Mathematically and biologically consistent framework for presence-absence pairwise indices of diversity
A 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 with each other. At the same time, the following issues have been largely overlooked: (i) requirement that a small re-arrangement of assemblages should only cause a small change in an index, (ii) inferences from the indices about diversity patterns, (iii) inter-dependence of indices based on their information value, (iv) overlap of the ecological phenomena that the indices aim to capture, and (v) incomparability of measures of different phenomena. Neglecting these issues has resulted in the invention or reinvention of indices without increasing their information value, although this value is crucial for correct interpretation of the indices. We offer a framework for pairwise diversity indices that accounts for these issues. We differentiate between statistical and information dependence of indices and show mathematical links between all indices, even those that have not yet been developed. Using linear algebra, we show (1) which set of indices carries complete information on assemblage arrangement, (2) how to calculate any index from two presence-absence indices, which can be used to standardize and compare different indices across the literature, and (3) what can be inferred about diversity phenomena from different informationally independent indices. It is impossible to purify an index of a single biodiversity phenomenon from the effects of other phenomena, because these phenomena inevitably constrain each other. Consequently, many recently proposed indices do not measure the phenomena that they were intended to measure. In contrast, a proper inference can be made by combining classical indices from different, information independent families.