CNTSELECT Procedure

Zero-Inflated Negative Binomial Regression

The zero-inflated negative binomial (ZINB) model in PROC CNTSELECT is based on the negative binomial model that has a quadratic variance function (when DIST=NEGBIN in the MODEL or PROC CNTSELECT statement). The ZINB model is obtained by specifying a negative binomial distribution for the data generation process referred to earlier as Process 2:

g left-parenthesis y Subscript i Baseline right-parenthesis equals StartFraction normal upper Gamma left-parenthesis y Subscript i Baseline plus alpha Superscript negative 1 Baseline right-parenthesis Over y Subscript i Baseline factorial normal upper Gamma left-parenthesis alpha Superscript negative 1 Baseline right-parenthesis EndFraction left-parenthesis StartFraction alpha Superscript negative 1 Baseline Over alpha Superscript negative 1 Baseline plus mu Subscript i Baseline EndFraction right-parenthesis Superscript alpha Super Superscript negative 1 Baseline left-parenthesis StartFraction mu Subscript i Baseline Over alpha Superscript negative 1 Baseline plus mu Subscript i Baseline EndFraction right-parenthesis Superscript y Super Subscript i

Thus the ZINB model is defined to be

StartLayout 1st Row 1st Column upper P left-parenthesis y Subscript i Baseline equals 0 vertical-bar bold x Subscript i Baseline comma bold z Subscript i Baseline right-parenthesis 2nd Column equals 3rd Column upper F Subscript i Baseline plus left-parenthesis 1 minus upper F Subscript i Baseline right-parenthesis left-parenthesis 1 plus alpha mu Subscript i Baseline right-parenthesis Superscript minus alpha Super Superscript negative 1 2nd Row 1st Column upper P left-parenthesis y Subscript i Baseline vertical-bar bold x Subscript i Baseline comma bold z Subscript i Baseline right-parenthesis 2nd Column equals 3rd Column left-parenthesis 1 minus upper F Subscript i Baseline right-parenthesis StartFraction normal upper Gamma left-parenthesis y Subscript i Baseline plus alpha Superscript negative 1 Baseline right-parenthesis Over y Subscript i Baseline factorial normal upper Gamma left-parenthesis alpha Superscript negative 1 Baseline right-parenthesis EndFraction left-parenthesis StartFraction alpha Superscript negative 1 Baseline Over alpha Superscript negative 1 Baseline plus mu Subscript i Baseline EndFraction right-parenthesis Superscript alpha Super Superscript negative 1 3rd Row 1st Column Blank 2nd Column times 3rd Column left-parenthesis StartFraction mu Subscript i Baseline Over alpha Superscript negative 1 Baseline plus mu Subscript i Baseline EndFraction right-parenthesis Superscript y Super Subscript i Superscript Baseline comma y Subscript i Baseline greater-than 0 EndLayout

In this case, the conditional expectation () and conditional variance (V) of are

upper E left-parenthesis y Subscript i Baseline vertical-bar bold x Subscript i Baseline comma bold z Subscript i Baseline right-parenthesis equals mu Subscript i Baseline left-parenthesis 1 minus upper F Subscript i Baseline right-parenthesis

upper V left-parenthesis y Subscript i Baseline vertical-bar bold x Subscript i Baseline comma bold z Subscript i Baseline right-parenthesis equals upper E left-parenthesis y Subscript i Baseline vertical-bar bold x Subscript i Baseline comma bold z Subscript i Baseline right-parenthesis left-bracket 1 plus mu Subscript i Baseline left-parenthesis upper F Subscript i Baseline plus alpha right-parenthesis right-bracket

Like the ZIP model, the ZINB model exhibits overdispersion because the conditional variance exceeds the conditional mean.

ZINB Model with Logistic Link Function

In this model, the probability is given by the logistic function, namely

phi Subscript i Baseline equals StartFraction exp left-parenthesis bold z prime Subscript i Baseline bold-italic gamma right-parenthesis Over 1 plus exp left-parenthesis bold z prime Subscript i Baseline bold-italic gamma right-parenthesis EndFraction

The log-likelihood function is

StartLayout 1st Row 1st Column script upper L 2nd Column equals 3rd Column sigma-summation Underscript StartSet i colon y Subscript i Baseline equals 0 EndSet Endscripts ln left-bracket exp left-parenthesis bold z prime Subscript i Baseline bold-italic gamma right-parenthesis plus left-parenthesis 1 plus alpha exp left-parenthesis bold x prime Subscript i Baseline bold-italic beta right-parenthesis right-parenthesis Superscript minus alpha Super Superscript negative 1 Superscript Baseline right-bracket 2nd Row 1st Column Blank 2nd Column plus 3rd Column sigma-summation Underscript StartSet i colon y Subscript i Baseline greater-than 0 EndSet Endscripts sigma-summation Underscript j equals 0 Overscript y Subscript i Baseline minus 1 Endscripts ln left-parenthesis j plus alpha Superscript negative 1 Baseline right-parenthesis 3rd Row 1st Column Blank 2nd Column plus 3rd Column sigma-summation Underscript StartSet i colon y Subscript i Baseline greater-than 0 EndSet Endscripts StartSet minus ln left-parenthesis y Subscript i Baseline factorial right-parenthesis minus left-parenthesis y Subscript i Baseline plus alpha Superscript negative 1 Baseline right-parenthesis ln left-parenthesis 1 plus alpha exp left-parenthesis bold x prime Subscript i Baseline bold-italic beta right-parenthesis right-parenthesis plus y Subscript i Baseline ln left-parenthesis alpha right-parenthesis plus y Subscript i Baseline bold x prime Subscript i Baseline bold-italic beta EndSet 4th Row 1st Column Blank 2nd Column minus 3rd Column sigma-summation Underscript i equals 1 Overscript upper N Endscripts ln left-bracket 1 plus exp left-parenthesis bold z prime Subscript i Baseline bold-italic gamma right-parenthesis right-bracket EndLayout

ZINB Model with Standard Normal Link Function

For this model, the probability is expressed by the standard normal distribution function (probit function): . The log-likelihood function is

For more information about the zero-inflated negative binomial regression model, see SAS/ETS User's Guide.

Last updated: July 09, 2026