Partial and Ecological Correlation: a Common Three Terms 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.