Data from: Diversity, disparity, and evolutionary rate estimation for unresolved Yule trees

The branching structure of biological evolution confers statistical dependencies on phenotypic trait values in related organisms. For this reason, comparative macroevolutionary studies usually begin with an inferred phylogeny that describes the evolutionary relationships of the organisms of interest. The probability of the observed trait data can be computed by assuming a model for trait evolution, such as Brownian motion, over the branches of this fixed tree. However, the phylogenetic tree itself contributes statistical uncertainty to estimates of other evolutionary quantities, and many comparative evolutionary biologists regard the tree as a nuisance parameter. In this paper, we present a framework for analytically integrating over unknown phylogenetic trees in comparative evolutionary studies by assuming that the tree arises from a continuous-time Markov branching model called the Yule process. To do this, we derive a closed-form expression for the distribution of phylogenetic diversity, which is the sum of branch lengths connecting a set of taxa. We then present a generalization of phylogenetic diversity which is equivalent to the expected trait disparity in a set of taxa whose evolutionary relationships are generated by a Yule process and whose traits evolve by Brownian motion. We derive expressions for the distribution of expected trait disparity under a Yule tree. Given one or more observations of trait disparity in a clade, we perform fast likelihood-based estimation of the Brownian variance for unresolved clades. Our method does not require simulation or a fixed phylogenetic tree. We conclude with a brief example illustrating Brownian rate estimation for thirteen taxonomic families in order Carnivora, in which the phylogenetic tree for each family is unresolved.

生物演化的分支结构使得近缘生物的表型性状值之间存在统计依存关系。因此，比较宏观演化研究通常以描述目标生物演化关系的推断系统发育树（phylogeny）为起点。通过在该固定树的分支上假定性状演化模型（如布朗运动（Brownian motion）），即可计算观测性状数据的概率。然而，系统发育树本身会为其他演化参数的估计引入统计不确定性，许多比较演化生物学家将该树视为冗余参数（nuisance parameter）。
本文提出了一套分析框架，可在比较演化研究中对未知系统发育树进行解析积分，前提是假设该树由名为尤尔过程（Yule process）的连续时间马尔可夫分支模型生成。为此，我们推导了系统发育多样性（phylogenetic diversity）分布的闭式表达式——系统发育多样性即连接一组分类群（taxa）的分支长度之和。
随后我们提出了系统发育多样性的推广形式，其等价于一组分类群的期望性状差异度（trait disparity）：该组分类群的演化关系由尤尔过程生成，且其性状按照布朗运动模型演化。我们推导了尤尔树框架下期望性状差异度的分布表达式。
若给定某一演化支（clade）内的一项或多项性状差异度观测值，我们可基于似然快速估计未解拓扑结构演化支的布朗运动方差。本方法无需进行模拟，也无需预先指定固定的系统发育树。
最后我们通过一则简短示例，展示了食肉目（Carnivora）下13个分类科的布朗运动速率估计场景，其中每个科的系统发育树均为未解拓扑结构。