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.

系统发育多样性（phylogenetic diversity）测度是系统生物学诸多研究中的基础工具。费思系统发育多样性指数（Faith’s PD，即连接所有目标物种的系统发育树（phylogenetic tree）的分支长度总和）是目前应用最广泛的系统发育多样性测度指标。与物种丰富度类似，基于采样数据计算的费思系统发育多样性指数高度依赖样本量与样本完整性。近期研究者基于样本量与样本覆盖度，开发了费思系统发育多样性指数的稀疏化（rarefaction）与外推（extrapolation）整合方法，以实现不同群落组合间的公平比较。然而，费思系统发育多样性指数未纳入物种多度信息。基于希尔数（Hill numbers）框架，费思系统发育多样性指数被推广为一类可纳入物种多度的系统发育多样性测度。本文中，我们针对这类可量化物种多度的系统发育多样性测度（包含系统发育熵与二次熵的简单变换形式），推导了无缝稀疏化与外推的理论公式与解析估计量。本研究既将原有费思系统发育多样性指数的稀疏化/外推模型推广至纳入物种多度的场景，也将原有的希尔数稀疏化/外推模型拓展为可纳入物种间系统发育差异的形式，由此可构建一套统一的方法，用于评估与比较物种/分类多样性及系统发育多样性。本文还提出了一种自助法（bootstrap method）以构建系统发育多样性的置信区间，便于开展多群落组合间的对比分析。本文所提出的公式与估计量可拓展至多采样单元获取的出现度数据场景。我们还利用人类远端食管的细菌序列数据与三类生境的叶口蝠科蝙蝠数据，对上述公式与估计量进行了实例演示。