Negative binomial distribution
Topic of Standard errors, and models for counts, part of Generalised linear models
Negative binomial distribution
| Type | Topic |
| Taught in | Standard errors, and models for counts |
| Part | Generalised linear models |
A count distribution with mean \mu and variance \mu+\mu^2/\nu, built as a gamma mixture of Poisson counts: draw a rate from a gamma law, then a Poisson count at that rate. It allows overdispersion, never underdispersion, and approaches the Poisson as \nu\to\infty.
In Standard errors, and models for counts
A quasi-Poisson variance is not, by itself, a distribution you can draw from. If you want a distribution, and you only need extra variation in the direction of overdispersion, the usual next model is the negative binomial. One construction is a gamma mixture of Poissons: draw a rate from a gamma distribution, then draw a Poisson count at that rate. The mean and variance are
\mathrm{E}(Y)=\mu,
\qquad
\mathrm{Var}(Y)=\mu+\frac{\mu^2}{\nu},
\qquad \nu>0.
This is quadratic in the mean, not a constant times the mean. It is always at least the Poisson variance. As \nu\to\infty the extra term vanishes and the Poisson returns. The parameter \nu is traditionally called a dispersion, but a larger \nu concentrates the counts, so it behaves as a concentration. The negative binomial is an exponential family in \mu only when \nu is held fixed. The fit in MASS::glm.nb is a profile: fix \nu, solve the GLM for the mean, move \nu, and repeat.
Used in
See also
Overdispersion · Underdispersion · Quasi-Poisson · Conway–Maxwell–Poisson distribution · Variance · Standard errors in a GLM · Binomial counts in a GLM · Count responses