An imputation approach for fitting two-part mixed effects models for longitudinal semi-continuous data

Abstract

Two-part mixed effects models are often used for analyzing longitudinal data with many zeros. Typically, these models are formulated with binary and continuous components separately with random effects that are correlated between the two components. Researchers have developed maximum-likelihood and Bayesian approaches for fitting these models that often require using particular software packages or very specialized software. We propose an imputation approach that will allow practitioners to separately use standard linear and generalized linear mixed models to estimate the fixed effects for two-part mixed effects models with complex random effects structures. An approximation to the conditional distribution of positive measurements given an individual’s pattern of non-zero measurements is proposed that can be easily estimated and then imputed from. We show that for a wide range of parameter values, the imputation approach results in nearly unbiased estimation and can be implemented with standard software. We illustrate the proposed imputation approach for the analysis of longitudinal clinical trial data with many zeros.

1. Introduction

Statistical outcomes, where measurements are either zero or a positive value, are often called semi-continuous. An example of this type of outcome is a dietary assessment where the amount is zero if the nutrient is not ingested and a positive value, otherwise. For modeling longitudinal data of this type, two-part mixed models have been proposed in which the binary (measurement is non-zero) and continuous (value if measurement is positive) components are modeled with a generalized linear mixed model and a linear mixed model, respectively. Numerous authors have proposed two-part mixed models for analyzing semi-continuous longitudinal data that have recognized the importance of incorporate random effects that are correlated between the two model components,^1,2 and most recently,³ where the large recent literature on these models has been reviewed. Various authors^4,5 have demonstrated bias in the estimation if dependence in the two processes is ignored.

Although the two-part mixed models are appealing from a practical viewpoint, particularly when the number of random effects is not small, fitting them can be computationally intensive. Although, directly maximizing the full likelihood by integrating over the random effects distribution can be done, it is only feasible for a small number of random effects (Liu et al.⁶ used PROC NLMIXED to conduct maximum-likelihood in SAS). Bayesian approaches require computationally intensive sampling approaches for estimation.^7–9 A major computational breakthrough that allows for exact likelihood inference¹⁰ has been recently applied for modeling semi-continuous autoregressive longitudinal data.¹¹ However, this methodology uses specialized software, requires a probit link for the binary in the generalized linear mixed model, and does not lend itself to easy implementation for the practitioner.

In this paper, we propose an imputation approach that allows practitioners to use standard linear and generalized linear mixed models to fit two-part mixed models with complex random effect structures that are correlated between the two processes. Conceptually, the continuous component is only present when the binary component is positive, thereby causing bias when the two processes are fit separately. The idea is to impute values for the missing continuous measurements when the binary values are zero, thereby eliminating bias in the separate modeling of both components. We develop an approximation to the conditional distribution of an individual’s positive measurements given their non-zero measurement pattern under the two-part mixed model with correlated random effects. Although this approximation has not been used for fitting two-part mixed models, a similar one has been used for analyzing informative missing data in longitudinal studies¹² and for joint modeling of longitudinal and survival data.¹³

We introduce the two-part mixed model and propose the imputation approach in Section 2. In Section 3, we quantify the asymptotic bias of the imputation approach and compare it to the naive approach, where we assume that the two processes are independent from each other. We conduct simulations to examine the small sample properties of the approach in Section 4. We illustrate the methodology in analyzing data from a longitudinal clinical trial in Section 5. A discussion follows in Section 6.

2. The conditional approximation-based imputation approach

In this section, we propose an imputation approach that approximates the maximum-likelihood estimators obtained with a full joint modeling approach. We begin by showing the approximation for a shared intercept model followed by the more complex cases of a random time-trend and two random effects where one is nested within the other.

2.1. Two-part mixed model

Let ${\mathit{Z}}_i=\left\{Z_1,Z_2,\cdots ,Z_{n_i}\right\}$ be a vector of indicator functions that are equal to 1 when the $j$th measurement on the ith subject is positive and equal to zero if the measurement is zero; $n_i$ is the number of measurements on the $i$th subject, and $N$ is the number of subjects. There is an associated positive value $Y_{ij}$ when $Z_{ij}=1$, that is undefined when $Z_{ij}=0$. We can construct a two-part mixed model by dividing data into a binary and continuous part, and separately modeling ${\mathit{Z}}_i$ and ${\mathit{Y}}_i\mid {\mathit{Z}}_i$. The model can be written as

$$logi{t}_r\left(Z_{ij}=1\mid {\mathit{U}}_i\right)={\text{X}}_{ij}\mathit{\theta }+W_{ij}{\text{U}}_i$$ (1)

$$Y_{ij}\mid Z_{ij}=1,{\mathit{V}}_i\sim N\left({\mathit{X}}_{ij}\mathit{\beta }+W_{ij}{\mathit{V}}_i,{\sigma }_e^2\right)$$ (2)

where $\mathit{\theta }$ and $\mathit{\beta }$ are $p$-dimensional vectors of fixed-effect coefficients for the binary part and the continuous part, respectively. Further, ${\mathit{U}}_i$ and ${\mathit{V}}_i$ are $q$ dimensional vectors of random effects for the two components, and ${\mathit{X}}_{ij}$ and ${\mathit{W}}_{ij}$ are design matrices for the fixed and random parts, respectively. We assume that all the random effect coefficients, ${\mathit{U}}_i$ and ${\mathit{V}}_i$ follow a multivariate normal distribution with a mean vector of zeros and a covariance matrix m effect coefficients, ing l joint modeling approach. We btween the binary and continuous parts.

The likelihood function for the two-part mixed model (equations (1) and (2)) is

$$L=\prod _{i=1}^N\underset{{\mathit{U}}_i}{\int }\cdots \underset{{\mathit{V}}_{ij}}{\int }\prod _i^{n_i}{\left\{1-P_r\left(Z_{ij}=1\mid \mathit{\theta },{\mathit{U}}_i\right)\right\}}^{I\left(Z_{ij}=0\right)}$$

$$\times {\left\{P_r\left(Z_{ij}=1\mid \mathit{\theta },{\mathit{U}}_i\right)f\left(Y_{ij}\mid Y_{ij}>0,\mathit{\beta },{\mathit{V}}_i\right)\right\}}^{I\left(Z_{ij}=1\right)}\times f\left({\mathit{U}}_i,{\mathit{V}}_i\mid \mathbf{\text{Ω}}\right)d{\mathit{V}}_{i}d{\mathit{U}}_i$$

which can be maximized with respect to the parameters to obtain the maximum-likelihood estimators. However, evaluating the likelihood is computationally burdensome due to the high-dimensional integration. Olsen and Schafer¹ proposed a higher order Laplace approximation for the likelihood to simplify the numerical integration. Tooze et al.² and Liu et al.,⁶ among others, have proposed using PROC NLMIXED in SAS for maximum-likelihood estimation. This approach uses adaptive Gaussian quadrature¹⁴ to directly integrate over the multivariate random effect distribution. We show in this article how estimation can be done by separately modeling the binary and continuous components using generalized linear mixed and linear mixed models, routines that are available with all standard statistical software packages. We introduce the proposed imputation approach with the case of two random intercepts.

2.2. Approximate approach for random intercept model

We fit the binary component with a generalized linear mixed model (e.g. glmer in R package lme4). For the continuous part, we use a conditional approximation-based imputation approach to obtain fixed effect estimates with reduced bias. We introduce the approach with random intercepts that are correlated between the two processes. If we knew the conditional distribution of the continuous values ($Y_{ij}$) given they are positive. ($Z_{ij}=1$), we could use it to impute values of $Y_{ij}$ that are undefined when $Z_{ij}=0$ to obtain unbiased estimates of the fixed effects for the continuous component. This conditional distribution can be written as

$$f\left({\mathit{Y}}_i\mid {\mathit{Z}}_i\right)=\underset{v_i}{\int }\left\{\prod _{j=1}^{n_i}f\left(Y_{ij}\mid Y_{ij}>0,v_i\right)\right\}h\left(v_i\mid {\mathit{Z}}_i\right)\text{d}{v}_i$$ (3)

where ${\mathit{Y}}_i$ is a vector of positive responses associated with $Z_{ij}=1$. The distribution given by equation (3) is difficult to evaluate, maximize to obtain parameter estimates, and to then impute from. Thus, we obtain an approximation to the conditional distribution that is easy for the practitioner to work with. Using sufficiency arguments similar to those presented in Wu and Follmann,¹⁵ it is easy to show that $h\left(v_i\mid {\mathit{Z}}_i\right)=h\left(v_i\mid d_i\right)$, where $d_i=\sum _j{Z}_{ij}$ when the binary process does not have a time-dependent fixed or random effect. However, the identification of these sufficient statistics is difficult to derive in most cases, and we propose approximating $h\left(v_i\mid {\mathit{Z}}_i\right)$ by $h\left(v_i\mid d_i\right)$.

We further approximate $h\left(v_i\mid d_i\right)$ as a normal distribution with a mean that is a simple quadratic function of $d_i$ and a variance that is constant. Simulations under the full joint model demonstrated the adequacy of this approximation under a wide range of parameter values. Assuming that $h\left(v_i\mid d_i\right)$ is normal with mean ${\beta }_0^{\ast }+{\delta }_0{d}_i+{\delta }_1{d}_i^2$ and variance ${\sigma }_v^{\ast 2}$, the conditional distribution can be expressed as the linear mixed model,

$$Y_{ij}=X_{ij}{\mathit{\beta }}^{\ast }+{\delta }_0{d}_i+{\delta }_1{d}_i^2+v_i^{\ast }+{ϵ}_{ij}^{\ast }$$ (4)

where $v_i^{\ast }=v_i-\left({\beta }_0^{\ast }+{\delta }_0{d}_i+{\delta }_1{d}_i^2\right)$ follows a normal distribution with mean zero and a constant variance ${\sigma }_v^{\ast 2}$, ${\mathit{\beta }}^{\ast }$ are the fixed effects characterizing the approximate conditional distribution, and ${ϵ}_{ij}^{\ast }$ is normal with mean 0 and variance ${\sigma }_{ϵ}^{2\ast }$. Thus, the parameters of this approximate conditional distribution can be estimated with standard linear mixed modeling software.

Estimation can be done with the following two-stage algorithm. The longitudinal binary responses are analyzed with a generalized linear mixed model (equation (1)) in the first stage. An analysis of the longitudinal continuous data is performed with Steps 1–4 in the following algorithm:

Step 1: Fit the approximate conditional model (equation (4)) using a linear mixed modeling software. Obtain individual random effects estimates using shrinkage estimation.

Step 2: Impute values of $Y_{ij}$ when $Z_{ij}=0$ by generating from the estimated approximate conditional distribution obtained in Step 1.

Step 3: Fit the linear mixed model (equation (2)) to the complete data containing the imputed values when $Z_{ij}=0$.

Step 4: Repeat Steps 2–3 $I$ times ($I=100$ in this study) to obtain a vector of fixed effects estimates by averaging $I$ sets of imputed complete datasets.

Rubin’s Rule¹⁶ can be used for estimating standard errors of the parameter estimates.

2.3. Random slope model

We consider variation in subject-specific time trend (slope). Thus, the random effect coefficients are now ${\mathit{U}}_i=\left\{u_{0i},u_{1i}\right\}$ for the binary part and for the continuous part. Then, extended from the random intercept model structure, the approximate conditional distribution of $Y_{ij}\mid Z_{ij}$ based on the random slope structure

$$Y_{ij}\mid Z_{ij}=1,v_i^{\ast },d_i=X_{ij}{\mathit{\beta }}^{\ast }+{\delta }_0{d}_i+{\delta }_1{d}_i^2+{\delta }_2{t}_{ij}{d}_i+{\delta }_3{t}_{ij}{d}_i^2+v_{0i}^{\ast }+v_{1i}^{\ast }{t}_{ij}+{ϵ}_{ij}^{\ast }$$ (5)

Compared with equation (4), equation (5) has two additional fixed-effect terms, ${\delta }_2{t}_{ij}{d}_i$ and ${\delta }_3{t}_{ij}{d}_i^2$ and one additional random effect term, $v_{1i}^{\ast }{t}_{ij}$, that results from approximating $h\left(v_{i0},v_{i1}\mid d_i\right)$ by a multivariate normal distribution with a constant conditional variance ${\text{Σ}}_v^{\ast }$. Then, based on the approximate conditional distribution (equation (5)), we can estimate the fixed effects of the continuous component in a similar way as described in Section 2.2.

2.4. Nested random effects model

We consider the case of two random effects with one nested within the other. Specifically, a two-part model of this type is

$$\text{logit}{P}_r\left(Z_{ijk}=1\mid {\mathit{u}}_i\right)={\theta }_0+{\theta }_1{t}_{ij}+{\theta }_2{G}_i+{\mathit{u}}_i$$ (6)

$$Y_{ijk}\mid Z_{ijk}=1,{\mathit{v}}_i\sim N\left({\beta }_0+{\beta }_1{t}_{ij}+{\beta }_2{G}_i+{\mathit{v}}_i,{\sigma }_{ϵ}^2\right)$$ (7)

where $Y_{ijk}$ and $Z_{ijk}$ indicate the $k$th replicate measurement at the $j$th time point within the $i$th subject. Further, ${\mathit{u}}_i=\left\{u_i,u_{ij}\right\}$ and ${\mathit{v}}_i=\left\{v_i,v_{ij}\right\}$ are the random effect components of the binary part and the continuous part, where $u_i$ and $v_i$ are subject-specific random effects, and $u_{ij}$ and $v_{ij}$ are random effects nested within individual; ${\sigma }_{u0}^2$ and ${\sigma }_{v0}^2$ are variances of $u_i$ and $v_i$, and ${\sigma }_{u1}^2$ and ${\sigma }_{v1}^2$ are variances of $u_{ij}$ and $v_{ij}$; ${\rho }_0$ is the correlation coefficient between $u_i$ and $v_i$, and ${\rho }_1$ is the correlation coefficient between $u_{ij}$ and $v_{ij}$.

The conditional distribution based on the nested random effects structure is

$$f(\mathit{Y}\mid \mathit{Z})=\prod _{i=1}^N\underset{v_i}{\int }\prod _{j=1}^{J_i}\underset{v_{ij}}{\int }\prod _{k=1}^{n_{ij}}f\left(Y_{ijk}\mid Y_{ijk}>0,\beta ,v_i,v_{ij},{\sigma }_e^2\right)h\left(v_i,v_{ij}\mid {\mathit{Z}}_i\right)\text{d}{v}_{ij}\text{d}{v}_i$$ (8)

where $N$ is the total number of subjects, $J_i$ is the total number of time points on the $i$th subject, and $n_{ij}$ is the total number of observations within the $j$th time point of the $i$th subject. Similar to equation (5), the approximate conditional expectation of $Y_{ijk}$ can be expressed as

$$E\left(Y_{ijk}\mid Z_{ijk}=1,v_i^{\ast },v_{ij}^{\ast },d_i\right)={\beta }_0^{\ast }+{\beta }_1^{\ast }{t}_{ij}+{\beta }_2^{\ast }{G}_i+{\delta }_0{d}_i+{\delta }_1{d}_i^2+{\delta }_2{m}_{ij}+{\delta }_3{m}_{ij}^2+v_i^{\ast }+v_{ij}^{\ast }$$ (9)

where $d_i=\sum _{j,k}{Z}_{ijk}$, and $m_{ij}=\sum _k{Z}_{ijk}$. The conditional model simply assumes that $v_i^{\ast }$ and $v_{ij}^{\ast }$ follow independent normal distributions with mean zero and variances that do not depend on $d_i$ or $m_{ij}$, respectively. The imputation approach for the nested design is then similar to the algorithm presented in Section 2.2.

3. Quantification of asymptotic bias

Su et al.⁵ demonstrated that separately modeling the two components (i.e. treating the two processes as independent) when the two processes are correlated results in asymptotically unbiased fixed effect estimation for the binary component and sizable bias for the continuous component. Thus, we evaluate the asymptotic bias for fixed effects estimation of the continuous model. As in Su et al.,⁵ we evaluate asymptotic bias under a random intercepts model. We consider the following model,

$$\begin{gathered} \text{logit}{P}_r\left(Z_{ij}=1\mid u_i\right)={\theta }_0+{\theta }_1{t}_{ij}+{\theta }_2{G}_i+u_i \\ {Y}_{ij}\mid Z_{ij}=1,v_i\sim N\left({\beta }_0+{\beta }_1{t}_{ij}+{\beta }_2{G}_i+v_i,{\sigma }_e^2\right) \end{gathered}$$ (10)

where $\mathit{\theta }=\left\{{\theta }_0,{\theta }_1,{\theta }_2\right\}$ and $\mathit{\beta }=\left\{{\beta }_0,{\beta }_1,{\beta }_2\right\}$ are fixed effect coefficients corresponding to an intercept, slope over time $t_{ij}$ from 0 to 19 $\left(n_i=20\right)$, and a group effect of $G_i$ generated from a Bernoulli distribution with size N and probability of 0.5 for both binary and continuous parts; ${\sigma }_e^2$ is the error variance from the continuous part; $u_i$ and $v_i$ are the two random intercepts following a multivariate normal distribution with a mean vector of zeros and a covariance matrix, ${\text{Σ}}_{u,v}$ given by

$${\text{Σ}}_{u,v}=\left[\begin{array}{cc}{\sigma }_u^2& \rho {\sigma }_u{\sigma }_v\\ \rho {\sigma }_u{\sigma }_v& {\sigma }_v^2\end{array}\right]$$

where $\rho$ denotes the correlation between the two random intercepts. Asymptotic bias is defined as the bias obtained for a fixed longitudinal design as the number of subjects gets large $(N\to \text{d}\infty )$. The approximate conditional distribution for the random intercepts model is characterized by the linear mixed model.

$$Y_{ij}={\beta }_0^{\ast }+{\beta }_1^{\ast }{t}_{ij}+{\beta }_2^{\ast }{G}_i+{\delta }_0{d}_i+{\delta }_1{d}_i^2+v_i^{\ast }+{ϵ}_{ij}^{\ast }$$ (11)

The asymptotic bias of the imputation approach starts with estimating parameters based on the approximate conditional model (equation (11)) given by

$${\mathit{\zeta }}^{\ast }=\underset{\zeta }{\max }\underset{F}{E}\left[log{L}_c(\mathit{Y};\mathit{\zeta })\right]$$ (12)

where $L_c$ is the likelihood corresponding to the linear mixed model (11), are parameters from the conditional model, and $E_F\left[log{L}_c(\mathit{Y};\mathit{\zeta })\right]$ is the expected value of the conditional likelihood under the full model (equation (10)). Maximizing equation (12) is analytically difficult since the calculation requires integrating over a complex likelihood. As an alternative, we numerically evaluate ${\zeta }^{\ast }$ by approximating $E_F\left[log{L}_c\right]\approx \frac{1}{M}\sum _{m=1}^{M}log{L}_c\left({\mathit{Y}}_m;\mathit{\zeta }\right)$ with simulating a large number of ${\mathit{Y}}^{\prime }$s (large $M$ under the full model) and performing the maximization. We then obtain the asymptotic bias of $\mathit{\beta }$ by $\underset{\mathit{\beta }}{\max }\frac{1}{M}\sum _{m=1}^{M}logL\left({\mathit{Y}}_m,\mathit{\beta }\right)-\mathit{\beta }$, where realizations are simulated under equation (10) with missing values of $\mathit{Y}$ (when $\mathit{Z}=0$) that are imputed based on the approximate conditional model with parameters ${\mathit{\zeta }}^{\ast }$, and with $M=500,000$.

We consider the case where the number of repeated measure, $n_i$, is 20 and examine the asymptotic bias as a function of $\rho$. Figure 1 compares the asymptotic bias of ${\beta }_0$, ${\beta }_1$, and ${\beta }_2$ for the imputation and naive approach (that assumes independence between the two processes, $\rho =0$) for a fixed set of parameter values with the varying of $\rho$. The figure shows that the proposed imputation approach substantially reduces asymptotic bias over a wide range of $\rho$.

Figure 1.

The relative asymptotic bias as a function of $\rho$. Under model (10), the parameters are $\mathit{\theta }=\{-0.39,-1.75,1.98\}$, $\mathit{\beta }=\{-2.21,-0.85,3.87\}$, ${\sigma }_u^2=1.77$, ${\sigma }_v^2=1.60$, and ${\sigma }_{ϵ}^2=0.02$.

In order to more fully evaluate the asymptotic bias of the imputation approach, we compute the bias over a wide range of parameter values. We do this by generating parameters under a normal distribution with mean given by $\mathit{\theta }=\{-0.5,-1,log(2)\}$, $\mathit{\beta }=\{-0.3,-0.5,2\}$, $log\left({\sigma }_u^2\right)=log\left({\sigma }_v^2\right)=log(2)$, and $log\left({\sigma }_{ϵ}^2\right)=log(0.1)$, and ${\rho }_F=0$, where ${\rho }_F$ is Fisher transformed for the correlation coefficient. Further, we assume that simulated random parameters are independent from each other with variances that result in 25th and 75th percentiles for ${\theta }_0$ of (1.18, 0.15), for ${\theta }_1$ of (1.68, −0.35), for ${\theta }_2$ of (0.02, 1.39), for ${\beta }_0$ of (0.96, 0.38), for ${\beta }_1$ of (1.16, 0.17), for ${\beta }_2$ of (1.31, 2.70), for $\rho$ of (0.26, 0.26), for ${\sigma }_u^2$ of (0.56, 0.82), for ${\sigma }_v^2$ of (1.75, 2.29), and for ${\sigma }_{ϵ}^2$ of (0.06, 0.17). Figure 2 shows a plot of the asymptotic bias from the naive approach versus the imputation approach. Each point in the figure shows both biases for a randomly generated parameter vector. The much smaller spread on the y axis than the x axis demonstrates the dramatic reduction in bias for the proposed approach as compared to the naive approach of fitting the models separately.

Figure 2.

The asymptotic bias comparison.

4. Finite sample properties

We evaluate the proposed imputation method by assessing finite-sample bias with two different random effect design structures, a random slope case and a nested random effects case. We generated data under equations (1) and (2) for the random slope and equations (6) and (7) for the nested random effects model. For each case, we consider a time effect and a group effect.

4.1. A random slope case

We evaluate the imputation approach with $u_i$ and $v_i$ in equation (11) being replaced by $u_{i0}+u_{i1}{t}_{ij}$ and $v_{i0}+v_{i1}{t}_{ij}$, respectively. The parameters $\mathit{\theta }$ and $\mathit{\beta }$ are set as $\{1,-1,log(2)\}$ and $\{-0.3,-0.5,2\}$, respectively. For a fixed set of variance components and with $corr\left(u_{i0},v_{i1}\right)=corr\left(u_{i1},v_{i0}\right)=0$, Table 1 presents the finite sample bias for the fixed effects of the continuous component model with random intercept and slope effects.

Table 1.

The simulation summary for the random slope structure with Monte-Carlo size of 500; $N=500$, $n_i=20$, $\mathit{\theta }=\left\{{\theta }_0,{\theta }_1,{\theta }_2\right\}=\{1,-1,log(2)\}$, ${\sigma }_{u_0}^2=0.5$, ${\sigma }_{u_1}^2=0.I$, ${\sigma }_{v_0}^2=0.05$, ${\sigma }_{v_1}^2=0.03$, ${\sigma }_{ϵ}^2=0.08$.

		${\rho }_0={\rho }_1=0.8$
		Imputation approach				Naive approach
Parameters	True	Bias	SEa	${\text{SE}}_{\text{rb}}^{\text{b}}$	MSEd	Bias	SEa	${\text{SE}}_{\text{H}}^{\text{c}}$	MSEd
${\beta }_0$	−0.3	0.005	0.0239	0.0270	0.0006	0.014	0.0236	0.0230	0.0007
${\beta }_1$	−0.5	0.015	0.0159	0.0183	0.0005	0.063	0.0128	0.0125	0.0041
${\beta }_2$	2.0	−0.018	0.0305	0.0326	0.0013	−0.019	0.0305	0.0301	0.0013
		${\rho }_0={\rho }_1=0.5$
		Imputation approach				Naive approach
Parameters	True	Bias	SEa	${\text{SE}}_{\text{rb}}^{\text{b}}$	MSEd	Bias	SEa	${\text{SE}}_{\text{H}}^{\text{c}}$	MSEd
${\beta }_0$	−0.3	0.004	0.0244	0.0273	0.0006	0.008	0.0239	0.0232	0.0006
${\beta }_1$	−0.5	0.008	0.0165	0.0191	0.0003	0.038	0.0125	0.0126	0.0016
${\beta }_2$	2.0	−0.011	0.0311	0.0334	0.0011	−0.012	0.0310	0.0306	0.0011
		${\rho }_0=0.8$, ${\rho }_1=0.2$
		Imputation approach				Naive approach
Parameters	True	Bias	SEa	${\text{SE}}_{\text{rb}}^b$	MSEd	Bias	SEa	${\text{SE}}_{\text{H}}^c$	MSEd
${\beta }_0$	−0.3	0.007	0.0245	0.0271	0.0006	0.029	0.0240	0.0231	0.0014
${\beta }_1$	−0.5	0.002	0.0170	0.0194	0.0003	0.015	0.0125	0.0126	0.0004
${\beta }_2$	2.0	−0.007	0.0312	0.0330	0.0010	−0.014	0.0310	0.0305	0.0012
		${\rho }_0=0.2$, ${\rho }_1=0.8$
		Imputation approach				Naive approach
Parameters	True	Bias	SEa	${\text{SE}}_{\text{rb}}^b$	MSEd	Bias	SEa	${\text{SE}}_{\text{H}}^c$	MSEd
${\beta }_0$	−0.3	0.002	0.0244	0.0272	0.0006	−0.012	0.0238	0.0233	0.0007
${\beta }_1$	−0.5	0.015	0.0166	0.0184	0.0005	0.062	0.0127	0.0126	0.0039
${\beta }_2$	2.0	−0.017	0.0306	0.0332	0.0012	−0.010	0.0305	0.0307	0.0010

MSE: mean squared errors.

^aMonte-Carlo standard errors.

^bStandard errors based on Rubin’s Rule.

^cHessian-based standard errors.

^dMean squared errors based on Monte-Carlo standard errors.

As expected, all the biases for both approaches are smaller as the correlations are lower, especially as the correlation coefficient for the random slope, ${\rho }_1$ is smaller. In general, the relative bias for the proposed imputation is smaller than for the naive approach. The relative bias for the fixed effect estimates using the imputation are lower than 5%, which is in contrast to the naive approach where the relative bias can be substantially more than 10%. Furthermore, the Rubin’s Rule-based standard errors are close to the Monte-Carlo estimates, suggesting their good performance in this setting.

The comparison of the mean squared errors (MSE) for the parameter estimates shows that even taking into account the increase in the standard errors with the naive approach, the imputation approach has a smaller MSE than the naive approach. The imputation approach, in general, has a smaller MSE than the naive approach.

4.2. A nested random effects case

For the nested random effects structure, we generate a dataset with three replicates $\left(n_{ij}=3\right)$ within each time point based on equations (6) and (7). Due to the small number of replicates, we only consider a linear term for $m_{ij}$ in the approximate conditional model, equivalent to ${\delta }_3=0$ in equation (9). The naive approach assumes that the two random effects from each components are all uncorrelated from each other. Table 2 shows the bias for the imputation versus the naive approaches under different degrees of correlation between the two processes. In general, the imputation approach performs much better when the naive approach is highly biased. The cases where the bias is larger for the imputation as compared to the naive approach are ones where the bias of both approaches are small.

Table 2.

The simulation summary for the nested random effect structure with Monte-Carlo size of 500; $N=500$, $n_i=20$, $n_{ij}=3$, $\mathit{\theta }=\left\{{\theta }_0,{\theta }_1,{\theta }_2\right\}=\{-0.5,-\text{I},log(2)\}$, ${\sigma }_{u_i}^2=1$, ${\sigma }_{u_{ij}}^2=0.1$, ${\sigma }_{v_i}^2=2$, ${\sigma }_{v_{ij}}^2=0.25$, ${\sigma }_{ϵ}^2=0.08$.

		${\rho }_0={\rho }_1=0.8$
		Imputation approach				Naive approach
Parameters	True	Bias	SEa	${\text{SE}}_{\text{rb}}^b$	MSEd	Bias	SEa	${\text{SE}}_{\text{H}}^c$	MSEd
${\beta }_0$	−0.3	−0.040	0.1134	0.1244	0.0145	0.297	0.1015	0.0981	0.0988
${\beta }_1$	−0.5	0.048	0.0179	0.0178	0.0026	0.038	0.0160	0.0157	0.0017
${\beta }_2$	2.0	−0.058	0.1295	0.1607	0.0201	−0.129	0.1300	0.1332	0.0337
		${\rho }_0={\rho }_1=0.5$
		Imputation approach				Naive approach
Parameters	True	Bias	SEa	${\text{SE}}_{\text{rb}}^b$	MSEd	Bias	SEa	${\text{SE}}_{\text{H}}^c$	MSEd
${\beta }_0$	−0.3	−0.023	0.1200	0.1348	0.0150	0.186	0.1055	0.1015	0.0457
${\beta }_1$	−0.5	0.030	0.0181	0.0181	0.0012	0.024	0.0160	0.0157	0.0008
${\beta }_2$	2.0	−0.036	0.1386	0.1757	0.0205	−0.082	0.1366	0.1380	0.0253
		${\rho }_0=0.8$, ${\rho }_1=0.2$
		Imputation approach				Naive approach
Parameters	True	Bias	SEa	${\text{SE}}_{\text{rb}}^b$	MSEd	Bias	SEa	${\text{SE}}_{\text{H}}^c$	MSEd
${\beta }_0$	−0.3	0.014	0.1150	0.1267	0.0134	0.265	0.1015	0.0990	0.0803
${\beta }_1$	−0.5	0.012	0.0180	0.0181	0.0005	0.020	0.0160	0.0157	0.0007
${\beta }_2$	2.0	−0.034	0.1296	0.1642	0.0179	−0.120	0.1304	0.1345	0.0314
		${\rho }_0=0.2$, ${\rho }_1=0.8$
		Imputation approach				Naive approach
Parameters	True	Bias	SEa	${\text{SE}}_{\text{rb}}^{\text{b}}$	MSEd	Bias	SEa	${\text{SE}}_{\text{H}}^{\text{c}}$	MSEd
${\beta }_0$	−0.3	−0.060	0.1217	0.1398	0.0184	0.108	0.1058	0.1032	0.0229
${\beta }_1$	−0.5	0.048	0.0185	0.0180	0.0026	0.027	0.0164	0.0157	0.0010
${\beta }_2$	2.0	−0.036	0.1398	0.1829	0.0209	−0.043	0.1378	0.1404	0.0208

MSE: mean squared errors.

^aMonte-Carlo standard errors.

^bStandard errors based on Rubin’s Rule.

^cHessian-based standard errors.

^dMean squared errors based on Monte-Carlo standard errors.

The estimated standard errors obtained with Rubin’s Rule are moderately inflated for the group effect ${\beta }_2$, while for the other two parameters, these standard errors are nearly unbiased (related to the Monte-Carlo standard errors). However, the Rubin’s Rule-based standard errors for the group effect became much closer to the MonteCarlo ones when the number of clusters increases (Table S3 in the supplement).

In the hope of improving the performance of the proposed imputation model, we investigated approaches that approximated the conditional distribution with models that allowed more flexible random effects distributions with variances that can vary with $d_i$ in the supplement (Tables S1 and S2). These approaches did not show improved estimation performance relative to the simple linear mixed model for approximating the conditional distribution.

For both the random slope and nested random effect cases, we fit a Bayesian model using Markov chain Monte Carlo (MCMC) on the simulated datasets (see Appendix C for a detailed explanation of the method and the results). The Bayesian approach is slightly more efficient at the expense of computational inefficiency (an average of one day per simulated dataset for the Bayesian and 11 minutes for the imputation approach). A Bayesian approach for fitting the nested case (see Appendix C) provided very similar results, but was much less computationally efficient.

5. Application study—CHEF data

We illustrate the proposed methodology using longitudinal data from the Cultivating Healthful Environments in Families of Youth with Type 1 Diabetes (CHEF) study.¹⁷ The CHEF study is a longitudinal study enrolling 8- to 16-year-old children with Type 1 diabetes where a primary objective was to evaluate a behavioral intervention on longitudinal measures of dietary intake. A total of 136 participants were randomly assigned to either a target behavioral intervention for improving diet or standard care. The intake of different types of foods was measured for each child at six time points with up to three replicates within each time point. In this application, we select the whole grain (WG) intake as a response variable where many of the repeated measure were zero (50% zeros).

In this analysis, the two-part model with a nested random effect structure is chosen in order to account for both correlations across replicates as well as across time points. Table 3 shows the results using the imputation and naive approaches. Although the results are similar, there were some important differences with the two approaches. Most notably, the $p$ value for the group effect is smaller for the imputation as compared with the naive approach. For the binary component, there were no significant effects of time or group on the probability of a non-zero whole grain intake measurement (Table 3). We also conducted a Bayesian approach with a nested random effect structure. The Bayesian approach leads to similar inferences as the imputation approach (see Appendix C).

Table 3.

Fixed effect estimation for whole grain intake. Inference for the continuous and binary components with the imputation and naive approaches.

	Parameters	Estimates	${SE}_{\text{bs}}^{\text{a}}$	${SE}_{\text{H}}^{\text{b}}$	$p_{\text{bs}}^{\text{c}}$	$p_{\text{H}}^{\text{d}}$
Binary part
Generalized linear mixed model	Intercept	–0.086	0.1853	0.1495	0.6512	0.5752
	Time	–0.035	0.0431	0.0343	0.4209	0.3118
	Sex	0.067	0.1495	0.1413	0.6632	0.6450
	Group	0.181	0.2141	0.1869	0.4196	0.3550
	Group × time	0.039	0.0596	0.0515	0.5017	0.4366
	Parameters	Estimates	${SE}_{\text{bs}}^{\text{a}}$	${SE}_{\text{rb}}^{\text{b}}$	$p_{\text{bs}}^{\text{c}}$	$p_{\text{rb}}^{\text{d}}$
Continuous part
Imputation approach	Intercept	−0.094	0.0730	0.0841	0.1980	0.2662
	Time	−0.020	0.0190	0.0194	0.2917	0.2971
	Sex	0.057	0.064	0.0746	0.3703	0.4420
	Group	0.195	0.0844	0.0913	0.0209	0.0333
	Group × time	−0.002	0.0283	0.0263	0.9436	0.9346
	Parameters	Estimates	${SE}_{\text{bs}}^{\text{a}}$	${SE}_{\text{H}}^{\text{e}}$	$p_{\text{bs}}^{\text{c}}$	$p_{\text{H}}^{\text{f}}$
Continuous part
Naive approach	Intercept	−0.009	0.0675	0.0710	0.8939	0.9014
	Time	−0.018	0.0184	0.0178	0.3280	0.3017
	Sex	0.051	0.0624	0.0637	0.4135	0.4219
	Group	0.180	0.0830	0.0884	0.0302	0.0421
	Group × time	−0.003	0.0280	0.0259	0.9148	0.9008

MSE: mean squared errors.

^aNonparametric bootstrap-based standard error.

^bRubin’s Rule-based standard error.

^cNonparametric bootstrap-based $p$ value.

^dRubin’s Rule-based $p$ value.¹⁸

^eHessian-based standard error.

^fHessian-based $p$ value.

^gFor the binary part, we obtain the fixed effect estimates as (−0.086, −0.035, 0.067, 0.181, 0.039) with the nonparametric bootstrap-based and the Hessian-based standard errors as (0.1853, 0.0431, 0.1495, 0.2141, 0.0596) and (0.1495, 0.0343, 0.1413, 0.1869, 0.0515) for the intercept, time, sex, group, and group × time effects, respectively.

6. Discussion

This article develops a simple imputation approach for fitting two-part mixed models without the need for specialized case-specific software. As reported by Su et al.,⁵ estimation of the continuous component are biased when the binary and continuous components are correlated, which is generally the case. They mentioned the connection with informative missingness that is induced by correlated random effects but do not propose an alternative to complex joint modeling for parameter estimation.

We propose an imputation approach where we impute based on a simple approximation to the conditional distribution of the continuous measurements given that it is positive. We show that this approach greatly reduces the bias involved in separately fitting the two components using standard statistical software. A similar type of imputation has been proposed for fitting joint models for longitudinal and time-to-event data,^13,19 and uses the original idea of approximating a shared random parameter model for informative dropout proposed by Wu and Bailey;¹² see a recent review of these methods by Albert.²⁰

A comparison of the imputation with a full Bayesian approach shows that both methods provide similar inferences. Further, the Bayesian approach was substantially more computationally expensive than the proposed imputation approach. However, in some contexts, a full likelihood Bayesian approach is advantageous in that models can be compared with well established techniques such as deviance information criterion (DIC).

We approximate the conditional distribution of the continuous measurements given the non-zero status by a normal distribution with a quadratic function of the number of positive measurements and a constant variance. We found that the polynomial mean function closely approximated the mean of the conditional distribution. However, we did observe that the conditional distribution did show some heteroscedasticity and minor skewness. Additional effort to improve the approximation using a non-parametric approach to random effects estimation or allowing the variance of the random effects to vary with the mean did not consistently improve the performance of the approach.

The proposed imputation approach can be extended to fitting multivariate two-part models. This would involve approximating the multivariate conditional distribution using multivariate linear mixed models which could be done with pseudo-likelihood approaches,²¹ even in high dimensions.