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A Multidimensional Item Response Model: Constrained Latent Class Analysis Using the Gibbs Sampler and Posterior Predictive Checks

Published online by Cambridge University Press:  01 January 2025

Herbert Hoijtink  and
Ivo W. Molenaar
Herbert Hoijtink*
Affiliation:
Department of Statistics and Measurement Theory, University of Groningen
Ivo W. Molenaar
Affiliation:
Department of Statistics and Measurement Theory, University of Groningen
*
Requests for reprints should be send to Herbert Hoijtink, Grote Kruisstraat 2/1, 9712 TS Groningen, THE NETHERLANDS.
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Abstract

In this paper it will be shown that a certain class of constrained latent class models may be interpreted as a special case of nonparametric multidimensional item response models. The parameters of this latent class model will be estimated using an application of the Gibbs sampler. It will be illustrated that the Gibbs sampler is an excellent tool if inequality constraints have to be taken into consideration when making inferences. Model fit will be investigated using posterior predictive checks. Checks for manifest monotonicity, the agreement between the observed and expected conditional association structure, marginal local homogeneity, and the number of latent classes will be presented.

Keywords

Gibbs samplerposterior predictive checksnonparametric item response theorymultidimensionalmanifest monotonicitylocal homogeneityconditional association
Type
Original Paper
Information
Psychometrika , Volume 62 , Issue 2 , June 1997 , pp. 171 - 189
Copyright
Copyright © 1997 The Psychometric Society

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Footnotes

This paper is supported by grant S40-645 of the Dutch Organization for Scientific Research (NWO).

References

Albert, J. H. (1992). Bayesian estimation of normal ogive item response curves using Gibbs sampling. Journal of Educational Statistics, 17, 251–269.CrossRefGoogle Scholar
Casella, G., George, E. (1992). Explaining the Gibbs sampler. American Statistican, 46, 167–174.CrossRefGoogle Scholar
Croon, M. A. (1990). Latent class analysis with ordered latent classes. British Journal of Mathematical and Statistical Psychology, 43, 171–192.CrossRefGoogle Scholar
Croon, M. A. (1991). Investigating Mokken scalability of dichotomous items by means of ordinal latent class analysis. British Journal of Mathematical and Statistical Psychology, 44, 315–331.CrossRefGoogle Scholar
Ellis, J. L., van den Wollenberg, A. L. (1993). Local homogeneity in latent trait models. A characterization of the homogeneous monotone IRT model. Psychometrika, 58, 417–429.CrossRefGoogle Scholar
Gelfand, A. E., Hills, S. E., Racine-Poon, A., Smith, A. F. M. (1990). Illustration of Bayesian inference in normal data models using Gibbs sampling. Journal of the American Statistical Association, 85, 972–985.CrossRefGoogle Scholar
Gelfand, A. E., Smith, A. F. M. (1990). Sampling-based approaches to calculating marginal densities. Journal of the American Statistical Association, 85, 398–409.CrossRefGoogle Scholar
Gelman, A., Carlin, J. B., Stern, H. S., Rubin, D. B. (1995). Bayesian Data Analysis, London: Chapman and Hall.CrossRefGoogle Scholar
Gelman, A., Meng, X., & Stern, H. S. (in press). Posterior predictive assessment of model fitness via realized discrepancies (with discussion). Statistica Sinica. to appear.Google Scholar
Gelman, A., Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7, 457–511.CrossRefGoogle Scholar
Grayson, D. A. (1988). Two-group classification in latent trait theory: scores with monotone likelihood ratio. Psychometrika, 53, 383–392.CrossRefGoogle Scholar
Holland, P. W. (1981). When are item response models consistent with observed data. Psychometrika, 46, 79–92.CrossRefGoogle Scholar
Holland, P. W., Rosenbaum, P. R. (1986). Conditional association and unidimensionality in monotone latent variable models. The Annals of Statistics, 14, 1523–1543.CrossRefGoogle Scholar
Holm, S. (1979). A simple sequentially rejective multiple test procedure. Scandinavian Journal of Statistics, 6, 65–70.Google Scholar
Junker, B. W. (1991). Essential independence and likelihood-based ability estimation for polytomous items. Psychometrika, 56, 255–278.CrossRefGoogle Scholar
Junker, B. W. (1993). Conditional association, essential independence and monotone unidimensional item response models. The Annals of Statistics, 3, 1359–1378.Google Scholar
Lindsay, B., Clogg, C. C., Grego, J. (1991). Semiparametric estimation in the Rasch model and related exponential response models, including a simple latent class model for item analysis. Journal of the American Statistical Association, 86, 96–107.CrossRefGoogle Scholar
Lord, F. M., Novick, M. R. (1968). Statistical theory of mental test scores, London: Addison-Wesley.Google Scholar
MacEachern, S. N., Berliner, L. M. (1994). Subsampling the Gibbs sampler. The American Statistician, 48, 188–190.CrossRefGoogle Scholar
Meng, X. L. (1994). Posterior Predictive p-Values. The Annals of Statistics, 22, 1142–1160.CrossRefGoogle Scholar
Mokken, R. J. (1971). A theory and procedure of scale analysis, The Hague/Berlin: Mouton/DeGruyler.CrossRefGoogle Scholar
Mokken, R. J., Lewis, C. (1982). A nonparametric approach to the analysis of dichotomous item responses. Applied Psychological Measurement, 6, 417–430.CrossRefGoogle Scholar
Molenaar, I. W. (1996). Nonparametric models for polytomous responses. In van der Linden, W. J., Hambleton, R. K. (Eds.), Handbook of Modern Item Response Theory (pp. 361–372). New York: Springer.Google Scholar
Molenaar, I. W., Debets, P., Sijtsma, K., Hemker, B. T. (1994). User's Manual MSP, Groningen: IecProgamma.Google Scholar
NAG (1992). Foundation library, Oxford: The Numerical Algorithms Group.Google Scholar
Narayanan, A. (1990). Computer generation of Dirichlet random vectors. Journal of Statistical Computations and Simulations, 36, 19–30.CrossRefGoogle Scholar
Ripley, B. D. (1987). Stochastic Simulation, New York: Wiley.CrossRefGoogle Scholar
Rosenbaum, P. R. (1984). Testing the conditional independence and monotonicity assumptions of item response theory. Psychometrika, 49, 425–435.CrossRefGoogle Scholar
Rubin, D. B. (1984). Bayesian justifiable and relevant frequency calculations for the applied statistician. The Annals of Statistics, 12, 1151–1172.CrossRefGoogle Scholar
Rubin, D. B., Stern, H. L. (1993). Testing in latent class models using a posterior predictive check distribution. In von Eye, A., Clogg, C. (Eds.), Latent variables Analysis. Applications for Developmental Research, London: SAGE.Google Scholar
Sanders, K., Hoijtink, H. (1992). Androgynie bestaat (Persons who are both psychologically masculine and feminine exist). Nederlands Tijdschrift voor de Psychologie, 47, 123–133.Google Scholar
Shaffer, J. P. (1994). Multiple hypothesis testing: A review, Research Triangle Park, NC: National Institute of Statistical Sciences.Google Scholar
Shealy, R., Stout, W. (1993). A model based standardization approach that separates true bias/DIF from group ability differences and detects test bias/DTF as well as item bias/DIF. Psychometrika, 58, 159–194.CrossRefGoogle Scholar
Sijtsma, K., Molenaar, I. W. (1987). Reliability of test scores in nonparametric item response theory. Psychometrika, 52, 79–97.CrossRefGoogle Scholar
Smith, A. F. M., Roberts, G. O. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. Journal of the Royal Statistical Society, Series B, 55, 2–23.CrossRefGoogle Scholar
Stout, W. (1987). A nonparametric approach for assessing the latent trait dimensionality. Psychometrika, 52, 589–618.CrossRefGoogle Scholar
Stout, W. F. (1990). A new item response theory modelling approach with applications to unidimensionality assessment and ability estimation. Psychometrika, 55, 293–326.CrossRefGoogle Scholar
Tanner, M. A. (1993). Tools for statistical inference, methods for the explorations of posterior distributions and likelihood functions, New York: Springer.Google Scholar
Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Annals of Statistics, 22, 1701–1762.Google Scholar
Thissen, D., Steinberg, L., Wainer, H. (1988). Use of item response theory in the study of group differences in trace lines. In Wainer, H., Braun, H. I. (Eds.), Test validity (pp. 147–169). Hillsdale, NJ: Erlbaum.Google Scholar
Zeger, S. L., Karim, M. R. (1991). Generalized linear models with random effect; a Gibbs sampling approach. Journal of the American statistical association, 86, 79–86.CrossRefGoogle Scholar
Zellner, A., Min, C. (1995). Gibbs sampler convergence criteria. Journal of the American Statistical Association, 90, 921–927.CrossRefGoogle Scholar