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Sufficient Statistics and Latent Trait Models

Published online by Cambridge University Press:  01 January 2025

Erling B. Andersen
Erling B. Andersen*
Affiliation:
University of Copenhagen
*
Requests for reprints should be sent to Dr. Erling B. Andersen, Department of Statistics, University of Copenhagen, Studiestraede 6, 1455 Copenhagen, Denmark.
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Abstract

For questionnaires with two answer categories, it has been proven in complete generality that if a minimal sufficient statistic exists for the individual parameter and if it is the same statistic for all values of the item parameters, then the raw score (or the number of correct answers) is the minimal sufficient statistic. It follows that the model must by of the Rasch type with logistic item characteristic curves and equal item-discriminating powers.

This paper extends these results to multiple choice questionnaires. It is shown that the minimal sufficient statistic for the individual parameter is a function of the so-called score vector. It is also shown that the so-called equidistant scoring is the only scoring of a questionnaire that allows for a real valued sufficient statistic that is independent of the item parameters, if a certain ordering property for the sufficient statistic holds.

Keywords

logistic latent traitminimal sufficient statisticRasch modelequidistant scoringsufficiency
Type
Original Paper
Information
Psychometrika , Volume 42 , Issue 1 , March 1977 , pp. 69 - 81
Copyright
Copyright © 1977 The Psychometric Society

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References

Reference Note

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