Rarefaction and extrapolation of phylogenetic diversity

Traditional species diversity measures do not make distinctions among species. Faith's phylogenetic diversity (PD), which is defined as the sum of the branch lengths of a phylogenetic tree connecting all species, takes into account phylogenetic differences among species and has found many applications in various research fields. In this paper, we extend Faith's PD to represent the total length of a phylogenetic tree from any fixed point on its main trunk. Like species richness, Faith's PD tends to be an increasing function of sampling effort and thus tends to increase with sample completeness. We develop in this paper the 'PD accumulation curve' (an extension of the species accumulation curve) to depict how PD increases with sampling size and sample completeness. To make fair comparisons of Faith's PD among several assemblages based on sampling data from each assemblage, we derive both theoretical formulae and analytic estimators for seamless rarefaction (interpolation) and extrapolation (prediction). We develop a lower bound of the undetected PD for an incomplete sample to guide the extrapolation; the PD estimator for an extrapolated sample is generally reliable up to twice the size of the empirical sample. We propose an integrated curve that smoothly links rarefaction and extrapolation to standardize samples on the basis of sample size or sample completeness. A bootstrap method is used to obtain the unconditional variances of PD estimators and to construct the confidence interval of the expected PD for a fixed sample size or fixed degree of sample completeness. This facilitates comparison of multiple assemblages of both rarefied and extrapolated samples. We illustrate our formulae and estimators using empirical data sets from Australian birds in two sites. We discuss the extension of our approach to the case of multiple incidence data and to incorporate species abundances.