Introduction

Modeling count variables is a common task in economics and the social sciences. The classical Poisson regression model for count data is often of limited use in these disciplines because empirical count data sets typically exhibit over-dispersion and/or an excess number of zeros. The former issue can be addressed by extending the plain Poisson regression model in various directions: e.g., using sandwich covariances or estimating an additional dispersion parameter (in a so-called quasi-Poisson model). Another more formal way is to use a negative binomial (NB) regression. All of these models belong to the family of generalized linear models (GLMs, see Nelder and Wedderburn 1972; McCullagh and Nelder 1989). However, although these models typically can capture over-dispersion rather well, they are in many applications not sufficient for modeling excess zeros. Since Mullahy (1986) and Lambert (1992) there is increased interest, both in the econometrics and statistics literature, in zero-augmented models that address this issue by a second model component capturing zero counts. Hurdle models (Mullahy 1986) combine a left-truncated count component with a right-censored hurdle component. Zero-inflation models (Lambert 1992) take a somewhat different approach: they are mixture models that combine a count component and a point mass at zero. An overview of count data models in econometrics, including hurdle and zero-inflated models, is provided in Cameron and Trivedi (1998),countreg:Cameron+Trivedi:2005.

In R (R Development Core Team 2008), GLMs are provided by the model fitting functions glm() (Chambers and Hastie 1992) in the stats package and glm.nb() in the MASS package (Venables and Ripley 2002) along with associated methods for diagnostics and inference. Here, we discuss the implementation of hurdle and zero-inflated models in the functions hurdle() and zeroinfl() in the countreg package (Zeileis and Kleiber 2013), available from the Comprehensive R Archive Network (CRAN) at http://CRAN.R-project.org/package=countreg. The design of both modeling functions as well as the methods operating on the associated fitted model objects follows that of the base R functionality so that the new software integrates easily into the computational toolbox for modeling count data in R.

The remainder of this article provides an overview of the supported count data models in countreg. More detailed discussions of the classical count data models and their zero-augmented counterparts can be found in “Basics” and “Zero-augmented models”, respectively. Both links contain applications of the respective models to a microeconomic cross-section data set on the demand for medical care.

Models and software

Table 1: Overview of discussed count regression models. All GLMs use the same log-linear mean function (\(\log(\mu) = x^\top \beta\)) but make different assumptions about the remaining likelihood. The zero-augmented models extend the mean function by modifying (typically, increasing) the likelihood of zero counts.

Type	Distribution	Method	Description
GLM	Poisson	ML	Poisson regression: classical GLM, estimated by maximum likelihood (ML)
		quasi	"quasi-Poisson regression": same mean function, estimated by quasi-ML (QML) or equivalently generalized estimating equations (GEE), inference adjustment via estimated dispersion parameter
		adjusted	"adjusted Poisson regression": same mean function, estimated by QML/GEE, inference adjustment via sandwich covariances
	NB	ML	NB regression: extended GLM, estimated by ML including additional shape parameter
zero-augmented	Poisson	ML	zero-inflated Poisson (ZIP), hurdle Poisson
	NB	ML	zero-inflated NB (ZINB), hurdle NB

In this section, we briefly outline the theory and its implementation in R (R Development Core Team 2008) for some basic count data regression models as well as their zero-augmented extensions (see Table 1 for an overview). The classical Poisson, geometric and negative binomial models are described in a generalized linear model (GLM) framework; they are implemented in R by the glm() function (Chambers and Hastie 1992) in the stats package and the glm.nb() function in the MASS package (Venables and Ripley 2002). The hurdle and zero-inflated extensions of these models are provided by the functions hurdle() and zeroinfl() in package countreg (Zeileis and Kleiber 2013). The original implementation of Jackman (2008) was improved by Kleiber and Zeileis (2008) for countreg to make the fitting functions and the fitted model objects more similar to their glm() and glm.nb() counterparts. The most important features of the new hurdle() and zeroinfl() functions are discussed below while some technical aspects are deferred to the appendix.

An alternative implementation of zero-inflated count models is available in the currently orphaned package zicounts (Mwalili 2007). Another extension of zero-inflated Poisson models is available in package ZIGP (Erhardt 2008) which allows dispersion—in addition to mean and zero-inflation level—to depend on regressors. However, the interfaces of both packages are less standard with fewer (or no) standard methods provided. Therefore, re-using generic inference tools is more cumbersome and hence these packages are not discussed here.

Two packages that embed zero-inflated models into more general implementations of GLMs and GAMs (generalized additive models) are gamlss (Stasinopoulos and Rigby 2007) and VGAM (Yee 2008). The latter also provides hurdle models (under the name zero-altered models). Both implementations allow specification of only one set of regressors.

In addition to zero-augmented models, there are many further extensions to the classical Poisson model which are not discussed here. Some important model classes include finite mixture models—implemented in R in package flexmix (Leisch 2004)—and generalized estimating equations (GEE)—provided in R by package geepack (Halekoh, Højsgaard, and Yan 2006)—and mixed-effects models—available in R in packages lme4 and nlme (see Pinheiro and Bates 2000). Further information about the models and alternative R implementations can be found in the respective references.

References

Cameron, A. Colin, and Pravin K. Trivedi. 1998. Regression Analysis of Count Data. Cambridge: Cambridge University Press.

Chambers, John M., and Trevor J. Hastie, eds. 1992. Statistical Models in S. London: Chapman & Hall.

Erhardt, Vinzenz. 2008. ZIGP: Zero-Inflated Generalized Poisson Regression Models. http://CRAN.R-project.org/package=ZIGP.

Halekoh, Ulrich, Søren Højsgaard, and Jun Yan. 2006. “The R Package geepack for Generalized Estimating Equations.” Journal of Statistical Software 15 (2): 1–11. http://www.jstatsoft.org/v15/i02/.

Jackman, Simon. 2008. pscl: Classes and Methods for R Developed in the Political Science Computational Laboratory, Stanford University. Stanford, California: Department of Political Science, Stanford University. http://CRAN.R-project.org/package=pscl.

Kleiber, Christian, and Achim Zeileis. 2008. Applied Econometrics with R. New York: Springer-Verlag.

Lambert, Diane. 1992. “Zero-Inflated Poisson Regression, with an Application to Defects in Manufacturing.” Technometrics 34: 1–14.

Leisch, Friedrich. 2004. “FlexMix: A General Framework for Finite Mixture Models and Latent Class Regression in R.” Journal of Statistical Software 11 (8): 1–18. http://www.jstatsoft.org/v11/i08/.

McCullagh, P., and John A. Nelder. 1989. Generalized Linear Models. 2nd ed. London: Chapman & Hall.

Mullahy, J. 1986. “Specification and Testing of Some Modified Count Data Models.” Journal of Econometrics 33: 341–65.

Mwalili, Samuel M. 2007. zicounts: Classical and Censored Zero-Inflated Count Data Models. http://CRAN.R-project.org/src/contrib/Archive/zicounts/.

Nelder, J. A., and R. W. M. Wedderburn. 1972. “Generalized Linear Models.” Journal of the Royal Statistical Society A 135: 370–84.

Pinheiro, Jose C., and Douglas M. Bates. 2000. "Mixed-Effects Models in S and S-PLUS. New York: Springer-Verlag.

R Development Core Team. 2008. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. http://www.R-project.org/.

Stasinopoulos, D. Mikis, and Robert A. Rigby. 2007. “Generalized Additive Models for Location Scale and Shape (GAMLSS) in R.” Journal of Statistical Software 23 (7). http://www.jstatsoft.org/v23/i07/.

Venables, William N., and Brian D. Ripley. 2002. Modern Applied Statistics with S. 4th ed. New York: Springer-Verlag.

Yee, Thomas W. 2008. VGAM: Vector Generalized Linear and Additive Models. http://CRAN.R-project.org/package=VGAM.

Zeileis, Achim, and Christian Kleiber. 2013. countreg: Count Data Regression. http://R-Forge.R-project.org/projects/countreg/.