Negative Binomial Regression

where m > 0 is the mean of Y and a > 0 is the heterogeneity parameter. Hilbe [1] derives this parametrization as a Poisson-gamma mixture, or alternatively as the number of failures before the H1 e aLth success, though we will not require 1 e a to be an integer. The traditional negative binomial regression model, designated the NB2 model in [1], is (2) ln m = b0 + b1 x1 + b2 x2 +o⋯+ bp xp, where the predictor variables x1, x2, ..., xp are given, and the population regression coefficients b0, b1, b2, ..., bp are to be estimated. Given a random sample of n subjects, we observe for subject i the dependent variable yi and the predictor variables x1i, x2i, ..., xpi. Utilizing vector and matrix notation, we let b = H b0 b1 b2 o⋯ bp L¬, and we gather the predictor data into the design matrix X as follows: