Categorical › Logistic regression

Negative binomial regression (overdispersed counts)

Negative binomial regression models overdispersed counts, fitting the log-mean and the dispersion parameter θ jointly.

What is Negative binomial regression (overdispersed counts)?

NB extends Poisson by adding a dispersion parameter θ that captures the extra variance: Var(Y) = μ + μ² / θ. As θ → ∞ the NB collapses to Poisson; small θ ⇒ heavy overdispersion.

When Poisson's residual deviance / df is > 1.5 the SEs and p-values are too small — false-positive risk. NB fixes that by inflating the variance to match reality. Coefficient estimates are similar to Poisson; SEs typically larger.

When should I use Negative binomial regression (overdispersed counts)?

Count outcomes where Poisson is overdispersed (residual dev / df > 1.5).
Counts with substantial unmeasured heterogeneity.

What data does it need?

Integer-valued response + numeric predictors + intercept toggle.

What does it report?

Same as Poisson + dispersion θ + SE(θ), plus a likelihood ratio test of the fit against Poisson.

What does it assume?

Independent observations.
Negative binomial conditional distribution.
Log-linear relationship.

How do I interpret the result?

θ < 5 is heavy overdispersion. Report θ alongside the coefficients.

Read the likelihood ratio test rather than θ on its own when deciding whether the dispersion parameter is needed. θ is unbounded above and its standard error grows with it, so a large θ can mean either 'no overdispersion' or 'θ is barely estimated' — θ = 85 ± 81 and θ = 85 ± 3 look identical otherwise. Because Poisson sits on the boundary of the negative binomial family (α = 1/θ = 0), the test halves the usual chi-squared p-value.