A Marginalized Zero-inflated Poisson Regression Model with Random Effects

Summary

Public health research often concerns relationships between exposures and correlated count outcomes. When counts exhibit more zeros than expected under Poisson sampling, the zero-inflated Poisson (ZIP) model with random effects may be used. However, the latent class formulation of the ZIP model can make marginal inference on the sampled population challenging. This article presents a marginalized ZIP model with random effects to directly model the mean of the mixture distribution consisting of ‘susceptible’ individuals and excess zeroes, providing straightforward inference for overall exposure effects. Simulations evaluate finite sample properties, and the new methods are applied to a motivational interviewing-based safer sex intervention trial, designed to reduce the number of unprotected sexual acts.

1. Introduction

Infectious disease researchers are often concerned with reducing risky sexual behavior among HIV-positive individuals. One measure of risky sexual behavior is the Unprotected Anal and Vaginal Intercourse (UAVI) count, the number of unprotected anal or vaginal intercourse acts with any partner over a specified period of time. The SafeTalk program was developed by Golin et al. (2012) to reduce the number of unprotected sexual acts through a multicomponent, motivational interviewing-based, safer sex intervention. Sexual behavior count data can display a distribution with excess zeros (Heilbron, 1994; Ghosh and Tu, 2009). To examine the efficacy of the SafeTalk program over time, a randomized controlled clinical trial collected risky sexual behavior data at baseline and up to three follow-up visits.

Several methods have been developed for modeling correlated count data with many zeros such as UAVI from the SafeTalk clinical trial. Building upon the zero-inflated Poisson (ZIP) regression model established by Mullahy (1986) and Lambert (1992), Hall (2000) extends the ZIP regression model to include random effects in the Poisson process. In order to account for overdispersion beyond the excess zeros, Yau, Wang and Lee (2003) modify the zero-inflated negative-binomial (ZINB) regression model to include random effects. Instead of using random effects to handle correlated data, Hall and Zhang (2004) employ GEE methodology for zero-inflated models in order to achieve population-averaged interpretations. For each of these zero-inflated methods, two sets of parameter estimates are produced, those associated with the excess zero process that models the probability of being non-susceptible for the disease or condition and those associated with the count process that models the mean count among susceptible individuals. In many applications, the two latent class interpretations are not clinically supported or simply not of interest, and the zero-inflated methodology is used as a convenient modeling technique to account for excess zeros in a population (Mwalili, et al., 2008).

While closely related to the zero-inflated methodology, hurdle models (including zero-altered models) specify a model for the probability of any zero in addition to the model for the mean of the untruncated distribution of the count data process (Mullahy, 1986; Heilbron, 1994). Dobbie and Welsh (2001) use the zero-altered Poisson model, modified to utilize GEE, to account for correlated observations. Min and Agresti (2005) extend the zero-altered model to include random effects.

The choice between the hurdle and zero-inflated model classes has been approached from various angles. Much of the literature pertaining to the analysis of count data with excess zeros focuses on model fit, using fit statistics to provide justification of model class choice. Gilthorpe, et al.(2009) argue that a priori knowledge of the data-generating mechanism could be used to identify the class of models from which to choose, supported by statements in Neelon et al.(2010) and Buu et al.(2012). Applications in which all zeros are considered as arising from an identical process indicate a hurdle model, rather than a zero-inflated model, where zeros can occur from the two different processes.

While many health-related fields are implementing zero-inflated techniques, sometimes health researchers wish to make inference upon an entire sampled population rather than the latent classes modeled by ZIP methodology (Preisser, et al., 2012). Albert et al.(2014) contend that interpretations for features of the marginal mixture distribution have been generally overlooked in the zero-inflated literature, such as the overall mean count, owing to the fact that ZIP models and hurdle models do not produce a direct overall estimate of exposure effect for the marginal mean count. In particular, transformation methods, with variance estimation by the delta method or resampling methods, may be used to make inference on overall estimates of a dichotomous exposure effect for ZIP and ZINB models (Albert, et al., 2014). However, such transformations can be tedious for many analysts, and the treatment of continuous covariates is not necessarily apparent.

Proposing the marginalized model for longitudinal binary data, Heagerty (1999) employs joint models by directly modeling the marginal mean and simultaneously using a linked random effects model to account for correlated responses. Through this joint model, marginalization over random effects achieves population-averaged parameters, while accounting for correlated measures. Extending the marginalized model approach, Lee et al.(2011) focus on the hurdle model formulation for Poisson and negative binomial data with excess zeros while marginalizing over random effects for clustering. Since Lee et al. focus on marginalizing over the random effects, the two sets of parameters from their marginalized hurdle models have the same interpretations as hurdle models for independent responses.

Adapting the marginalized model approach to achieve inference on the marginal mean for independent count responses with excess zeroes, Long et al.(2014) present a new marginalized ZIP model that jointly models the marginal mean and excess zero process to produce estimates for marginal mean inference while accounting for excess zeroes. Where as marginalized models often average over random effects to obtain population-average effect estimates, the marginalized ZIP model averages over the two ZIP model processes to achieve overall effect estimates for expected counts, providing parameter estimates with the same interpretation as Poisson regression. This article builds upon both the marginalized ZIP model and current ZIP methods for correlated data and proposes the marginalized ZIP model with random effects.

Sections 2 and 3 briefly review the ZIP model with random effects from Hall (2000) and the marginalized ZIP model from Long et al.(2014), respectively. Section 4 proposes the marginalized ZIP model with random effects, which has subject-specific parameters, and discusses the situation where those parameters have equivalent population-averaged interpretations. Section 5 presents simulation study results examining the finite sample performance of the new model. In Section 6, we consider data from the SafeTalk randomized controlled clinical trial. A discussion is provided in Section 7.

2. ZIP model with random effects

Extending Lambert's ZIP model to incorporate correlated zero-inflated count data, Hall (2000) developed the ZIP model with random effects. Let $\mathit{Y}=({\mathit{Y}}_1^{\mathit{\prime }},\dots ,{\mathit{Y}}_K^{\mathit{\prime }})$ where K is the number of independent clusters and Y_i = (Y_i1,…, Y_iTi)′, where T_i is the number of observations for the i^th cluster. Let s_ij = 1 if Y_ij is from the first process (i.e. Y_ij is an excess zero) and s_ij = 2 if Y_ij is from the second (Poisson) process; s_ij is unobserved when Y_ij = 0. Then

$$Y_{ij}~\{\begin{array}{cc}0& \text{with probability}P(s_{ij}=1)={\psi }_{ij}\\ \text{Poisson}({\mu }_{ij}^C)& \text{with probability}P(s_{ij}=2)=1-P(s_{ij}=1)=1-{\psi }_{ij}\end{array}$$ (1)

Where ${\mu }_{ij}^C=E(Y_{ij}|s_{ij}=2,b_i)$. The notation ${\mu }_{ij}^C$ indicates that the Poisson mean is conditional on the random effect b_i. The log-linear and logistic regression models are

$$\begin{array}{c}logit({\psi }_{ij})={\mathit{Z}}_{ij}^{\mathit{\prime }}\mathit{\gamma }\\ log({\mu }_{ij}^C)={\mathit{X}}_{ij}^{\mathit{\prime }}\mathit{\beta }+\sigma b_i,\end{array}$$ (2)

where $b_1,\dots b_K\stackrel{i.i.d.}{~}N(0,1)$, and Z_ij and X_ij are the covariate vectors for the logistic and Poisson processes, respectively. Note that γ and β are latent class parameters, providing separate inference for the excess zero and Poisson processes, respectively. The log-likelihood can be expressed

$$l(\mathbf{\Omega },\mathit{y})=\sum _{i=1}^{K}log{\int }_{-\infty }^{\infty }[\prod _{j=1}^{T_i}Pr(Y_{ij}=y_{ij}|b_i,\mathbf{\Omega })]\varphi (b_i)d{b}_i$$

where Ω = (γ′, β′, σ), ϕ is the standard normal probability density and

$$Pr(Y_{ij}=y_{ij}|b_i,\mathit{\theta })={[{\psi }_{ij}+(1-{\psi }_{ij})e^{-{\mu }_{ij}^C}]}^{u_{ij}}{\left[\frac{(1-{\psi }_{ij})e^{-{\mu }_{ij}^C}{({\mu }_{ij}^C)}^{y_{ij}}}{y_{ij}!}\right]}^{1-u_{ij}}={(1+e^{{\mathit{Z}}_{ij}^{\mathit{\prime }}\mathit{\gamma }})}^{-1}\{u_{ij}[e^{{\mathit{Z}}_{ij}^{\mathit{\prime }}\mathit{\gamma }}+exp(-e^{{\mathit{X}}_{ij}^{\mathit{\prime }}\mathit{\beta }+\sigma b_i}\left)\right]+(1-u_{ij})\frac{exp\left[y_{ij}\right({\mathit{X}}_{ij}^{\mathit{\prime }}\mathit{\beta }+\sigma b_i)-e^{{\mathit{X}}_{ij}^{\mathit{\prime }}\mathit{\beta }+\sigma b_i}]}{y_{ij}!}\},$$ (3)

where u_ij = I(y_ij = 0). Using the EM algorithm framework that Lambert (1992) proposed, Hall fits this ZIP model with random effects with the EM algorithm with Gaussian quadrature. Generally, the overall conditional mean $E(Y_{ij}|b_i)=(1-{\psi }_{ij}){\mu }_{ij}^C$ will depend on γ, β and b_i through a complicated function that does not permit easy and direct inference for overall effects, here defined as ratios of such means when a single covariate is allowed to vary. Although Hall (2000) used (2) to account for correlation within the Poisson process only, others have utilized correlated random effects in both processes of the ZIP and hurdle models (Dobbie and Welsh, 2002; Min and Agresti, 2005; Ghosh and Tu, 2009; Neelon et al., 2010).

3. Marginalized ZIP model for independent responses

Rather than jointly modeling the excess zero probability and the latent class Poisson mean μ_i, Long, et al.(2014) instead propose the marginalized ZIP regression model, which directly models the marginal mean of the mixture distribution in addition to the zero-inflation process. For independent outcomes Y_i, the marginalized ZIP model is given by

$$\begin{array}{c}logit({\psi }_i)={\mathit{Z}}_{\mathit{i}}^{\mathit{\prime }}\mathit{\gamma }\\ log({\nu }_i)={\mathit{X}}_{\mathit{i}}^{\mathit{\prime }}\mathit{\alpha }.\end{array}$$ (4)

where ν_i is the marginal mean, that is ν_i ≡ E(Y_i). The elements of γ provide inference on the probability of an excess zero, the same interpretations as ZIP models. However, the modeling of the marginal mean ν_i allows log-incidence density rate interpretations of the elements of α, providing the same interpretation as in Poisson regression. The marginalized ZIP model utilizes the ZIP likelihood framework and the concept of marginalized models to marginalized over the two processes. Specifically, the Poisson process mean is redefined as a general function of model parameters in (4). Solving ν_i = (1 − ψ_i)μ_i, with substitution for (4), provides

$${\mu }_i=(1+e^{{\mathit{Z}}_{\mathit{i}}^{\mathit{\prime }}\mathit{\gamma }})e^{{\mathit{X}}_{\mathit{i}}^{\mathit{\prime }}\mathit{\alpha }}.$$

This definition of μ_i reparameterizes the ZIP model, allowing for inference on the marginal mean. Using this redefined μ_i, ${\psi }_i={logit}^{-1}({\mathit{Z}}_{\mathit{i}}^{\mathit{\prime }}\mathit{\gamma })$ and the ZIP likelihood, the marginalized ZIP likelihood for (γ,α) is derived to be

$$L(\mathit{\gamma },\mathit{\alpha }|\mathit{y})=\mathbf{\prod }_{y_i}{(1+e^{{\mathit{Z}}_{\mathit{i}}^{\mathit{\prime }}\mathit{\gamma }})}^{-1}\prod _{y_i=0}(e^{{\mathit{Z}}_{\mathit{i}}^{\mathit{\prime }}\mathit{\gamma }}+e^{-(1+exp({\mathit{Z}}_{\mathit{i}}^{\mathit{\prime }}\mathit{\gamma }))exp({\mathit{X}}_{\mathit{i}}^{\mathit{\prime }}\mathit{\alpha })})\times \prod _{y_i>0}[e^{-(1+exp({\mathit{Z}}_{\mathit{i}}^{\mathit{\prime }}\mathit{\gamma }))exp({\mathit{X}}_{\mathit{i}}^{\mathit{\prime }}\mathit{\alpha })}{(1+e^{{\mathit{Z}}_{\mathit{i}}^{\mathit{\prime }}\mathit{\gamma }})}^{y_i}{e}^{{\mathit{X}}_{\mathit{i}}^{\mathit{\prime }}\mathit{\alpha }{y}_i}/(y_i!)].$$

Long et al.(2014) note that analysts may fit this marginalized ZIP model in SAS NLMIXED, providing sample code as well as details for robust (empirical) standard error estimation. Although derived from a reparameterization of the ZIP model, the marginalized ZIP parameters yield direct inference on the marginal mean rather than the latent classes and gives statistical analysts a new class of models to address marginal exposure effects.

4. Marginalized ZIP model with random effects

4.1. Subject-specific marginalized ZIP model

Building upon both Hall (2000) and Long et al.(2014), we present a marginalized adaptation of the ZIP model with random effects for repeated measures data. Rather than modeling the conditional Poisson process mean ${\mu }_{ij}^C$ as in (2), the marginalized ZIP model for clustered data directly models the overall subject-specific mean ${\nu }_{ij}^C=E(Y_{ij}|{\mathit{d}}_i)$ through

$$\begin{array}{c}logit({\psi }_{ij}^C)={\mathit{Z}}_{ij}^{\mathit{\prime }}\mathit{\gamma }+{\mathit{w}}_{1ij}^{\mathit{\prime }}{\mathit{c}}_i\\ log({\nu }_{ij}^C)={\mathit{X}}_{ij}^{\mathit{\prime }}\mathit{\alpha }+log(N_i)+{\mathit{w}}_{2ij}^{\mathit{\prime }}{\mathit{d}}_i,\end{array}$$ (5)

where ${\psi }_{ij}^C=P(s_{ij}=1|{\mathit{c}}_i)$ and b_i = (c_i, d_i)′ follows the multivariate normal distribution with mean zero and covariance matrix $\mathbf{\sum }=\left[\begin{array}{cc}{\sum }_{11}& {\sum }_{12}\\ {\sum }_{21}& {\sum }_{22}\end{array}\right]$. Above, N_i represents an off-set variable for situations where the incidence density ν_i/N_i is of interest. To account for clustering within each process, we propose correlated random effects c_i and d_i and corresponding column design vectors w_1ij, w_2ij, usually subsets of ${\mathit{Z}}_{ij}^{\mathit{\prime }}$ and ${\mathit{X}}_{ij}^{\mathit{\prime }}$, respectively. For many applications and focus of our subsequent simulation study and example, random intercepts may adequately model clustering. Note that for independent responses, this marginalized ZIP model with random effects reduces to the Long et al.(2014) marginalized ZIP model.

Because ${\nu }_{ij}^C$ is modeled directly in this marginalized ZIP with random effects model, the k^th parameter of α, α_k, is interpreted as the subject-specific log-incidence density ratio (IDR) for the k^th covariate; that is, for a one-unit increase in corresponding covariate x_k, exp(α_k) is the amount by which the mean ${\nu }_{ij}^C$ for a particular subject is multiplied, which is the same interpretation as in a Poisson random effects model. The direct modeling of ${\nu }_{ij}^C$ rather than the Poisson process mean ${\mu }_{ij}^C$ in Section 2 provides marginal mean inference often of interest to researchers.

For θ = (γ′, α′, Σ)′, the log-likelihood for this marginalized ZIP model with random effects can be written

$$l(\mathit{\theta };\mathit{y})=\sum _{i=1}^{K}log{\int }_{-\infty }^{+\infty }[\mathit{\prod }_{j=1}^{T_i}P(Y_{ij}=y_{ij}|{\mathit{b}}_i,\mathit{\theta })]\mathrm{\Phi }({\mathit{b}}_i)d{\mathit{b}}_i,$$ (6)

where Φ is the multivariate normal density (0, Σ). Augmenting the ZIP likelihood presented in (3) similar to the Long et al.(2014) reparameterization, the marginalized ZIP likelihood redefines ${\mu }_{ij}^C=exp({\delta }_{ij}^C)$, where ${\delta }_{ij}^C$ is not necessarily a linear function of covariates. Following from the ZIP likelihood specification in (3),

$$P(Y_{ij}=y_{ij}|{\mathit{b}}_i,\mathit{\theta })={[{\psi }_{ij}^C+(1-{\psi }_{ij}^C)e^{-exp({\delta }_{ij}^C)}]}^{u_{ij}}{\left[\frac{(1-{\psi }_{ij}^C)e^{-exp({\delta }_{ij}^C)}{e}^{{\delta }_{ij}^C{y}_{ij}}}{y_{ij}!}\right]}^{1-u_{ij}}.$$ (7)

Using (5) and the knowledge ${\nu }_{ij}^C=(1-{\psi }_{ij}^C){\mu }_{ij}^C$, solving for ${\delta }_{ij}^C=log({\mu }_{ij}^C)$ gives

$${\delta }_{ij}^C=log(N_i)+log[1+exp({\mathit{Z}}_{ij}^{\mathit{\prime }}\mathit{\gamma }+{\mathit{w}}_{1ij}^{\mathit{\prime }}{\mathit{c}}_i)]+{\mathit{X}}_{ij}^{\mathit{\prime }}\mathit{\alpha }+{\mathit{w}}_{2ij}^{\mathit{\prime }}{\mathit{d}}_i.$$ (8)

Rather than linking a linear function of covariates to the Poisson latent class mean, the form of ${\mu }_{ij}^C$ is derived to express a linear function of covariates on the marginal mean ${\nu }_{ij}^C$. Through substitution of (8) into (7), this subject-specific marginalized ZIP model with random effects may be fit using SAS NLMIXED (SAS Institute Inc, 2013), which employs an adaptive Gauss-Hermite quadrature to approximate the integral of the likelihood (6) over the random effects. For the simulation study, 25 quadrature points were used, and this was increased to 50 quadrature points for the analysis of the SafeTalk efficacy trial (Lesaffre and Spiessens, 2001). Additionally, SAS NLMIXED can provide robust (empirical) standard error estimates of the parameters, through the likelihood-based ‘sandwich’ estimator, to address model misspecification (White, 1982).

4.2. Population-averaged marginalized ZIP model for clustered data

The primary objective in the marginalized models literature (e.g. Heagerty, 1999) is to obtain parameters with marginalized (population-averaged) interpretations rather than parameters with subject-specific interpretations. In Section 4.1, we described the marginalized ZIP model with random effects, where the ‘marginalization’ is over the two latent classes of the ZIP model to achieve overall exposure effect estimates. However, because the marginalized ZIP with random effects models ${\nu }_{ij}^C=E(Y_{ij}|{\mathit{d}}_i)$, it yields parameters with subject-specific interpretations.

For data with repeated measures, statistical analysts usually choose between methods employing subject-specific (SS) parameters (mixed models) and methods having population-average (PA) parameters (GEE), though in a few notable cases (e.g. the Gaussian mixed model) parameters have both interpretations. However, Ritz and Spiegelman (2004) and Young et al. (2007) investigate the exact nature of the relationship between SS and PA parameters for Poisson count data, using well-established methods (e.g. McCulloch and Searle, 2001). For models with log links and normally distributed random effects, the mathematical relationships between SS and PA parameters can be quite straightforward.

To explore the connection between SS and PA parameters for the marginalized ZIP model with random effects, we restate model (5) as

$$\begin{array}{c}logit({\psi }_{ij}^C)={\mathit{Z}}_{ij}^{\mathit{\prime }}{\mathit{\gamma }}^{\mathit{\text{SS}}}+{\mathit{w}}_{1ij}^{\mathit{\prime }}{\mathit{c}}_i\\ log({\nu }_{ij}^C)={\mathit{X}}_{ij}^{\mathit{\prime }}{\mathit{\alpha }}^{\mathit{\text{SS}}}+log(N_i)+{\mathit{w}}_{2ij}^{\mathit{\prime }}{\mathit{d}}_i,\end{array}$$

where the SS superscript indicates that subject-specific interpretations are appropriate for these parameters. Then

$$E(Y_{ij}|{\mathit{d}}_i)=exp[{\mathit{X}}_{ij}^{\mathit{\prime }}{\mathit{\alpha }}^{\mathit{\text{SS}}}+log(N_i)+{\mathit{w}}_{2ij}^{\mathit{\prime }}{\mathit{d}}_i]$$

and

$$\begin{array}{c}E(Y_{ij})=E[E(Y_{ij}|{\mathit{d}}_i)]\\ =N_{i}exp({\mathit{X}}_{ij}^{\mathit{\prime }}{\mathit{\alpha }}^{\mathit{\text{SS}}})E(exp({\mathit{w}}_{2ij}^{\mathit{\prime }}{\mathit{d}}_i))\\ =N_{i}exp({\mathit{X}}_{ij}^{\mathit{\prime }}{\mathit{\alpha }}^{\mathit{\text{SS}}})exp(0.5{\mathit{w}}_{2ij}^{\mathit{\prime }}{\sum }_{22}{\mathit{w}}_{2ij})\end{array}$$ (9)

where d_i ∼ N(0, Σ₂₂). From (9), defining ${\nu }_{ij}^M=E(Y_{ij})$,

$$log({\nu }_{ij}^M)={\mathit{X}}_{ij}^{\mathit{\prime }}{\mathit{\alpha }}^{\mathit{\text{SS}}}+log(N_i)+0.5{\mathit{w}}_{2ij}^{\mathit{\prime }}{\sum }_{22}{\mathit{w}}_{2ij}.$$

Now consider the fully marginal model (10), where PA denotes population-averaged parameters

$$log({\nu }_{ij}^M)={\mathit{X}}_{ij}^{\mathit{\prime }}{\mathit{\alpha }}^{\mathit{\text{PA}}}+log(N_i).$$ (10)

The PA parameters in (10) are multiplicatively offset from the SS parameters by the function $exp(0.5{\mathit{w}}_{2ij}^{\mathit{\prime }}{\sum }_{22}{\mathit{w}}_{2ij})$ of the (ij)-th row of the model matrix for the random effects and respective covariance matrix. Thus, for all fixed effect covariates that do not have corresponding random effects, the respective parameters in α^SS are equivalent to corresponding parameters in α^PA. Consider the model with only a random intercept $({\mathit{w}}_{2ij}^{\mathit{\prime }}=1)$ and ${\sum }_{22}={\sigma }_b^2$; then

$$log({\nu }_{ij}^M)=[{\alpha }_0^{\mathit{\text{SS}}}+({\sigma }_b^2/2)]+{\stackrel{\sim }{\mathit{X}}}_{ij}^{\mathit{\prime }}{\stackrel{\sim }{\mathit{\alpha }}}^{\mathit{\text{SS}}}+log(N_i),$$

where ${\stackrel{\sim }{\mathit{X}}}_{ij}^{\mathit{\prime }}$ and α˜^SS contain all the covariates and corresponding parameters excluding the intercept. In this situation, α˜^SS also have population-averaged interpretations. While analysts may choose to include further normal random effects, such as a random slope over time, all parameters without a corresponding random effect have population-averaged as well as subject-specific interpretations because of the log link and normal random effects.

5. Simulation study

To examine the properties of the marginalized ZIP model with random effects, a simulation study was performed using SAS 9.3 NLMIXED. Let Y_ij be a zero-inflated Poisson outcome for the i^th participant at time j, and let g_i be a time-constant exposure variable of interest for each subject. The simulation scenario is motivated by the constant treatment assignment in the SafeTalk clinical trial. In the SafeTalk motivating example, Y_ij is the UAVI count outcome and g_i is an indicator of randomization to the SafeTalk intervention group. For this simulation study, three time points were used with I(j = 2) and I(j = 3) being the indicators of whether an observation occurs at follow-up time 2 or 3. Data were simulated using the marginalized ZIP model with random effects given by

$$\begin{array}{c}logit({\psi }_{ij}^C)={\gamma }_0+{\gamma }_{1}I(j=2)+{\gamma }_{2}I(j=2)g_i+{\gamma }_{3}I(j=3)+{\gamma }_{4}I(j=3)g_i+c_i\\ log({\nu }_{ij}^C)={\alpha }_0+{\alpha }_{1}I(j=2)+{\alpha }_{2}I(j=2)g_i+{\alpha }_{3}I(j=3)+{\alpha }_{4}I(j=3)g_i+d_i,\end{array}$$ (11)

where c_i, d_i are bivariate normal random intercepts with variances ${\sigma }_1^2,{\sigma }_2^2$ and correlation ρ used to account for correlated outcomes for the i^th participant. For a fixed sample, g_i was generated from a Bernoulli(0.5) and (c_i, d_i) were generated from a bivariate normal distribution with ${\sigma }_1^2={\sigma }_2^2=1$ and ρ = −0.25. In most scenarios, we expect that the probability of an excess zero will be negatively correlated with the marginal mean as in our motivating example.

The parameters ${\psi }_{ij}^C$ and ${\nu }_{ij}^C$ are calculated with the specified values of γ and α. Using the first model part in equation (11) and ${\mu }_{ij}^C={\nu }_{ij}^C/(1-{\psi }_{ij}^C))$, excess zeros and Poisson counts were randomly generated. Define ${\psi }_i^C=({\psi }_{i1}^C,{\psi }_{i2}^C,{\psi }_{i3}^C)$ and ${\nu }_i^C=({\nu }_{i1}^C,{\nu }_{i2}^C,{\nu }_{i3}^C)$. These simulations were performed for 100, 300, 500 and 1000 participants, respectively, with γ, α vectors chosen such that ${\psi }_i^C=\{0.45,0.50,0.50\}$, ${\nu }_i^C=\{1.75,1.70,1.70\}$ for g_i = 0 and ${\psi }_i^C=\{0.45,0.65,0.65\}$, ${\nu }_i^C=\{1.75,1.275,1.11\}$ for g_i = 1. These marginal mean specifications correspond to IDR values of (0.97,0.97) in the unexposed group and (0.75,0.65) in the exposed group across follow-up time 2 and 3. Across the combinations of g_i and time j, the total percent of zero counts ranged from 44% to 69%. For each cluster size, 1,000 simulations were attempted, but the SAS NLMIXED procedure failed to converge for 5% iterations. Others have reported difficulties in convergence of ZIP models with random effects (Min and Agresti, 2005).

Table 1 presents the raw and percent relative median bias, simulation standard deviation and median standard errors (model-based and robust) of each estimate from the marginalized ZIP model. The vectors of parameters to simulate the above values of ψ_ij and ν_ij are γ = {−0.2007, 0.2007, 0.8197, 0.2007, 0.8197} and α = {0.5596, -0.0290, -0.2877, -0.0290, −0.4263}.

Table 1. Marginalized ZIP with RE Performance with 1,000 Simulations and Varying Number of Subjects.

Parameter	K	Raw Median Bias	Percent Relative Median Bias	Simulation Std Dev	Median Std Error	Median Robust Std Error
γ0	100	-0.003	1.59	0.2061	0.2518	0.2521
	300	-0.035	17.26	0.1510	0.1554	0.1557
	500	-0.005	2.68	0.1057	0.1159	0.1155
	1000	0.013	-6.34	0.0777	0.0870	0.0870
γ1	100	0.006	3.19	0.3369	0.3888	0.3656
	300	-0.017	-8.51	0.2391	0.2379	0.2362
	500	-0.011	-5.50	0.1727	0.1798	0.1747
	1000	-0.004	-2.08	0.1340	0.1330	0.1324
γ2	100	-0.026	-3.18	0.4213	0.4808	0.4744
	300	0.011	1.30	0.2886	0.2924	0.2904
	500	0.000	-0.04	0.2068	0.2202	0.2179
	1000	0.001	0.11	0.1606	0.1628	0.1627
γ3	100	-0.010	-4.83	0.3604	0.3857	0.3667
	300	0.000	-0.18	0.2514	0.2384	0.2368
	500	-0.012	-5.89	0.1695	0.1792	0.1730
	1000	-0.009	-4.59	0.1333	0.1329	0.1330
γ4	100	0.003	0.33	0.4135	0.4868	0.4775
	300	0.000	0.02	0.3056	0.2944	0.2941
	500	-0.003	-0.35	0.2069	0.2220	0.2193
	1000	-0.002	-0.20	0.1664	0.1642	0.1640
α0	100	0.064	11.49	0.1264	0.1685	0.1657
	300	0.124	22.09	0.0976	0.1080	0.1074
	500	0.077	13.71	0.0661	0.0803	0.0782
	1000	0.056	10.08	0.0530	0.0617	0.0613
α1	100	-0.003	9.19	0.1661	0.1765	0.1582
	300	-0.002	5.39	0.1159	0.1207	0.1200
	500	0.005	-16.51	0.0812	0.0847	0.0820
	1000	-0.003	10.01	0.0680	0.0655	0.0652
α2	100	0.009	-3.03	0.2530	0.2799	0.2632
	300	0.005	-1.79	0.1826	0.1823	0.1822
	500	-0.004	1.27	0.1251	0.1302	0.1287
	1000	0.003	-0.88	0.0997	0.0999	0.0997
α3	100	0.000	0.32	0.1627	0.1726	0.1553
	300	-0.005	16.85	0.1243	0.1201	0.1197
	500	0.006	-22.00	0.0847	0.0847	0.0815
	1000	0.004	-13.75	0.0675	0.0654	0.0652
α4	100	0.007	-1.64	0.2617	0.2823	0.2640
	300	0.001	-0.22	0.1863	0.1848	0.1844
	500	-0.007	1.58	0.1244	0.1324	0.1295
	1000	-0.001	0.30	0.0987	0.1006	0.1005

True parameter values: γ = {−0.2007, 0.2007, 0.8197, 0.2007, 0.8197}

α = {0.5596, -0.0290, -0.2877, -0.0290, -0.4263}

The raw median bias is small for each cluster size K, and both the model-based and robust standard errors are close to the standard deviation of the parameter estimates, indicating adequate estimation of the variability in parameter estimates. The largest percent relative bias in estimating α occur for α₀, α₁ and α₃. The parameters α₁ and α₃ are the log-IDR for times 2 and 3 relative to time 1 for the unexposed groups and have true values very close to 0, inflating the relative bias. For K = 500, the true α₃ is −0.0290 and the median bias is 0.00638, yielding a percent relative median bias of -22.0%. Despite these inflated relative median biases for true parameters near zero, the marginalized ZIP with random effects model has low bias across the simulation scenarios.

In addition to the marginalized ZIP model with random effects, both a Poisson population-average model with GEE estimation and a Poisson random intercept model were fit in SAS 9.3 GENMOD and NLMIXED, respectively, for comparison in estimating the population-average IDR. The model for the Poisson population-average model is

$$log({\nu }_{ij}^M)={\alpha }_0^{\ast }+{\alpha }_{1}I(j=2)+{\alpha }_{2}I(j=2)g_i+{\alpha }_{3}I(j=3)+{\alpha }_{4}I(j=3)g_i,$$ (12)

with unstructured covariance and model-based standard errors scaled with Pearson's chi-square for potential overdispersion, as well as empirical (robust) standard errors; (11) expresses the model for the Poisson random intercept model with ${\nu }_{ij}^C$ representing the Poisson mean E(Y_ij|d_i). As discussed in Section 4.2, the parameters (α₁, α₂, α₃, α₄) from (11) have population-average interpretations (since intercept is the only random effect), so the parameters from the Poisson population-average model with GEE estimation in (12) are estimating the same quantities. For time 2, Table 2 presents the relative median bias in estimating both the log-IDR and IDR corresponding to {α₁, α₂, α₃, α₄} for all three models, as well as the 95% Wald-type coverage probabilities and power.

Table 2. Percent Relative Median Bias, Coverage & Power for Estimating IDR and log-IDR.

		Percent Relative Median Bias (IDR)†	Percent Relative Median Bias (Log-IDR)	Model-Based Coverage	Model-Based Power	Robust Coverage	Robust Power
α1
100	mZIP	-0.27	9.19	0.956	0.054	0.949	0.058
	Poisson PA	-1.84	64.01	0.944	0.058	0.928	0.074
	Poisson RI	-0.77	26.72	0.508	0.515	0.936	0.065
300	mZIP	-0.16	1.30	0.964	0.044	0.961	0.047
	Poisson PA	-0.74	25.69	0.952	0.047	0.940	0.071
	Poisson RI	0.99	-34.04	0.508	0.476	0.933	0.071
500	mZIP	0.48	-16.51	0.955	0.057	0.952	0.060
	Poisson PA	-0.08	2.66	0.961	0.049	0.939	0.071
	Poisson RI	1.39	-47.56	0.520	0.466	0.946	0.053
1000	mZIP	-0.29	10.01	0.935	0.088	0.938	0.088
	Poisson PA	-0.17	5.83	0.961	0.040	0.949	0.059
	Poisson RI	0.91	-31.18	0.506	0.525	0.943	0.057
α2
100	mZIP	0.87	-3.03	0.949	0.426	0.944	0.427
	Poisson PA	-0.37	1.28	0.913	0.294	0.923	0.294
	Poisson RI	-2.09	7.34	0.473	0.718	0.930	0.268
300	mZIP	0.52	5.39	0.942	0.333	0.945	0.341
	Poisson PA	-0.02	0.09	0.917	0.237	0.927	0.236
	Poisson RI	-2.36	8.29	0.472	0.711	0.933	0.216
500	mZIP	-0.36	1.27	0.952	0.675	0.946	0.682
	Poisson PA	0.27	-0.92	0.923	0.419	0.935	0.395
	Poisson RI	-2.33	8.21	0.492	0.846	0.941	0.375
1000	mZIP	0.25	-0.88	0.956	0.841	0.954	0.837
	Poisson PA	-0.36	1.26	0.940	0.525	0.946	0.498
	Poisson RI	-2.77	9.76	0.479	0.908	0.946	0.475
α3
100	mZIP	-0.01	0.32	0.948	0.058	0.948	0.070
	Poisson PA	-0.77	26.70	0.948	0.043	0.938	0.061
	Poisson RI	0.33	-11.22	0.538	0.477	0.952	0.052
300	mZIP	-0.49	16.85	0.943	0.066	0.943	0.071
	Poisson PA	-1.05	36.51	0.959	0.037	0.946	0.062
	Poisson RI	0.13	-4.44	0.515	0.483	0.939	0.061
500	mZIP	0.64	-22.00	0.935	0.068	0.933	0.074
	Poisson PA	-0.73	25.12	0.954	0.043	0.933	0.062
	Poisson RI	0.85	-29.07	0.497	0.499	0.928	0.059
1000	mZIP	0.40	-13.75	0.948	0.067	0.946	0.073
	Poisson PA	0.37	-12.70	0.941	0.067	0.953	0.047
	Poisson RI	1.48	-50.64	0.502	0.483	0.942	0.068
α4
100	mZIP	0.70	-1.64	0.957	0.572	0.952	0.575
	Poisson PA	-0.04	0.08	0.944	0.467	0.942	0.485
	Poisson RI	-1.42	3.34	0.488	0.792	0.940	0.442
300	mZIP	0.09	-0.22	0.952	0.651	0.948	0.646
	Poisson PA	-1.84	4.36	0.947	0.389	0.937	0.404
	Poisson RI	-2.89	6.88	0.485	0.831	0.937	0.378
500	mZIP	-0.67	1.58	0.945	0.925	0.943	0.931
	Poisson PA	0.41	-0.96	0.937	0.685	0.928	0.665
	Poisson RI	-2.09	4.95	0.488	0.943	0.939	0.641
1000	mZIP	-0.13	0.30	0.946	0.988	0.947	0.990
	Poisson PA	0.35	-0.83	0.940	0.811	0.936	0.805
	Poisson RI	-2.45	5.81	0.458	0.982	0.941	0.755

^*mZIP: Marginalized ZIP model with random effects;

Poisson PA: Poisson population average model with GEE estimation;

Poisson RI: Poisson random intercept model

^†Simulated scenario: True IDRs exp(α₁) = 0.97, exp(α₂) = 0.75, exp(α₃) = 0.97, exp(α₄) = 0.65

Note that the marginalized ZIP model with random effects has lower percent relative median bias for most scenarios, as well as appropriate coverage. With the model-based standard errors in the Poisson random intercept model, the coverage probabilities are much less than the expected 0.95, indicating these standard errors are underestimating the extra-Poisson variability in the ZIP data due to the excess zeros. The robust standard errors for both Poisson models provide appropriate coverage of the IDR, but the marginalized ZIP model has increased power to detect significance in IDR over both Poisson methods, particularly for α₂, α₄ where parameter estimates deviate further from 0. Using the Pearson-scaled model-based standard errors, the Poisson PA models have very similar absolute bias in many scenarios and only slightly less coverage than the marginalized ZIP model, but there is a marked difference in power with the Poisson PA model having significantly less ability to detect differences in mean IDR.

6. Analysis of the SafeTalk efficacy trial

In safer sex counseling for people living with HIV/AIDS, an outcome of interest is Unprotected Anal or Vaginal Intercourse acts (UAVI), defined as the number of unprotected sexual acts with any partner. Researchers developed the motivational interview-based intervention SafeTalk to reduce the number of unprotected sexual acts (Golin et al., 2007; Golin et al., 2010; Golin et al., 2012). For the clinical trial examining SafeTalk efficacy, participants were randomized to receive either SafeTalk intervention counseling or a control nutritional counseling. These participants completed questionnaires about both nutritional and sexual behavior at baseline as well as at three follow-up visits spaced at four-month intervals. After data cleaning, the sample sizes at each time point are 476, 399, 363 and 301. The overall percentage of zero UAVI counts across both treatment groups and all visits was 83.1%.

While some researchers may choose to focus on the latent class interpretations provided by the ZIP model with random effects, our collaborative researchers are interested in quantifying the effect of the SafeTalk intervention over time among the entire randomized population, leading to a choice of marginal mean inference provided by the marginalized ZIP model with random effects. In order to evaluate the efficacy of the SafeTalk intervention over time, the marginalized ZIP with random effects is fit to the UAVI counts at all four time points. The model of interest is

$$\begin{array}{c}logit({\psi }_{ij}^C)={\gamma }_0+{\gamma }_1{x}_{i1}+{\gamma }_2{x}_{i2}+{\gamma }_{3}I(j=2)+{\gamma }_{4}I(j=2)g_i+{\gamma }_{5}I(j=3)+{\gamma }_{6}I(j=3)g_i+{\gamma }_{7}I(j=4)+{\gamma }_{8}I(j=4)g_i+c_i\\ log({\nu }_{ij}^C)={\alpha }_0+{\alpha }_1{x}_{i1}+{\alpha }_2{x}_{i2}+{\alpha }_{3}I(j=2)+{\alpha }_{4}I(j=2)g_i+{\alpha }_{5}I(j=3)+{\alpha }_{6}I(j=3)g_i+{\alpha }_{7}I(j=4)+{\alpha }_{8}I(j=4)g_i+d_i,\end{array}$$

where c_i, d_i are bivariate normal random intercepts with covariance $\sum =\left[\begin{array}{cc}{\sigma }_{11}& {\sigma }_{12}\\ {\sigma }_{12}& {\sigma }_{22}\end{array}\right]$, j is the visit number, g_i is an indicator of randomization to SafeTalk intervention group, and x_i1 and x_i2 are fixed effects for study site.

Using SAS NLMIXED (for which the code is presented in the Appendix), the SafeTalk analysis results are presented in Table 3. The contrast testing treatment effect over time H₀ : (α₄, α₆, α₈)′ = (0, 0,0)′ is highly significant (Robust-Wald p = 0.0003), indicating that the SafeTalk intervention affects UAVI count. At the second follow-up visit, for which the IDR (and 95% Wald-type robust confidence interval) is 0.542 (0.260, 1.128), a participant randomized to SafeTalk has 46% fewer unprotected sexual acts with any partner than he or she would have if randomized to the nutritional intervention. Because the only random effect for the above model is a random intercept, the parameters associated with treatment effect from this analysis additionally have population-averaged interpretations. Thus, at the second follow-up visit, those participants randomized to SafeTalk had on average 46% fewer unprotected sexual acts with any partner than the participants randomized to the nutritional intervention. The SafeTalk intervention appears to have the largest effect on UAVI count at the first follow-up survey, where the estimated IDR (and 95% Wald-type robust confidence interval) of treatment effect is 0.280 (0.145, 0.542). By the third follow-up survey, we observe less reduction in UAVI count due to SafeTalk, with an IDR of 0.769 (0.307, 1.928). Figure 1 displays the predicted mean UAVI over time, as well as the IDR of treatment at each time point. The SafeTalk intervention appears to have a significant effect in reducing UAVI counts at the first follow-up visit, but the difference between the two treatment groups is reduced at each subsequent follow-up visit. From Figure 1 and Table 3, note that the nutritional control arm has a significant reduction in predicted UAVI count at the final visit, numerically represented through α₇. Additionally, note that the correlation between the random intercepts, estimated to be -0.79, is highly significant, indicating those participants with higher expected UAVI counts have lower odds of excess zero latent class membership. In fact, if independence of the random intercepts is assumed, individual parameter estimates from the marginalized ZIP model differ as much as 40%, leading us to recommend the inclusion of correlated random effects in the two processes.

Table 3. Marginalized ZIP Model with Random Effects Results: SafeTalk efficacy trial.

	Parameter	Parameter Estimate	Model-Based Std Error	Robust Std Error
Zero-Inflation Model
Intercept	γ0	2.1187	0.3581	0.3665
Site 2	γ1	0.1026	0.4311	0.4184
Site 3	γ2	0.2445	0.8782	0.9548
Follow-up 1	γ3	1.2709	0.3287	0.3468
Follow-up 1*Treatment	γ4	0.8849	0.4144	0.4627
Follow-up 2	γ5	1.7071	0.3611	0.7011
Follow-up 2*Treatment	γ6	-0.6021	0.5022	0.9185
Follow-up 3	γ7	1.0214	0.4577	0.6881
Follow-up 3*Treatment	γ8	-0.3331	0.6034	1.0968
Marginalized Mean Model
Intercept	α0	-0.8966	0.2803	0.2965
Site 2	α1	0.0362	0.2941	0.2893
Site 3	α2	-0.0220	0.6191	0.6442
Follow-up 1	α3	0.2011	0.1471	0.1969
Follow-up 1*Treatment	α4	-1.2725	0.2197	0.3365
Follow-up 2	α5	-0.1217	0.1632	0.2264
Follow-up 2*Treatment	α6	-0.6128	0.2082	0.3742
Follow-up 3	α7	-0.4762	0.2203	0.3521
Follow-up 3*Treatment	α8	-0.2630	0.2611	0.4691
Variance Parameters†
	σ11	9.7487	2.1328	2.4313
	σ12	-4.5957	0.8270	0.7345
	σ22	3.4461	0.6929	0.6599

^† $\widehat{\rho }={\widehat{\sigma }}_{12}/\left(\sqrt{{\widehat{\sigma }}_{11}{\widehat{\sigma }}_{22}}\right)=-0.79$

Fig 1.

Marginalized ZIP with random effects (c_i = d_i = 0) predicted UAVI means over time. Follow-up visits (FU1, FU2, FU3) are at four, eight and twelve months post-randomization.

When the SafeTalk data are examined using a Poisson population-average model with GEE estimation and empirical standard errors, the Wald contrast with 3 degrees of freedom testing treatment effect is non-significant (p=0.8259). At the second follow-up, the GEE model estimates the IDR to be 0.768 with 95% Wald-type model-based and empirical confidence intervals (0.391, 1.508) and (0.403, 1.466), respectively. Using the Poisson random intercept model, the treatment efficacy contrast is significant when using the model-based standard errors (p=0.0303) but non-significant when robust standard errors are used (p=0.8446). At the second follow-up, the random intercept model estimates the IDR to be 0.711 with model-based and robust 95% Wald-type confidence intervals of (0.556, 0.908) and (0.336, 1.502). Because the simulations in Section 5 suggest that the model-based standard errors in the Poisson random intercept model underestimate the variability due to the excess zero process, the conclusions of the robust methods are preferred.

To highlight the differences between the proposed marginalized ZIP model with random effects and the ZIP model with random effects from Section 2, the latter was also fit to the SafeTalk data, given by

$$\begin{array}{c}logit({\psi }_{ij}^C)={\gamma }_0+{\gamma }_1{x}_{i1}+{\gamma }_2{x}_{i2}+{\gamma }_{3}I(j=2)+{\gamma }_{4}I(j=2)g_i+{\gamma }_{5}I(j=3)+{\gamma }_{6}I(j=3)g_i+{\gamma }_{7}I(j=4)+{\gamma }_{8}I(j=4)g_i\\ log({\mu }_{ij}^C)={\beta }_0+{\beta }_1{x}_{i1}+{\beta }_2{x}_{i2}+{\beta }_{3}I(j=2)+{\beta }_{4}I(j=2)g_i+{\beta }_{5}I(j=3)+{\beta }_{6}I(j=3)g_i+{\beta }_{7}I(j=4)+{\beta }_{8}I(j=4)g_i+d_i,\end{array}$$

where d_i ∼ N(0, σ²). For this model, the contrast of treatment effect is highly significant (p<0.0001) with β₄ = −0.96, β₆ = −0.89, and β₈ = −0.42. In contrast to the marginalized ZIP model with random effects and the Poisson models which model the marginal mean directly, these traditional ZIP parameter estimates are the log-IDR for treatment among the non-excess zero latent class. Among the non-excess zero latent class, those participants randomized to SafeTalk had 62%, 59% and 35% fewer UAVI acts than those participants randomized to control at the first, second and third follow-up visits, respectively.

7. Conclusion

Motivated by the aim to estimate overall exposure effects for correlated count observations with excess zeroes, we have proposed a marginalized ZIP model with random effects. Since the overall subject-specific mean is modeled directly, the parameters from this new model allow subject-specific inference rather than inference on the latent class components of the subject-specific ZIP model. Additionally, when the log link is used for the marginal mean and normal random effects are used, those parameters without corresponding random effects have both subject-specific and population-average interpretations.

The new marginalized ZIP model with random effects was applied to repeated measures data from a clinical trial to reduce risky sexual behavior among HIV-positive individuals. We observed that the robust standard errors for intervention effect parameters were notably larger than their model-based counterparts, suggesting the counts are overdispersed. Future research could extend the marginalized ZIP model for random effects to handle overdispersion as well as excess zeros.

In the SafeTalk data, missing at random (MAR) is assumed, meaning that the probability of attending a visit and having UAVI recorded depends only on observed data. There is evidence that the assumption of missing completely at random (MCAR) is not valid because those participants with any risky baseline behavior have 54.1% retention at the final visit versus 65.6% retention in those with non-risky baseline behavior. Maximum likelihood estimation of the marginalized ZIP with random effects model described in Section 4.1 provides valid inference under MAR when the model is correctly specified (Ibrahim and Molenberghs, 2009).

In the simulation study, we experienced convergence issues similar to ZIP model instability occasionally associated with those effects in the excess zero portion of ZIP models (Min and Agresti, 2005). Future research includes exploring other optimization techniques with more stability for zero-inflated models, such as the Bayesian methods proposed in Neelon et al.(2010). In addition to other computational strategies, the relatively small number of simulation iterations with failed NLMIXED convergence could possibly be lessened by reducing the complexity of the excess zero model. In marginalized ZIP regression, the excess zero model parameters are considered nuisance parameters, as the primary hypotheses concern the marginalized mean. However, as unintended constraints on the marginal means can be introduced by the omission of covariates in the excess zero model, the reduction of the excess zero model should be carefully considered and rigorously justified.

In contrast to exclusive reliance on fit statistics or conjectures about data-generating mechanisms as a basis for selecting the type of count regression model for handling data with many zeros, we affirm that the choice between marginalized ZIP, ZIP and hurdle model classes should be motivated by the interpretations desired. When inference upon the overall marginal mean is desired, the marginalized ZIP model is preferred. The a priori choice of model class for zero-inflation is analogous to the a priori choice between PA and SS models for longitudinal data (Heagerty, 1999) where the interpretations of regressions parameters differ in models with non-identity link functions.

Rather than marginalizing over the two processes of the ZIP model, the ZIP model with random effects could be marginalized over the random effects, similar to the marginalized hurdle model in Lee et al. (2011). Additionally, one could marginalize over both the random effects and two ZIP processes to achieve a ‘doubly’ marginalized ZIP model. As shown in Section 4.2, the marginalized ZIP model can be used not only for subject-specific inference on overall conditional effects but also for population-average inference for overall effects in many problems.

8. Appendix

The following SAS NLMIXED code was used for the SafeTalk motivating example.

proc nlmixed data=safetalk seed=31415;

  parms b0 0 b1 0 b2 0 b3 0 b4 0 b5 0 b6 0 b7 0 b8 0

    a0 0 a1 0 a2 0 a3 0 a4 0 a5 0 a6 0 a7 0 a8 0

    sigma1 1 sigma12 0 sigma2 1;

   /* linear predictor for the zero-inflation probability */

   logit_psi = a0 + a1*site2 + a2*site3 + a3*v2 + a4*v2*st + a5*v3 + a6*v3*st

                  + a7*v4 + a8*v4*st + c1;

   *logit(\psi)=Z\gamma + c;

   /* useful functions of \psi */

   psi1 = exp(logit_psi)/(1+exp(logit_psi));

   *\psi = exp(Z\gamma+c)/(1+exp(Z\gamma+c));

   psi2 = 1/(1+exp(logit_psi));

   *1−\psi = (1+exp(Z\gamma+c))^−1;

   /* Overall mean \nu */

   log_nu = b0 + b1*site2 + b2*site3 + b3*v2 + b4*v2*st + b5*v3 + b6*v3*st

               + b7*v4 + b8*v4*st + d1;

   delta = log(psi2**(−1)) + log_nu;

   /* Build the mZIP + RE log likelihood */

   if outcome=0 then

        ll = log(psi1 + psi2*(exp(−exp(delta))));

   else ll = log(psi2) − exp(delta) + outcome*(delta) − lgamma(outcome + 1);

   model outcome ∼ general(ll);

   random c1 d1∼normal([0,0],[sigma1,sigma12,sigma2]) SUBJECT=urn;

   contrast “TX” b4, b6, b8;

run;