Abstract
Finding the greatest lower bound for the reliability of the total score on a test comprisingn 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 casesn=2 andn=3 are solved explicity. A computer search in the space of the θ i is developed for the general case. When Guttman's λ4 (maximum split-half coefficient alpha) is not the g.l.b., the maximizing set of θ i makes the rank of Σ T less thann − 1. Numerical examples of various bounds are given.
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Reference note
Woodhouse, B.Lower bounds for the reliability of a test. University of Wales M.Sc. thesis, 1976.
References
Bentler, P. M. A lower-bound method for the dimension-free measurement of internal consistency.Social Science Research, 1972,1, 343–357.
Guttman, L. A basis for analyzing test-retest reliability.Psychometrika, 1945,10, 255–282.
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Present affiliation of the first author: St. Hild's College of Education, Durham City, England.
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Woodhouse, B., Jackson, P.H. 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. Psychometrika 42, 579–591 (1977). https://doi.org/10.1007/BF02295980
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DOI: https://doi.org/10.1007/BF02295980