Null model analyses of presence-absence matrices need a definition of independence

Tests for species interactions that involve the comparison of a statistic calculated from observed matrix of species presences and absences with the distribution of the same statistic generated from a null model have been used by ecologists for about 30 years. We argue that the validity of these tests requires a specific definition of independence. In particular, we note that an assumption that is often made is that all presence-absence matrices with the same row and column totals are equally likely if there is no interaction. However, we show using a simple model for species presences and absences without any species interactions that, in general, this assumption should be made with caution. Our model incorporates a definition of independence, allowing the computation of probabilities of different patterns in the null matrices. Other definitions of independence are possible; one of them is outlined using a new generalized linear model approach for carrying out tests applicable to different null models with or without the assumption of keeping row and column totals fixed.