The shape of phylogenies under phase-type distributed times to speciation and extinction

Phylogenetic trees describe relationships between extant species, but beyond that their shape and their relative branch lengths can provide information on broader evolutionary processes of speciation and extinction. However, currently, many of the most widely used macro-evolutionary models make predictions about the shapes of phylogenetic trees that differ considerably from what is observed in empirical phylogenies. Here, we propose a flexible and biologically plausible macroevolutionary model for phylogenetic trees where times to speciation or extinction events are drawn from a Coxian phase-type (PH) distribution. First, we show that different choices of parameters in our model lead to a range of tree balances as measured by Aldous' $\beta$ statistic. In particular, we demonstrate that it is possible to find parameters that correspond well to empirical tree balance. Next, we provide a natural extension of the $\beta$ statistic to sets of trees. This extension produces less biased estimates of $\beta$ compared to using the median $\beta$ values from individual trees. Furthermore, we derive a likelihood expression for the probability of observing an edge-weighted tree under a model with speciation but no extinction. Finally, we illustrate the application of our model by performing both absolute and relative goodness-of-fit tests for two large empirical phylogenies (squamates and angiosperms) that compare models with Coxian PH distributed times to speciation with models that assume exponential or Weibull distributed waiting times. In our numerical analysis, we found that, in most cases, models assuming a Coxian PH distribution provided the best fit.系统发育树（phylogenetic tree）描述了现存物种之间的演化关系，除此之外，其拓扑结构与相对分支长度还能为物种形成与灭绝等更广泛的进化过程提供相关信息。然而，当前诸多主流宏观进化模型对系统发育树拓扑结构的预测，与经验系统发育树中观测到的结果存在显著差异。本文提出了一种灵活且符合生物学合理性的宏观进化模型，用于建模系统发育树，该模型将物种形成或灭绝事件的发生时间建模为服从考克斯相位型（Coxian phase-type, PH）分布的随机变量。首先，我们证明了模型中不同的参数选择会通过奥尔德斯β统计量（Aldous' β statistic）得到一系列不同的树平衡度。特别地，我们证实可以找到与经验树平衡度高度匹配的参数组合。其次，我们将β统计量自然拓展至多棵系统发育树的集合场景，相较于使用单棵树的β统计量中位数，该拓展方法得到的β估计值偏差更小。此外，我们推导了在仅存在物种形成、无灭绝事件的模型下，观测到带边权系统发育树的似然表达式。最后，我们通过对两组大型经验系统发育数据集（有鳞类动物与被子植物）开展绝对与相对拟合优度检验，展示了本模型的应用场景：该检验将考克斯PH分布的物种形成等待时间模型，与假设服从指数分布或威布尔分布的等待时间模型进行对比。数值分析结果表明，在绝大多数场景下，采用考克斯PH分布的模型均能取得最优拟合效果。