Robust hypothesis tests for independence in community assembly

The extent to which competition affects the distributions of species at large spatial scales is unclear. To evaluate this question, hypothesis tests that do not depend on parametric assumptions are needed. Here, we develop a broadly applicable test that requires only one parametric assumption. Letting i and j denote the ith and jth colonists to arrive at a site, respectively, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle ij \rangle}$$\end{document} the event that i and j belong to the same “unit” (e.g., functional group, genus), we show how colonists will be partitioned into units if for all i and j, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle ij \rangle}$$\end{document} is independent of whether i and j share unit membership with the other colonists, conditional on other information about shared units. Our distribution of partitions is useful for inferring competitive effects, because these effects predict that for at least one i and j, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${P(\langle ij \rangle)}$$\end{document} will be less when i and j share unit membership than when they do not.