The HPCOUNTREG Procedure
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Overview
- Getting Started
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Syntax
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DetailsMissing ValuesPoisson RegressionConway-Maxwell-Poisson RegressionNegative Binomial RegressionZero-Inflated Count Regression OverviewZero-Inflated Poisson RegressionZero-Inflated Conway-Maxwell-Poisson RegressionZero-Inflated Negative Binomial RegressionParameter Naming Conventions for the RESTRICT, TEST, BOUNDS, and INIT StatementsComputational ResourcesCovariance Matrix TypesDisplayed OutputOUTPUT OUT= Data SetOUTEST= Data SetODS Table Names
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Examples
- References
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
![\[ P(Y_{i}=y_{i}|\mathbf{x}_{i}) = \frac{e^{-\mu _{i}}\mu _{i}^{y_{i}}}{y_{i}!}, \quad y_ i = 0,1,2,\ldots \]](images/etshpug_hpcountreg0052.png)
and the mean parameter—that is, the mean number of events per period—is given by
![\[ \mu _{i} = \exp (\mathbf{x}_{i}^{\prime } \bbeta ) \]](images/etshpug_hpcountreg0053.png)
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
![\[ E(y_{i}|\mathbf{x}_{i}) = \mu _{i} = \exp (\mathbf{x}_{i}^{\prime } \bbeta ) \]](images/etshpug_hpcountreg0059.png)
Note that the conditional variance of the count random variable is equal to the conditional mean in the Poisson regression model:
![\[ V(y_{i}|\mathbf{x}_{i}) = E(y_{i}|\mathbf{x}_{i}) = \mu _{i} \]](images/etshpug_hpcountreg0060.png)
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
![\[ \mathcal{L} = \sum _{i=1}^{N}(-\mu _{i} + y_{i} \ln \mu _{i} - \ln y_{i}!) = \sum _{i=1}^{N}(-e^{\mathbf{x}_{i}^{\prime } \bbeta } + y_{i}\mathbf{x}_{i}^{\prime } \bbeta - \ln y_{i}!) \]](images/etshpug_hpcountreg0061.png)
For more information about the Poisson regression model, see the section Poisson Regression in SAS/ETS 14.1 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.