A Class of Joint Models for Multivariate Longitudinal Measurements and a Binary Event

Summary

Predicting binary events such as newborns with large birthweight is important for obstetricians in their attempt to reduce both maternal and fetal morbidity and mortality. Such predictions have been a challenge in obstetric practice, where longitudinal ultrasound measurements taken at multiple gestational times during pregnancy may be useful for predicting various poor pregnancy outcomes. The focus of this paper is on developing a flexible class of joint models for the multivariate longitudinal ultrasound measurements that can be used for predicting a binary event at birth. A skewed multivariate random effects model is proposed for the ultrasound measurements, and the skewed generalized t-link is assumed for the link function relating the binary event and the underlying longitudinal processes. We consider a shared random effect to link the two processes together. Markov chain Monte Carlo sampling is used to carry out Bayesian posterior computation. Several variations of the proposed model are considered and compared via the deviance information criterion, the logarithm of pseudomarginal likelihood, and with a training-test set prediction paradigm. The proposed methodology is illustrated with data from the NICHD Successive Small-for-Gestational-Age Births study, a large prospective fetal growth cohort conducted in Norway and Sweden.

1. Introduction

Predicting binary events such as the birth of newborns with large birthweight is important for obstetricians in their attempt to reduce both maternal and fetal morbidity and mortality. Such predictions have been a challenge in obstetric practice, where longitudinal ultrasound measurements taken at multiple irregularly spaced times in gestation may be useful for predicting various poor pregnancy outcomes. In the NICHD Successive Small-for-Gestational-Age Births (SGA) study, a large prospective study conducted in Norway and Sweden from 1986 to 1989, each pregnant woman was targeted to have four ultrasound examinations at 17, 25, 33 and 37 weeks of gestation. At each visit, sonography was conducted to measure the following fetal growth anthropometry variables: biparietal diameter (BPD), middle abdominal diameter (MAD) and femur length (FL). Figure 1 shows the longitudinal trajectories for BPD, MAD, and FL on the original scale. The figure shows that the trajectories are rather smooth and that, although observations were targeted at particular times in gestation, there was sizable variation in the observation times across subjects.

Figure 1.

Plots of longitudinal trajectory for BPD, MAD, and FL by gestational age in the study population: the solid lines represent a lowess smooth curve.

The development of accurate methods for predicting a newborn larger than 4,000 grams, defined as macrosomia, using longitudinal ultrasound measurements is important for the clinical management of both mother and baby. Recently, Albert (2012) proposed a shared random effects model to evaluate the predictive accuracy of longitudinal ultrasound measurement using a two-stage procedure. In the first stage, a linear mixed model is fitted to the longitudinal measurements, and in the second stage, the binary outcome is modeled with a probit link function where the predicted random effects are treated as covariates. The probit link was chosen since measurement error of the predicted random effects can be easily incorporated in closed form. This two stage approach can easily be extended to the multivariate setting as illustrated by Zhang et al. (2012).

The Gaussian assumption for the residual error in Albert (2012) may not hold. Figure 2 shows residuals obtained using linear mixed models with cubic spline fixed and random effects for BPD, MAD, and FL, fit on both the original and log scales. For both the original and log scales, the empirical residuals suggest that the residual error distributions are heavy tailed and possibly skewed. Other transformations also showed these features, suggesting that directly transforming the longitudinal measurements will not address this problem. Further, although computationally feasible, the probit link function may not ideally model the probability of a poor pregnancy outcome since it is symmetric around 0.5, a strong assumption in this analysis. Thus, new statistical methodology is necessary to incorporate flexibility in the error distributions as well as in the link function relating the longitudinal trajectories to the binary response.

Figure 2.

Q-Q plots of the residuals for BPD, MAD, and FL from fitting the linear mixed random effects model separately: (a), (b), and (c) for original scale; (d), (e), and (f) for log scale.

In this paper, we develop a flexible class of joint models for the longitudinal ultrasound measurements to predict a binary event at birth, such as macrosomia. A skewed multivariate random effects model is proposed for the ultrasound measurements, and skewed link is assumed for the link function relating the binary event and the underlying longitudinal processes. We consider a shared random effect to link the two processes together. We also use the polynomial regression spline based on truncated power basis to capture the nonlinear structure in the longitudinal mean trajectory. To this end, we develop efficient Bayesian computational methods for fitting this joint model via several modified collapsed Gibbs samplers (Chen et al., 2000; Liu, 1994). In addition, we derive the deviance information criterion (DIC) and logarithm of pseudomarginal likelihood (LPML) for comparing several variations of the proposed joint models.

We organize the rest of the paper as follows. We provide in Section 2 the methodological development of a flexible class of joint models for the longitudinal ultrasound measurements and a subsequent binary event, and show how this approach generalizes methodology already developed for this problem. In Section 3, we discuss the prediction of an adverse pregnancy outcome from a series of multiple ultrasound measurements taken at various gestational ages that can be irregularly spaced across individuals. The prior and posterior distributions as well as the goodness-of-fit criterion are discussed in Section 4. Section 5 presents an analysis of the SGA study data. A discussion follows in Section 6.

2. Model Framework

Let i denote individual, j denote time point, and k denote ultrasound measurement. We assume that each measurement is taken at repeated time points, which are potentially irregularly spaced times in gestation. Further, we assume that there are I individuals in the study, each contributing J_i time points, where J_i denotes the number of repeated time points on the ith individual. Let x_ijk = (x_ijk1, . . ., x_ijk,p1)′ and z_i = (z_i1, . . ., z_i,p2)′ denote the vectors of fixed effect covariates, where z_i may share common components with x_ijk. Also, let β_k = (β_k1, . . ., β_k,p1)′ and γ = (γ₁, . . ., γ_p2)′ denote the corresponding vector of regression coefficients for the longitudinal and binary model components, respectively, i = 1, . . ., I, j = 1, . . ., J_i, and k = 1, . . ., K. Note that β_k does not vary by i or j and γ is constant across i, j, and k. Also let $X_{ij}=\text{diag}({\mathit{x}}_{ij1}^{\prime },\dots ,{\mathit{x}}_{\mathit{ijK}}^{\prime }),X={(X_{11}^{\prime },\dots ,X_{I,J_I}^{\prime })}^{\prime },Z={({\mathit{z}}_1^{\prime },\dots ,{\mathit{z}}_I^{\prime })}^{\prime }$, and $\mathit{\beta }={({\mathit{\beta }}_1^{\prime },\dots ,{\mathit{\beta }}_K^{\prime })}^{\prime }$. Furthermore, we assume that t_ijk denotes the time (i.e., gestational age) of the kth type of ultrasound measurement (i.e., BPD, MAD, or FL) at the jth time point on the ith individual.

2.1 Joint models

Let y_ijk denote the kth type of longitudinal ultrasound measurement at the jth time point on the ith individual. Also let S_i denote an adverse binary event (e.g., macrosomia) for the ith individual. We propose the following joint models with shared random effects for the longitudinal measurements and binary outcome,

$$y_{\mathit{ijk}}={\mathit{x}}_{\mathit{ijk}}^{\prime }{\mathit{\beta }}_k+g(t_{\mathit{ijk}})+g_b(t_{\mathit{ijk}};{\mathit{b}}_{ik})+e_{\mathit{ijk}}^y,$$ (1)

$$\begin{array}{l}{S}_i=1\text{if}{R}_i\ge 0;0\text{if}{R}_i<0,\\ R_i={{\mathit{z}}_i}^{\prime }\mathit{\gamma }+\sum _k{\alpha }_{k}h({\mathit{b}}_{ik})+e_i^R.\end{array}$$ (2)

For the longitudinal data, g(t_ijk) and g_b(t_ijk; b_ik) in (1) are functions of time t_ijk corresponding to fixed and random effects, where b_ik is a vector of random effects. In addition, $e_{\mathit{ijk}}^y$ are random variables for the error distributions that will allow for long-tailed and skewed distributions, which are apparent in our longitudinal imaging data. These model components for the mean, random effects, and errors are flexible and are described in Sections 2.2 and 2.3. For the binary data, we introduce a latent variable R_i that characterizes the risk of the binary outcome (Albert and Chib, 1993) and is linked to the longitudinal processes through a function of random effects, h(b_ik), where the random effects are shared between the longitudinal and binary event processes. For each longitudinal variable (k), the parameters α_k’s explicitly introduce this dependence. Further, to provide a flexible link function, the error distribution $e_i^R$ incorporates the possibility of both long tails and skewness as described in Section 2.3. This approach generalized the work of Albert (2012) and Zhang et al. (2012), who, in the same setting, proposed a shared random effects model with Gaussian error structures and a probit link function. With these simplifying assumptions they show that a simple two-stage estimation procedure is possible. Specifically, they assumed quadratic fixed and random effects for g(t_ijk) and g_b(t_ijk; b_ik), a normal distribution for $e_{\mathit{ijk}}^y$, and a normal distribution for $e_i^R$, which results in a probit link function for the binary process. As is evident from Figure 2, the normal distribution is not adequate for the longitudinal fetal growth anthropometry data. Further, the probit link function in Albert (2012) might be too restrictive in this application.

Similar to the procedure used by Albert (2012) and Zhang et al. (2012), h(b_ik) is chosen so that the function relates the random effects to the underlying longitudinal processes close to birth. This is scientifically reasonable since, in the fetal growth example, the dependence between the longitudinal fetal growth and the binary birth outcome should be at a time close to birth. However, in other applications, we could introduce a parameter associated with each random effect component (e.g., linear, quadratic, and cubic terms) for each of the multivariate longitudinal measurements into the link function. The following sections describe each of the model components in more detail.

2.2 Longitudinal model with flexible mean structures

Longitudinal models have been proposed for characterizing fetal growth pattern (Deter, 2004; Slaughter et al., 2009). For fetal growth, the longitudinal profile for ultrasound measurements can be characterized to capture the nonlinear structure in the mean trajectory. To this end, we assume the following polynomial regression spline for g(t_ijk) and g_b(t_ijk; b_ik):

$$\begin{array}{l}g(t_{\mathit{ijk}})={\varphi }_{k0}+{\varphi }_{k1}{t}_{\mathit{ijk}}+\cdots +{\varphi }_{kq}{t}_{\mathit{ijk}}^q+\sum _{l=1}^m{\mathit{\varphi }}_{k,q+l}{(t_{\mathit{ijk}}-{\zeta }_{kl})}_{+}^q\equiv {\mathit{w}}_{\mathit{ijk}}^{\prime }{\mathit{\varphi }}_k\text{and}\\ g_b(t_{\mathit{ijk}};{\mathit{b}}_{ik})=b_{ik0}+b_{ik1}{t}_{\mathit{ijk}}+\cdots +b_{\mathit{ikq}}{t}_{\mathit{ijk}}^q+\sum _{l=1}^m{b}_{ik,q+l}{(t_{\mathit{ijk}}-{\zeta }_{kl})}_{+}^q\equiv {\mathit{w}}_{\mathit{ijk}}^{\prime }{\mathit{b}}_{ik},\end{array}$$ (3)

where q is a pre-specified degree of polynomial spline, m is the number of knots, ${(t_{\mathit{ijk}}-{\zeta }_{kl})}_{+}^q=max(0,{(t_{\mathit{ijk}}-{\zeta }_{kl})}^q)$, ζ_k = (ζ_k1, . . ., ζ_km)′ is the knot sequence with a_ζk < _ζk1 < ··· < ζ_km < b_ζk, ${\mathit{w}}_{\mathit{ijk}}={(1,t_{\mathit{ijk}},\dots ,t_{\mathit{ijk}}^q,{(t_{\mathit{ijk}}-{\zeta }_{kl})}_{+}^q,\dots ,{(t_{\mathit{ijk}}-{\zeta }_{km})}_{+}^q)}^{\prime }$ is a truncated polynomial basis functions of degree q, and ϕ_k = (ϕ_k0, . . ., ϕ_k,q+m)′ and b_ik = (b_ik0, . . ., b_ik,q+m)′ are corresponding vectors of parameters and random effects, respectively. Let ${\mathit{b}}_i={({\mathit{b}}_{i1}^{\prime },{\mathit{b}}_{i2}^{\prime },\dots ,{\mathit{b}}_{iK}^{\prime })}^{\prime }$. We then assume that the random effect b_i follows a multivariate normal distribution with mean 0 and the K(q+m+1)×K(q+m+1) unstructured variance-covariance matrix Ω. In (1), the random effects for the kth type of ultrasound measurement are interpreted as individual departures in an individual’s growth curve relative to the average fetal growth curve in the population. This correlated random effects structure allows for a flexible correlation in the longitudinal measurements over time and across type (i.e. BPD, MAD, and FL).

2.3 Error distributions for longitudinal measurements and link functions

We consider flexible distributions for $e_{\mathit{ijk}}^y$ in (1) and $e_i^R$ in (2) that allow for flexibility in the longitudinal error distribution as well as in the link function. In this paper, we propose to use long-tailed and skewed distributions to accomplish this goal. Let Δ_θ = diag(θ₁, θ₂,. . ., θ_K), ξ_ij = (ξ_ij1, . . ., ξ_ijK)′, ε_ij = (ε_ij1, . . ., ε_ijK)′, and ${\mathit{e}}_{ij}^y={(e_{ij1}^y,\dots ,e_{\mathit{ijK}}^y)}^{\prime }$. Specifically, we model

$${\mathit{e}}_{ij}^y={\mathrm{\Delta }}_{\theta }({\mathit{\xi }}_{ij}-E{\mathit{\xi }}_{ij})+{\mathit{\epsilon }}_{ij}\text{and}$$ (4)

$$e_i^R=\delta ({\psi }_i-E{\psi }_i)+{\eta }_i,$$ (5)

where the first terms in (4) and (5) reflect components for the skewness and the second terms reflect components for the long tails. In (4), we assume that (i) ξ_ijk and ε_ijk are independent; (ii) ξ_ijk ~ G_ξ, and ξ_ijk and ξ_i′j′k′ are independent, where G_ξ is the cumulative density function (cdf) of a skewed distribution defined on R⁺ = (0, ∞); and (iii) ε_ij follows a multivariate symmetric distribution with the K×K unstructured variance-covariance matrix Σ_j = (σ_jkk′), where k and k′ = 1, ···, K, and ε_ij and ε_i′j′ are independent. In (5), we assume that (i) ψ_i and η_i are independent; (ii) ψ_i ~ G_ψ are each independent, where G_ψ is the cdf of a skewed distribution defined on R⁺ = (0, ∞); and (iii) η_i follows a symmetric distribution, and η_i and η_i′ are independent. Further, θ_k in (4) and δ in (5) are skewness parameters. When θ_k = 0 (δ = 0), the distribution of y_ijk (R_i) is symmetric. Following Chen et al. (1999) and Kim et al. (2008), we assume that G_ξ and G_ψ are known cdfs to ensure model identifiability. In this paper, we first specify several different distributions for G_ξ and G_ψ. Then we adopt the DIC proposed by Spiegelhalter et al. (2002) and the LPML (Ibrahim et al., 2001) to determine which G_ξ and G_ψ fit the data the best.

For characterizing skewness (the first terms in (4) and (5)), we consider the following distributions for G_ξ and G_ψ: (a) G_ξ is degenerated at 0, denoted by Δ_{0}, yielding a multivariate symmetric distribution; G_ψ is degenerated at 0 for a symmetric link; (b) G_ξ is a standard exponential distribution (ℰ) with probability density function (pdf) f_ℰ(ξ_ijk) = exp(−ξ_ijk) if ξ_ijk > 0 and 0 otherwise; G_ψ is a ℰ; and (c) G_ξ is a half normal (ℋ𝒩) with pdf $f_{\mathcal{H}\mathcal{N}}({\xi }_{\mathit{ijk}})=\frac{2}{\sqrt{2\pi }}exp(-{\xi }_{\mathit{ijk}}^2/2)$ if ξ_ijk > 0 and 0 otherwise; G_ξ is a ℋ𝒩. Thus given G_ξ the model (4) yields a skewed multivariate distribution. Also, ℰ and ℋ𝒩 for G_ψ in (5) both lead to skewed links. For the symmetric component of the error terms in the longitudinal outcomes (second term in (4)), we assume the following multivariate scale-mixture normal distribution for ε_ij

$${\mathit{\epsilon }}_{ij}\mid {\lambda }_{ij}~N_K(\mathbf{0},{\mathrm{\sum }}_j/{\lambda }_{ij}),$$ (6)

where λ_ij’s are independent across j and each follows the distribution Gamma (ν₁/2, ν₂/2), where Gamma (a, b) is a Gamma distribution with mean a/b. The marginal distribution of ε_ij is then a multivariate generalized t-distribution with parameters ν₁ and ν₂. Specifically, the pdf of ε_ij is given by

$$f_{gt,{\nu }_1,{\nu }_2}({\mathit{\epsilon }}_{ij})=\frac{\mathrm{\Gamma }\left(\frac{{\nu }_1+K}{2}\right)}{\mathrm{\Gamma }\left(\frac{{\nu }_1}{2}\right)}\frac{1}{{(\pi {\nu }_2)}^{K/2}}{\mid {\mathrm{\sum }}_j\mid }^{1/2}{\left[1+\frac{1}{{\nu }_2}{\mathit{\epsilon }}_{ij}^{\prime }{\mathrm{\sum }}_j^{-1}{\mathit{\epsilon }}_{ij}\right]}^{-({\nu }_1+K)/2}$$ (7)

with

$$\text{Var}({\epsilon }_{\mathit{ijk}})={\sigma }_{\mathit{jkk}}\left(\frac{{\nu }_2}{{\nu }_1-2}\right)\text{for}{\nu }_1>2\text{and}{\nu }_2>0$$ (8)

and

$$\text{Cov}({\epsilon }_{\mathit{ijk}},{\epsilon }_{ij{k}^{\prime }})={\sigma }_{jk{k}^{\prime }}\times \frac{{\nu }_2}{2}\times \frac{\mathrm{\Gamma }{\left(\frac{{\nu }_1-1}{2}\right)}^2}{\mathrm{\Gamma }{\left(\frac{{\nu }_1}{2}\right)}^2}\text{for}{\nu }_1>1\text{and}{\nu }_2>0,$$ (9)

where ν₁ is a shape parameter (or degrees of freedom) and ν₂ is a scale parameter. When ν₁ = ν₂ = ν, (7) reduces to a multivariate t-distribution with ν degrees of freedom. Similar to a multivariate t-distribution, the pdf of a multivariate generalized t-distribution is symmetric about zero, with a small value of ν₁ corresponding to a heavy tailed distribution. For a multivariate generalized t-distribution, ν₂ is assumed to be fixed to ensure identifiability. Without loss of generality, we assume ν₂ to be 1 in this paper. For the symmetric component of the link function for binary outcome (second term in (5)), η_i, we assume a generalized t-distribution (Abranowitz and Stegun, 1972) that is a univariate version of (7) with variance 1 given by

$$f_{gt,{\nu }_1^{\ast },{\nu }_2^{\ast }}({\eta }_i)=\frac{1}{\sqrt{\pi }}\frac{\mathrm{\Gamma }\left(\frac{{\nu }_1^{\ast }+1}{2}\right)}{\sqrt{{\nu }_2^{\ast }}\mathrm{\Gamma }\left(\frac{{\nu }_1^{\ast }}{2}\right)}\times {\left(1+\frac{{\eta }_i^2}{{\nu }_2^{\ast }}\right)}^{-\frac{{\nu }_1^{\ast }+1}{2}}.$$ (10)

where ${\nu }_1^{\ast }$ is a shape parameter (or degrees of freedom) and ${\nu }_2^{\ast }$ is a scale parameter. To ensure identifiability, we assume ${\nu }_2^{\ast }$ to be 1 (Kim et al., 2008).

We note that based on (7) and (10), ${\mathit{e}}_{ij}^y$ in (4) is a multivariate skewed generalized t-distribution, and $e_i^R$ in (5) leads to a skewed generalized t-link for S_i. Web Figure 1 illustrates the skewed generalized t-distribution. The joint models defined in (1), (2), (3), (4), (5), (7), and (10) are general and flexible and include the normal/probit and skewed t/skewed t-link models as special cases for modeling the longitudinal measurements/binary outcome.

To complete the model specification, we need to define h(b_ik). For the fetal growth example, h(b_ik) is chosen so that the longitudinal process is linked with the binary process through an individual’s projected fetal growth at a point close to birth (Albert, 2012). Thus, h(b_ik) = u_k′b_ik in (2) is chosen, as ${\mathit{u}}_k={(1,t_{\ast },\dots ,t_{\ast }^q,{(t_{\ast }-{\zeta }_{k1})}_{+}^q,\dots ,{(t_{\ast }-{\zeta }_{km})}_{+}^q)}^{\prime }$ is a truncated polynomial basis functions of degree q with m knots and t_* a time point near the time of birth, such as 39 weeks of gestation.

2.4 Likelihood functions

For the longitudinal data, we let y = (y₁₁₁, . . ., y_I,JI,K)′, $W_{ij}=\text{diag}({\mathit{w}}_{ij1}^{\prime },\dots ,{\mathit{w}}_{\mathit{ijk}}^{\prime }),W={(W_{11}^{\prime },\dots ,W_{I,J_I}^{\prime })}^{\prime },\mathit{\varphi }={({\mathit{\varphi }}_1^{\prime },\dots ,{\mathit{\varphi }}_K^{\prime })}^{\prime }$, θ = (θ₁, . . .,θ_K)′, Σ = diag(Σ₁, . . ., Σ_JI), and $\mathit{b}={({\mathit{b}}_1^{\prime },\dots ,{\mathit{b}}_I^{\prime })}^{\prime }$. Also let $D_{\mathit{obs}}^y=(\mathit{y},X,W)$ denote the observed data. Given b and $D_{\mathit{obs}}^y$, the likelihood function of (β, ϕ, θ, Σ, ν₁) for longitudinal ultrasound measurements is given by

$$L_y(\mathit{\beta },\mathit{\varphi },\mathit{\theta },\mathrm{\sum },{\nu }_1\mid \mathit{b},D_{\mathit{obs}}^y)=\prod _{i=1}\prod _{j=1}\int \frac{\mathrm{\Gamma }\left(\frac{{\nu }_1+K}{2}\right)}{{\pi }^{K/2}\mathrm{\Gamma }\left(\frac{{\nu }_1}{2}\right)}{\mid {\mathrm{\sum }}_j\mid }^{-1/2}{\left[1+{({\mathit{y}}_{ij}-{\mathit{\mu }}_{ij})}^{\prime }{\mathrm{\sum }}_j^{-1}({\mathit{y}}_{ij}-{\mathit{\mu }}_{ij})\right]}^{-({\nu }_1+K)/2}d{G}_{\xi }({\mathit{\xi }}_{ij}),$$ (11)

where μ_ij = X_ijβ + W_ijϕ + W_ijb_i + Δ_θ(ξ_ij − Eξ_ij). For the binary data, we let S = (S₁, . . ., S_I)′, $U_{\ast }=\text{diag}({\mathit{u}}_1^{\prime },\dots ,{\mathit{u}}_K^{\prime })$, α = (α₁, . . ., α_K)′, and ψ = (ψ₁, . . ., ψ_I)′. Also let $D_{\mathit{obs}}^S=(\mathit{S},Z,U_{\ast })$ denote the observed data. The likelihood function of (γ, α, δ, ${\nu }_1^{\ast }$) for the binary outcome is

$$\begin{array}{l}{L}_S(\mathit{\gamma },\mathit{\alpha },\delta ,{\nu }_1^{\ast }\mid \mathit{b},D_{\mathit{obs}}^S)\\ =\prod _{i=1}^I\int {[F_{gt,{\nu }_1^{\ast },{\nu }_2^{\ast }=1}({{\mathit{z}}_i}^{\prime }\mathit{\gamma }+{\mathit{\alpha }}^{\prime }{U}_{\ast }{\mathit{b}}_i+\delta ({\psi }_i-E{\psi }_i))]}^{S_i}\\ \times {[1-F_{gt,{\nu }_1^{\ast },{\nu }_2^{\ast }=1}({{\mathit{z}}_i}^{\prime }\mathit{\gamma }+{\mathit{\alpha }}^{\prime }{U}_{\ast }{\mathit{b}}_i+\delta ({\psi }_i-E{\psi }_i))]}^{1-S_i}d{G}_{\psi }({\psi }_i).\end{array}$$ (12)

Furthermore, the observed joint likelihood function of (β, ϕ, θ, Σ, ν₁, γ, α, δ, ${\nu }_1^{\ast }$, Ω) is

$$L(\mathit{\beta },\mathit{\varphi },\mathit{\theta },\mathrm{\sum },{\nu }_1,\mathit{\gamma },\mathit{\alpha },\delta ,{\nu }_1^{\ast },\mathrm{\Omega }\mid D_{\mathit{obs}}^y,D_{\mathit{obs}}^S)=\int L_y(\mathit{\beta },\mathit{\varphi },\mathit{\theta },\mathrm{\sum },{\nu }_1\mid \mathit{b},D_{\mathit{obs}}^y)\times L_S(\mathit{\gamma },\mathit{\alpha },\delta ,{\nu }_1^{\ast }\mid \mathit{b},D_{\mathit{obs}}^S)\times \prod _{i=1}N({\mathit{b}}_i\mid 0,\mathrm{\Omega })d\mathit{b},$$ (13)

where $L_y(\mathit{\beta },\mathit{\varphi },\mathit{\theta },\mathrm{\sum },{\nu }_1\mid \mathit{b},D_{\mathit{obs}}^y)$ and $L_S(\mathit{\gamma },\mathit{\alpha },\delta ,{\nu }_1^{\ast }\mid \mathit{b},D_{\mathit{obs}}^S)$ are given in (11) and (13), respectively. Since it is difficult to work directly with the observed joint likelihood function of (β, ϕ, θ, Σ, ν₁, γ, α, δ, ${\nu }_1^{\ast }$, Ω) in (13), it is infeasible to develop an efficient Markov chain Monte Carlo sampling algorithm (MCMC) algorithm. Instead, we use the fact that the generalized t-distribution can be represented as a gamma mixture of normal distributions for ε_ij and $e_i^R$, and we introduce the complete data likelihood function of (β, ϕ, θ, Σ, ν₁, γ, α, δ, ${\nu }_1^{\ast }$, Ω) as described in Web Appendix A.

3. Predicting the Binary Outcome

The joint model in (1) and (2) relates the longitudinal fetal growth pattern to the probability of an abnormal birth outcome through an individual’s predicted measurement at time t^*, and it can be used to develop a predictor of the binary outcome from longitudinally collected measurements. We are interested in predicting an adverse pregnancy outcome such as macrosomia from a series of multivariate ultrasound measurements taken at various gestational ages. To predict the abnormal binary outcome at birth using the multivariate longitudinal ultrasound measurements, we let y^P = (y_t1, y_t2, . . ., y_tL) denote the longitudinal measurements taken at time points t₁, t₂, . . ., t_L, where L is the number of repeated measurements in the predictor. We also let S^P denote the binary outcome we wish to predict. Let $\mathrm{\Theta }=(\mathit{\beta },\mathit{\varphi },\mathit{\theta },\mathrm{\sum },{\nu }_1,\mathit{\gamma },\mathit{\alpha },\delta ,{\nu }_1^{\ast },\mathrm{\Omega })$. Then the posterior predictive probability for S^P based on longitudinal measurements y^P can be given by

$$P(S^P=1\mid {\mathit{y}}^P)=\int F_{gt,{\nu }_1^{\ast },{\nu }_2^{\ast }=1}\left({\mathit{z}}^{\prime }\mathit{\gamma }+{\mathit{\alpha }}^{\prime }{U}_{\ast }{\mathit{b}}^p+\delta (\psi -E\psi )\right)\pi \left(\mathrm{\Theta },{\mathit{b}}^P\mid {\mathit{y}}^P\right){dG}_{\psi }(\psi )d{\mathit{b}}^{P}d\mathrm{\Theta },$$ (14)

where b^P is a multivariate random effect (b^P ~ N (0, Ω)) and π(Θ, b^P |y^P) is the posterior predictive distribution for Θ and b^P based on y^P. To evaluate the prediction accuracy, we divide the data into training and test set data. To obtain the posterior predictive probability in (14), we sample b^P from the joint posterior distribution based on test set data (with parameter estimates obtained from the training set data). We consider two approaches to assess the predictive ability of the longitudinal classifiers. One approach is the receiver operator characteristic curve (ROC) used by Albert (2012), which is a standard approach for estimating the accuracy of a binary classification using a continuous marker. In particular, we compute the area under the ROC curve (AUC), where a value of 1 corresponds to perfect classification, while a value of 0.5 corresponds to completely random classification. Furthermore, the ROC is a plot of 1-specificity versus sensitivity for multiple cut-off values of the predictor. We also use the mean-squared error (MSE) of prediction, which is a measure for assessing absolute risk (Gail and Pfeiffer, 2005) and the average squared difference between the predicted probability and the binary outcome. To perform valid prediction assessment, we estimate the model parameters of the joint model in (1) and (2) and formulate the predictor using the training set data and then validate the predictor using the test set data.

4. Posterior Inference

4.1 Prior and Posterior Distributions

We assume that β, ϕ, θ, Σ, ν₁, γ, α, δ, ${\nu }_1^{\ast }$, and Ω are independent a priori. Thus, the joint prior for (β, ϕ, θ, Σ, ν₁, γ, α, δ, ${\nu }_1^{\ast }$, Ω) is of the form

$$\pi (\mathit{\beta },\mathit{\varphi },\mathit{\theta },\mathrm{\sum },{\nu }_1,\mathit{\gamma },\mathit{\alpha },\delta ,{\nu }_1^{\ast },\mathrm{\Omega })=\pi (\mathit{\beta })\pi (\mathit{\varphi })\pi (\mathit{\theta })\pi (\mathrm{\sum })\pi ({\nu }_1)\pi (\mathit{\gamma })\pi (\mathit{\alpha })\pi (\delta )\pi ({\nu }_1^{\ast })\pi (\mathrm{\Omega }).$$ (15)

We further assume that β_k ~ N_p1 (0, c₁I_p1), ϕ_k ~ N_q+m+1(0, c₂I_q+m+1), θ ~ N_K(0, c₃I_K), γ ~ N_p2 (0, c₄I_p2), α ~N_K(0, c₅I_K), δ ~ N (0, c₆), ν₁ ~ Gamma (a₁,b₁) with ${\nu }_1^{\ast }~\text{Gamma}(a_2,b_2)$ with ${\nu }_1^{\ast }>1,{\mathrm{\sum }}_j^{-1}~{\text{Wishart}}_K(d_0,V_0)$, and Ω⁻¹ ~ Wishart_K(q+m+1) (d₁, V₁), where c₁, c₂, c₃, c₄, c₅, c₆, a₁, b₁, a₂, b₂, d₀, V₀, d₁, and V₁ are the prespecified hyperparameters. Here, Wishart_K (d, V) denotes a Wishart prior distribution with d degrees of freedom and mean dV. We recommend choosing vague priors for all parameters except ${\nu }_1^{\ast }$, since posterior inference was not sensitive to the choice of these hyperparameters. For this application, an informative prior was necessary for ${\nu }_1^{\ast }$, since there was little information about the long tails in the link function for the binary outcome. However, assuming an informative prior for ${\nu }_1^{\ast }$ still provides a very flexible form for the link function and ensures convergence of the Gibbs sampler.

Based on the prior distributions specified above, the joint posterior distribution of β, ϕ, θ, Σ, ν₁, γ, α, δ, ${\nu }_1^{\ast }$, and Ω is

$$\begin{array}{l}\pi (\mathit{\beta },\mathit{\varphi },\mathit{\theta },\mathrm{\sum },{\nu }_1,\mathit{\gamma },\mathit{\alpha },\delta ,{\nu }_1^{\ast },\mathrm{\Omega }\mid D_{\mathit{obs}}^y,D_{\mathit{obs}}^S)\\ \propto L(\mathit{\beta },\mathit{\varphi },\mathit{\theta },\mathrm{\sum },{\nu }_1,\mathit{\gamma },\mathit{\alpha },\delta ,{\nu }_1^{\ast },\mathrm{\Omega }\mid D_{\mathit{obs}}^y,D_{\mathit{obs}}^S)\\ \times \pi (\mathit{\beta })\pi (\mathit{\varphi })\pi (\mathit{\theta })\pi (\mathrm{\sum })\pi ({\nu }_1)\pi (\mathit{\gamma })\pi (\mathit{\alpha })\pi (\delta )\pi ({\nu }_1^{\ast })\pi (\mathrm{\Omega }),\end{array}$$ (16)

where $L(\mathit{\beta },\mathit{\varphi },\mathit{\theta },\mathrm{\sum },{\nu }_1,\mathit{\gamma },\mathit{\alpha },\delta ,{\nu }_1^{\ast },\mathrm{\Omega }\mid D_{\mathit{obs}}^y,D_{\mathit{obs}}^S)$ is defined in (13). A description of the MCMC algorithm is given in Web Appendix B.

4.2 Model Comparison

To assess the goodness of fit of the models, we use the LPML given in Ibrahim et al. (2001) and the DIC proposed by Spiegelhalter et al. (2002). First, LPML is a well-established Bayesian model comparison criterion based on conditional predictive ordinate (CPO) statistics. As suggested in Ibrahim et al. (2001), a natural summary statistic of the CPO_is is the LPML defined as $\text{LPML}={\sum }_i^{I}log({\text{CPO}}_i)$, where the CPO statistic for the ith subject is the marginal posterior predictive density of y_i and S_i. Second, for computing the DIC, we are unable to easily integrate out b analytically in (1) and (2). We therefore take a different approach that uses an extension of the DIC (Huang et al., 2005), given in Web Appendix C. The larger the LPML value and smaller the DIC value, the better the model fits the data.

5. Analysis of the Successive Small-for-Gestational-Age Births Study Data

We used the proposed joint model in (1) and (2) to analyze data from the SGA study discussed in Section 1, and we focus on predicting macrosomia, defined as a newborn > 4, 000g, using the longitudinal ultrasound measurements. By considering macrosomia, we are examining the occurrence of large birthweight that has the potential to have pathological effects. Alternatively, we could have used other measures of large birthweight, including a commonly used measure called large for gestational age (LGA), that explicitly account for gestational age. An analysis of the actual birthweight is also possible but we think less relevant, since our interest is predominately on identifying (diagnosing) abnormally large neonates.

The response variable y_ij = (y_ij1, y_ij2, y_ij3)′ is the anthropomorphic ultrasound measurements for the ith woman at the jth time point: BPD (mm), MAD (mm), and FL (mm), where each pregnant woman has four ultrasound examinations at approximately 17, 25, 33 and 37 weeks of gestation. The time point t_ijk is the jth gestational age (GA, weeks) for the ith woman and kth type of ultrasound measurement. We consider the four covariates: maternal age (Age, years), pre-pregnancy body mass index (BMI, kg/m²), history of small-for-gestational age birth (SGA, yes/no), and smoking during pregnancy (Smoking, number of cigarettes per day). Furthermore, the adverse binary outcome S_i is macrosomia, which is defined as birthweight > 4000g (Zhang et al., 2012). We focus on 1474 women who had complete or partial longitudinal ultrasound measurements along with the birth outcome and relevant covariates in this analysis. Thus, I = 1474, J = 4 is the maximum number of repeated ultrasound measurements, and K = 3 is the number of ultrasound measurement types. Descriptive statistics for the SGA study data are presented in Web Table 1. In addition, Web Figure 2 illustrates the timing of ultrasound examinations and birth. We further use m = 3 for the number of knots and ζ_k = (18.86, 26.14, 33.71)′ for the locations of the knot points corresponding to the 25th, 50th, and 75th percentiles of all measurement times. As mentioned in Section 2.1, we use cubic splines for g(t_ijk) and g_b(t_ijk; b_ik) by setting q = 3 in (3) to incorporate a flexible mean structure. We also use q = 3 and t_* = 39 for h(b_ik) = u_k′b_ik in (2). In addition, we divide the whole data into two data sets with a 60% and 40% random split into training and test set data, respectively (884 in the training set and 590 in the test set). We use the training set data to develop the predictor (i.e., ROC, AUC, and MSE) by first fitting the joint model in (1) and (2), while the test set data are used to validate the predictor with different accuracy measures.

In all of the analyses, we standardized the covariates, in which each covariate was subtracted from its sample mean and divided by its sample standard deviation (SD). This was done to help the numerical stability in the posterior computation using the MCMC sampling algorithm in Web Appendix B. The means and standard deviations are (28.33, 4.26) for Age, (21.51, 3.17) for BMI, (0.24, 0.43) for SGA, and (6.78, 7.15) for Smoking. Furthermore, t_ijk is re-scaled to the unit interval, which is divided by the maximum value of t_ijk, so that 0 < t_ijk ≤ 1. The location of knots ζ_k is also re-scaled by the maximum value of t_ijk. The hyperparameters of the prior in (15) were specified as c₁ = 100, c₂ = 100, c₃ = 100, c₄ = 100, c₅ = 100, c₆ = 100, a₁ = 1, b₁ = 0.1, a₂ = 1, b₂ = 1, d₀ = K + 0.1, V₀ = 0.1, d₁ = q + m + 1.00001, and V₁ = 0.00001 in the analysis. For all of the posterior computations, we first generated 100,000 MCMC Gibbs samples with a burn-in of 20,000 iterations, and we then used 20,000 iterations obtained from every 5^th iteration for computing all the posterior estimates, including posterior means, posterior standard deviations, 95% highest posterior density (HPD) intervals, and the LPMLs and DICs for model comparison. The computer programs were written in FORTRAN 95 using IMSL subroutines with double precision accuracy. The convergence of the MCMC sampling algorithm for all the parameters was checked based on the recommendations of Cowles and Carlin (1996). All trace and autocorrelation plots showed good convergence and excellent mixing of the MCMC sampling algorithm.

We are interested in investigating how the goodness of fit might be affected by the distributions of ε_ij and η_i and by the choices of G_ξ and G_ψ for joint model in (1) and (2) using the DIC and LMPL discussed in Section 4.2. This investigation involves the following distributions: (i) normal, generalized t (GT), and skewed generalized t (SGT) for ε_ij and η_i; (ii) Δ_{0}, ℰ, and ℋ𝒩 for G_ξ and G_ψ. With the combination of those distributions, we have the following ten models for model comparison: (1) Probit-Normal model with symmetric normal for η_i and ε_ij; (2) SProbitE-Normal model with normal for η_i and ε_ij, and ℰ for G_ψ; (3) SProbitN-Normal model with normal for η_i and ε_ij, and ℋ𝒩 for G_ψ; (4) GT-GT model with symmetric GT for η_i and ε_ij; (5) SGTE-GT model with GT for η_i and ε_ij, and ℰ for G_ψ; (6) SGTE-SGTE model with GT for η_i and ε_ij, and ℰ for G_ψ and G_ξ; (7) SGTE-SGTN model with GT for η_i and ε_ij, ℰ for G_ψ and ℋ𝒩 for G_ξ; (8) SGTN-GT with GT for η_i and ε_ij, and ℋ𝒩for G_ψ; (9) SGTN-SGTE with GT for η_i and ε_ij, ℋ𝒩 for G_ψ and ℰ for G_ξ; (10) SGTN-SGTN model with GT for η_i and ε_ij, and ℋ𝒩 for G_ψ and G_ξ.

Table 1 shows the DIC and LPML values for the ten models under consideration, with the smallest value for DIC (33878.04) and largest value for LPML (−19351.22) corresponding to the SGTN-SGTN model. This demonstrates that the SGTN-SGTN model fits the training data the best among all models considered. This affirms the need for considering the heavy tail distributions for both η_i and ε_ij and skewed distributions with ℋ𝒩 for both G_ξ and G_ψ. The models with symmetric distributions for η_i and ε_ij have the larger DIC values and smaller LPML values, suggesting that these models fit data worse than the skewed models. Furthermore, between skewed models, the models with ℋ𝒩 for ε_ij have a better fit than models with ℰ for ε_ij. Importantly, all models considered show a better fit than the ProbitNormal model proposed by Albert (2012) and Zhang et al. (2012).

Table 1.

The values of DIC and LPML based on the training set data

Model	D(Θ̄)	PD	DIC	LPML
Probit-Normal	35138.35	2349.66	39837.67	−20648.07
SProbitE-Normal	34511.85	2581.42	39674.68	−20583.01
SProbitN-Normal	34407.83	2573.29	39554.40	−20531.47
GT-GT	31788.66	3419.64	38627.94	−20495.25
SGTE-GT	31409.85	3463.43	38336.71	−20424.44
SGTE-SGTE	26532.40	4234.55	35001.50	−19679.18
SGTE-SGTN	23645.86	5218.22	34082.31	−19469.08
SGTN-GT	31369.51	3464.70	38298.90	−20380.60
SGTN-SGTE	26124.64	4242.18	34608.99	−19513.09
SGTN-SGTN	23334.20	5271.92	33878.04	−19351.22

Tables 2 and 3 show the posterior means, standard deviations and 95% HPD intervals of the parameters under the best model (SGTN-SGTN model) based on the training set data. The results in Table 2 show that only maternal age is significant and has positive association with MAD, suggesting that older women have fetuses with larger abdominal diameter. Further, a women’s BMI is negatively and positively associated with BPD and MAD, respectively. The skewness parameters for MAD and FL were significantly different from zero, with MAD being negative and FL positive. The posterior estimate for ν₁ is 3.92, suggesting that ultrasound measurements have heavy tail distributions. Posterior estimates of g(t_ijk) and Σ under the best model (SGTN-SGTN model) are presented in Web Tables 2 and 3. The estimated longitudinal trajectory plots of BPD, MAD, and FL over gestational age are given in Web Figure 3. This plot adjusts for Age, BMI, SGA, and Smoking and takes full advantage of the specification of our flexible model. The estimated longitudinal trajectories for MAD and FL are steadily increasing across gestation, and the trajectory for BPD appears to level off at later gestational ages. Furthermore, ultrasound measurements are positively correlated with each other at each of the follow-up times, with the correlation being the lowest at the third follow-up time (Web Table 4).

Table 2.

Posterior estimates of the parameters for ultrasound measurements under the best model: fit to training set data

	Variable	Parameter	Mean	SD	95% HPD Interval
BPD	Age	β11	−0.029	0.033	(−0.094, 0.036)
	BMI	β12	−0.099	0.033	(−0.163, −0.033)
	SGA	β13	0.041	0.034	(−0.026, 0.106)
	Smoking	β14	0.051	0.034	(−0.013, 0.120)
	Skewness	θ1	−0.072	0.647	(−0.870, 0.815)
MAD	Age	β21	0.122	0.057	(0.011, 0.231)
	BMI	β22	0.126	0.058	(0.010, 0.237)
	SGA	β23	−0.048	0.057	(−0.157, 0.066)
	Smoking	β24	0.074	0.057	(−0.042, 0.182)
	Skewness	θ2	−1.983	0.122	(−2.215, −1.742)
FL	Age	β31	−0.069	0.052	(−0.171, 0.030)
	BMI	β32	0.067	0.052	(−0.034, 0.170)
	SGA	β33	0.042	0.052	(−0.059, 0.145)
	Smoking	β34	−0.064	0.052	(−0.163, 0.040)
	Skewness	θ3	1.243	0.139	(0.957, 1.500)
	d.f.	ν1	3.924	0.313	(3.303, 4.524)

Table 3.

Posterior estimates for macrosomia under the best model: fit to training set data

Variable	Parameter	Mean	SD	95% HPD Interval
Intercept	γ1	−17.299	3.733	(−24.357, −10.989)
Age	γ2	1.062	0.711	(−0.296, 2.495)
BMI	γ3	0.986	0.721	(−0.358, 2.475)
SGA	γ4	−3.368	1.119	(−5.676, −1.410)
Smoking	γ5	−1.173	0.717	(−2.661, 0.187)
	α1	0.989	0.316	(0.435, 1.594)
	α2	1.583	0.419	(0.835, 2.404)
	α3	−0.022	0.453	(−0.998, 0.798)
	δ	−21.619	5.278	(−31.637, −12.171)
	${\nu }_1^{\ast }$	1.461	0.606	(1.000, 2.570)

Furthermore, the results in Table 3 show that only SGA is significant and has negative association with macrosomia, demonstrating that the probability of macrosomia is lower for women with a history of small-for-gestational-age birth. The parameters α₁ for BPD and α₂ for MAD, which link the two processes, are positive and highly statistically significant, while α₃ for FL is not significant, suggesting that the trajectories for BPD and MAD are positively associated with macrosomia, while the trajectory for FL is not related to macrosomia. Further, the posterior estimate of skewness δ is negative and highly significant (−21.62), demonstrating that the negative skewed link is needed for appropriately modeling macrosomia. In addition, the negative skewed link has a small value of ${\nu }_1^{\ast }$ (1.46). The posterior estimates of Ω for b_i are presented in Web Tables 4 to 9.

Using the test set data, we estimated the overall assessments of diagnostic accuracy for predicting macrosomia. Specifically, we estimated AUC and MSE using the ten models under consideration. Table 4 presents the posterior means, standard deviations and 95% HPD intervals of the AUC and MSE. The results from Table 4 show that the skewed models have higher AUC and smaller MSE than the models with symmetric distributions. The model with the highest AUC and lowest MSE was SGTN-SGTN, demonstrating the importance of incorporating long-tailed skewed error distributions in the longitudinal error distribution as well as the link function formulation. ROC curves corresponding to these AUCs are presented in Web Figure 4. Interestingly, all extended models had sizable increases in diagnostic accuracy as compared to the Probit-Normal model, the special case that reduces to the model developed by Albert (2012) and Zhang et al. (2012). The estimated individual probabilities of prediction for the Probit-Normal model, SGTN-SGTE model, and SGTN-SGTN model (the best model) for the test set data are given in Web Figure 5. These plots suggest that the Probit-Normal model provides an overestimate when the probability for the SGTN-SGTN model is greater than about 0.2. The individual probabilities of prediction are close between skewed models (e.g. SGTN-SGTE and SGTN-SGTN model).

Table 4.

AUC and MSE values based on the test set data

	AUC			MSE
Model	Mean	Std	95% HPD	Mean	Std	95% HPD
Probit-Normal	0.817	0.014	(0.790, 0.843)	0.123	0.004	(0.114, 0.132)
SProbitE-Normal	0.838	0.013	(0.812, 0.864)	0.106	0.004	(0.098, 0.114)
SProbitN-Normal	0.838	0.013	(0.812, 0.863)	0.106	0.004	(0.098, 0.114)
GT-GT	0.847	0.012	(0.823, 0.871)	0.102	0.004	(0.095, 0.111)
SGTE-GT	0.864	0.008	(0.848, 0.880)	0.098	0.003	(0.093, 0.104)
SGTE-SGTE	0.868	0.007	(0.854, 0.881)	0.098	0.002	(0.093, 0.103)
SGTE-SGTN	0.868	0.007	(0.854, 0.881)	0.097	0.002	(0.093, 0.102)
SGTN-GT	0.865	0.008	(0.850, 0.881)	0.098	0.003	(0.093, 0.103)
SGTN-SGTE	0.868	0.007	(0.854, 0.881)	0.097	0.002	(0.093, 0.102)
SGTN-SGTN	0.869	0.006	(0.857, 0.881)	0.096	0.002	(0.092, 0.099)

6. Discussion

This paper presents a new class of models that can be used to predict a binary outcome (e.g. macrosomia) from multivariate longitudinal data (e.g. ultrasound measurements of fetal growth). The models are flexible in that they allow for skewed long-tailed distributions for the ultrasound measurements and a very flexible link function for relating the longitudinal trajectories to the binary outcome of macrosomia. We demonstrate with a fetal growth study that this flexible modeling improves diagnostic and prediction accuracy relative to more standard approaches. The model extends the work of Albert (2012) and Zhang et al. (2012), who proposed a similar analytical framework with the assumption of a normal longitudinal error structure and a probit link function. In this paper, we show that a more general model provides improved model fit as well as substantially improved diagnostic accuracy as compared with the simpler analysis. We recognize that this improved performance comes at the cost of increased computational expense, since a Bayesian approach is needed to handle the multivariate random effects in this complex setting.

The methodology was developed specifically to address the important medical/epidemiologic question of predicting poor pregnancy outcomes from longitudinal ultrasound data. However, the methodology can be applied more generally to any situation where we are predicting a binary event where the timing of this event is of secondary interest and does not cause the longitudinal data to be censored. In the latter case, a joint model of longitudinal and time-to-event would be more appropriate.

In this approach, we incorporate a dependence between the longitudinal and binary processes with shared random effects that link the two process through a function h(b_ik). We specified h(b_ik) to be the projection of the fetal growth process to a time close to birth. In our analyses, we chose this time to be 39 gestational weeks, but results were not sensitive to this assumption. Alternatively, one could link the two processes with a linear combination of all the random effects and unknown parameters. However, in our situation where we have three longitudinal outcomes and 7 random effects per outcome, this would be problematic (21 coefficients to estimate). The current approach requires us to estimate only one parameter for each longitudinal outcome (a total of three coefficients).

We incorporated a flexible mean structure for the longitudinal trajectories using cubic splines. A priori, we choose three knot points at gestational times corresponding to each of three quartiles. For the fetal growth application, this was sensible, since knot point locations were chosen close to the targeted measurements. However, in other situations where observations are more irregularly spaced and the mean trajectories follow more complex patterns, a larger number of knot points may be necessary. Future research may focus on estimating the optimal number of knot points in these situations.

The model induced dependence between the longitudinal and binary outcomes using shared random effects. We recognize that this does place a constraint on the correlation between these processes. However, including separate correlated random effects for the longitudinal and binary components is not possible, since random effects cannot be incorporated for a single binary response. For applications with a repeated binary response, such an extension could be considered.