跨分布和样本量比较 Pearson 和 Spearman 相关系数：基于模拟和实证数据的教程
Comparing the Pearson and Spearman Correlation Coefficients Across Distributions and Sample Sizes: A Tutorial Using Simulations and Empirical Data
AI总结：
通过模拟与实证数据，在变异性、偏差和异常值稳健性三方面比较 Pearson 和 Spearman 相关系数，发现前者适合轻尾分布，后者更适用于重尾分布或存在异常值的情况。
摘要：
Pearson 积差相关系数（rp）和 Spearman 等级相关系数（rs）在心理学研究中被广泛使用。我们在 3 个标准上比较了 rp 和 rs：变异性、相对于总体值的偏差以及对异常值的稳健性。通过对低（N = 5）到高（N = 1,000）样本量进行模拟，我们表明，对于正态分布变量，rp 和 rs 具有相似的期望值，但 rs 变异性更大，尤其是在相关性较强时。然而，当变量具有高峰度时，rp 的变异性大于 rs。接下来，我们对一个以轻尾对称分布数据为特征的心理测量数据集，以及 2 个 Likert 型调查数据集（1 个为轻尾分布，另 1 个为重尾分布）进行了抽样研究。与模拟结果一致，在心理测量数据集中 rp 的变异性低于 rs。在具有重尾变量的调查数据集中，rs 的变异性低于 rp，并且通常比 rp 更准确地对应于总体 Pearson 相关系数（Rp）。模拟和抽样研究表明，通过选择 rs 而不是 rp，以标准差衡量的变异性可以降低约 20%。相比之下，将样本量增加 2 倍会使 rs 和 rp 的标准差降低 41%。总之，rp 适用于轻尾分布，而当变量呈现重尾分布或存在异常值时（心理学研究中经常如此），rs 是更优的选择。
英文摘要：
The Pearson product-moment correlation coefficient (rp) and the Spearman rank correlation coefficient (rs) are widely used in psychological research. We compare rp and rs on 3 criteria: variability, bias with respect to the population value, and robustness to an outlier. Using simulations across low (N = 5) to high (N = 1,000) sample sizes we show that, for normally distributed variables, rp and rs have similar expected values but rs is more variable, especially when the correlation is strong. However, when the variables have high kurtosis, rp is more variable than rs. Next, we conducted a sampling study of a psychometric dataset featuring symmetrically distributed data with light tails, and of 2 Likert-type survey datasets, 1 with light-tailed and the other with heavy-tailed distributions. Consistent with the simulations, rp had lower variability than rs in the psychometric dataset. In the survey datasets with heavy-tailed variables in particular, rs had lower variability than rp, and often corresponded more accurately to the population Pearson correlation coefficient (Rp) than rp did. The simulations and the sampling studies showed that variability in terms of standard deviations can be reduced by about 20% by choosing rs instead of rp. In comparison, increasing the sample size by a factor of 2 results in a 41% reduction of the standard deviations of rs and rp. In conclusion, rp is suitable for light-tailed distributions, whereas rs is preferable when variables feature heavy-tailed distributions or when outliers are present, as is often the case in psychological research.
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