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Nonunique Solutions to the Likelihood Equation for the Three-Parameter Logistic Model

Published online by Cambridge University Press:  01 January 2025

Wendy M. Yen ,
George R. Burket  and
Robert C. Sykes
Wendy M. Yen*
Affiliation:
CTB Macmillan/McGraw-Hill
George R. Burket
Affiliation:
CTB Macmillan/McGraw-Hill
Robert C. Sykes
Affiliation:
CTB Macmillan/McGraw-Hill
*
Requests for reprints should be sent to Wendy M. Yen, CTB Macmillan/McGraw-Hill, 2500 Garden Road, Monterey, CA 93940.
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Abstract

Samejima identified the possibility of multiple solutions to the likelihood equation (multiple maxima in the likelihood function) for estimating an examinee's trait value for the three-parameter logistic model. In the practical applications that Lord studied, he found that multiple solutions did not occur when the number of items was ≥20. In the present paper, fourteen multiple-choice achievement tests with from 20 to 50 items were examined to see if it was possible for them to produce item response vectors with multiple maxima; such vectors were found for all the tests. Examination of response vectors for large groups of real examinees found that from 0 to 3.1% of them had response vectors with multiple maxima. The implications of these results for multiple-choice tests are discussed.

Keywords

item response theorymaximum likelihood estimationtests
Type
Original Paper
Information
Psychometrika , Volume 56 , Issue 1 , March 1991 , pp. 39 - 54
Copyright
Copyright © 1991 The Psychometric Society

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