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Estimating Latent Distributions

Published online by Cambridge University Press:  01 January 2025

Robert J. Mislevy
Robert J. Mislevy*
Affiliation:
National Opinion Research Center
*
Requests for reprints should be addressed to Dr. Robert Mislevy, Educational Testing Service, Princeton, N.J. 08540
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Abstract

Consider vectors of item responses obtained from a sample of subjects from a population in which ability θ is distributed with density g (θ‖α), where the α are unknown parameters. Assuming the responses depend on θ through a fully specified item response model, this paper presents maximum likelihood equations for the estimation of the population parameters directly from the observed responses; i.e., without estimating an ability parameter for each subject. Also provided are asymptotic standard errors and tests of fit, computing approximations, and details of four special cases: a non-parametric approximation, a normal solution, a resolution of normal components, and a beta-binomial solution.

Keywords

latent distributionsmaximum likelihoodEM algorithmempirical Bayes estimationGaussian resolutionbeta-binomial modelitem response theory
Type
Original Paper
Information
Psychometrika , Volume 49 , Issue 3 , September 1984 , pp. 359 - 381
Copyright
Copyright © 1984 The Psychometric Society

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Footnotes

The author would like to thank R. Darrell Bock for his comments, suggestions, and encouragement during the course of this work.

References

Andersen, E. B. (1972). The numerical solution of a set of conditional estimation equations. Journal of the Royal Statistical Society, 34, 42–54.CrossRefGoogle Scholar
Andersen, E. B. and Madsen, M. (1977). Estimating the parameters of a latent population distribution. Psychometrika, 42, 357–374.CrossRefGoogle Scholar
Balker, J. T., Krasnoff, A. G. and Peaco, D. (1981). Visuo-spatial perception in adolescents and their parents: theX-linked recessive hypothesis. Behavior Genetics, 11, 403–413.Google Scholar
Bock, R. D. (1983). The Discrete Bayesian. In Wainer, H. and Messick, S. (Eds.), Principals of Modern Psychological Measurement, Hillsdale, New Jersey: Erlbaum.Google Scholar
Bock, R. D. and Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: an application of an EM algorithm. Psychometrika, 46, 443–459.CrossRefGoogle Scholar
Bock, R. D. and Kolakowski, D. F. (1973). Further evidence of sex-linked major gene influence on human spatial visualizing ability. American Journal of Human Genetics, 25, 1–14.Google ScholarPubMed
Bock, R. D. and Lieberman, M. (1970). Fitting a response model forn dichotomously scored items. Psychometrika, 35, 179–197.CrossRefGoogle Scholar
Christofferson, A. (1975). Factor analysis of dichotomized variables. Psychometrika, 40, 5–32.CrossRefGoogle Scholar
Day, N. E. (1969). Estimating the components of a mixture of normal distributions. Biometrika, 56, 463–473.CrossRefGoogle Scholar
Deely, J. J. and Lindley, D. V. (1981). Bayes empirical Bayes. Journal of the American Statistical Association, 76, 833–841.CrossRefGoogle Scholar
Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of Royal Statistical Society, 39, 1–38.CrossRefGoogle Scholar
Dempster, A. M., Rubin, D. B. and Tsutakawa, R. K. (1981). Estimation in Covariance Components Models. Journal of the American Statistical Association, 76, 341–353.CrossRefGoogle Scholar
Kendall, M. and Stuart, A. (1979). The Advanced Theory of Statistics, Volume II, New York: Macmillan.Google Scholar
Laird, N. M. (1978). Nonparametric maximum likelihood estimation of a mixing distribution. Journal of the American Statistical Association, 73, 805–811.CrossRefGoogle Scholar
Lord, F. M. (1969). Estimating true-score distributions in psychological testing (An empirical Bayes problem). Psychometrika, 34, 259–299.CrossRefGoogle Scholar
Lord, F. M. and Novick, M. R. (1968). Statistical Theories of Mental Test Scores, Reading, Mass.: Addison-Wesley.Google Scholar
Louis, T. A. (1982). Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society, 44, 226–233.CrossRefGoogle Scholar
Muthén, B. (1978). Contributions to factor analysis of dichotomous variables. Psychometrika, 43, 551–560.CrossRefGoogle Scholar
O'Connor, J. (1948). Structural Visualization, Boston: Human Engineering Laboratory.Google Scholar
Ramsay, J. O. (1975). Solving implicit equations in psychometric data analysis. Psychometrika, 40, 361–372.CrossRefGoogle Scholar
Rasch, G. (1980). Probabilistic Models for Some Intelligence and Attainment Tests, CopenhagenChicago: Danish Institute for Educational Research University of Chicago Press.Google Scholar
Rigdon, S. E. and Tsutakawa, R. K. (1983). Parameter estimation in latent trait models. Psychometrika, 48, 567–574.CrossRefGoogle Scholar
Sanathanan, L. and Blumenthal, N. (1978). The logistic model and estimation of latent structure. Journal of the American Statistical Association, 73, 794–798.CrossRefGoogle Scholar
Sörbom, D. (1974). A general method for studying differences in factor means and factor structures between groups. British Journal of Mathematical and Statistical Psychology, 37, 222–239.Google Scholar
Stroud, A. H. and Sechrest, D. (1966). Gaussian Quadrature Formulas, Englewood Cliffs, New Jersey: Prentice-Hall.Google Scholar
Zimowski, M. (in progress). Implications of item difficulties for visuo-spatial information processing. Doctoral dissertation, Department of Behavioral Sciences, University of Chicago.Google Scholar