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

Carlson, James E.
Publication Year:
Report Number:
ETS Research Report
Document Type:
Page Count:
Subject/Key Words:
Subscores, Variance, Subtests, Vector Geometry, Weighted Scores, Test Reliability, Test Validity


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.

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