where fNB(k;μ,α) is the probability mass function of the negative binomial distribution with mean μ and dispersion α, and the second term provides the Cox–Reid bias adjustment [47]. This adjustment, first used in the context of dispersion estimation for SAGE data [48] and then for HTS data [3] in edgeR, corrects for the negative bias of dispersion estimates from using the MLEs for the fitted values \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\hat \mu _{\textit {ij}}^{0}$ \end{document}μ^ij0 (analogous to Bessel’s correction in the usual sample variance formula; for details, see [49], Section 10.6). It is formed from the Fisher information for the fitted values, which is here calculated as det(XtWX), where W is the diagonal weight matrix from the standard iteratively reweighted least-squares algorithm. As the GLM’s link function is g(μ)= log(μ) and its variance function is V(μ;α)=μ+αμ2, the elements of the diagonal matrix Wi are given by: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$w_{jj} = \frac{1}{g^{\prime}(\mu_{j})^{2} V(\mu_{j})} = \frac{1}{1/\mu_{j} + \alpha}. $$ \end{document}wjj=1g′(μj)2V(μj)=11/μj+α.
