The negative binomial distribution GLM with log link function can be expressed in eq. (2), where μs,b,l represents the expected mean for Ns,b,l and θ is the dispersion parameter (the shape parameter of the gamma mixing distribution). The mean E(Ns,b,l) = μs,b,l and variance VAR(Ns,b,l) = μs,b,l + θμs,b,l2 can be estimated from GLM shown below.2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \log \left({\mu}_{s,b,l}\right)= \log \left(E\left({N}_{s,b,l}\Big|{\boldsymbol{c}}_{s,b,l}\right)\right)= \log \left({d}_{s,l}\right)+{\boldsymbol{\beta}}^{\boldsymbol{\hbox{'}}}{\boldsymbol{c}}_{s,b,l} $$\end{document}logμs,b,l=logENs,b,l|cs,b,l=logds,l+β'cs,b,l
