The main motivation for zero-inflated count models is that real-life data frequently display overdispersion and excess zeros. Zero-inflated count models provide a way of modeling the excess zeros in addition to allowing for overdispersion. In particular, for each observation, there are two possible data generation processes. The result of a Bernoulli trial is used to determine which of the two processes is used. For observation i, Process 1 is chosen with probability and Process 2 with probability . Process 1 generates only zero counts. Process 2 generates counts from either a Poisson or a negative binomial model. In general,
Therefore, the probability of can be described as
where follows either the Poisson or the negative binomial distribution. You can specify the probability by using the PROBZERO= option in the OUTPUT statement.
When the probability depends on the characteristics of observation i, is written as a function of , where is the vector of zero-inflation covariates and is the vector of zero-inflation coefficients to be estimated. (The zero-inflation intercept is ; the coefficients for the q zero-inflation covariates are .) The function that relates the product (which is a scalar) to the probability is called the zero-inflation link function,
In the COUNTREG procedure, the zero-inflation covariates are indicated in the ZEROMODEL statement. Furthermore, the zero-inflation link function can be specified as either the logistic function,
or the standard normal cumulative distribution function (also called the probit function),
The zero-inflation link function is indicated in the LINK option in ZEROMODEL statement. The default ZI link function is the logistic function.