The parametrization (6) is based on reports by us and others of decreasing dependence of dispersion on the mean in many datasets [3-6,51]. Some caution is warranted to disentangle true underlying dependence from effects of estimation bias that can create a perceived dependence of the dispersion on the mean. Consider a negative binomial distributed random variable with expectation μ and dispersion α. Its variance v=μ+αμ2 has two components, v=vP+vD, the Poisson component vP=μ independent of α, and the overdispersion component vD=αμ2. When μ is small, μ≲1/α (vertical lines in Additional file 1: Figure S1), the Poisson component dominates, in the sense that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $v_{\text {P}}/v_{\text {D}} = 1/(\alpha \mu) \gtrsim 1$ \end{document}vP/vD=1/(αμ)≳1, and the observed data provide little information on the value of α. Therefore the sampling variance of an estimator for α will be large when μ≲1/α, which leads to the appearance of bias. For simplicity, we have stated the above argument without regard to the influence of the size factors, sj, on the value of μ. This is permissible
