Desirable mathematical properties of indicators for biodiversity change

Numerous indicator approaches are found in the scientific literature to describe changes in biodiversity. It is however far from clear which indicators are most appropriate and which are less suitable to summarize trends in biodiversity. One reason for this lack of clarity is that so far the mathematical properties of indicator approaches have had little attention. In this paper, we derive a number of desirable mathematical properties of indicators from economic price-index theory and apply these in the form of tests to 10 metrics to summarize changes in biodiversity. The metrics species richness, Simpson index, Shannon index, Buckland's modified Shannon index and Sørensen's similarity coefficient violate the monotonicity and proportionality test. The percentage of increasing minus declining species also fails the proportionality test, and in the case where trends are assessed relative to the preceding year, this metric also violates the identity test. Most of these indicators are sensitive to spatial scale. The arithmetic and geometric mean of population indices and the mean abundance have better mathematical performance, but the first two are sensitive to appearing and disappearing species in the system surveyed. The metric mean abundance however can only be applied under particular conditions and has some undesirable properties. Unlike the arithmetic mean, the geometric mean is not sensitive to the base year chosen and has the most favourable mathematical properties of the indicators evaluated. The geometric mean can be straightforwardly extended to take into account unequal values of species if desired.