The COUNTREG Procedure

Zero-Inflated Negative Binomial Regression

Subsections:

The zero-inflated negative binomial (ZINB) model in PROC COUNTREG is based on the negative binomial model with quadratic variance function (p=2). The ZINB model is obtained by specifying a negative binomial distribution for the data generation process referred to earlier as Process 2:

\[ g(y_{i}) = \frac{\Gamma (y_{i}+\alpha ^{-1})}{y_{i}! \Gamma (\alpha ^{-1})}\left(\frac{\alpha ^{-1}}{\alpha ^{-1}+\mu _{i}} \right)^{\alpha ^{-1}}\left(\frac{\mu _{i}}{\alpha ^{-1}+\mu _{i}} \right)^{y_{i}} \]

Thus the ZINB model is defined to be

\begin{eqnarray*} P(y_{i}=0|\mathbf{x}_{i},\mathbf{z}_{i}) & =& F_{i} + \left(1 - F_{i}\right)(1+\alpha \mu _{i})^{-\alpha ^{-1}} \\ P(y_{i}|\mathbf{x}_{i},\mathbf{z}_{i}) & =& \left(1- F_{i} \right) \frac{\Gamma (y_{i}+\alpha ^{-1})}{y_{i}! \Gamma (\alpha ^{-1})}\left(\frac{\alpha ^{-1}}{\alpha ^{-1}+\mu _{i}} \right)^{\alpha ^{-1}} \\ & \times & \left(\frac{\mu _{i}}{\alpha ^{-1}+\mu _{i}} \right)^{y_{i}} , \quad y_{i}>0 \end{eqnarray*}

In this case, the conditional expectation and conditional variance of $y_{i}$ are

\[ E(y_{i}|\mathbf{x}_{i},\mathbf{z}_{i}) = \mu _{i}(1 -F_{i}) \]
\[ V(y_{i}|\mathbf{x}_{i},\mathbf{z}_{i}) = E(y_{i}|\mathbf{x}_{i},\mathbf{z}_{i})\left[1+\mu _{i} (F_{i}+\alpha ) \right] \]

Like the ZIP model, the ZINB model exhibits overdispersion because the conditional variance exceeds the conditional mean.

ZINB Model with Logistic Link Function

In this model, the probability $\varphi _{i}$ is given by the logistic function—namely,

\[ \varphi _{i}=\frac{\exp (\mathbf{z}_{i}'\bgamma )}{1+\exp (\mathbf{z}_{i}'\bgamma )} \]

The log-likelihood function is

\begin{eqnarray*} \mathcal{L} & = & \sum _{\{ i: y_{i}=0\} } w_ i\ln \left[\exp (\mathbf{z}_{i}’\bgamma )+(1+\alpha \exp (\mathbf{x}_{i}’\bbeta ))^{-\alpha ^{-1}} \right] \\ & + & \sum _{\{ i: y_{i}>0\} } w_ i\sum _{j=0}^{y_{i}-1}\ln (j+\alpha ^{-1}) \\ & + & \sum _{\{ i: y_{i}>0\} } w_ i\left\{ -\ln (y_{i}!) - (y_{i}+\alpha ^{-1}) \ln (1+\alpha \exp (\mathbf{x}_{i}^{\prime }\bbeta )) +y_{i}\ln (\alpha ) + y_{i}\mathbf{x}_{i}^{\prime }\bbeta \right\} \\ & - & \sum _{i=1}^{N}w_ i\ln \left[ 1 + \exp (\mathbf{z}_{i}’\bgamma ) \right] \end{eqnarray*}

See the section Poisson Regression for the definition of $w_ i$.

The gradient for this model is given by

\begin{eqnarray*} \frac{\partial \mathcal{L}}{\partial \bgamma } & = & \sum _{\{ i: y_{i}=0\} } w_ i\left[\frac{\exp (\mathbf{z}_{i}'\bgamma )}{\exp (\mathbf{z}_{i}'\bgamma ) + (1+\alpha \exp (\mathbf{x}_{i}'\bbeta ))^{-\alpha ^{-1}}}\right] \mathbf{z}_{i} \\ & - & \sum _{i=1}^{N} w_ i\left[\frac{\exp (\mathbf{z}_{i}'\bgamma )}{1 + \exp (\mathbf{z}_{i}'\bgamma )} \right] \mathbf{z}_{i} \end{eqnarray*}
\begin{eqnarray*} \frac{\partial \mathcal{L}}{\partial \bbeta } & = & \sum _{\{ i: y_{i}=0\} } w_ i\left[\frac{-\exp (\mathbf{x}_{i}'\bbeta ) (1+\alpha \exp (\mathbf{x}_{i}'\bbeta ))^{-\alpha ^{-1}-1}}{\exp (\mathbf{z}_{i}'\bgamma ) + (1+\alpha \exp (\mathbf{x}_{i}'\bbeta ))^{-\alpha ^{-1}}}\right] \mathbf{x}_{i} \\ & + & \sum _{\{ i: y_{i}>0\} } w_ i\left[ \frac{y_{i} - \exp (\mathbf{x}_{i}'\bbeta )}{1 + \alpha \exp (\mathbf{x}_{i}'\bbeta )} \right] \mathbf{x}_{i} \end{eqnarray*}
\[ \frac{\partial \mathcal{L}}{\partial \alpha } = \sum _{\{ i: y_{i}=0\} } w_ i\frac{ \alpha ^{-2} \left[(1+\alpha \exp (\mathbf{x}_{i}'\bbeta )) \ln (1+\alpha \exp (\mathbf{x}_{i}'\bbeta )) - \alpha \exp (\mathbf{x}_{i}'\bbeta )\right]}{\exp (\mathbf{z}_{i}'\bgamma ) (1+\alpha \exp (\mathbf{x}_{i}'\bbeta ))^{(1+\alpha )/\alpha } + (1+\alpha \exp (\mathbf{x}_{i}'\bbeta ))} \]
\[ + \sum _{\{ i: y_{i}>0\} } w_ i\left\{ - \alpha ^{-2} \sum _{j=0}^{y_{i}-1} \frac{1}{(j + \alpha ^{-1})} + \alpha ^{-2} \ln (1+\alpha \exp (\mathbf{x}_{i}’\bbeta )) + \frac{y_{i}-\exp (\mathbf{x}_{i}'\bbeta )}{\alpha (1+\alpha \exp (\mathbf{x}_{i}'\bbeta ))}\right\} \]

ZINB Model with Standard Normal Link Function

For this model, the probability $\varphi _{i}$ is specified using the standard normal distribution function (probit function): $\varphi _{i}= \Phi (\mathbf{z}_{i}’\bgamma )$. The log-likelihood function is

\begin{eqnarray*} \mathcal{L} & = & \sum _{\{ i: y_{i}=0\} } w_ i\ln \left\{ \Phi (\mathbf{z}_{i}’\bgamma ) + \left[ 1 - \Phi (\mathbf{z}_{i}’\bgamma ) \right] (1+\alpha \exp (\mathbf{x}_{i}’\bbeta ))^{-\alpha ^{-1}} \right\} \\ & + & \sum _{\{ i: y_{i}>0\} } w_ i\ln \left[ 1 - \Phi (\mathbf{z}_{i}’\bgamma ) \right] \\ & + & \sum _{\{ i: y_{i}>0\} } w_ i\sum _{j=0}^{y_{i}-1} \left\{ \ln (j+\alpha ^{-1})\right\} \\ & - & \sum _{\{ i: y_{i}>0\} } w_ i\ln (y_{i}!) \\ & - & \sum _{\{ i: y_{i}>0\} } w_ i(y_{i}+\alpha ^{-1}) \ln (1+\alpha \exp (\mathbf{x}_{i}^{\prime }\bbeta )) \\ & + & \sum _{\{ i: y_{i}>0\} } w_ iy_{i}\ln (\alpha ) \\ & + & \sum _{\{ i: y_{i}>0\} } w_ iy_{i} \mathbf{x}_{i}^{\prime }\bbeta \end{eqnarray*}

See the section Poisson Regression for the definition of $w_ i$.

The gradient for this model is given by

\[ \frac{\partial \mathcal{L}}{\partial \bgamma } = \sum _{\{ i: y_{i}=0\} } w_ i\left[\frac{\varphi (\mathbf{z}_{i}'\bgamma ) \left[1-(1+\alpha \exp (\mathbf{x}_{i}'\bbeta ))^{-\alpha ^{-1}} \right]}{ \Phi (\mathbf{z}_{i}'\bgamma ) + \left[1- \Phi (\mathbf{z}_{i}'\bgamma )\right] (1+\alpha \exp (\mathbf{x}_{i}'\bbeta ))^{-\alpha ^{-1}}} \right] \mathbf{z}_{i} \]
\[ - \sum _{\{ i: y_{i}>0\} } w_ i\left[\frac{\varphi (\mathbf{z}_{i}'\bgamma )}{1 - \Phi (\mathbf{z}_{i}'\bgamma )} \right] \mathbf{z}_{i} \]
\[ \frac{\partial \mathcal{L}}{\partial \bbeta } = \sum _{\{ i: y_{i}=0\} } w_ i\frac{-\left[1-\Phi (\mathbf{z}_{i}'\bgamma )\right] \exp (\mathbf{x}_{i}'\bbeta ) (1+\alpha \exp (\mathbf{x}_{i}'\bbeta ))^{-(1+\alpha )/\alpha }}{\Phi (\mathbf{z}_{i}'\bgamma ) + \left[ 1 - \Phi (\mathbf{z}_{i}'\bgamma ) \right] (1+\alpha \exp (\mathbf{x}_{i}'\bbeta ))^{-\alpha ^{-1}} } \mathbf{x}_{i} \]
\[ + \sum _{\{ i: y_{i}>0\} } w_ i\left[ \frac{y_{i} - \exp (\mathbf{x}_{i}'\bbeta )}{1 + \alpha \exp (\mathbf{x}_{i}'\bbeta )} \right] \mathbf{x}_{i} \]
\[ \frac{\partial \mathcal{L}}{\partial \alpha } = \sum _{\{ i: y_{i}=0\} } w_ i\frac{\left[ 1-\Phi (\mathbf{z}_{i}'\bgamma ) \right]\alpha ^{-2} \left[(1 + \alpha \exp (\mathbf{x}_{i}'\bbeta )) \ln (1 + \alpha \exp (\mathbf{x}_{i}'\bbeta ))-\alpha \exp (\mathbf{x}_{i}'\bbeta )\right]}{\Phi (\mathbf{z}_{i}'\bgamma ) (1 + \alpha \exp (\mathbf{x}_{i}'\bbeta ))^{(1+\alpha )/\alpha } + \left[1 -\Phi (\mathbf{z}_{i}'\bgamma ) \right] (1 + \alpha \exp (\mathbf{x}_{i}'\bbeta ))}\\ \]
\[ + \sum _{\{ i: y_{i}>0\} } w_ i\left\{ - \alpha ^{-2} \sum _{j=0}^{y_{i}-1} \frac{1}{(j + \alpha ^{-1})} + \alpha ^{-2} \ln (1+\alpha \exp (\mathbf{x}_{i}’\bbeta )) + \frac{y_{i}-\exp (\mathbf{x}_{i}'\bbeta )}{\alpha (1+\alpha \exp (\mathbf{x}_{i}'\bbeta ))}\right\} \]