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Partial and ecological correlation: a common three-term covariance decomposition

机译：局部和生态相关：常见的三项协方差分解

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摘要

Let (X, Y, Z) be a trivariate statistical variable observed at individual level. We propose a three-term decomposition of covariance between variables X and Y conditionally on the effects induced by the existence of a variable Z. The three terms are called residual covariance, covariance lack of fit and covariance fit, respectively. Partial covariance, between X and Y after removing the linear effects of Z, σ_Z (X, Y), is the sum of the first two terms while ecological covariance, between the two regression functions μ_X(Z) and μ_Y(Z), Cov(μ_X(Z), μ_Y(Z)), is the sum of the last two terms and, consequently, covariance lack of fit is the common additive term. Simple examples are given in two contexts: in ecological fallacy problems arising in linear modelling with aggregate level analysis contrasted with partial correlation step-wise procedures performed at individual level and in the special case of a two-level nested model. Previous basic decomposition is extended to a multivariate-multiple framework. Distinction between descriptive and stochastic approaches is not essential.
机译：令（X，Y，Z）为在个体水平上观察到的三变量统计变量。我们提出了变量X和Y之间的协方差的三项分解，条件是根据变量Z的存在而引起的影响。这三个项分别称为残差协方差，缺乏拟合的协方差和协方差拟合。去除Z的线性影响后X和Y之间的偏协方差σ_Z（X，Y）是前两个项的总和，而两个回归函数μ_X（Z）和μ_Y（Z）之间的生态协方差Cov （μ_X（Z），μ_Y（Z））是最后两项的总和，因此，缺乏拟合的协方差是常见的累加项。在以下两种情况下给出了简单的示例：在生态谬误问题中，聚合级别分析的线性建模与在个别级别执行的部分相关逐步过程形成对比，在特殊情况下，则是两级嵌套模型。先前的基本分解扩展到了多元多元框架。描述性方法和随机性方法之间的区别不是必需的。

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