γ	–0.029	0.148	–0.331	0.253	–0.044	0.102	–0.247	0.152	–0.024	0.141	–0.307	0.250
V0	0.349	0.070	0.211	0.489					0.358	0.071	0.211	0.492
Vl	0.660	0.050	0.563	0.760					0.665	0.050	0.571	0.771
V2					0.097	0.108	–0.122	0.308	0.021	0.160	–0.293	0.338

Table 5 also suggests that patients with worse disease severity (u_hi0) and higher disease progression rate (higher u_hi1) tend to need levodopa therapy earlier, as indicated by the significance of the estimates of parameters ν₀ (0.349, 95% CI: [0.211, 0.489] in reduced model A and 0.358, 95% CI: [0.211, 0.492] in full model) and ν₁ (0.660, 95% CI: [0.563, 0.760] in reduced model A and 0.665, 95% CI: [0.571, 0.771]). However, there is no significant association between disease severity and survival at the research center level, as indicated by the insignificance of ${\widehat{\nu }}_2$ (0.097, 95% CI: [−0.122, 0.308] in reduced model B and 0.021, 95% CI: [−0.293, 0.338] in full model).

To provide parameter interpretation on clinically interpretable scales, we tabulate in Table 6 the change from baseline to each follow-up visit for the outcome UPDRS, and the odds ratio of the cumulative probability at certain thresholds of the outcomes SEADL and HY, all estimated from reduced model A. At month 3, the placebo patients are expected to increase 3.035 (95% CI: [2.758, 3.339]) units in UPDRS, and the odds ratio of the cumulative probabilities at certain thresholds of SEADL and HY are expected to be 0.596 (95% CI: [0.563, 0.628]) and 0.678 (95% CI: [0.649, 0.705]), respectively, compared with baseline, while the treatment patients are expected to increase 2.880 (95% CI: [2.573, 3.161]) units in UPDRS, and the odds ratio of the cumulative probabilities at certain thresholds of SEADL and HY are expected to be 0.612 (95% CI: [0.582, 0.647]) and 0.692 (95% CI: [0.664, 0.722]), respectively, compared with baseline. At month 15, the placebo patients are expected to increase 15.177 (95% CI: [13.790, 16.697]) units in UPDRS, and the odds ratio of the cumulative probabilities at certain thresholds of SEADL and HY are expected to be 0.076 (95% CI: [0.056, 0.097]) and 0.144 (95% CI: [0.115, 0.174]), respectively, compared with baseline, while the treatment patients are expected to increase 14.400 (95% CI: [12.865, 15.804]) units in UPDRS, and the odds ratio of the cumulative probabilities at certain thresholds of SEADL and HY are expected to be 0.087 (95% CI: [0.067, 0.113]) and 0.159 (95% CI: [0.129, 0.196]), respectively, compared with baseline. At the end of the study at month 24, the placebo patients are expected to increase 24.283 (95% CI: [22.065, 26.716]) units in UPDRS, and the odds ratio of the cumulative probabilities at certain thresholds of SEADL and HY are expected to be 0.016 (95% CI: [0.010, 0.024]) and 0.045 (95% CI: [0.031, 0.061]), respectively, compared with baseline, while the treatment patients are expected to increase 23.040 (95% CI: [20.583, 25.287]) units in UPDRS, and the odds ratio of the cumulative probabilities at certain thresholds of SEADL and HY are expected to be 0.020 (95% CI: [0.013, 0.031]) and 0.053 (95% CI: [0.038, 0.074]), respectively, compared with baseline. Similar conclusions can be made at months 9 and 21.

Table 6.

Change from baseline to each follow-up visit for the outcome UPDRS and the odds ratio (OR) of the cumulative probability at certain thresholds of the outcomes SEADL and HY, estimated from reduced model A. The numbers in the subscript are the standard deviations (SD). The numbers within the square brackets are 95% equal-tailed credible intervals.

	UPDRS		OR{p(SEADL ≤ l)}		OR{p(HY ≤ I)}
	Placebo	Treatment	Placebo	Treatment	Placebo	Treatment
Month 3	3.0350.150	2.8800.150	0.5960.017	0.6120.017	0.6780.015	0.6920.015
	[2.758, 3.339]	[2.573, 3.161]	[0.563, 0.628]	[0.582, 0.647]	[0.649, 0.705]	[0.664, 0.722]
Month 9	9.1060.451	8.6400.449	0.2130.018	0.2300.019	0.3120.020	0. 3320.021
	[8.274, 10.018]	[7.719, 9.482]	[0.178, 0.247]	[0.197, 0.271]	[0.273, 0.351]	[0.293, 0.376]
Month 15	15.1770.752	14.4000.748	0.0760.011	0.0870.012	0.1440.015	0.1590.017
	[13.790, 16.697]	[12.865, 15.804]	[0.056, 0.097]	[0.067, 0.113]	[0.115, 0.174]	[0.129, 0.196]
Month 21	21.2481.053	20.1601.047	0.0270.005	0.0330.006	0.0670.010	0.0770.011
	[19.307, 23.376]	[18.010, 22.126]	[0.018, 0.038]	[0.022, 0.047]	[0.049, 0.087]	[0.057, 0.102]
Month 24	24.2831.203	23.0401.196	0.0160.004	0.0200.004	0.0450.008	0.0530.009
	[22.065, 26.716]	[20.583, 25.287]	[0.010, 0.024]	[0.013, 0.031]	[0.031, 0.061]	[0.038, 0.074]

Note that the standard error of the random intercept (u_hi1) from reduced model A is 2.190 (95% CI: [1.987, 2.413]), and the estimate of the correlation coefficient ρ between two subject-specific random effects (u_hi0 and u_hi1) is 0.450 (95% CI: [0.372, 0.523]). The significant positive correlation indicates that the patients with worse PD disease severity (u_hi0) tend to have faster disease progression rate (u_hi1) and vice versa. To gain further insight into u_hi0, u_hi1, and ρ, we plot in Web Figure 1 the rankings of patients’ subject-specific disease severity (u_hi0, upper panel) and disease progression rate (u_hi1, lower panel). Each patient is ordered by his or her rank: patients at the bottom left corner show milder disease severity (lower rank) and slower disease progression rate (lower rank), while patients at the upper right corner have poorer disease severity (higher rank) and faster disease progression rate (higher rank). To visualize the effect of the correlation coefficient ρ, we have selected two patients as examples. Patient 213 has the worst disease severity and he/she ranks No. 9 in disease progression rate. Patient 105 has the fastest disease progression rate and he/she ranks No. 30 in disease severity.

Web Figure 2 displays the ranking of the posterior estimates (with 95% equal-tail credible intervals) of the center-specific random effects (v_h) on disease severity. We observe large variability of the center-specific random effects, in addition to different lengths of the credible intervals as a result of different numbers of patients in each center. Center 1 has the most patients with severe PD, while center 18 has the most patients with mild PD. It is interesting to note that the estimate of v_h of center 31 (the “artificial” center) ranks No. 17 and its length of credible interval is the smallest because the number of patients in this center is much larger than all other centers.

6 Discussion

In this article, we propose several hierarchical joint models that consists of a MLIRT submodel for the multivariate longitudinal outcomes and a Cox submodel for the dependent terminal event, while accounting for the possible correlation within each research center. We assume that the survival time may be correlated with the multivariate longitudinal outcomes at various subject and center levels. We adopt Bayesian inference framework based on Markov chain Monte Carlo (MCMC) simulation for parameter estimation. This Bayesian framework work provides accurate parameter estimates, in additional to various subject-specific PD disease severity estimation and center profiling. It also provides additional insight into the correlation between the multivariate longitudinal outcomes and the dependent terminal event at both subject and center levels. It has been shown in the extensive simulation studies that incorrect assumption on the correlation structure between longitudinal and survival outcomes results in considerable biases and poor coverage probabilities in many parameters of interest. We apply the hierarchical joint models to analyze the motivating DATATOP study dataset and use various model selection criteria to select among all the candidate models. All proposed hierarchical joint models have better fit than the model without accounting for the within-center correlation. The treatment tocopherol has insignificant effects in either reducing the PD disease severity or slowing the PD disease progression. We identify significant correlation between the longitudinal and survival outcomes at the subject level, but not at the center level. We provide subject-specific disease severity and disease progression rate estimates for all patients at each visit and the center profiling information, in addition to the figures for clear visualization. The proposed models have the capability to accommodate hierarchically structured data, to estimate latent disease severity at different levels, and to investigate the relationships between disease severity and predictors at different levels. The proposed model can be easily implemented using the publicly available BUGS language and can be readily accessible to, modified, and extended by applied researchers.

Our modeling framework provides great flexibility for extensions. For example, this article only considers a single type terminal event. In the presence of multiple failure types, e.g., levodopa therapy and dropout due to disease-related causes, the proposed joint model can be extended to accommodate competing risks survival data. Further, the treatment main effect (β₀) is assumed constant over the follow-up period. This convenient assumption may not be realistic and may lead to bias in the coefficient estimates [41, 42]. Varying-coefficient models have been studied in the literature [43–51]. Developing a formal nonparametric statistical model in the framework of MLIRT models to define and estimate the time-dependent treatment effect is part of our ongoing research. Moreover, to develop statistical inference tools that account for covaraite measurement errors, continuous outcome outliers, and departure from normal distributional assumptions for random effects is an interesting direction of future research.

Inference from the proposed Bayesian MLIRT modeling framework is valid when the model fits the data and the model assumptions are met. In this paper, we have assumed unidimensionality (i.e., there exists a single latent variable θ_hij to measure the disease severity). But there may be multiple latent variables representing multidimensional (e.g., sensoria, functions, and cognition) impairment caused by PD. Expanding the unidimensional MLIRT model to the multidimensional one is an interesting direction of future research. Also, we have assumed local dependence (i.e., condition on random effect u_hi and v_h, all measurements y_hijk and time t_hi are independent. However, there may be multiple variables that are constructed and implemented as a unit of measurement (i.e., measuring the same ability such as activities of daily living). Therefore, these variables may have “testlet” effect. How to detect the testlet effect and testlet dimension in the proposed model warrants further investigation. Moreover, differential item functioning (DIF, people with the same latent disease severity have different outcome measurements) detection in this complex hierarchical model is also part of our future research.