CNTSELECT Procedure

Poisson Regression

The most widely used model for count data analysis is Poisson regression. Poisson regression assumes that , given the vector of covariates , is independently Poisson distributed with

upper P left-parenthesis upper Y Subscript i Baseline equals y Subscript i Baseline vertical-bar bold x Subscript i Baseline right-parenthesis equals StartFraction e Superscript minus mu Super Subscript i Superscript Baseline mu Subscript i Superscript y Super Subscript i Superscript Baseline Over y Subscript i Baseline factorial EndFraction comma y Subscript i Baseline equals 0 comma 1 comma 2 comma ellipsis

and the mean parameter—that is, the mean number of events per period—is given by

mu Subscript i Baseline equals exp left-parenthesis bold x prime Subscript i Baseline bold-italic beta right-parenthesis

where is a parameter vector. (The intercept is ; the coefficients for the k regressors are .) Taking the exponential of ensures that the mean parameter is nonnegative. It can be shown that the conditional mean is given by

upper E left-parenthesis y Subscript i Baseline vertical-bar bold x Subscript i Baseline right-parenthesis equals mu Subscript i Baseline equals exp left-parenthesis bold x prime Subscript i Baseline bold-italic beta right-parenthesis

Note that the conditional variance of the count random variable is equal to the conditional mean in the Poisson regression model:

upper V left-parenthesis y Subscript i Baseline vertical-bar bold x Subscript i Baseline right-parenthesis equals upper E left-parenthesis y Subscript i Baseline vertical-bar bold x Subscript i Baseline right-parenthesis equals mu Subscript i

The equality of the conditional mean and variance of is known as equidispersion.

The standard estimator for the Poisson model is the maximum likelihood estimator (MLE). Because the observations are independent, the log-likelihood function is written as

script upper L equals sigma-summation Underscript i equals 1 Overscript upper N Endscripts left-parenthesis minus mu Subscript i Baseline plus y Subscript i Baseline ln mu Subscript i Baseline minus ln y Subscript i Baseline factorial right-parenthesis equals sigma-summation Underscript i equals 1 Overscript upper N Endscripts left-parenthesis minus e Superscript bold x prime Super Subscript i Superscript bold-italic beta Baseline plus y Subscript i Baseline bold x prime Subscript i Baseline bold-italic beta minus ln y Subscript i Baseline factorial right-parenthesis

For more information about the Poisson regression model, see SAS/ETS User's Guide.

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 because 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. Conway-Maxwell-Poisson regression enables you to model both overdispersion and underdispersion.

Last updated: July 09, 2026