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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

Information

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

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