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Statistical Inference for Multiple Choice Tests

Published online by Cambridge University Press:  01 January 2025

John S. J. Hsu ,
Tom Leonard  and
Kam-Wah Tsui
John S. J. Hsu*
Affiliation:
Department of Statistics and Applied Probability, The University of California, Santa Barbara
Tom Leonard
Affiliation:
Department of Statistics, The University of Wisconsin, Madison
Kam-Wah Tsui
Affiliation:
Department of Statistics, The University of Wisconsin, Madison
*
Requests for reprints should be sent to John S.J. Hsu, Department of Statistics and Applied Probability, University of California-Santa Barbara, Santa Barbara, CA 93106.
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Abstract

Finite sample inference procedures are considered for analyzing the observed scores on a multiple choice test with several items, where, for example, the items are dissimilar, or the item responses are correlated. A discrete p-parameter exponential family model leads to a generalized linear model framework and, in a special case, a convenient regression of true score upon observed score. Techniques based upon the likelihood function, Akaike's information criteria (AIC), an approximate Bayesian marginalization procedure based on conditional maximization (BCM), and simulations for exact posterior densities (importance sampling) are used to facilitate finite sample investigations of the average true score, individual true scores, and various probabilities of interest. A simulation study suggests that, when the examinees come from two different populations, the exponential family can adequately generalize Duncan's beta-binomial model. Extensions to regression models, the classical test theory model, and empirical Bayes estimation problems are mentioned. The Duncan, Keats, and Matsumura data sets are used to illustrate potential advantages and flexibility of the exponential family model, and the BCM technique.

Keywords

multiple choice testexponential familylikelihoodAkaike's information criteriongeneralized linear modelBayesian marginalizationimportance samplingregression of true score upon observed scoreclassical test theory model

Information

Type
Article
Information
Psychometrika , Volume 56 , Issue 2 , June 1991 , pp. 327 - 348
Copyright
Copyright © 1991 The Psychometric Society

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Footnotes

The authors wish to thank Ella Mae Matsumura for her data set and helpful comments, Frank Baker for his advice on item response theory, Hirotugu Akaike and Taskin Atilgan, for helpful discussions regarding AIC, Graham Wood for his advice concerning the class of all binomial mixture models, Yiu Ming Chiu for providing useful references and information on tetrachoric models, and the Editor and two referees for suggesting several references and alternative approaches.

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