  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 paper proposes
inference for negative binomial regression based on adjustments of the score
function aimed at mean and median bias reduction. The resulting estimating
equations are similar to those available for improved inference in generalized
linear models and, in particular, can be solved using a suitable extension of
iterative weighted least squares. Simulation studies show a remarkable
performance of the new methods, which are also found to solve in many cases
numerical problems of maximum likelihood estimates. 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 be
generally preferable to explicit bias correction.
