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Characterizing the manifest probabilities of latent trait models

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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.

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  • 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), pp. 443–459.

    Article  Google Scholar 

  • Cressie, N. A useful empirical Bayes identity.The Annals of Statistics. (in press)

  • Holland, P. W. When are item response theory models consistent with observed data?Psychometrika, 1981,46(1), pp. 79–92.

    Article  Google 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, Surveys No. 1, 1943.

    Google 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, Preprint No. 6, 1980.

    Google Scholar 

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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.

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Cressie, N., Holland, P.W. Characterizing the manifest probabilities of latent trait models. Psychometrika 48, 129–141 (1983).

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