Lower bounds for the reliability of the total score on a test composed of non-homogeneous items: I: Algebraic lower bounds

Abstract

Let Σ _x be the (population) dispersion matrix, assumed well-estimated, of a set of non-homogeneous item scores. Finding the greatest lower bound for the reliability of the total of these scores is shown to be equivalent to minimizing the trace of Σ _x by reducing the diagonal elements while keeping the matrix non-negative definite. Using this approach, Guttman's bounds are reviewed, a method is established to determine whether his λ₄ (maximum split-half coefficient alpha) is the greatest lower bound in any instance, and three new bounds are discussed. A geometric representation, which sheds light on many of the bounds, is described.