Zeileis-Kleiber-Jackman (2008) Regression Models for Count Data in R

Summary

This Journal of Statistical Software article reviews the standard regression models for count data and introduces a reference R implementation of the zero-augmented models. The classical Poisson, quasi-Poisson, geometric, and negative-binomial regressions are generalized linear models (GLMs) already available in base R; the paper adds hurdle and zero-inflated models — via the hurdle() and zeroinfl() functions in the pscl package — that reuse the design and methods of the base GLM functions. These zero-augmented models handle the two problems endemic to count data in economics and the social sciences — over-dispersion and an excess of zeros — better than their classical counterparts. The methods are illustrated end-to-end (fitting, inspection, testing) on cross-section data for the demand for medical care. (JSS 27(8): 1–25.)

Key Claims

• Count data break plain Poisson two ways. Empirical counts typically show over-dispersion (variance > mean, violating Poisson) and/or excess zeros (more zeros than any Poisson/NB predicts). The classical Poisson GLM handles neither well.
• The classical GLM ladder. All share the log-linear mean $\log\mu=x^\top\beta$ but differ in the rest of the likelihood: Poisson (ML); quasi-Poisson (same mean, dispersion estimated — equivalently a GEE / quasi-ML fit); adjusted Poisson (inference via sandwich covariances); negative binomial (NB) (an extra shape parameter that lets the variance exceed the mean). NB and quasi-Poisson address over-dispersion but often not excess zeros.
• Hurdle models (Mullahy 1986). A two-part model: a binary "hurdle" component decides zero vs. positive, and a left-truncated (at zero) count model governs the positive counts, with the hurdle itself a right-censored count/binary process. The two parts are estimated separately and can have different covariates.
• Zero-inflated models (Lambert 1992). A mixture of a count component and a point mass at zero: with some probability an observation is a "structural zero," otherwise it is drawn from a Poisson/NB (which may itself yield a "sampling zero"). Gives ZIP (Poisson) and ZINB (negative binomial) variants; the zero-inflation probability is modeled by its own (logit) regression.
• Hurdle vs. zero-inflated — different zero mechanisms. Both add a second component for zeros, but hurdle treats all zeros as coming from one process (never crossing the hurdle) while zero-inflation splits zeros into structural vs. count zeros — a substantively different story about how zeros arise.
• Software design. hurdle() and zeroinfl() in pscl mirror the interface and S3 methods of base glm()/glm.nb() (summary, predict, coef, logLik, AIC, vcov, sandwich covariances), so the zero-augmented models slot into the existing R modeling toolbox; the econometric reference for the models is Cameron–Trivedi (1998, 2005).

Concepts Introduced or Extended

• Count-Data Regression
• Generalized Linear Mixed Model
• Mixture Model

Entities Mentioned

• Achim Zeileis
• Christian Kleiber
• Simon Jackman

Quotes

"Empirical count data sets typically exhibit over-dispersion and/or an excess number of zeros. … Hurdle models combine a left-truncated count component with a right-censored hurdle component. Zero-inflation models … are mixture models that combine a count component and a point mass at zero."

"It re-uses design and functionality of the basic R functions just as the underlying conceptual tools extend the classical models."

My Take

As a software artifact this paper's value is that it made hurdle and zero-inflated regression routine and inspectable in R with a base-glm()-compatible interface, which is why pscl::hurdle/zeroinfl became the default way applied researchers fit these models. Its conceptual contribution to this wiki is the clean taxonomy it lays out: the Poisson → quasi-Poisson/sandwich → negative-binomial ladder for over-dispersion, and the orthogonal hurdle vs. zero-inflated split for excess zeros — with the genuinely useful reminder that hurdle and zero-inflation encode different stories about where zeros come from (one gate vs. a mixture of structural and sampling zeros), so the choice is substantive, not just a fit contest. It is the count-data companion to the wiki's other limited-dependent-variable material (Tobit, discrete choice) and, like the GEE work, sits at the GLM/quasi-likelihood interface.