Mathematically and biologically consistent framework for presence-absence pairwise indices

Aim Large number of indices for presence-absence data that compare two assemblages have been proposed or reinvented. Interpretation of these indices varies across the literature, despite efforts for clarification and unification. Most effort has focused on the mathematics behind the indices, their relationships with diversity, and between each other. At the same time, following issues have been largely overlooked: (i) inter-dependence of indices based on their informational value, (ii) overlap of the ecological phenomena that the indices aim to capture, (iii) requirement that a small re-arrangement of assemblages should only cause a small change in an index, and (iv) inferences from the indices about diversity patterns. Underappreciation of these issues has led to invention or reinvention of indices without increasing their information value. We offer a framework for pairwise diversity indices that accounts for these issues. Methods We present a framework that links different ecological phenomena to pairwise indices, and show mathematical links between the indices. We distinguish statistical dependence of indices from their informational (mathematical) dependence. Using linear algebra, we found the minimum number of indices needed to detect different patterns. Results We (1) classified existing indices into three major and four minor mutually independent families, (2) demonstrated how assemblage interrelation confuses conceptually different patterns, (3) showed what can be inferred about diversity phenomena from different indices, (4) demonstrated problems with most of the indices of nestedness, (5) provided formula linking mathematically (informationally) dependent indices, and (6) showed which combinations of indices can be used for meaningful ecological inference. Additionally, (7) we showed how to calculate any index from two presence-absence indices, which can be used to standardize and compare different indices across the literature. Main conclusions It is impossible to purify an index of a single biodiversity phenomenon from the effects of other phenomena, because phenomena inevitably bound each other (e.g. a species richness gradient bounds possible values of Jaccard index of community similarity). Consequently, inventing indices which seemingly purify these effects (e.g. pure turnover or nestedness) leads to misleading inference. In contrast, a proper inference is obtained by using a combination of classical indices from different, information-independent families. ### Competing Interest Statement The authors have declared no competing interest.