In this chapter we describe how count response data can be modeled using the NB2 negative binomial regression. NB2 is the traditional parameterization of the negative binomial model, and is the one with which most statisticians are familiar. For this chapter, then, any reference to negative binomial regression will be to the NB2 model unless otherwise indicated.

Poisson versus negative binomial

We have earlier stated that, given the direct relationship in the negative binomial variance between α and the fitted value, μ, the model becomes Poisson as the value of α approaches zero. A negative binomial with α = 0 will not converge because of division by zero, but values close to zero allow convergence. Where α is close to zero, the model statistics displayed in Poisson output are nearly the same as those of a negative binomial.

The relationship can be observed using simulated data. The code below constructs a 50,000 observation synthetic Poisson model with an intercept value of 2 and parameter values of x1 = 0.75 and x2 = 1.25. Each predictor is generated as a synthetic random normal variate. The Poisson data are then modeled using a negative binomial where the value of α is estimated prior to the calculation of parameter estimates and associated statistics.