Independent filtering does not compromise type-I error control as long as the distribution of the test statistic is marginally independent of the filter statistic under the null hypothesis [22], and we argue in the following that this is the case in our application. The filter statistic in DESeq2 is the mean of normalized counts for a gene, while the test statistic is p, the P value from the Wald test. We first consider the case where the size factors are equal and where the gene-wise dispersion estimates are used for each gene, i.e. without dispersion shrinkage. The distribution family for the negative binomial is parameterized by θ=(μ,α). Aside from discreteness of p due to low counts, for a given μ, the distribution of p is Uniform(0,1) under the null hypothesis, so p is an ancillary statistic. The sample mean of counts for gene i, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\bar {K}_{i}$ \end{document}K¯i, is boundedly complete sufficient for μ. Then from Basu’s theorem, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\bar {K}_{i}$ \end{document}K¯i and p are independent.
