Random binary matrices in biogeographical ecology—Instituting a good neighbor policy

Abstract

Binary matrices originating from presence/absence data on species (rows) distributed over sites (columns) have been a subject of much controversy in ecological biogeography. Under the null hypothesis that every matrix is equally likely, the distributions of some test statistics measuring co-occurrences between species are sought, conditional on the row and column totals being fixed at the values observed for some particular matrix. Many ad hoc methods have been proposed in the literature, but at least some of them do not provide uniform random samples of matrices. In particular, some “swap” algorithms have not accounted for the number of neighbors each matrix has in the universe of matrices with a set of fixed row and column sums. We provide a Monte-Carlo method using random walks on graphs that gives correct estimates for the distributions of statistics. We exemplify its use with one statistic.