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

Abstract

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.