On the comparison of the strength of morphological integration across morphometric datasets

Evolutionary morphologists frequently wish to understand the extent to which organisms are integrated, and whether the strength of morphological integration among subsets of phenotypic variables differ among taxa or other groups. However, comparisons of the strength of integration across datasets are difficult, in part because the summary measures that characterize these patterns (RV and rPLS) are dependent both on sample size and on the number of variables. As a solution to this issue we propose a standardized test statistic (a z-score) for measuring the degree of morphological integration between sets of variables. The approach is based on a partial least squares analysis of trait covariation, and its permutation-based sampling distribution. Under the null hypothesis of a random association of variables, the method displays a constant expected value and confidence intervals for datasets of differing sample sizes and variable number, thereby providing a consistent measure of integration suitable for comparisons across datasets. A two-sample test is also proposed to statistically determine whether levels of integration differ between datasets, and an empirical example examining cranial shape integration in Mediterranean wall lizards illustrates its use. Some extensions of the procedure are also discussed.

进化形态学家（evolutionary morphologists）常常希望探明生物体的形态整合程度，以及表型变量（phenotypic variables）子集间的形态整合（morphological integration）强度在不同类群（taxa）或其他分组间是否存在差异。然而，跨数据集比较整合强度颇具挑战，部分原因在于表征此类模式的汇总统计量RV和rPLS（RV and rPLS）同时依赖于样本量与变量数目。为解决这一问题，我们提出一种用于衡量变量集间形态整合程度的标准化检验统计量z分数（z-score）。该方法基于性状协变异的偏最小二乘分析（partial least squares analysis）及其基于置换的抽样分布（permutation-based sampling distribution）。在变量随机关联的原假设（null hypothesis）下，对于样本量与变量数目各异的数据集，该方法的期望取值与置信区间均保持恒定，从而提供了适用于跨数据集比较的一致化整合强度度量。我们还提出了两样本检验（two-sample test），用于统计学上判断不同数据集间的整合水平是否存在差异；并以地中海壁蜥（Mediterranean wall lizards）的颅骨形状整合（cranial shape integration）分析为例，展示了该方法的实际应用。此外，本文还讨论了该分析流程的若干扩展方向。