Zero-Inflated Negative Binomial Regression :: SAS/ETS(R) 12.3 User's Guide: High-Performance Procedures

Zero-Inflated Negative Binomial Regression

ZINB Model with Logistic Link Function
ZINB Model with Standard Normal Link Function

The zero-inflated negative binomial (ZINB) model in PROC HPCOUNTREG is based on the negative binomial model that has a quadratic variance function (when DIST=NEGBIN in the MODEL or PROC HPCOUNTREG statement). 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

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

In this case, the conditional expectation ( $\Emph{E}$ ) and conditional variance (V) 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

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

The gradient for this model is given by

$\displaystyle \frac{\partial \mathcal{L}}{\partial \bgamma }$	$\displaystyle =$	$\displaystyle \sum _{\{ i: y_{i}=0\} } \left[\frac{\exp (\mathbf{z}_{i}\bgamma )}{\exp (\mathbf{z}_{i}\bgamma ) + (1+\alpha \exp (\mathbf{x}_{i}\bbeta ))^{-\alpha ^{-1}}}\right] \mathbf{z}_{i}$
$\displaystyle$	$\displaystyle -$	$\displaystyle \sum _{i=1}^{N} \left[\frac{\exp (\mathbf{z}_{i}\bgamma )}{1 + \exp (\mathbf{z}_{i}\bgamma )} \right] \mathbf{z}_{i}$

$\displaystyle \frac{\partial \mathcal{L}}{\partial \bbeta }$	$\displaystyle =$	$\displaystyle \sum _{\{ i: y_{i}=0\} } \left[\frac{-\exp (\mathbf{x}_{i}\bbeta ) (1+\alpha \exp (\mathbf{x}_{i}\bbeta ))^{-\alpha ^{-1}-1}}{\exp (\mathbf{z}_{i}\bgamma ) + (1+\alpha \exp (\mathbf{x}_{i}\bbeta ))^{-\alpha ^{-1}}}\right] \mathbf{x}_{i}$
$\displaystyle$	$\displaystyle +$	$\displaystyle \sum _{\{ i: y_{i}>0\} } \left[ \frac{y_{i} - \exp (\mathbf{x}_{i}\bbeta )}{1 + \alpha \exp (\mathbf{x}_{i}\bbeta )} \right] \mathbf{x}_{i}$

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

$+ \sum _{\{ i: y_{i}>0\} } \left\{ - \alpha ^{-2} \sum _{j=0}^{y_{i}-1} \frac{1}{(j + \alpha ^{-1})} + \alpha ^{-2} \ln (1+\alpha \exp (\mathbf{x}_{i}’\bbeta )) + \frac{y_{i}-\exp (\mathbf{x}_{i}\bbeta )}{\alpha (1+\alpha \exp (\mathbf{x}_{i}\bbeta ))}\right\}$

ZINB Model with Standard Normal Link Function

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

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

The gradient for this model is given by

$\frac{\partial \mathcal{L}}{\partial \bgamma } = \sum _{\{ i: y_{i}=0\} } \left[\frac{\varphi (\mathbf{z}_{i}\bgamma ) \left[1-(1+\alpha \exp (\mathbf{x}_{i}\bbeta ))^{-\alpha ^{-1}} \right]}{ \Phi (\mathbf{z}_{i}\bgamma ) + \left[1- \Phi (\mathbf{z}_{i}\bgamma )\right] (1+\alpha \exp (\mathbf{x}_{i}\bbeta ))^{-\alpha ^{-1}}} \right] \mathbf{z}_{i}$

$- \sum _{\{ i: y_{i}>0\} } \left[\frac{\varphi (\mathbf{z}_{i}\bgamma )}{1 - \Phi (\mathbf{z}_{i}\bgamma )} \right] \mathbf{z}_{i}$

$\frac{\partial \mathcal{L}}{\partial \bbeta } = \sum _{\{ i: y_{i}=0\} } \frac{-\left[1-\Phi (\mathbf{z}_{i}\bgamma )\right] \exp (\mathbf{x}_{i}\bbeta ) (1+\alpha \exp (\mathbf{x}_{i}\bbeta ))^{-(1+\alpha )/\alpha }}{\Phi (\mathbf{z}_{i}\bgamma ) + \left[ 1 - \Phi (\mathbf{z}_{i}\bgamma ) \right] (1+\alpha \exp (\mathbf{x}_{i}\bbeta ))^{-\alpha ^{-1}} } \mathbf{x}_{i}$

$+ \sum _{\{ i: y_{i}>0\} } \left[ \frac{y_{i} - \exp (\mathbf{x}_{i}\bbeta )}{1 + \alpha \exp (\mathbf{x}_{i}\bbeta )} \right] \mathbf{x}_{i}$

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

The HPCOUNTREG Procedure