Data from: How many more? Sample size determination in studies of morphological integration and evolvability

The variational properties of living organisms are an important component of current evolutionary theory. As a consequence, researchers working on the field of multivariate evolution have increasingly used integration and evolvability statistics as a way of capturing the potentially complex patterns of trait association and their effects over evolutionary trajectories. Little attention has been paid, however, to the cascading effects that inaccurate estimates of trait covariance have on these widely used evolutionary statistics.
Here, we analyze the relationship between sampling effort and inaccuracy in evolvability and integration statistics calculated from 10-trait matrices with varying patterns of covariation and magnitudes of integration. We then extrapolate our initial approach to different numbers of traits and different magnitudes of integration and estimate general equations relating the inaccuracy of the statistics of interest to sampling effort. We validate our equations using a dataset of cranial traits, and use them to make sample size recommendations.
Our results suggest that highly inaccurate estimates of evolvability and integration statistics resulting from small sample sizes are likely common in the literature, given the sampling effort necessary to properly estimate them. We also show that patterns of covariation have no effect on the sampling properties of these statistics, but overall magnitudes of integration interact with sample size and lead to varying degrees of bias, imprecision, and inaccuracy.
Finally, we provide R functions that can be used to calculate recommended sample sizes or to simply estimate the level of inaccuracy that should be expected in these statistics, given a sampling design.

生物体的变异特性是现代进化理论的核心组成部分。有鉴于此，深耕多变量进化领域的研究者愈发倾向于采用性状整合（integration）与可进化性（evolvability）统计量，以捕捉性状关联的潜在复杂模式及其对进化轨迹的作用效应。然而，学界迄今对性状协方差估计失准给这类广泛应用的进化统计量所造成的连锁效应关注甚少。

本研究首先针对具备不同协变模式与整合程度的10性状矩阵，计算可进化性与整合性统计量，进而剖析采样投入与该类统计量估计偏差之间的关联。随后，我们将初始分析框架推广至不同性状数量与不同整合程度的场景，并构建起目标统计量估计偏差与采样投入间的通用关系式。我们借助颅骨性状数据集对所推导出的关系式进行验证，并据此提出采样量建议。

研究结果表明：若要准确估计可进化性与整合性统计量，所需的采样投入颇高；当前文献中因样本量过小导致的这类统计量严重估计失准情况，或许相当普遍。此外我们发现，协变模式并不会对这类统计量的采样特性产生影响，但整体整合程度会与采样量产生交互作用，进而引发不同程度的偏差、不精确性与估计误差。

最后，我们提供了可用于计算推荐采样量，或基于给定采样设计直接预估这类统计量预期偏差水平的R语言函数。