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The Asymptotic Bias of Minimum Trace Factor Analysis, with Applications to the Greatest Lower Bound to Reliability

Published online by Cambridge University Press:  01 January 2025

Alexander Shapiro  and
Jos M. F. Ten Berge
Alexander Shapiro*
Affiliation:
School of Industrial and Systems Engineering, Georgia Institute of Technology
Jos M. F. Ten Berge
Affiliation:
Heymans Institute of Psychological Research, University of Groningen
*
Requests for reprints should be sent to Alexander Shapiro, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0205. E-Mail: ashapiro@isye.gatech.edu
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Abstract

In theory, the greatest lower bound (g.l.b.) to reliability is the best possible lower bound to the reliability based on single test administration. Yet the practical use of the g.l.b. has been severely hindered by sampling bias problems. It is well known that the g.l.b. based on small samples (even a sample of one thousand subjects is not generally enough) may severely overestimate the population value, and statistical treatment of the bias has been badly missing. The only results obtained so far are concerned with the asymptotic variance of the g.l.b. and of its numerator (the maximum possible error variance of a test), based on first order derivatives and the asumption of multivariate normality. The present paper extends these results by offering explicit expressions for the second order derivatives. This yields a closed form expression for the asymptotic bias of both the g.l.b. and its numerator, under the assumptions that the rank of the reduced covariance matrix is at or above the Ledermann bound, and that the nonnegativity constraints on the diagonal elements of the matrix of unique variances are inactive. It is also shown that, when the reduced rank is at its highest possible value (i.e., the number of variables minus one), the numerator of the g.l.b. is asymptotically unbiased, and the asymptotic bias of the g.l.b. is negative. The latter results are contrary to common belief, but apply only to cases where the number of variables is small. The asymptotic results are illustrated by numerical examples.

Keywords

asymptotic biasasymptotic normalityreliabilityminimum trace factor analysislarge sample theorysemidefinite programming
Type
Original Paper
Information
Psychometrika , Volume 65 , Issue 3 , September 2000 , pp. 413 - 425
Copyright
Copyright © 2000 The Psychometric Society

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Footnotes

This research was supported by grant DMI-9713878 from the National Science Foundation.

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