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.