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

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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.

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Present affiliation of the second author: Department of Statistics, University of Nigeria (Nsukka Campus). Work on this paper was carried out while on study leave in Aberystwyth.

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Jackson, P.H., Agunwamba, C.C. Lower bounds for the reliability of the total score on a test composed of non-homogeneous items: I: Algebraic lower bounds. Psychometrika 42, 567–578 (1977).

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