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Logistic Positive Exponent Family of Models: Virtue of Asymmetric Item Characteristic Curves

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, Department of Psychology, 405 Austin Peay Bldg., University of Tennessee, Knoxville, TN 37996-0900. E-mail: samejima@psych1.psych.utk.edu
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Abstract

The paper addresses and discusses whether the tradition of accepting point-symmetric item characteristic curves is justified by uncovering the inconsistent relationship between the difficulties of items and the order of maximum likelihood estimates of ability. This inconsistency is intrinsic in models that provide point-symmetric item characteristic curves, and in this paper focus is put on the normal ogive model for observation. It is also questioned if in the logistic model the sufficient statistic has forfeited the rationale that is appropriate to the psychological reality. It is observed that the logistic model can be interpreted as the case in which the inconsistency in ordering the maximum likelihood estimates is degenerated.

The paper proposes a family of models, called the logistic positive exponent family, which provides asymmetric item chacteristic curves. A model in this family has a consistent principle in ordering the maximum likelihood estimates of ability. The family is divided into two subsets each of which has its own principle, and includes the logistic model as a transition from one principle to the other. Rationale and some illustrative examples are given.

Keywords

latent trait modelsitem characteristic curvesdichotomous responsesitem response theorylogistic model
Type
Original Paper
Information
Psychometrika , Volume 65 , Issue 3 , September 2000 , pp. 319 - 335
Copyright
Copyright © 2000 The Psychometric Society

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References

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (Eds.), Statistical theories of mental test scores. Reading, MA: Addison WesleyGoogle Scholar
Levine, M. (1984). An introduction to multilinear formula scoring theory. Champaign, IL: University of ChicagoGoogle Scholar
Lord, F. M., Novick, M. R. (1968). Statistical theories of mental test scores. Reading, MA: Addison WesleyGoogle Scholar
Ramsay, J. O. (1991). Kernel smoothing approaches to nonparametric item characteristic curve estimation. Psychometrika, 56, 611–630CrossRefGoogle Scholar
Ramsay, J. O., Abrahamowicz, M. (1989). Binomial regression with monotone spines: A psychometric application. Journal of the American Statistical Association, 84, 906–915CrossRefGoogle Scholar
Samejima, F. (1969). Estimation of ability using a response pattern of graded scores. Psychometrika Monograph, No. 17.CrossRefGoogle Scholar
Samejima, F. (1972). A general model for free-response data. Psychometrika Monograph, No. 18.Google Scholar
Samejima, F. (1973). A comment on Birnbaum's three-parameter logistic model in the latent trait theory. Psychometrika, 38, 221–233CrossRefGoogle Scholar
Samejima, F. (1995). Acceleration model in the heterogeneous case of the general graded response model. Psychometrika, 60, 549–572CrossRefGoogle Scholar
Samejima, F. (1996). Evaluation of mathematical models for ordered polychotomous responses. Behaviormetrika, 23, 17–35CrossRefGoogle Scholar
Samejima, F. (1998). Efficient nonparametric approaches for estimating the operating characteristics of discrete item responses. Psychometrika, 63, 111–130CrossRefGoogle Scholar
Samejima, F. (1999, April). Usefulness of the logistic positive exponent family of models in educational measurement. Paper presented at the 1999 Meeting of the American Educational Research Association, Montreal, Quebec, Canada.Google Scholar