CrimeStat_Appendix_D_Neg_Bin_Regression

CrimeStat_Appendix_D_Neg_Bin_Regression - Appendix D:...

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D 1 Appendix D: Negative Binomial Regression Models and Estimation Methods By Dominique Lord Texas A&M University Byung-Jung Park Korea Transport Institute This appendix presents the characteristics of Negative Binomial regression models and discusses their estimating methods. Probability Density and Likelihood Functions The properties of the negative binomial models with and without spatial intersection are described in the next two sections. Poisson-Gamma Model The Poisson-Gamma model has properties that are very similar to the Poisson model discussed in Appendix C, in which the dependent variable i y is modeled as a Poisson variable with a mean i where the model error is assumed to follow a Gamma distribution. As it names implies, the Poisson-Gamma is a mixture of two distributions and was first derived by Greenwood and Yule (1920). This mixture distribution was developed to account for over-dispersion that is commonly observed in discrete or count data (Lord et al., 2005). It became very popular because the conjugate distribution (same family of functions) has a closed form and leads to the negative binomial distribution. As discussed by Cook (2009), “the name of this distribution comes from applying the binomial theorem with a negative exponent.” There are two major parameterizations that have been proposed and they are known as the NB1 and NB2, the latter one being the most commonly known and utilized. NB2 is therefore described first. Other parameterizations exist, but are not discussed here (see Maher and Summersgill, 1996; Hilbe, 2007). NB2 Model Suppose that we have a series of random counts that follows the Poisson distribution:  ; ! i i ii i e gy y ( D - 1 ) where i y is the observed number of counts for 1, 2, i n ; and i is the mean of the Poisson distribution. If the Poisson mean is assumed to have a random intercept term and this term enters the conditional mean function in a multiplicative manner, we get the following relationship (Cameron and Trivedi, 1998):
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D 2  1 0 0 1 0 1 exp K ij j j i K ij j j i K ii j j i j x i x i i x ee     ( D - 2 ) where, 0 exp i is defined as a random intercept;   0 1 exp K j j j x is the log-link between the Poisson mean and the covariates or independent variables xs ; and s are the regression coefficients. As discussed in Appendix C, the relationship can also be formulated using vectors, such that ) exp( β x ' i i . The marginal distribution of i y can be obtained by integrating the error term, i , ;; , , iii i i o f yg y h d fy E gy     (D-3) where i h is a mixing distribution. In the case of the Poisson-Gamma mixture, ;, gy is the Poisson distribution and i h is the Gamma distribution. This distribution has a closed form and leads to the NB distribution.
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CrimeStat_Appendix_D_Neg_Bin_Regression - Appendix D:...

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