A Bayesian shared parameter model for joint modeling of longitudinal continuous and binary outcomes

Abstract

Joint modeling of associated mixed biomarkers in longitudinal studies leads to a better clinical decision by improving the efficiency of parameter estimates. In many clinical studies, the observed time for two biomarkers may not be equivalent and one of the longitudinal responses may have recorded in a longer time than the other one. In addition, the response variables may have different missing patterns. In this paper, we propose a new joint model of associated continuous and binary responses by accounting different missing patterns for two longitudinal outcomes. A conditional model for joint modeling of the two responses is used and two shared random effects models are considered for intermittent missingness of two responses. A Bayesian approach using Markov Chain Monte Carlo (MCMC) is adopted for parameter estimation and model implementation. The validation and performance of the proposed model are investigated using some simulation studies. The proposed model is also applied for analyzing a real data set of bariatric surgery.

1. Introduction

In many medical studies, more than one biomarker, with different types of outcomes, may be collected for the same patient. In most of the applications, these biomarkers are highly associated. They may also show different patterns for the effect of treatment. Separate modeling of these biomarkers may lead to misleading results, since the associations among biomarkers are not taken into account.

Simultaneous or joint modeling of mixed biomarkers is an effective approach for accounting associations among different biomarkers and more likely to provide more valid results and conclusions.

The motivation for our study is based on the data collected from people with morbid obesity undergoing a weight loss surgery recorded in an Iranian hospital. This data set contains one longitudinal binary response and one continuous longitudinal outcome. The continuous longitudinal outcome (BMI in our study) was collected from the first visit of the study while the longitudinal binary outcome (having complication due to surgery) was collected after the surgery. The existence of missing values is an inevitable part of each longitudinal study. In our study, the pattern of missingness of two longitudinal outcomes is different. Therefore, we consider two missingness mechanisms for our two longitudinal outcomes.

Rubin [21] distinguished among three important missingness mechanisms. When missingness mechanism is unrelated to the missing or observed values, it is called missing completely at random (MCAR). When missingness mechanism depends only on the observed data, that is, given the observed data, missingness does not depend on unobserved values, missing mechanism is called missing at random mechanism (MAR). A mechanism wherein missingness depends on the unobserved data, perhaps in addition to the observed data, is called missing not at random (MNAR). In the likelihood and Bayesian paradigms and when mild regularity conditions are satisfied, the MCAR and MAR mechanisms are ignorable, in the sense that inferences can proceed by analyzing the observed data only, without explicitly addressing a parametric form for the missing mechanism. But, MNAR mechanism is non-ignorable that is for analyzing the data set with MNAR mechanism modeling missingness mechanism in addition to longitudinal measurements modeling is essential. In this status, ignoring the missingness mechanism leads to biased parameter estimates. Therefore, we need some methods that can be jointly modeled missingness mechanisms and longitudinal measurements.

Joint modeling of longitudinal mixed outcomes has been studied by many authors. Usually, a random effects model, a marginal model or a conditional model is used for accounting association between the response variables (vide [6–11,14,22,24]). Li et al. [16] proposed the use of a random effects model for joint modeling of longitudinal continuous, binary and ordinal variables in a Bayesian paradigm. Kurum et al. [15] propose a semivarying joint model for longitudinal binary and continuous outcome in a frequentist paradigm. Liu et al. [17] proposed a joint model for modeling longitudinal binary and continuous data in a Bayesian paradigm and they used a data augmentation approach for handling missing values considering an ignorable missingness mechanism. Also, Gaskins et al. [12] propose a non-ignorable drop-out model via using a pattern mixture model for joint modeling of two longitudinal outcomes.

In this paper, we propose a joint modeling of binary and longitudinal continuous data using a conditional model. As the patterns of misingness for two longitudinal outcomes are different, we use two shared parameter models for missingness mechanisms. Also, dependence of two missingness mechanisms is considered by a conditional model. We develop the Bayesian paradigm for parametric inference. To adopt the Bayesian approach, Markov Chain Monte Carlo (MCMC) methods are used. The iterative simulation of the conditional posterior distribution for each parameter is done using the Gibbs sampler, where the OpenBUGS software is used for model implementation. We use the proposed approach to analyze a real data with morbid obesity recorded in Rasul-e-Akram hospital of Tehran in Iran. In this study, all the patients who have a weight loss surgery between July 2008 and April 2017 are considered. The surgery is categorized on three levels: Mini-gastric bypass, RY gastric bypass and some other laparoscopic surgeries. The simultaneous effect of surgery on BMI and the indicator of complications due to surgery, as two associated mixed longitudinal responses, is investigated.

The rest of the paper is organized as follows. In Section 2, we describe the real data set that would be analyzed in this paper. In Section 3, the formulation of joint model for analyzing of longitudinal mixed continuous and binary measurements is discussed. Section 4 includes a Bayesian implementation of the proposed method. Some simulation studies are implemented for investigating the performance of the proposed method in Section 5. Section 6 includes our application where the proposed model is considered for analyzing the data set. Some conclusions are also given in the last section.

2. Motivation: applied example

The motivation of our study is based on the data collected from people with morbid obesity in Rasul-e-Akram hospital of Tehran which is a referral hospital for bariatric surgery (weight loss surgery) in Iran. The data are recorded between July 2008 and April 2017 for people who undergo Mini-gastric bypass, RY gastric bypass and some other laparoscopic surgery. These people were enrolled and followed up for 9 months. The data on these patients were drawn from Iran national obesity surgery data base. The evaluation of the patients was conducted at three phases: preoperation, peri-operation and postoperation follow-up at 1-, 3-, 6- and 9-month after surgery.

In the pre-operative phase, the basic information includes demographic characteristics (age, sex) and medical history [fatty liver, type 2 Diabetes, dyslipidemia, hypothyroidism (also called underactive or low thyroid)]. These are recorded and considered as predictors in our study. Table 1 summarizes these basic characteristics for each type of surgery. In this table, the mean of age and the proportion of having each characteristic are reported. Also, p-values are given for testing whether samples are from the same distribution for binary outcomes and for testing the equality of means for three types of surgeries. In the peri-operation phase which includes after surgery, after 10 days from surgery and at each follow-up period, the patients were examined for complications due to surgery such as vomiting, constipation, diarrhea, and dumping and the results were recorded. Also, BMI at each visit was recorded. This cohort study was conducted on 554 male patients and 2336 female patients. In this cohort study, 1558 out of 2890 patients have Mini-gastric bypass surgery, 1031 out of 2890 patients have RY gastric bypass surgery and the 301 remaining patients have other laparoscopic surgeries. The mean±sd of BMI for entry, before surgery, after surgery, 10th days after surgery and at 1-, 3-, 6- and 9-month after surgery are given by $44.958\pm 6.562$, $44.857\pm 6.236$, $45.362\pm 6.338$, $42.302\pm 6.002$, $40.581\pm 5.999$, $36.130\pm 5.570$, $32.561\pm 5.241$ and $30.559\pm 5.105$, respectively. The behavior of BMI for three types of surgeries during time are plotted in Figure 1. Also, patterns of missingness for BMI and indicator for complications due to surgery are plotted in Figure 2. In each panel of this figure, each row represents an existing pattern of missingness. The red color represents observed values and the gray color represents missingness for that time.

Table 1. Basic characteristics for each type of surgery and total patients ( $\mathrm{mean}\pm \mathrm{sd}$).

	Mini-gastric bypass	RY gastric bypass	Other laparoscopic	p-value	Total
Age	$38.132\pm 10.538$	$38.488\pm 10.065$	$36.827\pm 10.010$	0.063	$38.123\pm 10.324$
Gender	$0.799\pm 0.010$	$0.830\pm 0.011$	$0.777\pm 0.024$	0.055	$0.808\pm 0.007$
Fatty liver	$0.817\pm 0.009$	$0.729\pm 0.013$	$0.724\pm 0.025$	<0.001	$0.776\pm 0.007$
Helicobacter	$0.438\pm 0.012$	$0.424\pm 0.015$	$0.372\pm 0.027$	0.102	$0.426\pm 0.009$
Type 2 Diabetes	$0.212\pm 0.010$	$0.147\pm 0.011$	$0.089\pm 0.016$	<0.001	$0.176\pm 0.007$
Dyslipidemia	$0.384\pm 0.010$	$0.441\pm 0.011$	$0.335\pm 0.016$	<0.001	$0.399\pm 0.007$
Hypothyroidism	$0.198\pm 0.010$	$0.192\pm 0.012$	$0.153\pm 0.021$	0.176	$0.191\pm 0.007$

Figure 1.

Profiles of BMI measurements over time for different types of laparoscopic surgery superimposed by mean over all observed individuals at each time shown by black line. (a) Mini-gastric bypass surgery, (b) RY gastric bypass surgery, (c) other laparoscopic surgery.

Figure 2.

Pattern of missingness for BMI (left panel) and complication due to surgery (right panel). Each row represents a pattern of missingness at each time point. Red color represents observed values and gray color represents missingness for the occasion time.

3. Model specification

3.1. Joint model

The data structure can be described by four components. The first component contains continuous repeated measurements that are assumed to follow a linear mixed-effects model. The second component contains repeated measurements in a binary scale that are assumed to follow a mixed-effects logistic regression model. The last two components contain the missingness mechanisms for the first two components.

Let $Y_{ij}$ be the longitudinal continuous response for the ith, $i=1,2,\dots ,n,$ subject at time $t_{ij},j=1,2,\dots ,m^y$ and ${\mathit{Y}}_i=(Y_{i1},\dots ,Y_{i{m}^y}{)}^{\mathrm{\prime }}$ be the vector of longitudinal measurements for the ith subject at time ${\mathit{t}}_i=(t_{i1},\dots ,t_{i{m}^y}{)}^{\mathrm{\prime }}$. Assume that $Y_{ij}$ follows a linear mixed-effects model as

$$Y_{ij}={{\mathit{x}}_{ij}^y}^{\mathrm{\prime }}{\mathit{\beta }}_1+{{\mathit{w}}_{ij}^y}^{\mathrm{\prime }}{\mathit{b}}_{1i}+{\epsilon }_{ij},$$ (1)

where ${\mathit{\beta }}_1$ is a vector of $p^y$ unknown fixed-effects parameters, ${\mathit{x}}_{ij}^y$ is a $p^y$-vector of the explanatory variables, ${\mathit{b}}_{1i}$ is a vector of $q^y$ unobservable random effects, ${\mathit{w}}_{ij}^y$ is an $q^y$-dimensional design vector for random effects and ${\epsilon }_{ij}$ is the random error variable. We assume that ${\mathit{b}}_{1i}$s are independent and identically normally distributed with zero-mean vector and covariance matrix ${\mathit{D}}^y$. Typically, the error terms, ${\epsilon }_{ij}$'s, are assumed to be normal with zero mean and variance ${\sigma }_{\epsilon }^2$.

Also, assume that in addition to $m^y$-dimensional vector of longitudinal continuous measurements ${\mathit{Y}}_i$, longitudinal binary measurements are also observed for the ith subject. Let $Z_{ij}$ be the longitudinal binary response for the ith subject, $i=1,2,\dots ,n$ at time $s_{ij},j=1,2,\dots ,m^z$. Also, let ${\mathit{Z}}_i=(Z_{i1},\dots ,Z_{i{m}^z}{)}^{\mathrm{\prime }}$ be the vector of longitudinal binary measurements at time ${\mathit{s}}_i=(s_{i1},\dots ,s_{i{m}^z}{)}^{\mathrm{\prime }}$. Assume that $Z_{ij}$ follows the logistic mixed-effects regression model as

$$\begin{array}{rl}{Z}_{ij}|{\mathit{b}}_{2i},y_{ij}& \sim \mathrm{Ber}({\pi }_{ij}),\\ \text{logit}({\pi }_{ij})& ={{\mathit{x}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\beta }}_2+{{\mathit{w}}_{ij}^z}^{\mathrm{\prime }}{\mathit{b}}_{2i}+{\gamma }_j{y}_{ij},\end{array}$$ (2)

where ${\mathit{\beta }}_2$ is a vector of $p^z$ unknown fixed-effects parameters, ${\mathit{x}}_{ij}^z$ is a $p^z$-vector of the explanatory variables, ${\mathit{b}}_{2i}$ is a vector of $q^z$ unobservable random effects, ${\mathit{w}}_{ij}^z$ is a $q^z$-dimensional design vector for random effects and ${\gamma }_j$ is the effect of the first response at time $t_{ij}$ (i.e. $Y_{ij}$) on the second response on the time $s_{ij}$ (i.e. $Z_{ij}$), when $t_{ij}=s_{ij}$. This parameter can take into account the association between two responses at the same time. We assume that ${\mathit{b}}_{2i}$s are independent and identically normally distributed with zero-mean vector and covariance matrix ${\mathit{D}}^z$. Note that, for simplicity and our scientific question of interest in application, we assume that the longitudinal continuous response at a time point may affect the longitudinal binary response at the same time point.

Let $R_{ij}^y$ and $R_{ij}^z$ be the missingness indicators for the continuous and the binary response at time $t_{ij}$ and $s_{ij}$, respectively, such that $R_{ij}^y=1$ if $Y_{ij}$ is missed and $R_{ij}^z=1$ if $Z_{ij}$ is missed. We consider a shared parameter model [3] for considering non-ignorable mechanisms for ${\mathit{Y}}_i$ and ${\mathit{Z}}_i$. Assume that $R_{ij}^y$ follows the logistic mixed-effects regression model as

$$\begin{array}{rl}{R}_{ij}^y|{\mathit{b}}_{1i}& \sim \mathrm{Ber}({\pi }_{ij}^{R^y}),\\ \text{logit}({\pi }_{ij}^{R^y})& ={{\mathit{u}}_{ij}^y}^{\mathrm{\prime }}{\mathit{\alpha }}_1+{{\mathit{\psi }}^y}^{\mathrm{\prime }}{\mathit{b}}_{1i},\end{array}$$ (3)

where ${\mathit{\alpha }}_1$ is a vector of $p_r^y$ unknown fixed-effects parameters, ${\mathit{u}}_{ij}^y$ is a $p_r^y$-vector of the explanatory variables, ${\mathit{b}}_{1i}$ is the vector of unobservable random effects in model (1), ${\mathit{\psi }}^y$ is an $q^y$-dimensional vector of parameters that relate the missingness to the continuous outcome data by the shared random effects. Also, we assume that $R_{ij}^z$ follows the logistic mixed-effects regression model as

$$\begin{array}{rl}{R}_{ij}^z|{\mathit{b}}_{2i},r_{ij}^y& \sim \mathrm{Ber}({\pi }_{ij}^{R^z}),\\ \text{logit}({\pi }_{ij}^{R^z})& ={{\mathit{u}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\alpha }}_2+{{\mathit{\psi }}^z}^{\mathrm{\prime }}{\mathit{b}}_{2i}+{\kappa }_j{r}_{ij}^y,\end{array}$$ (4)

where ${\mathit{\alpha }}_2$ is a vector of $p_r^z$ unknown fixed-effects parameters, ${\mathit{u}}_{ij}^z$ is a $p_r^z$-vector of the explanatory variables, ${\mathit{b}}_{2i}$ is the vector of unobservable random effects in model (2), ${\mathit{\psi }}^z$ is an $q^z$-dimensional vector of the parameters that relates the missingness to the binary outcome data by the shared random effects and ${\kappa }_j$ is the effect of the first missingness mechanism at time $t_{ij}$ $(R_{ij}^y)$ on the second missingness mechanism at time $s_{ij}$ $(R_{ij}^z)$, when $t_{ij}=s_{ij}$. Let ${\mathit{R}}_i^y=(R_{i1}^y,R_{2i}^y,\dots ,R_{i{m}^y}^y{)}^{\mathrm{\prime }}$ and ${\mathit{R}}_i^z=(R_{i1}^z,R_{2i}^z,\dots ,R_{i{m}^z}^z{)}^{\mathrm{\prime }}$.

The joint model, given the random effects (for simplicity conditioning on the explanatory variables is dropped throughout) is given by

$$\begin{array}{rl}f({\mathit{y}}_i,{\mathit{z}}_i,{\mathit{r}}_i^y,{\mathit{r}}_i^z|{\mathit{b}}_{1i},{\mathit{b}}_{2i})& =f({\mathit{z}}_i|{\mathit{y}}_i,{\mathit{r}}_i^y,{\mathit{r}}_i^z,{\mathit{b}}_{1i},{\mathit{b}}_{2i})\times f({\mathit{y}}_i|{\mathit{r}}_i^y,{\mathit{r}}_i^z,{\mathit{b}}_{1i})\\ & \times f({\mathit{r}}_i^z|{\mathit{r}}_i^y,{\mathit{b}}_{2i},{\mathit{b}}_{2i})\times f({\mathit{r}}_i^y|{\mathit{b}}_{1i},{\mathit{b}}_{2i})\\ & =f({\mathit{z}}_i|{\mathit{y}}_i,{\mathit{r}}_i^z,{\mathit{b}}_{2i})\times f({\mathit{y}}_i|{\mathit{r}}_i^y,{\mathit{b}}_{1i})\times f({\mathit{r}}_i^z|{\mathit{r}}_i^y,{\mathit{b}}_{2i})\times f({\mathit{r}}_i^y|{\mathit{b}}_{1i})\\ & =f({\mathit{z}}_i|{\mathit{y}}_i,{\mathit{b}}_{2i})\times f({\mathit{y}}_i|{\mathit{b}}_{1i})\times f({\mathit{r}}_i^z|{\mathit{r}}_i^y,{\mathit{b}}_{2i})\times f({\mathit{r}}_i^y|{\mathit{b}}_{1i}).\end{array}$$ (5)

Also, given the random effects, we have

$$\begin{array}{rl}f({\mathit{y}}_i|{\mathit{b}}_{1i})& =\prod _{j=1}^{m^y}f(y_{ij}|{\mathit{b}}_{1i}),\end{array}$$ (6)

$$\begin{array}{rl}f({\mathit{z}}_i|{\mathit{b}}_{2i},{\mathit{y}}_i)& =\prod _{j=1}^{m^z}f(z_{ij}|{\mathit{b}}_{1i},y_{ij}),\end{array}$$ (7)

$$\begin{array}{rl}f({\mathit{r}}_i^y|{\mathit{b}}_{1i})& =\prod _{j=1}^{m^y}f(r_{ij}^y|{\mathit{b}}_{1i}),\end{array}$$ (8)

$$\begin{array}{rl}f({\mathit{r}}_i^z|{\mathit{b}}_{2i},{\mathit{r}}_i^y)& =\prod _{j=1}^{m^z}f(r_{ij}^z|{\mathit{b}}_{2i},r_{ij}^y).\end{array}$$ (9)

Therefore, the joint distribution function of $({\mathit{y}}_i,{\mathit{z}}_i,{\mathit{r}}_i^y,{\mathit{r}}_i^z)$ is given by

$$\begin{array}{rl}f({\mathit{y}}_i,{\mathit{z}}_i,{\mathit{r}}_i^y,{\mathit{r}}_i^z)& ={\int }_{{\mathit{b}}_{1i}}{\int }_{{\mathit{b}}_{2i}}f({\mathit{y}}_i,{\mathit{z}}_i,{\mathit{r}}_i^y,{\mathit{r}}_i^z|{\mathit{b}}_{1i},{\mathit{b}}_{2i})g({\mathit{b}}_{1i},{\mathit{b}}_{2i})\mathrm{d}{\mathit{b}}_{1i}\mathrm{d}{\mathit{b}}_{2i}\\ & ={\int }_{{\mathit{b}}_{1i}}{\int }_{{\mathit{b}}_{2i}}\prod _{j=1}^{m^y}f(y_{ij}|{\mathit{b}}_{1i})\prod _{j=1}^{m^z}f(z_{ij}|{\mathit{b}}_{2i},y_{ij})\\ & \times \prod _{j=1}^{m^y}f(r_{ij}^y|{\mathit{b}}_{1i})\prod _{j=1}^{m^z}f(r_{ij}^z|{\mathit{b}}_{2i},r_{ij}^y)g({\mathit{b}}_{1i},{\mathit{b}}_{2i})\mathrm{d}{\mathit{b}}_{1i}\mathrm{d}{\mathit{b}}_{2i}.\end{array}$$ (10)

Note that ${\mathit{y}}_i$ and ${\mathit{z}}_i$ can be partitioned to ${\mathit{y}}_i=({\mathit{y}}_i^{\mathrm{obs}},{\mathit{y}}_i^{\mathrm{mis}})$ and ${\mathit{z}}_i=({\mathit{z}}_i^{\mathrm{obs}},{\mathit{z}}_i^{\mathrm{mis}})$, respectively. Therefore,

$$\begin{array}{rl}f({\mathit{y}}_i^{\mathrm{obs}},{\mathit{z}}_i^{\mathrm{obs}},{\mathit{r}}_i^y,{\mathit{r}}_i^z)& ={\int }_{{\mathit{y}}_i^{\mathrm{mis}}}\sum _{{\mathit{z}}_i^{\mathrm{mis}}}{\int }_{{\mathit{b}}_{1i}}{\int }_{{\mathit{b}}_{2i}}f({\mathit{y}}_i,{\mathit{z}}_i,{\mathit{r}}_i^y,{\mathit{r}}_i^z|{\mathit{b}}_{1i},{\mathit{b}}_{2i})g({\mathit{b}}_{1i},{\mathit{b}}_{2i})\mathrm{d}{\mathit{b}}_{1i}\mathrm{d}{\mathit{b}}_{2i}\mathrm{d}{\mathit{y}}_i^{\mathrm{mis}}\\ & ={\int }_{{\mathit{b}}_{1i}}{\int }_{{\mathit{b}}_{2i}}f({\mathit{y}}_i^{\mathrm{obs}},{\mathit{z}}_i^{\mathrm{obs}},{\mathit{r}}_i^y,{\mathit{r}}_i^z|{\mathit{b}}_{1i},{\mathit{b}}_{2i})g({\mathit{b}}_{1i},{\mathit{b}}_{2i})\mathrm{d}{\mathit{b}}_{1i}\mathrm{d}{\mathit{b}}_{2i}.\end{array}$$

In this paper, we propose Bayesian approach via MCMC to estimate the parameters of the model. The approach is described in the next section.

4. Estimation via MCMC

The model of the previous section can be written as the following hierarchical model:

$$\begin{array}{rl}{Y}_{ij}|{\mathit{b}}_{1i}& \sim N({{\mathit{x}}_{ij}^y}^{\mathrm{\prime }}{\mathit{\beta }}_1+{{\mathit{w}}_{ij}^y}^{\mathrm{\prime }}{\mathit{b}}_{1i},{\sigma }^2),\\ Z_{ij}|{\mathit{b}}_{2i},y_{ij}& \sim \mathrm{Ber}({\pi }_{ij}),\\ R_{ij}^y|{\mathit{b}}_{1i}& \sim \mathrm{Ber}({\pi }_{ij}^{R^y}),\\ R_{ij}^z|{\mathit{b}}_{2i},r_{ij}^y& \sim \mathrm{Ber}({\pi }_{ij}^{R^z}),\end{array}$$ (11)

where ${\pi }_{ij}$, ${\pi }_{ij}^{R^y}$ and ${\pi }_{ij}^{R^z}$ are given in (2), (3) and (4), respectively. We now consider the following prior distributions for the unknown parameters:

$$\begin{array}{rl}{\mathit{\beta }}_1& \sim N_{p^y}({\mathit{\mu }}_{{\mathit{\beta }}_1},{\mathbf{\Sigma }}_{{\mathit{\beta }}_1}),\\ {\sigma }^2& \sim \mathrm{I}\mathrm{\Gamma }(a_{{\sigma }^2},b_{{\sigma }^2}),\\ {\mathit{\beta }}_2& \sim N_{p^z}({\mathit{\mu }}_{{\mathit{\beta }}_2},{\mathbf{\Sigma }}_{{\mathit{\beta }}_2}),\\ {\mathit{D}}_1& \sim IWishart({\mathbf{\Psi }}_{{\mathit{D}}_1},{\nu }_{{\mathit{D}}_1}),\\ {\mathit{D}}_2& \sim IWishart({\mathbf{\Psi }}_{{\mathit{D}}_2},{\nu }_{{\mathit{D}}_2}),\\ {\mathit{\alpha }}_1& \sim N_{p_r^y}({\mathit{\mu }}_{{\mathit{\alpha }}_1},{\mathbf{\Sigma }}_{{\mathit{\alpha }}_1}),\\ {\mathit{\alpha }}_2& \sim N_{p_r^z}({\mathit{\mu }}_{{\mathit{\alpha }}_2},{\mathbf{\Sigma }}_{{\mathit{\alpha }}_2}),\\ {\mathit{\psi }}^y& \sim N_{q^y}({\mathit{\mu }}_{{\mathit{\psi }}^y},{\mathbf{\Sigma }}_{{\mathit{\psi }}^y}),\\ {\mathit{\psi }}^z& \sim N_{q^z}({\mathit{\mu }}_{{\mathit{\psi }}^z},{\mathbf{\Sigma }}_{{\mathit{\psi }}^z}),\\ {\kappa }_j& \sim N({\mu }_{{\kappa }_j},{\sigma }_{{\kappa }_j}^2),j=1,2,\dots ,m^y,\\ {\gamma }_j& \sim N({\mu }_{{\gamma }_j},{\sigma }_{{\gamma }_j}^2),j=1,2,\dots ,m^z,\end{array}$$ (12)

where $\mathrm{I}\mathrm{\Gamma }(a,b)$ denotes the inverse gamma distribution with shape parameter a, scale parameter b, and mean a/b, $Iwishart(\mathit{\psi },\nu )$ denotes an inverse Wishart distribution with scale parameter ν and matrix parameter $\mathit{\psi }$. Also, $N_p(\mathit{\mu },\mathbf{\Sigma })$ represents the p-variate normal distribution with mean vector $\mathit{\mu }$ and covariance matrix $\mathbf{\Sigma }$.

The hyperparameters of (12) are assumed to be known and the priors are proper. In practice, the elicitation of hyperparameters may be difficult, they can be chosen based on strong prior knowledge or diffuse prior information. In this paper, we have no prior information from historical data or previous experience, so we prefer to assign the hyperparameters such that they lead to low-informative prior distribution for the parameters.

Suppose that ${\mathit{y}}^{\mathrm{obs}}=({{\mathit{y}}_1^{\mathrm{obs}}}^{\mathrm{\prime }},\dots ,{{\mathit{y}}_n^{\mathrm{obs}}}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$, ${\mathit{y}}^{\mathrm{mis}}=({{\mathit{y}}_1^{\mathrm{mis}}}^{\mathrm{\prime }},\dots ,{{\mathit{y}}_n^{\mathrm{mis}}}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$, ${\mathit{z}}^{\mathrm{obs}}=({{\mathit{z}}_1^{\mathrm{obs}}}^{\mathrm{\prime }},\dots ,{{\mathit{z}}_n^{\mathrm{obs}}}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$, ${\mathit{z}}^{\mathrm{mis}}=({{\mathit{z}}_1^{\mathrm{mis}}}^{\mathrm{\prime }},\dots ,{{\mathit{z}}_n^{\mathrm{mis}}}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$, ${\mathit{r}}^y=({{\mathit{r}}_1^y}^{\mathrm{\prime }},\dots ,{{\mathit{r}}_n^y}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$, ${\mathit{r}}^z=({{\mathit{r}}_1^z}^{\mathrm{\prime }},\dots ,{{\mathit{r}}_n^z}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$, ${\mathit{b}}_1=({\mathit{b}}_{11}^{\mathrm{\prime }},\dots ,{\mathit{b}}_{1n}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$ and ${\mathit{b}}_2=({\mathit{b}}_{21}^{\mathrm{\prime }},\dots ,{\mathit{b}}_{2n}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$. We further assume $\mathit{\theta }=\{{\mathit{\beta }}_1,{\mathit{\beta }}_2,{\sigma }^2,{\mathit{D}}_1,{\mathit{D}}_2,{\mathit{\alpha }}_1,{\mathit{\alpha }}_2,{\mathit{\psi }}^y,{\mathit{\psi }}^z,{\kappa }_k,{\gamma }_l\}$, for $k=1,2,,\dots ,m^y$ and $l=1,2,,\dots ,m^z$, to be the vector of unknown parameters. Also, elements of the parameter vector to be independent are assumed. The joint posterior density of all unobservable components is given by

$$\begin{array}{rl}\pi (\mathit{\theta },{\mathit{b}}_1,{\mathit{b}}_2,{\mathit{y}}^{\mathrm{mis}},{\mathit{z}}^{\mathrm{mis}}|{\mathit{y}}^{\mathrm{obs}},{\mathit{z}}^{\mathrm{obs}},{\mathit{r}}^y,{\mathit{r}}^z)& \propto \prod _{i=1}^n\prod _{j=1}^{m^y}\varphi (y_{ij};{{\mathit{x}}_{ij}^y}^{\mathrm{\prime }}{\mathit{\beta }}_1+{{\mathit{w}}_{ij}^y}^{\mathrm{\prime }}{\mathit{b}}_{1i},{\sigma }_{\epsilon }^2)\\ & \times {({\pi }_{ij}^{R^y})}^{r_{ij}^y}{(1-{\pi }_{ij}^{R^y})}^{r_{ij}^y}\\ & \times \prod _{i=1}^n\prod _{k=1}^{m^z}{{\pi }_{ik}}^{z_{ik}}{(1-{\pi }_{ik})}^{z_{ik}}\\ & \times {({\pi }_{ik}^{R^z})}^{r_{ik}^z}{(1-{\pi }_{ik}^{R^z})}^{r_{ik}^z}\times \pi (\mathit{\theta }),\end{array}$$ (13)

where $\pi (\mathit{\theta })$ is the joint prior distribution of the parameters i.e. the product of the priors given in (12). This joint posterior distribution is analytically intractable but MCMC methods such as the Gibbs sampler and the Metropolis–Hastings algorithm may be used to draw samples, from which features of the marginal posterior distribution of interest can be inferred. The Gibbs sampler works by drawing samples iteratively from conditional posterior distributions derived from (13). For this purpose, we need full conditional distributions. Let $\mathit{\vartheta }=(\mathit{\theta },{\mathit{b}}_1,{\mathit{b}}_2,{\mathit{y}}^{\mathrm{mis}},{\mathit{z}}^{\mathrm{mis}})$, and ω be one of its components. We define ${\mathit{\vartheta }}_{(-\omega )}$ for the above-mentioned vector when ω is omitted from it. Based on this notation, the full conditional distributions for the unknown parameters are given by:

$$\mathit{\beta }|{\mathit{\vartheta }}_{(-\mathit{\beta })},{\mathit{y}}^{\mathrm{obs}},{\mathit{z}}^{\mathrm{obs}},{\mathit{r}}^y,{\mathit{r}}^z\sim N_{p^y}({\mathbf{\Sigma }}_{\mathit{\beta }|\cdots }\left(\sum _{i=1}^n{{\mathit{x}}_i^y}^{\mathrm{\prime }}\frac{{\mathit{y}}_i-{{\mathit{w}}_i^y}^{\mathrm{\prime }}{\mathit{b}}_{1i}}{{\sigma }_{\epsilon }^2}\right),{\mathbf{\Sigma }}_{\mathit{\beta }|\cdots })$$ (14)

where ${\mathbf{\Sigma }}_{\mathit{\beta }|\cdots }=(\frac{\sum _{i=1}^n{{\mathit{x}}_i^y}^{\mathrm{\prime }}{\mathit{x}}_i^y}{{\sigma }_{\epsilon }^2}+{\mathbf{\Sigma }}_{{\mathit{\beta }}_1}^{-1}{)}^{-1}$.

$$\begin{array}{rl}{\sigma }_{\epsilon }^2|{\mathit{\vartheta }}_{(-{\sigma }_{\epsilon }^2)},{\mathit{y}}^{\mathrm{obs}},{\mathit{z}}^{\mathrm{obs}},{\mathit{r}}^y,{\mathit{r}}^z& \sim \mathrm{I}\mathrm{\Gamma }(\phantom{\sum _{i=1}^{n^{\frac{1}{2}}}}\frac{m_{y}n}{2}+a_{{\sigma }^2},b_{{\sigma }^2}\\ & +\frac{1}{2}\sum _{i=1}^n\sum _{j=1}^{m^y}(y_{ij}-{{\mathit{x}}_{ij}^y}^{\mathrm{\prime }}{{\mathit{\beta }}_1+{\mathit{w}}_{ij}^y}^{\mathrm{\prime }}{\mathit{b}}_{1i}{)}^2)\end{array}$$ (15)

$$\begin{array}{rl}\pi ({\mathit{\beta }}_2|{\mathit{\vartheta }}_{(-{\mathit{\beta }}_2)},{\mathit{y}}^{\mathrm{obs}},{\mathit{z}}^{\mathrm{obs}},{\mathit{r}}^y,{\mathit{r}}^z)& \propto \prod _{i=1}^n\prod _{j=1}^{m^z}\frac{\exp ({{\mathit{x}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\beta }}_2+{{\mathit{w}}_{ij}^z}^{\mathrm{\prime }}{\mathit{b}}_{2i}+{\gamma }_j{y}_{ij}{)}^{z_{ij}}}{1+\exp ({{\mathit{x}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\beta }}_2+{{\mathit{w}}_{ij}^z}^{\mathrm{\prime }}{\mathit{b}}_{2i}+{\gamma }_j{y}_{ij})}\\ & \times {\varphi }_{p^z}({\mathit{\beta }}_2;{\mathit{\mu }}_{{\mathit{\beta }}_2},{\mathbf{\Sigma }}_{{\mathit{\beta }}_2}),\end{array}$$ (16)

$$\begin{array}{rl}\pi ({\mathit{\alpha }}_1|{\mathit{\vartheta }}_{(-{\mathit{\alpha }}_1)},{\mathit{y}}^{\mathrm{obs}},{\mathit{z}}^{\mathrm{obs}},{\mathit{r}}^y,{\mathit{r}}^z)& \propto \prod _{i=1}^n\prod _{j=1}^{m^z}\frac{\exp ({{\mathit{u}}_{ij}^y}^{\mathrm{\prime }}{\mathit{\alpha }}_1+{{\mathit{\psi }}^y}^{\mathrm{\prime }}{\mathit{b}}_{1i}{)}^{r_{ij}^y}}{1+\exp ({{\mathit{u}}_{ij}^y}^{\mathrm{\prime }}{\mathit{\alpha }}_1+{{\mathit{\psi }}^y}^{\mathrm{\prime }}{\mathit{b}}_{1i})}\\ & \times {\varphi }_{p_r^y}({\mathit{\alpha }}_1;{\mathit{\mu }}_{{\mathit{\alpha }}_1},{\mathbf{\Sigma }}_{{\mathit{\alpha }}_1}),\end{array}$$ (17)

$$\begin{array}{rl}\pi ({\mathit{\alpha }}_2|{\mathit{\vartheta }}_{(-{\mathit{\alpha }}_2)},{\mathit{y}}^{\mathrm{obs}},{\mathit{z}}^{\mathrm{obs}},{\mathit{r}}^y,{\mathit{r}}^z)& \propto \prod _{i=1}^n\prod _{j=1}^{m^z}\frac{\exp ({{\mathit{u}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\alpha }}_2+{{\mathit{\psi }}^z}^{\mathrm{\prime }}{\mathit{b}}_{2i}+{\kappa }_j{r}_{ij}^y{)}^{r_{ij}^z}}{1+\exp ({{\mathit{u}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\alpha }}_2+{{\mathit{\psi }}^z}^{\mathrm{\prime }}{\mathit{b}}_{2i}+{\kappa }_j{r}_{ij}^y)}\\ & \times {\varphi }_{p_r^z}({\mathit{\alpha }}_2;{\mathit{\mu }}_{{\mathit{\alpha }}_2},{\mathbf{\Sigma }}_{{\mathit{\alpha }}_2}),\end{array}$$ (18)

$$\begin{array}{rl}{\mathit{D}}_1|{\mathit{\vartheta }}_{(-{\mathit{D}}_1)},{\mathit{y}}^{\mathrm{obs}},{\mathit{z}}^{\mathrm{obs}},{\mathit{r}}^y,{\mathit{r}}^z& \sim Iwishart({\mathit{\psi }}^y+\sum _{i=1}^n{\mathit{b}}_{1i}{\mathit{b}}_{1i}^{\mathrm{\prime }},\nu +q^y+n)\end{array}$$ (19)

$$\begin{array}{rl}{\mathit{D}}_2|{\mathit{\vartheta }}_{(-{\mathit{D}}_2)},{\mathit{y}}^{\mathrm{obs}},{\mathit{z}}^{\mathrm{obs}},{\mathit{r}}^y,{\mathit{r}}^z& \sim Iwishart({\mathit{\psi }}^z+\sum _{i=1}^n{\mathit{b}}_{2i}{\mathit{b}}_{2i}^{\mathrm{\prime }},\nu +q^z+n)\end{array}$$ (20)

$$\begin{array}{rl}\pi ({\mathit{\psi }}^y|{\mathit{\vartheta }}_{(-{\mathit{\psi }}^y)},{\mathit{y}}^{\mathrm{obs}},{\mathit{z}}^{\mathrm{obs}},{\mathit{r}}^y,{\mathit{r}}^z)& \propto \prod _{i=1}^n\prod _{j=1}^{m^z}\frac{\exp ({{\mathit{u}}_{ij}^y}^{\mathrm{\prime }}{\mathit{\alpha }}_1+{{\mathit{\psi }}^y}^{\mathrm{\prime }}{\mathit{b}}_{1i}{)}^{r_{ij}^y}}{1+\exp ({{\mathit{u}}_{ij}^y}^{\mathrm{\prime }}{\mathit{\alpha }}_1+{{\mathit{\psi }}^y}^{\mathrm{\prime }}{\mathit{b}}_{1i})}\\ & \times {\varphi }_{q^y}({\mathit{\psi }}^y;{\mathit{\mu }}_{{\mathit{\psi }}^y},{\mathbf{\Sigma }}_{{\mathit{\psi }}^y}),\end{array}$$ (21)

$$\begin{array}{rl}\pi ({\mathit{\psi }}^z|{\mathit{\vartheta }}_{(-{\mathit{\psi }}^z)},{\mathit{y}}^{\mathrm{obs}},{\mathit{z}}^{\mathrm{obs}},{\mathit{r}}^y,{\mathit{r}}^z)& \propto \prod _{i=1}^n\prod _{j=1}^{m^z}\frac{\exp ({{\mathit{u}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\alpha }}_2+{{\mathit{\psi }}^z}^{\mathrm{\prime }}{\mathit{b}}_{2i}+{\kappa }_j{r}_{ij}^y{)}^{r_{ij}^z}}{1+\exp ({{\mathit{u}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\alpha }}_2+{{\mathit{\psi }}^z}^{\mathrm{\prime }}{\mathit{b}}_{2i}+{\kappa }_j{r}_{ij}^y)}\\ & \times {\varphi }_{q^y}({\mathit{\psi }}^z;{\mathit{\mu }}_{{\mathit{\psi }}^z},{\mathbf{\Sigma }}_{{\mathit{\psi }}^z}),\end{array}$$ (22)

$$\begin{array}{rl}\pi ({\kappa }_j|{\mathit{\vartheta }}_{(-{\kappa }_j)},{\mathit{y}}^{\mathrm{obs}},{\mathit{z}}^{\mathrm{obs}},{\mathit{r}}^y,{\mathit{r}}^z)& \propto \prod _{i=1}^n\frac{\exp ({{\mathit{u}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\alpha }}_2+{{\mathit{\psi }}^z}^{\mathrm{\prime }}{\mathit{b}}_{2i}+{\kappa }_j{r}_{ij}^y{)}^{r_{ij}^z}}{1+\exp ({{\mathit{u}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\alpha }}_2+{{\mathit{\psi }}^z}^{\mathrm{\prime }}{\mathit{b}}_{2i}+{\kappa }_j{r}_{ij}^y)}\varphi ({\kappa }_j;{\mu }_{{\kappa }_j},{\sigma }_{{\kappa }_j}^2),\end{array}$$ (23)

$$\begin{array}{rl}\pi ({\gamma }_j|{\mathit{\vartheta }}_{(-{\gamma }_j)},{\mathit{y}}^{\mathrm{obs}},{\mathit{z}}^{\mathrm{obs}},{\mathit{r}}^y,{\mathit{r}}^z)& \propto \prod _{i=1}^n\frac{\exp ({{\mathit{x}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\beta }}_2+{{\mathit{w}}_{ij}^z}^{\mathrm{\prime }}{\mathit{b}}_{2i}+{\gamma }_j{y}_{ij}{)}^{z_{ij}}}{1+\exp ({{\mathit{x}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\beta }}_2+{{\mathit{w}}_{ij}^z}^{\mathrm{\prime }}{\mathit{b}}_{2i}+{\gamma }_j{y}_{ij})}\\ & \times \varphi ({\gamma }_j;{\mu }_{{\gamma }_j},{\sigma }_{{\gamma }_j}^2),\end{array}$$ (24)

$$\begin{array}{rl}\pi ({\mathit{b}}_{1i}|{\mathit{\vartheta }}_{(-{\mathit{b}}_{1i})},{\mathit{y}}^{\mathrm{obs}},{\mathit{z}}^{\mathrm{obs}},{\mathit{r}}^y,{\mathit{r}}^z)& \propto \prod _{j=1}^{m^y}\varphi (y_{ij};{{\mathit{x}}_{ij}^y}^{\mathrm{\prime }}{\mathit{\beta }}_1+{{\mathit{w}}_{ij}^y}^{\mathrm{\prime }}{\mathit{b}}_{1i},{\sigma }_{\epsilon }^2)\\ & \times \prod _{j=1}^{m^z}\frac{\exp ({{\mathit{u}}_{ij}^y}^{\mathrm{\prime }}{\mathit{\alpha }}_1+{{\mathit{\psi }}^y}^{\mathrm{\prime }}{\mathit{b}}_{1i}{)}^{r_{ij}^y}}{1+\exp ({{\mathit{u}}_{ij}^y}^{\mathrm{\prime }}{\mathit{\alpha }}_1+{{\mathit{\psi }}^y}^{\mathrm{\prime }}{\mathit{b}}_{1i})}\\ & \times {\varphi }_{q^y}({\mathit{b}}_{1i};\mathbf{0},{\mathit{D}}_1),\end{array}$$ (25)

$$\begin{array}{rl}\pi ({\mathit{b}}_{2i}|{\mathit{\vartheta }}_{(-{\mathit{b}}_{2i})},{\mathit{y}}^{\mathrm{obs}},{\mathit{z}}^{\mathrm{obs}},{\mathit{r}}^y,{\mathit{r}}^z)& \propto \prod _{j=1}^{m^z}\frac{\exp ({{\mathit{x}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\beta }}_2+{{\mathit{w}}_{ij}^z}^{\mathrm{\prime }}{\mathit{b}}_{2i}+{\gamma }_j{y}_{ij}{)}^{z_{ij}}}{1+\exp ({{\mathit{x}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\beta }}_2+{{\mathit{w}}_{ij}^z}^{\mathrm{\prime }}{\mathit{b}}_{2i}+{\gamma }_j{y}_{ij})}\\ & \times \prod _{j=1}^{m^z}\frac{\exp ({{\mathit{u}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\alpha }}_2+{{\mathit{\psi }}^z}^{\mathrm{\prime }}{\mathit{b}}_{2i}+{\kappa }_j{r}_{ij}^y{)}^{r_{ij}^z}}{1+\exp ({{\mathit{u}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\alpha }}_2+{{\mathit{\psi }}^z}^{\mathrm{\prime }}{\mathit{b}}_{2i}+{\kappa }_j{r}_{ij}^y)}\\ & \times {\varphi }_{q^z}({\mathit{b}}_{2i};\mathbf{0},{\mathit{D}}_2),\end{array}$$ (26)

$$\begin{array}{rl}{Y}_{ij}^{\mathrm{mis}}|{\mathit{\vartheta }}_{(-Y_{ij}^{\mathrm{mis}})},{\mathit{y}}^{\mathrm{obs}},{\mathit{z}}^{\mathrm{obs}},{\mathit{r}}^y,{\mathit{r}}^z& \sim N({{\mathit{x}}_{ij}^y}^{\mathrm{\prime }}{\mathit{\beta }}_1+{{\mathit{w}}_{ij}^y}^{\mathrm{\prime }}{\mathit{b}}_{1i},{\sigma }_{\epsilon }^2)\end{array}$$ (27)

and

$$Z_{ij}^{\mathrm{mis}}|{\mathit{\vartheta }}_{(-Z_{ij}^{\mathrm{mis}})},{\mathit{y}}^{\mathrm{obs}},{\mathit{z}}^{\mathrm{obs}},{\mathit{r}}^y,{\mathit{r}}^z\sim \mathrm{Ber}({\pi }_{ij}).$$ (28)

5. Simulation studies

5.1. Evaluation of the model and its performance comparison with that of using separate models

To examine the performance of the proposed joint model and to compare it with those obtained by using separate models, we conducted some simulation studies. We simulated N = 50 data sets with different sample sizes. The numbers of repeated measurements for individuals are different in each outcome and they obtained under a non-random missingness and ranged from 1 to $m^y=7$ for the continuous outcome and from 1 to $m^z=5$ for binary outcome. We consider the following joint model:

$$\begin{array}{rl}{Y}_{ij}& ={{\mathit{x}}_{ij}^y}^{\mathrm{\prime }}{\mathit{\beta }}_1+b_{1i}+{\epsilon }_{ij},i=1,2,\dots ,n,j=1,\dots ,m^y,\end{array}$$ (29)

$$\begin{array}{rl}{Y}_{ij}^{\ast }& ={{\mathit{x}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\beta }}_2+b_{2i}+{\gamma }_j{y}_{ij}+{\epsilon }_{ij}^{\ast },i=1,2,\dots ,n,j=1,\dots ,m^z,\end{array}$$ (30)

where ${\epsilon }_{ij}^{\ast }\sim \mathrm{logistic}(0,1)$, $Z_{ij}=1$ if $Y_{ij}^{\ast }\ge 0$ and $Z_{ij}=0$ if $Y_{ij}^{\ast }<0$.

$$R_{ij}^y|b_{1i}\sim \mathrm{Ber}({\pi }_{ij}^{R^y}),$$ (31)

where $\mathrm{logit}({\pi }_{ij}^{R^y})={\mathit{u}}_i^y{\mathit{\alpha }}_1+{\psi }^y{b}_{1i}$.

$$R_{ij}^z|b_{2i}\sim \mathrm{Ber}({\pi }_{ij}^{R^z}),$$ (32)

where $\mathrm{logit}({\pi }_{ij}^{R^z})={\mathit{u}}_i^z{\mathit{\alpha }}_2+{\psi }^z{b}_{2i}+{\kappa }_j{r}_{ij}^y$. In this simulation study, ${{\mathit{x}}_{ij}^y}^{\mathrm{\prime }}{\mathit{\beta }}_1={\beta }_{10}+{\beta }_{11}{x}_{1i}+{\beta }_{12}{x}_{2i}+{\beta }_{13}{t}_j$, ${{\mathit{x}}_{ij}^z}^{\mathrm{\prime }}{\mathit{\beta }}_2={\beta }_{20}+{\beta }_{21}{x}_{1i}+{\beta }_{22}{x}_{2i}+{\beta }_{23}{s}_j$, ${{\mathit{u}}_i^y}^{\mathrm{\prime }}{\mathit{\alpha }}_1={\alpha }_{01}+{\alpha }_{11}{x}_{1i}$, ${{\mathit{u}}_i^z}^{\mathrm{\prime }}{\mathit{\alpha }}_2={\alpha }_{02}+{\alpha }_{12}{x}_{2i}$, ${\epsilon }_{ij}\sim N(0,{\sigma }_{\epsilon }^2)$, $b_{1i}\sim N(0,{\sigma }_{b_1}^2)$, $b_{2i}\sim N(0,{\sigma }_{b_2}^2)$, $t_j=j,j=0,0.5,1,2,\dots ,m^y$, $s_j=1,2,\dots ,m^z$, ${\mathit{x}}_1$ and ${\mathit{x}}_2$ are generated from a $\mathrm{Ber}(0.3)$ and a $\mathrm{Ber}(0.4)$, respectively.

5.1.1. The effect of large sample size

We consider simulation studies with sample sizes n = 500, 1000, 2000 and 3000. Also, different association parameters are considered. The first is ${\gamma }_j=-1.000,j=1,\dots ,5$ and ${\kappa }_j=1.000,j=1,\dots ,5$, the second set of choices for parameters is ${\gamma }_j=0.000,j=1,\dots ,5$ and ${\kappa }_j=0.000,j=1,\dots ,5$ and the third set of choices for parameters is ${\gamma }_j=-0.500,j=1,\dots ,5$ and ${\kappa }_j=0.500,j=1,\dots ,5$. Also, the true values of parameters are ${\sigma }_{\epsilon }^2=4$, ${\sigma }_{b_1}^2={\sigma }_{b_2}^2=1$. The other true values are considered as ${\mathit{\beta }}_1=(2,1,-1,-2{)}^{\mathrm{\prime }}$, ${\mathit{\beta }}_2=(0,2,-1,-1{)}^{\mathrm{\prime }}$, ${\mathit{\alpha }}_1={\mathit{\alpha }}_2=(2,-2{)}^{\mathrm{\prime }}$ and $({\psi }^y,{\psi }^z{)}^{\mathrm{\prime }}=(2,2{)}^{\mathrm{\prime }}$. Another simulation study with ${\mathit{\beta }}_1=(8,-2,1,-2{)}^{\mathrm{\prime }}$, ${\mathit{\beta }}_2=(0,-1,-1,1{)}^{\mathrm{\prime }}$, ${\mathit{\alpha }}_1=(4,0{)}^{\mathrm{\prime }},{\mathit{\alpha }}_2=(0,2{)}^{\mathrm{\prime }}$, ${\gamma }_j=-0.500,j=1,\dots ,5$ and ${\kappa }_j=2.000,j=1,\dots ,5$ and $({\psi }^y,{\psi }^z{)}^{\mathrm{\prime }}=(1,2{)}^{\mathrm{\prime }}$, ${\sigma }_{\epsilon }^2=9$, ${\sigma }_{b_1}^2=4,{\sigma }_{b_2}^2=1$ is also performed.

The proposed joint model and separate models are fitted to each generated data. The results of these simulation studies are reported in Tables B.1 and B.2 of the Supplementary Materials B for the first set of association parameters, in Tables B.3 and B.4 for zero association parameters, in Tables B.5 and B.6 for the third association parameters and in Tables B.7 and B.8 for the last set of parameters. For more information about computational times of these models see Supplementary Materials E. In these tables, posterior mean, empirical standard errors, root of mean squared errors and relative biases are reported. The two letter criteria are given by $\mathrm{RMSE}(\theta )=\sqrt{\frac{1}{N}\sum _{i=1}^N({\theta }_i-\overline{\theta }{)}^2}$ and $\mathrm{Rel}.\mathrm{bias}(\theta )=\frac{\overline{\theta }}{\theta }-1$ where ${\theta }_i$, $i=1,2,\dots ,N$, is the estimate of θ for the ith sample. For the MCMC sampling, we ran 2 chains with 30,000 iterations, and 10,000 iterations of each chain are considered as burn-in period. Gelman-Rubin diagnosis test [13] are checked to ensure convergence of the MCMC chains. We consider the following prior distributions: ${\beta }_{jk}\sim N(0,1000)$, j = 1, 2, k = 0, 1, 2, 3, ${\gamma }_k\sim N(0,1000)$, k = 1, 2, 3, 4, 5, ${\kappa }_k\sim N(0,1000)$, k = 1, 2, 3, 4, 5, ${\sigma }^2\sim \mathrm{I}\mathrm{\Gamma }(0.01,0.01)$, ${\sigma }_{b_k}^2\sim \mathrm{I}\mathrm{\Gamma }(0.01,0.01)$, k = 1, 2, ${\alpha }_{jk}\sim N(0,1000)$, j = 1, 2, k = 0, 1 and ${\psi }^y,{\psi }^z\sim N(0,1000)$. The ‘R2OpenBUGS’ package is used for the implementation of these models.

Based on the results of these tables, the performance of the proposed joint model is well known for analyzing generated data under either zero or non-zero association parameters. The results of the simulation study for the non-zero association parameters model show that the increase in sample size is an effective way of decreasing biases and standard deviations of parameters. As shown in tables B.1, B.3, B.5 and B.7, relative biases and RMSEs are reduced by increasing sample sizes. This suggests that the proposed model is consistent. Based on the results of Tables B.2, B.6 and B.8, the performance of the separate models when data are generated under non-zero association parameters are not well, specially, the values of RMSEs and relative biases for the components of the variances are large. Also, based on the results of Tables B.3 and B.4, the performance of the proposed joint model when data are generated under the separate models is as good as that of the separate models. In general, these tables show that the proposed method performs well.

5.1.2. The effect of small sample size

For checking the performance of the model using small sample size, three different small sample sizes n = 50, 80, 100 are also considered. We use the above-mentioned model with ${\gamma }_j=-1.000,j=1,\dots ,5$ and ${\kappa }_j=1.000,j=1,\dots ,5$, ${\sigma }_{\epsilon }^2=4$, ${\sigma }_{b_1}^2={\sigma }_{b_2}^2=1$. The other true values are considered as ${\mathit{\beta }}_1=(2,1,-1,-2{)}^{\mathrm{\prime }}$, ${\mathit{\beta }}_2=(0,2,-1,-1{)}^{\mathrm{\prime }}$, ${\mathit{\alpha }}_1={\mathit{\alpha }}_2=(2,-2{)}^{\mathrm{\prime }}$ and $({\psi }^y,{\psi }^z{)}^{\mathrm{\prime }}=(2,2{)}^{\mathrm{\prime }}$. The Bayesian implementation including prior distributions and the number of iterations, etc. are the same as those of the previous simulation. The results of this simulation study for the joint and separate models are given in Tables B.9 and B.10, respectively. Note that, the number of parameters for joint model is 27 and N = 50 sample size seems to be too small for having accurate parameter estimations. Also, increasing sample size to n = 100 leads to more accurate estimations. The results of fitting separate models for small sample sizes do not show acceptable results and the value of RMSE and relative biases for some parameters are too large.

5.2. Evaluation of distributional assumption for random effects

In this section, another simulation study is performed to investigate the performance of the proposed model with respect to different distributional assumption for random effects. For this purpose, the mentioned model of Section 5.1.1 with n = 500 and three different distributional assumptions for $b_{1i}$ and $b_{2i},i=1,2,\dots ,n$, are considered and the proposed joint modeling is used for analyzing the data. These distributional assumptions are given as follows:

• Student's t distribution: $b_{1i},b_{2i}\stackrel{\mathrm{iid}}{\sim }t(2)$, that is, standard Student's t distribution with 2 degrees of freedom.

• Skew-normal distribution: $b_{1i},b_{2i}\stackrel{\mathrm{iid}}{\sim }SN(-\sqrt{\frac{2}{pi}}\delta ,1,\lambda )$, that is, zero-mean skew-normal distribution with scale parameter 1 and skewness parameter $\lambda =2$, such that, $\delta =\lambda /\sqrt{(1+{\lambda }^2)}$.

• Mixture of two normal distributions: $b_{1i},b_{2i}\stackrel{\mathrm{iid}}{\sim }0.5N(2,1)+0.5N(-2,1)$, that is a mixture of two normal distributions with mean zero.

The other true values are considered as ${\gamma }_j=-1.000,j=1,\dots ,5$, ${\kappa }_j=1.000,j=1,\dots ,5$, ${\sigma }_{\epsilon }^2=4$, ${\mathit{\beta }}_1=(2,1,-1,-2{)}^{\mathrm{\prime }}$, ${\mathit{\beta }}_2=(0,2,-1,-1{)}^{\mathrm{\prime }}$, ${\mathit{\alpha }}_1={\mathit{\alpha }}_2=(2,-2{)}^{\mathrm{\prime }}$ and $({\psi }^y,{\psi }^z{)}^{\mathrm{\prime }}=(2,2{)}^{\mathrm{\prime }}$. The Bayesian implementation including prior distributions and the number of iterations, etc. are the same as those of the previous simulations.

The results of this simulation study are given in Table B.11. In comparison to the results of the normal distributional assumption, results of this simulation (see Table B.1) show that the regression coefficients of continuous longitudinal model are not very sensitive with respect to different distributional assumptions, but the regression coefficients of binary longitudinal model are sensitive, specially for Student's t and skew-normal distributions. Also, the regression coefficients of missingness mechanism for binary outcome of Student's t and skew-normal distributions are far from the real values.

6. Analysis of morbid obesity data

The main objective of the analysis of the morbid obesity data is to find the simultaneous effect of type of surgery on having any complication due to surgery and on BMI. Also, the association between BMI and complication due to surgery is in particular interest. The data were collected from n = 2890 patients who had at least 2 measurements of both BMI and complication due to surgery and other measurements are ranged from 1 to 8 for BMI and from 1 to 6 for complication due to surgery. Let ${\mathrm{BMI}}_{ij},i=1,2,\dots ,n,j=1,\dots ,8$ be the BMI for patient i at time $t_j$. Also, ${\mathrm{Complicat}}_{ik},i=1,2,\dots ,n,k=1,\dots ,6$ be complication due to surgery for patient i at time $s_k$. Also, let ${\mathrm{Age}}_i$, ${\mathrm{Gender}}_i$, ${\mathrm{Helicob}}_i$, ${\mathrm{T}2\mathrm{Diab}}_i$, ${\mathrm{Fliver}}_i$, ${\mathrm{Dyslip}}_i$, ${\mathrm{Hypoth}}_i$, ${\mathrm{Mini}}_i$, ${\mathrm{Ryg}}_i$, denote the age, gender and having Helicobacter, Type 2 Diabetes, Fatty liver, Dyslipidemia, Hypothyroidism, Mini-gastric bypass and RY gastric bypass, respectively. We consider the following joint model:

$$\begin{array}{rl}{\mathrm{BMI}}_{ij}|b_{1i}& \sim N({\mu }_{ij},{\sigma }^2),\\ {\mathrm{Complicat}}_{ik}|b_{2i},{\mathrm{BMI}}_{ik}& \sim \mathrm{Ber}({\pi }_{ik}),\end{array}$$ (33)

where ${\mu }_{ij}={\beta }_{10}+{\beta }_{11}{t}_j+{\beta }_{12}{\mathrm{Age}}_i+{\beta }_{13}{\mathrm{Gender}}_i+{\beta }_{14}{\mathrm{Mini}}_i+{\beta }_{15}{\mathrm{RYG}}_i+{\beta }_{16}{\mathrm{Fliver}}_i+{\beta }_{17}{\mathrm{Helicob}}_i+{\beta }_{18}{\mathrm{T}2\mathrm{Diab}}_i+{\beta }_{19}{\mathrm{Dyslip}}_i+{\beta }_{1,10}{\mathrm{Hypoth}}_i+b_{1i}$ and $\mathrm{logit}({\pi }_{ik})={\beta }_{20}+{\beta }_{21}{s}_k+{\beta }_{22}{\mathrm{Age}}_i+{\beta }_{23}{\mathrm{Gender}}_i+{\beta }_{24}{\mathrm{Mini}}_i+{\beta }_{25}{\mathrm{RYG}}_i+{\beta }_{26}{\mathrm{Fliver}}_i+{\beta }_{27}{\mathrm{Helicob}}_i+{\beta }_{28}{\mathrm{T}2\mathrm{Diab}}_i+{\beta }_{29}{\mathrm{Dyslip}}_i+{\beta }_{2,10}{\mathrm{Hypoth}}_i+{\gamma }_k{\mathrm{BMI}}_{i{s}_k}+b_{2i}$. Also, the following models are used for missingness mechanisms: $R_{ij}^{\mathrm{BMI}}\sim \mathrm{Ber}({\pi }_{ij}^{\mathrm{BMI}})$, where

$$\mathrm{logit}({\pi }_{ij}^{\mathrm{BMI}})={\alpha }_{10}+{\alpha }_{11}{\mathrm{Mini}}_i+{\alpha }_{12}{\mathrm{RYG}}_i+{\psi }^{\mathrm{BMI}}{b}_{1i}$$

and $R_{ik}^{\mathrm{Complicat}}\sim \mathrm{Ber}({\pi }_{ik}^{\mathrm{Complicat}})$, where

$$\mathrm{logit}({\pi }_{ik}^{\mathrm{Complicat}})={\alpha }_{20}+{\alpha }_{21}{\mathrm{Mini}}_i+{\alpha }_{22}{\mathrm{RYG}}_i+{\kappa }_k{r}_{i{s}_k}^{\mathrm{BMI}}+{\psi }^{\mathrm{Complicat}}{b}_{2i},$$

where $b_{1i}\sim N(0,{\sigma }_{b_1}^2)$ and $b_{2i}\sim N(0,{\sigma }_{b_2}^2)$.

In the Bayesian method, two parallel MCMC chains are run with different initial values for 80,000 iterations each. Then, we have discarded the first 40,000 iterations as pre-convergence burn-in and retained 40,000 for the posterior inference. For checking the convergence of the MCMC chains, we have used the Gelman–Rubin diagnostic test. Also, trace plots of unknown parameters are shown in Figures D.1–D.6 of Supplementary Materials D. The prior distributions for the unknown parameters are the same as those of the simulation study section and are given by: ${\beta }_{jk}\sim N(0,1000),j=1,2,k=0,1,2,\dots ,11$, ${\gamma }_k\sim N(0,1000),k=1,2,3,\dots ,6$, ${\kappa }_k\sim N(0,1000),k=1,2,3,\dots ,6$, ${\sigma }^2\sim \mathrm{I}\mathrm{\Gamma }(0.01,0.01)$, ${\psi }^{\mathrm{BMI}},{\psi }^{\mathrm{Complicat}}\sim N(0,1000)$, ${\sigma }_{b_k}^2\sim \mathrm{I}\mathrm{\Gamma }(0.01,0.01),k=1,2$ and ${\alpha }_{jk}\sim N(0,1000),j=1,2,k=0,1,2$. Table 2 shows the results of the Bayesian joint modeling and separate models on the data set. These two models are compared using EAIC, EBIC, DIC and LPML. These criteria are given by Baghfalaki et al. [4,5]. Unlike EAIC, EBIC and DIC, larger values of LPML indicate a better fitting model. These comparison criteria show that the joint model has a better fit to the data.

Table 2. Parameter estimates (Est.), standard deviation (S.D.) and 95 $\mathrm{\%}$ credible interval for analyzing morbid obesity data.

	Joint model				Separate model
Parameters	Est.	S.D.	CI 2.5 $\mathrm{\%}$	CI 97.5 $\mathrm{\%}$	Est.	S.D.	CI 2.5 $\mathrm{\%}$	CI 97.5 $\mathrm{\%}$
	Continuous submodel (BMI)
${\beta }_{10}$ (Intercept)	50.850	3.027	45.270	55.150	52.430	1.691	48.870	55.100
${\beta }_{11}$ (Time)	−2.092	0.013	−2.116	−2.067	−2.092	0.012	−2.117	−2.068
${\beta }_{12}$ (Age)	0.001	0.012	−0.024	0.027	−0.001	0.012	−0.025	0.021
${\beta }_{13}$ (Gender; female)	−1.988	0.344	−2.623	−1.308	−2.143	0.298	−2.691	−1.519
${\beta }_{14}$ (Mini-gastric bypass)	0.704	0.353	0.051	1.450	0.693	0.357	−0.049	1.353
${\beta }_{15}$ (RY gastric bypass)	0.118	0.362	−0.544	0.877	0.127	0.363	−0.603	0.805
${\beta }_{16}$ (Fatty liver)	1.060	0.264	0.572	1.588	1.065	0.245	0.572	1.530
${\beta }_{17}$ (Helicobacter)	0.049	0.204	−0.356	0.441	0.036	0.208	−0.377	0.454
${\beta }_{18}$ (Type 2 Diabetes)	−0.895	0.288	−1.474	−0.333	−0.880	0.289	−1.442	−0.308
${\beta }_{19}$ (Dyslipidemia)	−0.545	0.222	−0.976	−0.108	−0.566	0.218	−0.996	−0.140
${\beta }_{110}$ (Hypothyroidism)	0.281	0.262	−0.237	0.788	0.281	0.259	−0.230	0.781
${\alpha }_{10}$ (Intercept)	0.775	0.044	0.689	0.862	0.774	0.044	0.689	0.861
${\alpha }_{11}$ (Mini-gastric bypass)	0.597	0.049	0.500	0.694	0.598	0.049	0.502	0.695
${\alpha }_{12}$ (RY gastric bypass)	0.047	0.050	−0.052	0.146	0.048	0.050	−0.051	0.146
${\sigma }_{b_1}^2$	27.840	0.789	26.320	29.410	27.820	0.785	26.310	29.391
${\sigma }^2$	11.000	0.129	10.750	11.250	11.000	0.130	10.740	11.250
${\psi }^{\mathrm{BMI}}$	−0.007	0.003	−0.013	0.000	−0.007	0.003	−0.013	0.000
	Binary submodel (Complication due to surgery)
${\beta }_{20}$ (Intercept)	1.554	0.938	−0.080	3.696	2.376	0.528	1.397	3.411
${\beta }_{21}$ (Time)	−0.136	0.098	−0.324	0.054	−0.277	0.013	−0.303	−0.250
${\beta }_{22}$ (Age)	−0.001	0.003	−0.007	0.004	−0.002	0.002	−0.006	0.002
${\beta }_{23}$ (Gender; female)	−0.373	0.086	−0.549	−0.211	−0.326	0.067	−0.461	−0.195
${\beta }_{24}$ (Mini-gastric bypass)	0.244	0.095	0.057	0.428	0.095	0.074	−0.051	0.242
${\beta }_{25}$ (RY gastric bypass)	0.100	0.099	−0.097	0.290	0.090	0.078	−0.064	0.243
${\beta }_{26}$ (Fatty liver)	−0.182	0.060	−0.299	−0.065	−0.224	0.050	−0.324	−0.125
${\beta }_{27}$ (Helicobacter)	−0.079	0.049	−0.175	0.016	−0.058	0.041	−0.140	0.022
${\beta }_{28}$ (Type 2 Diabetes)	−0.077	0.068	−0.211	0.057	−0.060	0.058	−0.173	0.054
${\beta }_{29}$ (Dyslipidemia)	0.145	0.051	0.045	0.246	0.134	0.043	0.050	0.220
${\beta }_{210}$ (Hypothyroidism)	0.008	0.061	−0.112	0.129	−0.045	0.052	−0.146	0.056
${\alpha }_{20}$ (Intercept)	−0.594	0.118	−0.826	−0.361	0.571	0.099	0.375	0.766
${\alpha }_{21}$ (Mini-gastric bypass)	0.887	0.127	0.634	1.136	0.936	0.110	0.723	1.154
${\alpha }_{22}$ (RY gastric bypass)	−0.092	0.130	−0.346	0.164	−0.059	0.112	−0.278	0.163
${\sigma }_{b_2}^2$	0.506	0.068	0.382	0.647	0.093	0.027	0.042	0.146
${\psi }^{\mathrm{Complicat}}$	2.258	0.177	1.940	2.629	4.679	0.826	3.569	6.718
	Shared parameters
${\gamma }_1$	0.070	0.008	0.056	0.085	–	–	–	–
${\gamma }_2$	−0.015	0.006	−0.026	−0.004	–	–	–	–
${\gamma }_3$	−0.021	0.004	−0.030	−0.012	–	–	–	–
${\gamma }_4$	−0.012	0.005	−0.021	−0.003	–	–	–	–
${\gamma }_5$	0.001	0.006	−0.011	0.013	–	–	–	–
${\gamma }_6$	0.005	0.009	−0.012	0.022	–	–	–	–
${\kappa }_1$	4.059	0.103	3.859	4.263	–	–	–	–
${\kappa }_2$	2.373	0.081	2.215	2.533	–	–	–	–
${\kappa }_3$	1.803	0.072	1.664	1.943	–	–	–	–
${\kappa }_4$	1.449	0.076	1.302	1.599	–	–	–	–
${\kappa }_5$	1.083	0.081	0.922	1.241	–	–	–	–
${\kappa }_6$	0.854	0.096	0.666	1.043	–	–	–	–
EAIC	142,190				147966
EBIC	142,458.6				148,163
DIC	4604				4717
LPML	−59,119.75				−61,951.82

This table shows that time, gender, type of surgery, fatty liver, type 2 diabetes and dyslipidemia are significant predictors in the continuous model (BMI), also, gender, type of surgery (mini-gastric bypass), fatty liver and dyslipidemia are significant predictors in the binary model (indicator of complication due to surgery). Based on the results, the increase in time reduces BMI index, females have smaller BMI and smaller probability to have complication due to surgery. The persons with fatty liver have larger values of BMI and smaller probability to have complication due to surgery. In addition, the persons with Dyslipidemia have smaller BMI but have the larger probability to have complication due to surgery. The conditional model shows a statistically significant relationship between missing time points and complication due to surgery ( ${\omega }_2$ is significantly different from zero), but not for BMI because, this dependence was not detected with the shared parameter model, where ${\omega }_1$ was not significantly different from zero. Therefore, a significant positive relationship between complication due to surgery and intermittent missingness (estimate ${\omega }_2$ was highly significant), suggesting that the missing-data mechanism is non-ignorable. Also, parameters ${\gamma }_k$ and ${\kappa }_k$, $k=1,2,\dots ,6$, show that the continuous and binary longitudinal measurements are dependent, also, two missingness mechanisms are dependent. Figures C.1–C.6 of supplementary material C show the posterior densities for the unknown parameters in the models. Figures C.1–C.4 compare the posterior densities of parameters using separate and joint models. Figure C.1 shows that the behavior of the predictors on modeling BMI in the two competing models is similar. Also, Figure C.2 shows that the performance of two models in estimating parameters of the binary model is a little different. Based on Figure C.3, except ${\alpha }_{20}$, the regression coefficients of missingness mechanisms in the two approaches are equivalent. Also, Figure C.4 shows that posterior densities of components of variance in the two approaches are different. Figures C.5 and C.6 show the posterior densities for the association parameters of the joint model. These figures show that most of the association parameters have non-zero means.

As other criteria for comparing two models, we use true positive rate (TPR), false positive rate (FPR) and true discovery rate (TDR) of the observed values of indicators of complication due to surgery and their predicted values by two approaches. Let ${\mathrm{Complicat}}_{ik}^{\mathrm{Pred}}=1$ if $(i,k)$th component of the indicator of complication due to surgery are predicted to be one. Also, for the observed values of situation $Complica{t}_{ik}=1$ if $(i,k)$th component of the indicator of complication due to surgery is in fact one. The TPR, the FPR and the TDR of the method can be calculated as follows:

$$\begin{array}{rl}\mathrm{TPR}& =\frac{\sum _{i=1}^n\sum _k{\mathrm{Complicat}}_{ik}{\mathrm{Complicat}}_{ik}^{\mathrm{Pred}}}{\sum _{i=1}^n\sum _k{\mathrm{Complicat}}_{ik}},\\ \mathrm{FPR}& =\frac{\sum _{i=1}^n\sum _k(1-{\mathrm{Complicat}}_{ik}){\mathrm{Complicat}}_{ik}^{\mathrm{Pred}}}{\sum _{i=1}^n\sum _k(1-{\mathrm{Complicat}}_{ik})},\\ \mathrm{TDR}& =\frac{\sum _{i=1}^n\sum _k{\mathrm{Complicat}}_{ik}{\mathrm{Complicat}}_{ik}^{\mathrm{Pred}}}{\sum _{i=1}^n\sum _k{\mathrm{Complicat}}_{ik}^{\mathrm{Pred}}}.\end{array}$$

A receiver operating characteristic (ROC) curve plots FPR versus TPR for the possible cutoffs. A common method of comparing the methods is the area under the ROC curve (often for simplicity referred to as the AUC). AUC is a portion of the area of the unit square; its value is always between 0 and 1. The larger the value of AUC, the better is the performance of the approach. The values of TPR, FPR and TDR for cutoff point equal to 0.5 for two competing models are reported in Table 3. TPR for the joint model is a bit smaller than that of separate models, but the difference of FPR between two models is a large value. Other criteria of this table also show the better performance of the joint model in comparison with that of using separate models. Furthermore, the values of AUC for two competing models are reported in this table and the larger value belongs to the joint model. Figure 3 shows ROC curve for indicator of complication due to surgery for two competing models which shows the AUC for joint model is larger than that of separate model. A sensitivity analysis with respect to the prior distributions of regression coefficients and association parameters is also performed. For this purpose, the sensitivity of the posterior mean to different values of the variance of the prior distribution is investigated. The results show that the parameter estimates are robust with respect to the normal priors when variances are considered to be larger than 100.

Table 3. The values of TPR, FPR and TDR for cutoff point equal to 0.5 and AUC for two competing models in analyzing morbid obesity data.

	Joint model	Separate models
TPR	0.875	0.909
FPR	0.469	0.864
TDR	0.697	0.527
AUC	0.817	0.660

Figure 3.

ROC curves of observed and predicted values by two competing approaches of complication due to surgery for the morbid obesity data.

7. Conclusions

In this paper, we propose a joint model for a mixed continuous and binary outcomes with different missing patterns. We use an MCMC procedure via the Gibbs sampler and Metropolis-Hastings algorithm to draw samples for parameters estimation. This process is facilitated using the freely available OpenBUGS software [23].

Two shared parameter models are considered for missingness mechanisms of two responses which jointly modeled using a conditional model. The usefulness of shared-parameter models for analyzing longitudinal data subject to non-random missing data is shown by many authors [1–3]. It should be recognized that, although these models account for NMAR data, it is impossible to distinguish between MAR and NMAR data within this class of models [3]. Also, one can use pattern mixture model and selection model frameworks for modeling missing mechanism [18–20].

In this paper, a normality assumption is used for the random effects and errors. One can use more flexible distributional assumption such as t, skew-normal, skew-t or other distributional assumptions for random effects. Also, a logistic mixed-effects model is used for analyzing binary data including binary longitudinal data and missingness mechanisms. As a future work, one can use the probit mixed-effects model for modeling the binary outcomes.

Also, we consider non-informative prior distributions for the unknown parameters, one can use informative priors or empirical Bayes approach for choosing hyper parameters in the Bayesian paradigm.

Funding Statement

This work has been supported by Iranian National Science Foundation (INSF) [grant number 97014144].

Disclosure statement

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