We seek to identify the credibly associated covariates XNxM to observed cell counts YNxK of K cell types measured in a single-cell experiment with N samples and M covariates. We address this question with a Bayesian generalized linear multivariate regression framework using a Dirichlet-Multinomial model with a log-link function to account for the compositional nature and uncertainties in the observed data. Effects between covariates m and cell types k are hierarchically modeled using individual, normally distributed effects \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\gamma }_{m,k}$$\end{document}γm,k with a covariate-specific scaling factor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{\rm{\sigma }}}}}}}_{m}^{2}$$\end{document}σm229,30. For automatic model selection and identification of credibly associated covariates and affected cell types, we utilize a logit-normal prior as a continuous relaxation of the spike-and-slab prior10 resulting in the following hierarchical model:1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y \sim {{{{{\rm{DirMult}}}}}}(\phi ,\bar{y})$$\end{document}Y~DirMult(ϕ,ȳ)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\log (\phi )={{{{{\boldsymbol{\alpha }}}}}}+X\beta$$\end{document}log(ϕ)=α+Xβ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha }_{k} \sim {{{{{\rm{N}}}}}}(0,5)\quad \forall k\in [1,{{{{\mathrm{.}}}}}.,K]$$\end{document}αk~N(0,5)∀k∈[1,..,K]4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}
