![]() This capability is rarely available with models estimated using full maximum likelihood or full quasi-likelihood methods. This gives the analyst the means to quantitatively test different modeling strategies with tools built into the GLM algorithm. The advantage of the GLM approach rests in its ability to utilize the specialized GLM fit and residual statistics that come with the majority of GLM software. Regardless of the manner in which the negative binomial is estimated, it is nevertheless nearly always used to model Poisson overdispersion. It is also the form of the negative binomial found in Stata’s glm command as well as in the SAS/STAT GENMOD procedure in SPSS’s GLZ command, and in GENSTAT’s GLM program. This form of negative binimoal was called the log-negative binomial by Hilbe (1993a), and was the basis of a well-used SAS negative binomial macro (Hilbe, 1994b). When this is done, the amended GLM-based negative binomial produces identical estimates and standard errors to that of the mixture-based negative binomial. The GLM negative binomial algorithm may be amended though to allow production of standard errors based on observed information. The standard GLM algorithm uses Fisher scoring to produce standard errors based on the expected information matrix – hence the difference in standard errors between the two versions of negative binomial. The latter uses by default the observed information matrix to produce standard errors. As a non-canonical linked model, however, the standard errors will differ slightly from the mixture model, which is typically estimated using a full maximum likelihood procedure. So doing produces a GLM-based negative binomial that yields identical parameter estimates to those calculated by the mixture-based model. Rather, one must convert the canonical link and inverse canonical link to log form. ![]() The straightforward derivation of the model from the negative binomial probability distribution function (PDF) does not, however, equate with the Poisson–gamma mixture-based version of the negative binomial. It may be derived as a generalized linear model, but only if its ancillary or heterogeneity parameter is entered into the distribution as a constant. The original derivation of the negative binomial regression model stems from this manner of understanding it, and has continued to characterize the model to the present time.Īs mentioned above, the negative binomial has recently been thought of as having an origin other than as a Poisson–gamma mixture. Certainly, when the negative binomial is derived as a Poisson–gamma mixture, thinking of it in this way makes perfect sense. When the negative binomial is used to model overdispersed Poisson count data, the distribution can be thought of as an extension to the Poisson model. Negative binomial regression is a standard method used to model overdispersed Poisson data. Data that have greater variance than the mean are termed Poisson overdispersed, but are more commonly designated as simply overdispersed. ![]() However, the Poisson distribution assumes the equality of its mean and variance – a property that is rarely found in real data. Poisson regression is the standard method used to model count response data. ![]() Such interpretation allows statisticians to apply to the negative binomial model the various goodness-of-fit tests and residual analyses that have been developed for GLMs. Most importantly, the characterization is applicable to the negative binomial. This family of distributions admits a characterization known as generalized linear models (GLMs), which summarizes each member of the family. However, the negative binomial may also be thought of as a member of the single parameter exponential family of distributions. The negative binomial is traditionally derived from a Poisson–gamma mixture model. 9780521857727 - Negative Binomial Regression - by Joseph M.
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