# The COUNTREG Procedure

### Zero-Inflated Conway-Maxwell-Poisson Regression

In the Conway-Maxwell-Poisson regression model, the data generation process is defined as

where the normalization factor is

and

The zero-inflated Conway-Maxwell-Poisson model can be written as

The conditional expectation and conditional variance of are given by

General form of the log-likelihood function for the Conway-Maxwell-Poisson zero-inflated model is

#### Zero-Inflated Conway-Maxwell-Poisson Model with Logistic Link Function

In this model the probability is expressed with a logistic link function as

The log-likelihood function is

#### Zero-Inflated Conway-Maxwell-Poisson Model with Normal Link Function

For this model, the probability is specified with the standard normal distribution function (probit function): .

The log-likelihood function is written as