Rarefaction and Extrapolation: Making Fair Comparison of Abundance-Sensitive Phylogenetic Diversity among Multiple Assemblages

Measures of phylogenetic diversity are basic tools in many studies of systematic biology. Faith’s PD (sum of branch lengths of a phylogenetic tree connecting all focal species) is the most widely used phylogenetic measure. Like species richness, Faith’s PD based on sampling data is highly dependent on sample size and sample completeness. The sample-size- and sample-coverage-based integration of rarefaction and extrapolation of Faith’s PD was recently developed to make fair comparison across multiple assemblages. However, species abundances are not considered in Faith’s PD. Based on the framework of Hill numbers, Faith’s PD was generalized to a class of phylogenetic diversity measures that incorporates species abundances. In this article, we develop both theoretical formulae and analytic estimators for seamless rarefaction and extrapolation for this class of abundance-sensitive phylogenetic measures, which includes simple transformations of phylogenetic entropy and of quadratic entropy. This work generalizes the previous rarefaction/extrapolation model of Faith’s PD to incorporate species abundance, and also extends the previous rarefaction/extrapolation model of Hill numbers to include phylogenetic differences among species. Thus a unified approach to assessing and comparing species/taxonomic diversity and phylogenetic diversity can be established. A bootstrap method is suggested for constructing confidence intervals around the phylogenetic diversity, facilitating the comparison of multiple assemblages. Our formulation and estimators can be extended to incidence data collected from multiple sampling units. We also illustrate the formulae and estimators using bacterial sequence data from the human distal esophagus and phyllostomid bat data from three habitats.