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Characterizing the Manifest Probabilities of Latent Trait Models

Published online by Cambridge University Press:  01 January 2025

Noel Cressie  and
Paul W. Holland
Noel Cressie
Affiliation:
Educational Testing Service
Paul W. Holland*
Affiliation:
Educational Testing Service
*
Requests for reprints should be sent to Paul W. Holland, ETS, Rosedale Road, Princeton, NJ 08541.
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Abstract

The problem of characterizing the manifest probabilities of a latent trait model is considered. The item characteristic curve is transformed to the item passing-odds curve and a corresponding transformation is made on the distribution of ability. This results in a useful expression for the manifest probabilities of any latent trait model. The result is then applied to give a characterization of the Rasch model as a log-linear model for a 2J- contingency table. Partial results are also obtained for other models. The question of the identifiability of “guessing” parameters is also discussed.

Keywords

Item Response TheoryRasch modelICClog-linear
Type
Article
Information
Psychometrika , Volume 48 , Issue 1 , March 1983 , pp. 129 - 141
Copyright
Copyright © 1983 The Psychometric Society

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Footnotes

The research reported here is collaborative in every respect and the order of authorship is alphabetical. Dr. Cressie was a Visiting Research Scientist at ETS during the Fall of 1980. His current address is: School of Mathematical Sciences, The Flinders University of South Australia, Bedford Park SA, 5042, AUSTRALIA. The preparation of this paper was supported, in part, by the Program Statistics Research Project in the Research Statistics Group at ETS.

References

Bishop, Y. M. M., Fienberg, S. E. & Holland, P. W. Discrete Multivariate Analysis: Theory and Practice, Cambridge, MA: MIT Press, 1975.Google Scholar
Bock, R. D. & Aitkin, M. Marginal maximum likelihood estimation of item parameters: an application of the EM algorithm. Psychometrika, 1981, 46(4), 443–459.CrossRefGoogle Scholar
Cressie, N. A useful empirical Bayes identity. The Annals of Statistics. (in press)Google Scholar
Holland, P. W. When are item response theory models consistent with observed data?. Psychometrika, 1981, 46(1), 79–92.CrossRefGoogle Scholar
Karlin, S. & Studden, W. J. Tchebycheff Systems: with applications in analysis and statistics, New York: John Wiley and Sons, 1966.Google Scholar
Lord, F. M. Applications of Item Response Theory to Practical Testing Problems, Hillsdale, N.J.: Lawrence Erlbaum Associates, 1980.Google Scholar
Lord, F. M. & Novick, M. R. Statistical Theories of Mental Test Scores, Reading, MA: Addison-Wesley, 1968.Google Scholar
Rasch, G. Probabilistic models for some intelligence and attainment tests, Copenhagen, Denmark: Danish Institute for Educational Research, 1960.Google Scholar
Shohat, J. H. & Tamarkin, J. D. The Problem of Moments, New York: American Mathematics Society, 1943.CrossRefGoogle Scholar
Tjur, T. A connection between Rasch’s item analysis model and a multiplicative Poisson model, Copenhagen, Denmark: University of Copenhagen, Institute of Mathematical Statistics, 1980.Google Scholar