Abstract
Rao’s quadratic diversity index is one of the most widely applied diversity indices in functional and phylogenetic ecology. The standard way of computing Rao’s quadratic diversity index for an ecological assemblage with a group of species with varying abundances is to sum the functional or phylogenetic distances between a pair of species in the assemblage, weighted by their relative abundances. Here, using both theoretically derived and observed empirical datasets, we show that this standard calculation routine in practical applications will statistically underestimate the true value, and the bias magnitude is derived accordingly. The underestimation will become worse when the studied ecological community contains more species or the pairwise species distance is large. For species abundance data measured using the number of individuals, we suggest calculating the unbiased Rao’s quadratic diversity index.
Highlights
If one had only a single data set in hand, the unbiased estimator could provide him with the highest probability to have a small empirical bias which was half smaller than that of Qðp^Þ
To cope with the situation that the empirical biases of two estimators can vary with the selection of the sampling abundance patterns, we calculated the corresponding overall statistical bias induced by Q^ U ðpÞ and Qðp^Þ as zero and -0.153, respectively
In addition to the statistical bias, the mean-squared-error (MSE; an effective measure when comparing the accuracy of different estimators) of Q^ U ðpÞ and Qðp^Þ were computed and given by 0.0499 and 0.0514, respectively
Summary
Measures of biodiversity should take into account species richness and abundance as well as other characteristics (like abundance evenness) quantified by information metrics, which are valuable and should be incorporated. Rao’s quadratic diversity index is one of the most important biodiversity metrics that is widely applied to studies of functional and phylogenetic ecology (Rao, 1982, 2010; Mouchet et al, 2010). Species’ distance can be very flexible, ranging from phylogenetic to functional (or trait) distances (Ricotta, 2005b). Ecologists do not normally consider the statistical bias of Rao’s quadratic diversity index when applying it to practical research questions.
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