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Bayes Modal Estimation in Item Response Models

Published online by Cambridge University Press:  01 January 2025

Robert J. Mislevy
Robert J. Mislevy*
Affiliation:
Educational Testing Service
*
Requests for reprints should be sent to Robert J. Mislevy, Educational Testing Service, Princeton, NJ 08541.
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Abstract

This article describes a Bayesian framework for estimation in item response models, with two-stage prior distributions on both item and examinee populations. Strategies for point and interval estimation are discussed, and a general procedure based on the EM algorithm is presented. Details are given for implementation under one-, two-, and three-parameter binary logistic IRT models. Novel features include minimally restrictive assumptions about examinee distributions and the exploitation of dependence among item parameters in a population of interest. Improved estimation in a moderately small sample is demonstrated with simulated data.

Keywords

Bayesian estimationEM algorithmhierarchical prior distributionsitem response modelsmarginal maximum likelihood
Type
Original Paper
Information
Psychometrika , Volume 51 , Issue 2 , June 1986 , pp. 177 - 195
Copyright
Copyright © 1986 The Psychometric Society

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Footnotes

This research was supported by a grant from the Spencer Foundation, Chicago, IL. Comments and suggestions on earlier drafts by Charles Lewis, Frederic Lord, Rosenbaum, James Ramsey, Hiroshi Watanabe, the editor, and two anonymous referees are gratefully acknowledged.

References

Andersen, E. B. (1973). Conditional inference and models for measuring, Copenhagen: Danish Institute for Mental Health.Google Scholar
Andersen, E. B., Madsen, M. (1977). Estimating the parameters of a latent population distribution. Psychometrika, 42, 357–374.CrossRefGoogle Scholar
Ando, A., Kaufman, O. M. (1965). Bayesian analysis of the independent normal process—neither mean nor precision known. Journal of the American Statistical Association, 60, 347–358.Google Scholar
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (Eds.), Statistical theories of mental test scores, Reading, MA: Addison-Wesley.Google Scholar
Bock, R. D., Aitken, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443–459.CrossRefGoogle Scholar
Bock, R. D., Mislevy, R. J. (1982). Adaptive EAP estimation of ability in a microcomputer environment. Applied Psychological Measurement, 6, 431–444.CrossRefGoogle Scholar
Dempster, A. P., Laird, N. M., Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society (Series B), 39, 1–38.CrossRefGoogle Scholar
Dunsmore, I. R. (1976). Asymptotic predictor analysis. Biometrika, 63, 627–630.CrossRefGoogle Scholar
Efron, B., Norris, C. (1975). Data analysis using Stein's estimator and its generalizations. Journal of the American Statistical Association, 70, 311–319.CrossRefGoogle Scholar
Fisher, G. (1973). The linear logistic test model as an instrument in educational research. Acta Psychologica, 37, 359–374.CrossRefGoogle Scholar
Hartigan, J. A. (1983). Bayes theory, New York: Springer-Verlang.CrossRefGoogle Scholar
James, W., Stein, C. (1961). Estimation with quadratic loss. Proceedings of the Fourth Berkeley Symposium on Mathematical Probability and Statistics (Vol. 1), Berkeley: University of California Press.Google Scholar
Jeffreys, H. (1961). Theory of probability 3rd ed.,, Oxford: Clarendon Press.Google Scholar
Kelley, T. L. (1927). The interpretation of educational of educational measurements, New York: World Press.Google Scholar
Leonard, T., Novick, M. R. (1985). Bayesian inference and diagnostics for the three-parameter logistic model, Iowa City, IA: University of Iowa.Google Scholar
Lewis, C. (1985). Estimating individual abilities with imperfectly known item response functions. Paper presented at the meeting of the Psychometric Society, Nashville, TN.Google Scholar
Lindley, D. V., Smith, A. F. M. (1972). Bayes estimates for the linear model. Journal of the Royal Statistical Society (Series B), 34, 1–41.CrossRefGoogle Scholar
Lord, F. M. (1952). A theory of test scores. Psychometrika Monograph No. 7, 17(4, Pt. 2).Google Scholar
Lord, F. M. (1975). Evaluation with artificial data of a procedure for estimating ability and item characteristic curve parameters (RB-75-33), Princeton, NJ: Educational Testing Service.Google Scholar
Lord, F. M. (1980). Applications of item response theory to practical testing problems, Hillsdale, NJ: Erlbaum.Google Scholar
Mislevy, R. J. (1984). Estimating latent distributions. Psychometrika, 49, 359–381.CrossRefGoogle Scholar
Mislevy, R. J. (in preparation). Stochastic test designs (Research Report). Princeton, NJ: Educational Testing Service.Google Scholar
Mislevy, R. J., Bock, R. D. (1982). BILOG: Item analysis and test scoring with binary logistic models [Computer program], Mooresville, IN: Scientific Software.Google Scholar
Mislevy, R. J., Bock, R. D. (1985). Implementation of anEM algorithm in the estimation of item parameters. In Weiss, D. J. (Eds.), Proceedings of the IRT/CAT Conference, Minneapolis, MN: Computerized Adaptive Testing Laboratory, University of Minnesota.Google Scholar
Morgan, G. (1985). Computing approximate marginal distributions in the Rasch logistic test model using prior information. Paper presented at the annual meeting of the American Educational Research Association in Chicago, IL.Google Scholar
Neyman, J., Scott, E. L. (1948). Consistent estimates based on partially consistent observations. Econometrica, 16, 1–32.CrossRefGoogle Scholar
Novick, M. R., Jackson, P. H., Thayer, D. T., Cole, N. S. (1972). Estimating multiple regressions inm-groups: A cross-validational study. British Journal of Mathematical and Statistical Psychology, 5, 33–50.CrossRefGoogle Scholar
O'Hagan, A. (1976). On posterior joint and marginal modes. Biometrika, 63, 329–333.CrossRefGoogle Scholar
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests, Copenhagen: Danish Institute for Educational Research.Google Scholar
Reiser, M. R. (1981, June). Bayesian estimation of item parameters in the two-parameter logistic model. Paper presented at the annual meeting of the Psychometric Society in Chapel Hill, NC.Google Scholar
Rigdon, S., Tsutakawa, R. K. (1983). Parameter estimation in latent trait models. Psychometrika, 48, 567–574.CrossRefGoogle Scholar
Rubin, D. B. (1976). Inference and missing data. Biometrika, 63, 581–592.CrossRefGoogle Scholar
Rubin, D. B. (1980). Using empirical Bayes techniques in the law school validity studies. Journal of the American Statistical Society, 75, 801–827.CrossRefGoogle Scholar
Sanathanan, L., Blumenthal, N. (1978). The logistic model and latent structure. Journal of the American Statistical Association, 73, 794–798.CrossRefGoogle Scholar
Stroud, A. H., Sechrest, D. (1966). Gaussian quadrature formulas, Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
Swaminathan, H., & Gifford, J. A. (1981). Bayesian estimation in the three-parameter logistic model. Paper presented at the annual meeting of the Psychometric Society, Chapel Hill, NC.Google Scholar
Swaminathan, H., Gifford, J. A. (1982). Bayesian estimation in the Rasch model. Journal of Educational Statistics, 7, 175–192.CrossRefGoogle Scholar
Swaminathan, H., Gifford, H. A. (1985). Bayesian estimation in the two-parameter logistic model. Psychometrika, 50, 349–364.CrossRefGoogle Scholar
Thissen, D. (1982). Marginal maximum likelihood estimation for the one-parameter logistic model. Psychometrika, 47, 175–186.CrossRefGoogle Scholar
Wainer, H., Thissen, D. (1982). Some standard errors in item response theory. Psychometrika, 47, 397–412.Google Scholar