Data from: A comment on the use of stochastic character maps to estimate evolutionary rate variation in a continuously valued trait

Phylogenetic comparative biology has progressed considerably in recent years. One of the most important developments has been the application of likelihood-based methods to fit alternative models for trait evolution in a phylogenetic tree with branch lengths proportional to time. An important example of this type of method is O’Meara et al.’s (2006) “noncensored” test for variation in the evolutionary rate for a continuously valued character trait through time or across the branches of a phylogenetic tree. According to this method, we first hypothesize evolutionary rate regimes on the tree (called “painting” in Butler and King, 2004); and then we fit an evolutionary model, specifically the popular Brownian model, in which the instantaneous variance of the Brownian random diffusion process has different values in different parts of the phylogeny. The authors suggest that to test a hypothesis that the state of a discrete character influenced the rate of a continuous character, one could use the approach of Neilsen (2002) to first stochastically map the discretely valued trait, and then “test to see whether the portions of the tree with one state for the discrete character have a different rate of evolution for the continuous character than portions of the tree to which the other discrete state has been mapped” (O’Meara et al., 2006, p. 931). Indeed, this has become common practice for this and other closely related methods. Here, I examine this practice. In particular, I show that evolutionary rates estimated this way (i.e., by using maximum likelihood [ML] to fit a multirate model on each stochastically mapped tree; and then averaging across trees) are systematically biased to be more similar to each other than are the underlying generating parameters. My analysis also reveals that this effect is dependent on the rate of evolution for the discrete trait. Specifically, if the rate of evolution for the discrete character is low then the difference between the true history and any stochastically mapped 1 history is generally small. This results in evolutionary rates for the continuous trait that are estimated with little bias. Conversely, if the rate of evolution for the discrete character is very high, then the true and hypothesized character histories are often extremely dissimilar, evolutionary rate estimates are biased to be more similar to each other than their underlying generating values, and we lose power to distinguish evolutionary rates on the tree.系统发育比较生物学（phylogenetic comparative biology）近年来取得了长足进展。其中最为重要的进展之一，便是将基于似然的方法应用于拟合分支长度与时间成正比的系统发育树中的性状演化替代模型。此类方法的一个重要范例，便是O'Meara等人（2006）提出的、用于检测连续型性状（continuously valued character trait）演化速率随时间或系统发育树分支变化的非删失检验（noncensored test）。依据该方法，研究者首先需在系统发育树上设定演化速率模式（Butler与King于2004年将其称为“着色（painting）”）；随后拟合演化模型，具体为当下流行的布朗模型（Brownian model），该模型中布朗随机扩散过程的瞬时方差在系统发育的不同区域取值各异。作者提出，若要检验“离散性状的状态会影响连续性状的演化速率”这一假说，可采用Neilsen（2002）的方法：首先对离散型性状进行随机映射（stochastically map），随后“检验系统发育树上携带某一离散性状状态的分支，其连续性状的演化速率是否与携带另一离散性状状态的分支存在差异”（O'Meara等人，2006，第931页）。事实上，这一流程已成为该类方法及其他密切相关方法的通用操作范式。本文即对该操作流程展开探讨。具体而言，本文证明：通过该流程估算的演化速率（即先在每一棵经随机映射的系统发育树上使用最大似然（maximum likelihood，ML）拟合多速率模型（multirate model），随后对各树的估算结果取平均），会系统性地出现偏差，使得估算出的速率彼此间比潜在生成参数更为相似。本文分析还表明，该偏差效应取决于离散性状的演化速率。具体而言，若离散性状的演化速率较低，则真实演化历史与任意经随机映射得到的历史之间的差异通常较小，此时连续性状的演化速率估算偏差也会极低。反之，若离散性状的演化速率极高，则真实性状演化历史与假设的历史往往差异极大，此时演化速率估算值会系统性地偏向于彼此比潜在生成值更为相似，同时我们也会丧失区分系统发育树上不同分支演化速率的统计效力。