Partial and Ecological Correlation: a Common Three Term Covariance Decomposition

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, σZ (X, Y ), is the sum of the first two terms while ecological covariance, between the two regression functions μX (Z) and μY (Z), Cov(μX (Z),μY (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.