# Negative Binomial Regression

This is a regression method that is preferable in certain situations to Poisson Regression.

Typically, Poisson regression is used to model counts of various kinds. However, Poisson random variables are expected to have a mean equal to its variance, which is often not the case in real-world situations. An example of this is the maximum diffusion length of Facebook fan pages in Sun, E., I. Rosenn, C. A Marlow, and T. M Lento. Gesundheit! Modeling Contagion through Facebook News Feed. Proc. ICWSM 9.

Formally, a Poisson distribution can be characterized as:

${\displaystyle f(k,\lambda )={\frac {\lambda ^{k}e^{-\lambda }}{k!}},\,\!}$

Whereas the negative binomial distribution can be defined as:

${\displaystyle f(k)\equiv \Pr(X=k)={k+r-1 \choose k}(1-p)^{r}p^{k}\quad {\text{for }}k=0,1,2,\dots }$

Negative binomial regression is appropriate when variance >> mean. A variation of this method, called zero-inflation correction, allows one to control for excess zeros. Zero-inflation binomial regression allows one, in a single regression, to select variables that contribute to the true content in the response variable (“count model coefficients”) and also a (potentially different) set of variables that contribute to the excess zeros in the response (“zero-inflation model coefficients”).