The COUNTREG Procedure
- Overview
- Getting Started
-
Syntax
-
Details
Specification of Regressors Missing Values Poisson Regression Negative Binomial Regression Zero-Inflated Count Regression Overview Zero-Inflated Poisson Regression Zero-Inflated Negative Binomial Regression Computational Resources Nonlinear Optimization Options Covariance Matrix Types Displayed Output OUTPUT OUT= Data Set OUTEST= Data Set ODS Table Names
-
Examples
- References
| Poisson Regression |
The most widely used model for count data analysis is Poisson regression. This assumes that 
, given the vector of covariates 
, is independently Poisson-distributed with
 |
and the mean parameter (that is, the mean number of events per period) is given by
 |
where 
is a 
parameter vector. (The intercept is 
; the coefficients for the 
regressors are 
.) Taking the exponential of 
ensures that the mean parameter 
is nonnegative. It can be shown that the conditional mean is given by
 |
The name log-linear model is also used for the Poisson regression model since the logarithm of the conditional mean is linear in the parameters:
 |
Note that the conditional variance of the count random variable is equal to the conditional mean in the Poisson regression model:
 |
The equality of the conditional mean and variance of is known as equidispersion.
The marginal effect of a regressor is given by
 |
Thus, a one-unit change in the 
th regressor leads to a proportional change in the conditional mean 
of 
.
The standard estimator for the Poisson model is the maximum likelihood estimator (MLE). Since the observations are independent, the log-likelihood function is written as
 |
where 
is defined as follows:
- 1
if neither the WEIGHT nor the FREQ statement is used.
where
are the nonnormalized values of the variable specified in the WEIGHT statement in which the NONORMALIZE option is specified. 
where
are the nonnormalized values of the variable specified in the WEIGHT statement. 
where
are the values of the variable specified in the FREQ statement. 
if both the WEIGHT statement, without the NONORMALIZE option, and the FREQ statement are specified.
if both the FREQ and the WEIGHT statements are specified.
The gradient and the Hessian are, respectively,
 |
 |
The Poisson model has been criticized for its restrictive property that the conditional variance equals the conditional mean. Real-life data are often characterized by overdispersion (that is, the variance exceeds the mean). Allowing for overdispersion can improve model predictions since the Poisson restriction of equal mean and variance results in the underprediction of zeros when overdispersion exists. The most commonly used model that accounts for overdispersion is the negative binomial model.