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Negative binomial pdf1/12/2024 ![]() Therefore, due to its widespread use, it is of interest to provide the quantities required for mean and median bias reduction, together with an efficient implementation, and to assess whether the general theoretical properties of the methods produce appreciable improvements over standard maximum likelihood. Negative binomial regression does not fall into the generalized linear models class when the shape parameter is unknown, as is the case in practical applications. A major effort has been devoted to generalized linear modelsĪdditional effort was needed for other specific models, such as beta regression Obtaining the quantities required for mean and median bias reduction, as well as development of efficient software, is not always straightforward, but is necessary in order to make these methods available to practitioners. Unlike traditional bias correction, that subtracts an estimate of the bias from the maximum likelihood estimate, see for instance section 9.2 of Cox and Hinkley,īoth mean and median bias reduction methods do not rely on finiteness of the maximum likelihood estimate and have the advantage of solving practical issues related to boundary estimates that can occur with positive probability in models for discrete data. Mean bias reduction is invariant under linear transformation of the parameters, while median bias reduction is invariant under monotone component‐wise parameter transformations. While mean bias reduction yields an estimator with reduced bias, median bias reduction is such that each component of the estimator is, with high accuracy, median unbiased, that is, it has the same probability of underestimating and overestimating the corresponding parameter component. Resulting in mean or median bias reduction, respectively. General improved estimation methods based on adjustments of the likelihood equations have been proposed starting from the contributions of Firth With moderate sample sizes, the maximum likelihood estimator of the shape parameter may be subject to a substantial bias that can influence the inferential conclusions also about regression coefficients. Maximum likelihood has been studied starting from Fisherįor independent and identically distributed data and from Lawlessįor the regression setting. It is not uncommon that empirical counts display substantial overdispersion and a popular modeling approach is negative binomial regression, see for example, section 7.3 of the book by Agrestiįrequentist inference about mean and shape parameters in negative binomial regression is typically based on the likelihood and this is the method of choice for standard software, such as the glm.nb function of the R package MASS. Regression models for count data are employed in many contexts, especially in social sciences, economics, biology, and epidemiology. 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. The methods are illustrated and evaluated using two case studies: an Ames salmonella assay data set and data on epileptic seizures. 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 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. This article proposes inference for negative binomial regression based on adjustments of the score function aimed at mean or median bias reduction. ![]() 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. Negative binomial regression is commonly employed to analyze overdispersed count data.
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