A geometric interpretation of the validity and reliability of difference scores

Abstract

For several decades, psychometricians have been concerned about the unreliability and meagre validity of difference scores and gain scores. The present paper explores a geometric interpretation, in which observed scores are identified with vectors in a function space of random variables, and true and error components of scores are identified with orthogonal projections onto complementary subspaces. This point of view provides a simple and easily visualized geometric explanation for the widespread belief that differences are inherently unreliable. Furthermore, it discloses conditions under which difference scores are highly reliable and have substantial correlations with other measures.