Zero-Inflated Poisson and Zero-Inflated Negative Binomial Models Using the COUNTREG Procedure

Real-life count data are frequently characterized by overdispersion and excess zeros. Zero-inflated count models provide a parsimonious yet powerful way to model this type of situation. Such models assume that the data are a mixture of two separate data generation processes: one generates only zeros, and the other is either a Poisson or a negative binomial data-generating process. The result of a Bernoulli trial is used to determine which of the two processes generates an observation. OVERVIEW The COUNTREG (count regression) procedure analyzes regression models in which the dependent variable takes nonnegative integer or count values. The dependent variable is usually the number of times an event occurs. Some examples of event counts are: number of claims per year on a particular car owner’s auto insurance policy number of workdays missed due to sickness of a dependent in a 4-week period number of papers published per year by a researcher In count regression, the conditional mean E.yi jxi / of the dependent variable, yi , is assumed to be a function of a vector of covariates, xi . Possible covariates for the auto insurance example are: age of the driver type of car daily commuting distance MARGINAL EFFECTS IN COUNT REGRESSION Marginal effects provide a way to measure the effect of each covariate on the dependent variable. The marginal effect of one covariate is the expected instantaneous rate of change in the dependent variable as a function of the change in that covariate, while keeping all other covariates constant. Unlike in linear models, the derivative of the conditional expectation with respect to xi;j is no longer equal to ˇj—that is, @E.yi jxi /=@xi;j ¤ ˇj . For example, for the Poisson regression with E.yi jxi / D ex 0 i ˇ is @E.yi jxi / @xi;j D ˇj e x i ˇ D ˇjE.yi jxi / (1) Therefore the marginal effect of the change in covariate xi;j depends not only on ˇj , but also on all other estimated coefficients, and on all other covariate values. Another interpretation is that a one-unit change in the j th covariate leads to a proportional change in the conditional mean E.yi jxi / of ˇj . BASIC MODELS: POISSON AND NEGATIVE BINOMIAL REGRESSION MODELS The Poisson (log-linear) regression model is the most basic model that explicitly takes into account the nonnegative integer-valued aspect of the dependent count variable. In this model, the probability of an event count yi , given the vector of covariates xi , is given by the Poisson distribution: P.Yi D yi jxi / D e i yi i yi Š ; yi D 0; 1; 2; : : :