Zimmerman, D.W. (1975). Probability spaces, Hilbert spaces, and the axioms of test theory. Psychometrika, 40, 395-412.

A branch of probability theory that has been studied extensively in recent years, the theory of conditional expectation, provides just the concepts needed for mathematical derivation of the main results of the classical test theory with minimal assumptions and greatest economy in the proofs. The collection of all random variables with finite variance defined on a given probability space is a Hilbert space; the function that assigns to each random variable its conditional expectation is a linear operator; and the properties of the conditional expectation needed to derive the usual test-theory formulas are general properties of linear operators in Hilbert space. Accordingly, each of the test-theory formulas has a simple geometric interpretation that holds in all Hilbert spaces.

Zimmerman, D.W. (1979). A simple duality principle in test theory. Journal of Mathematical Psychology, 20, 256-262.

It is demonstrated that theorems in test theory have corresponding dual theorems which are obtained by exchanging true scores and error scores, as well as reliability coefficients and their complements, in both the hypothesis and the conclusion. A formula that does not conform to the principle cannot be an identity in the classical test theory model, but must be based on additional assumptions in the hypothesis that perhaps are not immediately apparent. The usefulness of the principle is indicated, and its origin in the mathematical formalism underlying the theory is discussed.

Zimmerman, D.W. (1997). A geometric interpretation of the validity and reliability of difference scores. British Journal of Mathematical and Statistical Psychology, 50, 73-80.

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.