where y is the expression of a single gene across all samples, X j is the matrix of j th fixed effect with coefficients β j,Z k is the matrix corresponding to the k th random effect with coefficients α k drawn from a normal distribution with variance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sigma ^{2}_{\alpha _{k}}$\end{document}σαk2. The noise term, ε, is drawn from a normal distribution with variance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sigma ^{2}_{\varepsilon }$\end{document}σε2. All parameters are estimated with maximum likelihood [29] as simulations under a range of experimental designs indicate that this approach gives the most accurate FVE estimates (Additional file 1: Figures S1–S4).
