PROC COUNTREG: Zero-Inflated Poisson Regression

The COUNTREG Procedure

In the zero-inflated Poisson (ZIP) regression model, the data generation process referred to earlier as Process 2 is

$\text{[math]}$

where $\text{[math]}$ . Thus the ZIP model is defined as

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

The conditional expectation and conditional variance of $\text{[math]}$ are given by

$\text{[math]}$

Note that the ZIP model (as well as the ZINB model) exhibits overdispersion since $\text{[math]}$ .

In general, the log-likelihood function of the ZIP model is

$\text{[math]}$

Once a specific link function (either logistic or standard normal) for the probability $\text{[math]}$ is chosen, it is possible to write the exact expressions for the log-likelihood function and the gradient.

ZIP Model with Logistic Link Function

First, consider the ZIP model in which the probability $\text{[math]}$ is expressed with a logistic link function, namely

$\text{[math]}$

The log-likelihood function is

$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

The gradient for this model is given by

$\text{[math]}$

ZIP Model with Standard Normal Link Function

Next, consider the ZIP model in which the probability $\text{[math]}$ is expressed with a standard normal link function: $\text{[math]}$ . The log-likelihood function is

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

The gradient for this model is given by

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

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