The CNTSELECT Procedure
Zero-Inflated Poisson Regression
In the zero-inflated Poisson (ZIP) regression model, the data generation process that is referred to earlier as Process 2 is
where . Thus the ZIP model is defined as
The conditional expectation and conditional variance of are given by
Note that the ZIP model (in addition to the ZINB model) exhibits overdispersion because .
In general, the log-likelihood function of the ZIP model is
After a specific link function (either logistic or standard normal) for the probability 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 is expressed by a logistic link function, namely
The log-likelihood function is
ZIP Model with Standard Normal Link Function
Next, consider the ZIP model in which the probability is expressed by a standard normal link function: . The log-likelihood function is
For more information about the zero-inflated Poisson regression model, see SAS/ETS User's Guide.