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A Note on the Asymptotic Distribution of the Greatest Lower Bound to Reliability

Published online by Cambridge University Press:  01 January 2025

Alexander Shapiro
Alexander Shapiro*
Affiliation:
Department of Mathematics, University of South Africa
*
Requests for reprints should be sent to Alexander Shapiro, Department of Mathematics, University of South Africa, Pretoria 0001, SOUTH AFRICA.
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Extract

In a recent article Bentler and Woodward (1983) discussed computational and statistical issues related to the greatest lower bound ρ+ to reliability. Although my work (Shapiro, 1982) was cited frequently some results presented were misunderstood. A sample estimate + of ρ + was considered and it was claimed (Bentler & Woodward) that: “Since + is not a closed form expression ... an exact analytic expression for h has not been found” (p. 247). (h is a vector of partial derivatives of ρ+ as a function of the covariance matrix.) Therefore Bentler and Woodward proposed to use numerical derivatives in order to evaluate the asymptotic variance avar ( +) of +.

Type
Notes And Comments
Information
Psychometrika , Volume 50 , Issue 2 , June 1985 , pp. 243 - 244
Copyright
Copyright © 1985 The Psychometric Society

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References

Bentler, P. M., & Woodward, J. A. (1983). The greatest lower bound to reliability. In Wainer, H. & Messick, S. (Eds.), Principals of modern phychological measurement: A festschrift for Frederic M. Lord (pp. 237–253). Hillsdale, NJ: Erlbaum.Google Scholar
Shapiro, A. (1982). Rank-reducibility of a symmetrix matrix and sampling theory of minimum trace factor analysis. Psychometrika, 47, 187–199.CrossRefGoogle Scholar