Count-Data Regression

Definition

Count-data regression models a non-negative integer response $y\in\{0,1,2,\dots\}$ (event counts) as a function of covariates. The baseline is Poisson regression, a generalized linear model (GLM) with a log-linear mean $\log\mu = x^\top\beta$ and the Poisson variance restriction $\operatorname{Var}(y)=\mu$ . Because real count data — especially in economics and the social sciences — routinely violate that restriction through over-dispersion ( $\operatorname{Var}(y)>\mu$ ) and an excess of zeros, a ladder of richer models relaxes it: quasi-Poisson, negative binomial, and the zero-augmented hurdle and zero-inflated models.

Key Ideas

Two failures of plain Poisson. (1) Over-dispersion: the variance exceeds the mean, so Poisson standard errors are too small. (2) Excess zeros: far more zero counts than any Poisson/NB fit implies.
A shared mean, different likelihoods. The models below all keep the log-linear mean $\log\mu=x^\top\beta$ ; they differ only in how they model dispersion and the probability of a zero.
Two orthogonal fixes. Over-dispersion is handled by dispersion adjustment or an extra shape parameter (quasi-Poisson, NB); excess zeros are handled by a second model component (hurdle or zero-inflation). A model can need both.
Hurdle ≠ zero-inflation. Both add a component for zeros but tell different stories: hurdle treats all zeros as a single "didn't cross the hurdle" event; zero-inflation splits zeros into structural (always zero) and sampling (a count that happened to be zero) types.

How It Works

The over-dispersion ladder.

Poisson GLM — $y\mid x\sim\text{Poisson}(\mu)$ , $\log\mu=x^\top\beta$ , estimated by ML. Assumes $\operatorname{Var}=\mu$ .
Quasi-Poisson — same mean, a free dispersion parameter $\phi$ with $\operatorname{Var}=\phi\mu$ , estimated by quasi-ML / GEE; "adjusted Poisson" instead keeps Poisson estimates but uses sandwich standard errors.
Negative binomial (NB) — an extended GLM with an extra shape parameter $\theta$ giving $\operatorname{Var}=\mu+\mu^2/\theta$ (a gamma-mixed Poisson), estimated by ML; captures over-dispersion within a full likelihood.

The excess-zeros extensions (zero-augmented models).

Hurdle model (Mullahy 1986) — a two-part model: a binary component for zero-vs-positive (the "hurdle", a right-censored count/binary process) and a zero-truncated count model (Poisson or NB) for the positive counts. The two parts may use different covariates and are estimated separately.
Zero-inflated model (Lambert 1992) — a mixture of a point mass at zero (with probability $\pi(z)$ , typically a logit in covariates $z$ ) and a count distribution (Poisson → ZIP, NB → ZINB) that itself can produce zeros. So $P(y=0)=\pi+(1-\pi)\,f(0)$ .
Sample selection for counts (Greene 1994) — the count is observed only for a selected subsample, and if the selection error correlates with the count's unobservables, naive estimates are biased. Greene ports the Heckman selection correction from continuous outcomes to Poisson/NB counts (a probit selection equation jointly modelled with the count), extending the selection model to count data.

Why It Matters

Ubiquity. Counts (doctor visits, patents, accidents, roll-call events, defaults) are everywhere in applied micro-econometrics and the social sciences, and the Poisson restriction almost never holds.
Substantive zero mechanisms. Choosing hurdle vs. zero-inflation is a modeling statement about why zeros occur (a single participation gate vs. a mix of never-users and occasional-zero users), not just a goodness-of-fit tweak.
Standard tooling. Zeileis-Kleiber-Jackman (2008) gave these models a base-glm()-compatible implementation (pscl::hurdle, zeroinfl), making them routine in applied work; Cameron–Trivedi (1998) is the econometrics reference.

Open Questions