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An Approximation for the Bias Function of the Maximum Likelihood Estimate of a Latent Variable for the General Case where the Item Responses are Discrete

Published online by Cambridge University Press:  01 January 2025

Fumiko Samejima
Fumiko Samejima*
Affiliation:
University of Tennessee
*
Requests for reprints should be sent to Fumiko Samejima, 310B Austin Peay Building, University of Tennessee, Knoxville, TN 37996-0900.
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Abstract

Lord developed an approximation for the bias function for the maximum likelihood estimate in the context of the three-parameter logistic model. Using Taylor's expansion of the likelihood equation, he obtained an equation that includes the conditional expectation, given true ability, of the discrepancy between the maximum likelihood estimate and true ability. All terms of orders higher than n−1 are ignored where n indicates the number of items. Lord assumed that all item and individual parameters are bounded, all item parameters are known or well-estimated, and the number of items is reasonably large. In the present paper, an approximation for the bias function of the maximum likelihood estimate of the latent trait, or ability, will be developed using the same assumptions for the more general case where item responses are discrete. This will include the dichotomous response level, for which the three-parameter logistic model has been discussed, the graded response level and the nominal response level. Some observations will be made for both dichotomous and graded response levels.

Keywords

biasmaximum likelihood estimationlatent trait modelsdiscrete item responsesdichotomous responsesgraded responses
Type
Original Paper
Information
Psychometrika , Volume 58 , Issue 1 , March 1993 , pp. 119 - 138
Copyright
Copyright © 1993 The Psychometric Society

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Footnotes

This research was supported in part by the Office of Naval Research (N00014-81-C-0569). I would like to thank my assistant Barbara Livingston for help with this manuscript.

References

Lord, F. M. (1983). Unbiased estimators of ability parameters, of their variance, and of their parallel-forms reliability. Psychometrika, 48, 233–245.CrossRefGoogle Scholar
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monography No. 17, 34 (4, Pt. 2).CrossRefGoogle Scholar
Samejima, F. (1972). A general model for free-response data. Psychometrika Monograph No. 18, 37 (1, Pt. 2).Google Scholar
Samejima, F. (1973). Homogeneous case of continuous response level. Psychometrika, 38, 203–219.CrossRefGoogle Scholar
Samejima, F. (1977). A method of estimating item characteristic functions using the maximum likelihood estimate of ability. Psychometrika, 42, 163–191.CrossRefGoogle Scholar
Samejima, F. (1979). Constant information model: A new, promising item characteristic function, Washington, DC: Office of Naval Research.Google Scholar
Samejima, F. (1981). Final Report: EFficient methods of estimating the operating characteristics of item response categories and challenge to a new model for the multiple-choice item, Washington, DC: Office of Naval Research.CrossRefGoogle Scholar
Samejima, F. (1984). Plausibility functions of Iowa Vocabulary Test items estimated by the simple sum procedure of the conditional p.d.f. approach, Washington, DC: Office of Naval Research.Google Scholar
Samejima, F. (1984). Comparison of the estimated item parameters of Shiba's Word/Phrase Comprehension Tests obtained by Logist 5 and those by the tetrachoric method, Washington, DC: Office of Naval Research.Google Scholar
Samejima, F. (1987). Bias function of the maximum likelihood estimate of ability for discrete item responses, Washington, DC: Office of Naval Research.Google Scholar
Samejima, F. (1990). Final Report: Validity study in multidimensional latent space and efficient computerized adaptive testing, Washington, DC: Office of Naval Research.Google Scholar
Samejima, F. (in press). The bias function of the maximum likelihood estimate of ability for the dichotomous response level. Psychometrika.Google Scholar
Samejima, F., Livingston, P. S. (1979). Method of moments as the least squares solution for fitting a polynomial, Washington, DC: Office of Naval Research.Google Scholar
Shiba, S. (1978). Construction of a scale for acquisition of word meanings. Bulletin of Faculty of Education, University of Tokyo, 17. (in Japanese)Google Scholar