Variance Components: Partialled vs. Common

Abstract

Conventional partialling methods and the variance components they assume or generate were shown to be inconsistent with sound conceptual models of correlation and variance components. A different approach was taken, and newer partialling and variance-isolation procedures were employed. Like conventional partialling, this approach orthogonalizes variables by partitioning the scores/observations. Unlike conventional partialling, it yields a “common” component and two “unique” components. That is, two correlated variables are broken down into three orthogonal components, as depicted by a Venn diagram of two overlapping circles. The common component, or overlap area, C, has maximum variance—as a proportion of variance it equals the correlation, r, between the variables. And the variance of each unique component is 1 – r, as opposed to 1 – r ² which is the variance of the conventional residual. In addition to being consistent with fundamental conceptualizations of correlation/association and variance components, the three independent components can be used to maximally predict a third variable, Y. As a three-part composite, they relate to Y as highly as does the conventional multiple-regression composite. And their individual correlations with Y provide improved estimates of the underlying “contributions to Y variance.”