What Differential Weighting of Subsets of Items Does and Does Not Accomplish: Geometric Explanation

Abstract

A little-known theorem, a generalization of Pythagoras's theorem, due to Pappus, is used to present a geometric explanation of various definitions of the contribution of component tests to their composite. I show that an unambiguous definition of the unique contribution of a component to the composite score variance is present if and only if the component scores are uncorrelated. I further show the effect of differentially weighting the composites on the definitions of unique contributions and discuss some of the implications for composite score reliability and validity.