Two general methods are used to estimate count response models: (1) an iteratively re-weighted least squares (IRLS) algorithm based on the method of Fisher scoring, and (2) a full maximum likelihood Newton–Raphson type algorithm. Although the maximum likelihood approach was first used with both the Poisson and negative binomial, we shall discuss it following our examination of IRLS. We do this for strictly pedagogical purposes, which will become evident as we progress.

It should be noted at the outset that IRLS is a type or subset of maximum likelihood which can be used for estimation of generalized linear models (GLM). Maximum likelihood methods in general estimate model parameters by solving the derivative of the model log-likelihood function, termed the gradient, when set to zero. The derivative of the gradient with respect to the parameters is called the Hessian matrix, upon which model standard errors are based. Owing to the unique distributional structure inherent to members of GLM, estimation of model parameters and standard errors can be achieved using IRLS, which in general is a computationally simplier method of maximum likelihood estimation. Both methods are derived, described, and related in this chapter.

Derivation of the IRLS algorithm

The traditional generalized linear models (GLM) algorithm, from the time it was implemented in GLIM (generalized linear interactive modeling) through its current implementations in Stata, R, SAS, SPSS, GenStat, and other statistical software, uses some version of an IRLS estimating algorithm.