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A MCMC-Method for Models with Continuous Latent Responses

Published online by Cambridge University Press:  01 January 2025

Gunter Maris  and
Eric Maris
Gunter Maris*
Affiliation:
Department of Mathematical Psychology, Nijmegen Institute for Cognition and Information (NICI), University of Nijmegen
Eric Maris
Affiliation:
Department of Mathematical Psychology, Nijmegen Institute for Cognition and Information (NICI), University of Nijmegen
*
Requests for reprints should be sent to Gunter Maris, Department of Mathematical Psychology, Nijmegen Institute for Cognition and Information, University of Nijmegen, P.O. Box 9104, 6500 HE Nijmegen, THE NETHERLANDS. E-Mail: Gunter.Maris@citogroep.nl
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Abstract

This paper introduces a new technique for estimating the parameters of models with continuous latent data. Using the Rasch model as an example, it is shown that existing Bayesian techniques for parameter estimation, such as the Gibbs sampler, are not always easy to implement. Then, a new sampling-based Bayesian technique, called the DA-T-Gibbs sampler, is introduced. The DA-T-Gibbs sampler relies on the particular latent data structure of latent response models to simplify the computations involved in parameter estimation.

Keywords

MCMC methodsGibbs sampling

Information

Type
Articles
Information
Psychometrika , Volume 67 , Issue 3 , September 2002 , pp. 335 - 350
Copyright
Copyright © 2002 The Psychometric Society

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Footnotes

This research was supported by the Dutch National Research Council (NWO) (grant number 575-30-001).

References

Albert, J.H. (1992). Bayesian estimation of normal ogive item response curves using Gibbs sampling. Journal of Educational Statistics, 17(3), 251–269.CrossRefGoogle Scholar
Albert, J.H., & Chib, S. (1993). Bayesian analysis of binary and polytomous response data. Journal of the American Statistical Association, 88(422), 669–679.CrossRefGoogle Scholar
Chib, S., & Greenberg, E. (1995). Understanding the Metropolis-Hastings algorithm. The American Statistician, 49(4), 327–335.CrossRefGoogle Scholar
Devroye, L. (1986). Non-uniform random variate generation. New York, NY: Springer-Verlag.CrossRefGoogle Scholar
Embretson, S. (1980). Multicomponent latent trait models for ability tests. Psychometrika, 45, 479–494.Google Scholar
Embretson, S. (1984). A general latent trait model for response processes. Psychometrika, 49, 175–186.CrossRefGoogle Scholar
Gelman, A., Carlin, J.B., Stern, H.S., & Rubin, D.B. (1995). Bayesian data analysis. Chapman & Hall.CrossRefGoogle Scholar
Gelman, A., & Rubin, D.B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7(4), 457–472.CrossRefGoogle Scholar
Geman, S., & Geman, D. (1984). Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.CrossRefGoogle ScholarPubMed
Gilks, W., & Wild, P. (1992). Adaptive rejection sampling for Gibbs sampling. Applied Statistics, 41, 337–348.CrossRefGoogle Scholar
Hastings, W. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97–109.CrossRefGoogle Scholar
Hoijtink, H. (1990). PARELLA, measurement of latent traits by proximity items. Leiden, Netherlands: DSWO Press.Google Scholar
Liu, J. (1991). Correlation structure and convergence rate of the Gibbs sampler. Unpublished doctoral dissertation, The University of Chicago.Google Scholar
Maris, E. (1995). Psychometric latent response models. Psychometrika, 60, 523–547.CrossRefGoogle Scholar
Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., & Teller, E. (1953). Equations of state calculations by fast computing machines. Journal of Chemical Physics, 21, 1087–1091.CrossRefGoogle Scholar
Molenaar, I.W. (1995). Estimation of item parameters. In Fischer, G.H., & Molenaar, I.W. (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 39–51). New York, NY: Springer-Verlag.CrossRefGoogle Scholar
Mood, A.M., Graybill, F.A., & Boes, D.C. (1974). Introduction to the theory of statistics 3rd ed., Singapore: McGraw-Hill.Google Scholar
Patz, R.J., & Junker, B.W. (1999). A straightforward approach to Markov Chain Monte Carlo methods for item response models. Journal of Educational and Behavioral Statistics, 24(2), 146–178.CrossRefGoogle Scholar
Rasch, G. (1980). Probabilistic models for some intelligence and attainment tests. Chicago, IL: The University of Chicago Press.Google Scholar
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph Number, 17 4,Pt.2.Google Scholar
Tanner, M.A. (1996). Tools for statistical inference 3rd ed., New York, NY: Springer-Verlag.CrossRefGoogle Scholar
Thissen, D. (1982). Marginal maximum likelihood estimation for the one-parameter logistic model. Psychometrika, 47, 175–186.CrossRefGoogle Scholar
Tierney, L. (1994). Markov chains for exploring posterior distributions. The Annals of Statistics, 22(4), 1701–1762.Google Scholar
Thurstone, L.L. (1927). A law of comparative judgment. Psychological Review, 34, 273–286.CrossRefGoogle Scholar