1. 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.
2. Here, we analyse 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 data set of cranial traits and use them to make sample size recommendations.
3. 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.
4. 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.

中文翻译：

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还有多少？在形态整合和进化研究中确定样本量。

1. 生命有机体的变异特性是当前进化论的重要组成部分。结果，从事多元进化领域的研究人员越来越多地使用积分和可进化性统计数据来捕捉特征关联的潜在复杂模式及其对进化轨迹的影响。然而，很少关注的是对这些广泛使用的进化统计数据的性状协方差估算不准确所产生的级联效应。
2. 在这里，我们分析了抽样工作与可演化性和集成统计量之间的不准确性之间的关系，该统计量是由具有十种特征的协方差和积分幅度的10个特征矩阵计算得出的。然后，我们将初始方法外推到不同数量的特征和不同的整合程度，并估计将感兴趣的统计数据的不准确性与抽样工作相关的一般方程。我们使用颅骨特征数据集验证方程式，并使用其提出样本量建议。
3. 我们的结果表明，考虑到适当估计样本量所需的抽样工作，由于样本量较小而导致的可发展性和积分统计数据的高度不准确估计很可能在文献中普遍存在。我们还表明，协变模式对这些统计数据的采样属性没有影响，但是整体的总体大小与样本大小相互作用，并导致不同程度的偏差，不精确性和不准确性。
4. 最后，我们提供了r个函数，这些函数可用于计算建议的样本量或简单地估计给定样本设计的这些统计数据中应预期的不准确度。