Lower Bounds for the Reliability of the Total Score on a Test Composed of Non-Homogeneous Items: II: A Search Procedure to Locate the Greatest Lower Bound

Abstract

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.