Overdispersion in Count Models¶

Definition¶

A conditional count model is overdispersed relative to Poisson if $\operatorname{Var}(Y\mid x)>\mathbb{E}[Y\mid x]$$x]$. A quasi-Poisson specification uses mean $\mu$ and variance $\phi\mu$. A common negative binomial specification has mean $\mu$ and variance $\mu+\mu^2/r$${\mu }^2/r$, with size parameter $r>0$$0$.

Interpretation¶

Claims from apparently similar policyholders can still have different underlying rates. A gamma mixture of Poisson rates gives a negative binomial response and an extra variance term.

The figure compares the variance functions of the example below. The grey Poisson line has variance equal to the mean. Quasi-Poisson with $\phi=1.5$ scales it to $1.5\mu$, while the negative binomial with $r=4$ adds the quadratic term $\mu^2/4$. Both lie above the Poisson line, so both are overdispersed, and both give variance 3 at $\mu=2$$2$. Away from $\mu=2$ they differ: the negative binomial excess grows faster for larger means.

Poisson regression assumes equality of conditional mean and variance at fixed predictors. A mixture distribution over unobserved claim rates adds variation between those rates, explaining why a Poisson mixture can have a larger conditional variance.

Key Results¶

• Quasi-Poisson has overdispersion when $\phi>1$ and underdispersion when $0<\phi<1$$1$.
• Quasi-Poisson is a mean–variance specification, not generally the literal count distribution of $\phi\operatorname{Pois}(\mu/\phi)$.
• In the negative binomial model, larger $r$ reduces excess variance; the Poisson limit is $r\to\infty$ at fixed $\mu$.
• Compare dispersion conditional on predictors and exposure, rather than raw pooled mean and variance alone.

Examples¶

For $\mu=2$ and $r=4$$4$, negative binomial conditional variance is $2+4/4=3$$3$. A quasi-Poisson fit with $\phi=1.5$ has the same mean and variance but does not specify the same full law.

Prerequisites¶

• Poisson Regression
• Negative Binomial Distribution
• Mixture Distribution

Related Concepts¶

• Generalized Linear Model
• Information Criteria for Risk Models
• Exposure Offset in Claim-Frequency Models

Sources¶

• Risk Insurance · Lecture 6

Aliases: overdispersion, over-dispersion, extra-Poisson variation, extra-Poisson variability Keywords: quasi-Poisson model, negative binomial regression, Var(Y) > E[Y], dispersion parameter φ, gamma mixture of Poisson, underdispersion Symbols: infinity, mu, phi

Last updated: 17 October 2026 21:53