Poisson and Negative Binomial Regression
Also known as: count regression, log-linear count model, negative binomial regression, Poisson / Negatif Binom Regresyon
Poisson regression is a generalized linear model for count outcomes — events tallied as non-negative integers such as hospital admissions, accidents, or article counts. It models the log of the expected count as a linear function of the predictors, and is developed in the standard count-data treatment of Cameron and Trivedi (1998); when the counts are over-dispersed, the closely related negative binomial model (Hilbe, 2011) is preferred.
Key highlights
- Purpose-built for count outcomes: the log link keeps predicted counts non-negative and gives multiplicatively interpretable effects.
- Coefficients exponentiate into incidence rate ratios (IRR), a directly interpretable change in the event rate per unit of a predictor.
- Extends naturally to the negative binomial model for over-dispersed counts without changing the log-linear structure.
Intuition
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How it works
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When to use it
Use Poisson regression when the outcome is a count of events (non-negative integers) with a sample of at least about 30 observations, and you want to relate the event rate to one or more predictors. The core Poisson assumption is equidispersion: the conditional variance equals the conditional mean. Check this first — if a dispersion test shows variance greater than mean, switch to negative binomial regression. Also screen for excess zeros (a zero share above roughly 30% suggests a zero-inflated model) and for multicollinearity (VIF below 10). It is well suited to cross-sectional, panel, and longitudinal count data.
Strengths & limitations
- Purpose-built for count outcomes: the log link keeps predicted counts non-negative and gives multiplicatively interpretable effects.
- Coefficients exponentiate into incidence rate ratios (IRR), a directly interpretable change in the event rate per unit of a predictor.
- Extends naturally to the negative binomial model for over-dispersed counts without changing the log-linear structure.
- The basic Poisson model assumes the variance equals the mean; real count data are often over-dispersed, which understates the standard errors.
- Excess zeros beyond what the Poisson distribution predicts require a zero-inflated (ZIP/ZINB) model instead.
- Like other regressions, it is distorted by strong multicollinearity among predictors and needs an adequate sample size.
Common pitfalls
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Applications
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Frequently asked
When should I use negative binomial instead of Poisson?
When the counts are over-dispersed — the conditional variance exceeds the mean. A dispersion test (variance/mean > 1) flags this. Plain Poisson then understates the standard errors; the negative binomial model adds a dispersion parameter and corrects the inference.
What is an incidence rate ratio (IRR)?
It is the exponentiated coefficient, exp(β). An IRR of 1.2 means a one-unit increase in the predictor multiplies the expected event rate by 1.2, i.e. a 20% increase. IRRs are the standard way to report Poisson and negative binomial effects.
What if my data have too many zeros?
If the zero share is very high (roughly above 30%), the Poisson and negative binomial models may underfit the zeros. A zero-inflated model (ZIP or ZINB) separates the process that generates structural zeros from the count process and is usually more appropriate.
How do I check whether the model fits?
Examine the Pearson and deviance residuals for goodness of fit, and use the Vuong test to compare the Poisson model against the negative binomial alternative. A poor fit or significant over-dispersion points to the negative binomial or a zero-inflated specification.
Sources
- 1.Cameron, A. C. & Trivedi, P. K. (1998). Regression Analysis of Count Data. Cambridge University Press.
- 2.Hilbe, J. M. (2011). Negative Binomial Regression (2nd ed.). Cambridge University Press.
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Cite this page
ScholarGate. (2026, June 1). Poisson Regression. ScholarGate. https://scholargate.app/econometrics/poisson-regression