Département Génétique et Ecologie Evolutives, unité Ecologie, Systématique et Evolution, Université Paris-Sud 11, Orsay cedex, France.
Syst Biol. 2011 Dec;60(6):826-32. doi: 10.1093/sysbio/syr066. Epub 2011 Jul 29.
Phylogenies are fundamental to comparative biology as they help to identify independent events on which statistical tests rely. Two groups of phylogenetic comparative methods (PCMs) can be distinguished: those that take phylogenies into account by introducing explicit models of evolution and those that only consider phylogenies as a statistical constraint and aim at partitioning trait values into a phylogenetic component (phylogenetic inertia) and one or multiple specific components related to adaptive evolution. The way phylogenetic information is incorporated into the PCMs depends on the method used. For the first group of methods, phylogenies are converted into variance-covariance matrices of traits following a given model of evolution such as Brownian motion (BM). For the second group of methods, phylogenies are converted into distance matrices that are subsequently transformed into Euclidean distances to perform principal coordinate analyses. Here, we show that simply taking the elementwise square root of a distance matrix extracted from a phylogenetic tree ensures having a Euclidean distance matrix. This is true for any type of distances between species (patristic or nodal) and also for trees harboring multifurcating nodes. Moreover, we illustrate that this simple transformation using the square root imposes less geometric distortion than more complex transformations classically used in the literature such as the Cailliez method. Given the Euclidean nature of the elementwise square root of phylogenetic distance matrices, the positive semidefinitiveness of the phylogenetic variance-covariance matrix of a trait following a BM model, or related models of trait evolution, can be established. In that way, we build a bridge between the two groups of statistical methods widely used in comparative analysis. These results should be of great interest for ecologists and evolutionary biologists performing statistical analyses incorporating phylogenies.
系统发育是比较生物学的基础，因为它们有助于确定统计检验所依赖的独立事件。可以区分两类系统发育比较方法（PCM）：一类通过引入明确的进化模型来考虑系统发育，另一类仅将系统发育视为统计约束，并旨在将性状值分为系统发育成分（系统发育惯性）和一个或多个与适应进化相关的特定成分。PCM 将系统发育信息纳入的方式取决于所使用的方法。对于第一组方法，系统发育根据给定的进化模型（如布朗运动（BM））转换为性状的方差协方差矩阵。对于第二组方法，系统发育转换为距离矩阵，随后转换为欧几里得距离以执行主坐标分析。在这里，我们表明，只需从系统发育树上提取的距离矩阵的元素取平方根，即可确保获得欧几里得距离矩阵。对于物种之间的任何类型的距离（亲缘或节点）以及包含多叉节点的树都是如此。此外，我们说明，与文献中经典使用的更复杂的变换（如 Cailliez 方法）相比，这种使用平方根的简单变换会引起较少的几何失真。由于系统发育距离矩阵的元素取平方根具有欧几里得性质，因此可以建立遵循 BM 模型或相关性状进化模型的性状的系统发育方差协方差矩阵的正半定性质。通过这种方式，我们在比较分析中广泛使用的两类统计方法之间架起了一座桥梁。这些结果对于进行包含系统发育的统计分析的生态学家和进化生物学家应该具有重要意义。
