Improved estimation in negative binomial regression

Negative binomial regression is commonly employed to analyze overdispersed count data. With small to moderate sample sizes, the maximum likelihood estimator of the dispersion parameter may be subject to a significant bias, that in turn affects inference on mean parameters. This article proposes inference for negative binomial regression based on adjustments of the score function aimed at mean or median bias reduction. The resulting estimating equations generalize those available for improved inference in generalized linear models and can be solved using a suitable extension of iterative weighted least squares. Simulation studies confirm the good properties of the new methods, which are also found to solve in many cases numerical problems of maximum likelihood estimation. The methods are illustrated and evaluated using two case studies: an Ames salmonella assay data set and data on epileptic seizures. Inference based on adjusted scores turns out to generally improve on maximum likelihood, and even on explicit bias correction, with median bias reduction being overall preferable.