Methodology and Statistics, Faculty of Social and Behavioural Sciences, Utrecht University, Utrecht, The Netherlands.
Multivariate Behav Res. 2024 Nov-Dec;59(6):1111-1122. doi: 10.1080/00273171.2022.2155930. Epub 2023 Jan 4.
The cross-sectional correlation is frequently used to summarize psychological data, and can be considered the basis for many statistical techniques. However, the work of Peter Molenaar on has raised concerns about the meaning and utility of this measure, especially when the interest is in discovering general laws that apply to (all) individuals. Through using Cattell's databox and adopting a multilevel perspective, this paper provides a closer look at the cross-sectional correlation, with the goal to better understand its meaning when ergodicity is absent. An analytical expression is presented that shows the cross-sectional correlation is a function of the between-person correlation (based on person-specific means), and the within-person correlation (based on individuals' temporal deviations from their person-specific means). Two curiosities related to this expression of the cross-sectional correlation are elaborated on, that is: a) the difference between the within-person correlation and the (average) person-specific correlation; and b) the unexpected scenarios that can arise because the cross-sectional correlation is a weighted sum rather than a weighted average of the between-person and within-person correlations. Seven specific examples are presented to illustrate various ways in which these two curiosities may combine; R code is provided, which allows researchers to investigate additional scenarios.
横断面相关经常被用来总结心理学数据，可以被认为是许多统计技术的基础。然而，Peter Molenaar 在 上的工作引起了人们对该度量的意义和实用性的关注，尤其是当人们的兴趣在于发现适用于（所有）个体的一般规律时。本文通过使用 Cattell 的 databox 并采用多层次的视角，更仔细地研究了横断面相关，目的是在非遍历性情况下更好地理解其意义。提出了一个解析表达式，表明横断面相关是个体间相关（基于个体特定的平均值）和个体内相关（基于个体特定平均值的个体时间偏差）的函数。对这个横断面相关的表达式进行了两个有趣的阐述，即：a）个体内相关与（平均）个体特定相关之间的差异；b）由于横断面相关是个体间和个体内相关的加权和而不是加权平均值，可能会出现意想不到的情况。本文还提出了七个具体的例子来说明这两个有趣的方面可能结合的各种方式；提供了 R 代码，允许研究人员研究其他场景。
