The Wald test compares the beta estimate βir divided by its estimated standard error SE(βir) to a standard normal distribution. The estimated standard errors are the square root of the diagonal elements of the estimated covariance matrix, Σi, for the coefficients, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\operatorname {SE}(\beta _{\textit {ir}})=\sqrt {\Sigma _{i,rr}}$ \end{document}SE(βir)=Σi,rr. Contrasts of coefficients are tested similarly by forming a Wald statistics using (3) and (4). We use the following formula for the coefficient covariance matrix for a GLM with normal prior on coefficients [56, 58]: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\Sigma_{i} = \operatorname{Cov}(\vec{\beta}_{i}) = (X^{t} W X + \vec{\lambda} I)^{-1} (X^{t} W X) (X^{t} W X + \vec{\lambda} I)^{-1}. $$ \end{document}Σi=Cov(β→i)=(XtWX+λ→I)−1(XtWX)(XtWX+λ→I)−1. The tail integrals of the standard normal distribution are multiplied by 2 to achieve a two-tailed test. The Wald test P values from the subset of genes that pass the independent filtering step are adjusted for multiple testing using the procedure of Benjamini and Hochberg [21].
