Marginalized Zero-Altered Models for Longitudinal Count Data

Abstract

Count data often exhibit more zeros than predicted by common count distributions like the Poisson or negative binomial. In recent years, there has been considerable interest in methods for analyzing zero-inflated count data in longitudinal or other correlated data settings. A common approach has been to extend zero-inflated Poisson models to include random effects that account for correlation among observations. However, these models have been shown to have a few drawbacks, including interpretability of regression coefficients and numerical instability of fitting algorithms even when the data arise from the assumed model. To address these issues, we propose a model that parameterizes the marginal associations between the count outcome and the covariates as easily interpretable log relative rates, while including random effects to account for correlation among observations. One of the main advantages of this marginal model is that it allows a basis upon which we can directly compare the performance of standard methods that ignore zero inflation with that of a method that explicitly takes zero inflation into account. We present simulations of these various model formulations in terms of bias and variance estimation. Finally, we apply the proposed approach to analyze toxicological data of the effect of emissions on cardiac arrhythmias.

Appendices

Appendix 1: Calculation of $\Delta _{ij}$

In solving for $\Delta _{ij}$, we need to solve the convolution equation that links the marginal (i.e, $\mu _{ij}^{Y} = \hbox {E}(Y_{ij}|X_{ij}) = \hbox {exp} (\beta X_{ij})$) and conditional means, where, assuming the MZAP setting,

$$\begin{aligned} \mu _{ij}^{Y}= & {} \hbox {exp}(X_{ij} \beta ) \\= & {} \int [1-P(Y_{ij}=0|X_{ij},b_{i})] \frac{\mu _{ij}^{b}}{[1 - \hbox {exp}(-\mu _{ij}^{b})]} \phi (b_{i}|\sigma ) \mathrm{d}b_{i} \\= & {} \int \frac{[1-\hbox {exp}[-\hbox {exp}(\gamma _{1} + \gamma _{2} (\Delta _{ij} + b_{i}))]]}{[1 - \hbox {exp}[-\hbox {exp}(\Delta _{ij} + b_{i})]]} \hbox {exp}(\Delta _{ij} + b_{i}) \phi (b_{i}|\sigma ) \mathrm{d}b_{i} . \end{aligned}$$

Estimates of $\Delta _{ij}$ can be obtained using a Newton–Raphson algorithm, such that

$$\begin{aligned} \Delta _{ij}^{(t+1)}= & {} \Delta _{ij}^{(t)} - \left( \frac{\partial f(\Delta _{ij})}{\partial \Delta _{ij}}\right) ^{-1} \times f(\Delta _{ij}) , \end{aligned}$$

where $\Delta _{ij} = \Delta _{ij} (\beta , \gamma _{1}, \gamma _{2}, \sigma )$ and $f(\Delta _{ij})$ refers to the convolution equation above. The derivative needed for the Newton–Raphson algorithm is as follows

$$\begin{aligned} \frac{\partial }{\partial \Delta _{ij}} \mu _{ij}^{Y}= & {} \frac{\partial }{\partial \Delta _{ij}} \int \frac{[1-\hbox {exp}[-\hbox {exp}(\gamma _{1} + \gamma _{2} (\Delta _{ij} + b_{i}))]]}{[1 - \hbox {exp}[-\hbox {exp}(\Delta _{ij} + b_{i})]]} \hbox {exp}(\Delta _{ij} + b_{i}) \phi (b_{i}|\sigma ) \mathrm{d}b_{i} \\= & {} \int \left\{ \frac{\partial }{\partial \Delta _{ij}} \frac{[1-\hbox {exp}[-\hbox {exp}(\gamma _{1} + \gamma _{2} (\Delta _{ij} + b_{i}))]]}{[1 - \hbox {exp}[-\hbox {exp}(\Delta _{ij} + b_{i})]]} \hbox {exp}(\Delta _{ij} + b_{i}) \right\} \phi (b_{i}|\sigma ) \mathrm{d}b_{i} . \end{aligned}$$

After using the chain rule, $\diamond = \left\{ \frac{\partial }{\partial \Delta _{ij}} \frac{[1-\hbox {exp}[-\hbox {exp}(\gamma _{1} + \gamma _{2} (\Delta _{ij} + b_{i}))]]}{[1 - \hbox {exp}[-\hbox {exp}(\Delta _{ij} + b_{i})]]} \hbox {exp}(\Delta _{ij} + b_{i}) \right\}$ results in

$$\begin{aligned} \diamond= & {} \{ 1 - e^{-e^{\gamma _{1} + \gamma _{2} (\Delta _{ij} + b_{i})}} \} * \left\{ \frac{[1 - e^{-e^{\Delta _{ij} + b_{i}}}]e^{\Delta _{ij} + b_{i}} - e^{\Delta _{ij} + b_{i}}[-e^{-e^{\Delta _{ij} + b_{i}}}][-e^{\Delta _{ij} + b_{i}}]}{[1 - e^{-e^{\Delta _{ij} + b_{i}}}]^{2}} \right\} \\&+ \left\{ \frac{e^{\Delta _{ij} + b_{i}}}{1 - e^{-e^{\Delta _{ij} + b_{i}}}} \right\} * \{ e^{-e^{\gamma _{1} + \gamma _{2} (\Delta _{ij} + b_{i})}} [e^{\gamma _{1} + \gamma _{2} (\Delta _{ij} + b_{i})}] \gamma _{2} \} \\= & {} \frac{e^{\Delta _{ij} + b_{i}}}{1 - e^{-e^{\Delta _{ij} + b_{i}}}} * [ \{ 1 - e^{-e^{\gamma _{1} + \gamma _{2} (\Delta _{ij} + b_{i})}} \}* \left\{ 1 - \frac{[e^{-e^{\Delta _{ij} + b_{i}}}] [e^{\Delta _{ij} + b_{i}}]}{1 - e^{-e^{\Delta _{ij} + b_{i}}}} \right\} \\&+ \{ e^{-e^{\gamma _{1} + \gamma _{2} (\Delta _{ij} + b_{i})}} [e^{\gamma _{1} + \gamma _{2} (\Delta _{ij} + b_{i})}] \gamma _{2} \} ] . \end{aligned}$$

Gauss–Hermite quadrature can be used to evaluate this one-dimensional integral.

Appendix 2: SAS Execution via PROC NLMIXED

Marginalized zero-altered Poisson (MZAP) and marginalized zero-altered negative binomial (MZANB) models