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Finding the greatest lower bound for the reliability of the total score on a test comprising n non-homogenous items with dispersion matrix Σ x is equivalent to maximizing the trace of a diagonal matrix Σ E with elements θ I , subject to Σ E and Σ T =Σ x − Σ E being non-negative definite. The cases n =2 and n =3 are solved explicity. A computer search in the space of the θ i is developed for the general case. When Guttman's λ 4 (maximum split-half coefficient alpha) is not the g.l.b., the maximizing set of θ i makes the rank of Σ T less than n − 1. Numerical examples of various bounds are given.
Woodhouse et al. (1977) studied this question.