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Fitting and Testing Conditional Multinormal Partial Credit Models

Published online by Cambridge University Press:  01 January 2025

David J. Hessen
David J. Hessen*
Affiliation:
Utrecht University
*
Requests for reprints should be sent to David J. Hessen, Department of Methodology and Statistics, Utrecht University, Heidelberglaan 1, PO Box 80.140, 3508 TC Utrecht, The Netherlands. E-mail: [email protected]
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Abstract

A multinormal partial credit model for factor analysis of polytomously scored items with ordered response categories is derived using an extension of the Dutch Identity (Holland in Psychometrika 55:5–18, 1990). In the model, latent variables are assumed to have a multivariate normal distribution conditional on unweighted sums of item scores, which are sufficient statistics. Attention is paid to maximum likelihood estimation of item parameters, multivariate moments of latent variables, and person parameters. It is shown that the maximum likelihood estimates can be found without the use of numerical integration techniques. More general models are discussed which can be used for testing the model, and it is shown how models with different numbers of latent variables can be tested against each other. In addition, multi-group extensions are proposed, which can be used for testing both measurement invariance and latent population differences. Models and procedures discussed are demonstrated in an empirical data example.

Keywords

partial credit modelmultivariate normal distributionfactor analysismulti-group modelmeasurement invariance
Type
Original Paper
Information
Psychometrika , Volume 77 , Issue 4 , October 2012 , pp. 693 - 709
Copyright
Copyright © 2012 The Psychometric Society

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References

Agresti, A. (1993). Computing conditional maximum likelihood estimates for generalized Rasch models using simple loglinear models with diagonals parameters. Scandinavian Journal of Statistics, 20, 63–71.Google Scholar
Andersen, E.B. (1972). The numerical solution of a set of conditional estimation equations. Journal of the Royal Statistical Society. Series B, 34, 42–54.CrossRefGoogle Scholar
Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 561–574.CrossRefGoogle Scholar
Bartholomew, D.J., & Knott, M. (1999). Latent variable models and factor analysis. London: Arnold.Google Scholar
Bock, R.D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: application of an EM algorithm. Psychometrika, 46, 443–459.CrossRefGoogle Scholar
Bock, R.D., Gibbons, R., & Muraki, E. (1988). Full-information item factor analysis. Applied Psychological Measurement, 12, 261–280.CrossRefGoogle Scholar
Bock, R.D., & Lieberman, M. (1970). Fitting a response model for n dichotomously scored items. Psychometrika, 35, 179–197.CrossRefGoogle Scholar
Byrne, B.M., Shavelson, R.J., & Muthén, B. (1989). Testing for the equivalence of factor covariance and mean structures: the issue of partial measurement invariance. Psychological Bulletin, 105, 456–466.CrossRefGoogle Scholar
Cressie, N., & Holland, P.W. (1983). Characterizing the manifest probabilities of latent trait models. Psychometrika, 48, 129–141.CrossRefGoogle Scholar
Dekovic, M. (2003). Aggressive and nonaggressive antisocial behavior in adolescence. Psychological Reports, 93, 610–616.CrossRefGoogle ScholarPubMed
Holland, P.W. (1990). The Dutch Identity: a new tool for the study of item response models. Psychometrika, 55, 5–18.CrossRefGoogle Scholar
Kelderman, H. (1992). Computing maximum likelihood estimates of loglinear models from marginal sums with special attention to loglinear item response theory. Psychometrika, 57, 437–450.CrossRefGoogle Scholar
Kelderman, H., & Rijkes, C.P.M. (1994). Loglinear multidimensional IRT models for polytomously scored items. Psychometrika, 59, 149–176.CrossRefGoogle Scholar
Kiefer, J., & Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Annals of Statistics, 27, 805–811.Google Scholar
Li, Z. (2010). Loglinear models as item response models. Unpublished doctoral dissertation. University of Illinois at Urbana-Champaign. .Google Scholar
Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.CrossRefGoogle Scholar
Meredith, W. (1993). Measurement invariance, factor analysis and factorial invariance. Psychometrika, 58, 525–543.CrossRefGoogle Scholar
Meredith, W., & Millsap, R.E. (1992). On the misuse of manifest variables in the detection of measurement bias. Psychometrika, 57, 289–311.CrossRefGoogle Scholar
Millsap, R.E., & Meredith, W. (1992). Inferential conditions in the statistical detection of measurement bias. Applied Psychological Measurement, 16, 389–402.CrossRefGoogle Scholar
Muraki, E. (1992). A generalized partial credit model: application of an EM algorithm. Applied Psychological Measurement, 16, 159–176.CrossRefGoogle Scholar
Muraki, E. (1993). Information functions of the generalized partial credit model. Applied Psychological Measurement, 17, 351–363.CrossRefGoogle Scholar
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.Google Scholar
Rasch, G. (1966). An item analysis that takes individual differences into account. British Journal of Mathematical & Statistical Psychology, 19, 49–57.CrossRefGoogle ScholarPubMed
Reckase, M.D. (1985). The difficulty of test items that measure more than one ability. Applied Psychological Measurement, 9, 401–412.CrossRefGoogle Scholar
Reckase, M.D. (2009). Multidimensional item response theory. New York: Springer.CrossRefGoogle Scholar
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Richmond: The Psychometric Society.CrossRefGoogle Scholar
Samejima, F. (1997). Graded response model. In van der Linden, W.J., & Hambleton, R.K. (Eds.), Handbook of modern item response theory (pp. 85–100). New York: Springer.CrossRefGoogle Scholar
Schilling, S., & Bock, R.D. (2005). High-dimensional maximum marginal likelihood item factor analysis by adaptive quadrature. Psychometrika, 70, 533–555.Google Scholar
Silvapulle, M.J., & Sen, P.K. (2005). Constrained statistical inference: inequality, order, and shape restrictions. Hoboken: Wiley.Google Scholar
Tiet, Q.Q., Wasserman, G.A., Loeber, R., McReynolds, L.S., & Miller, L.S. (2001). Development and sex differences in types of conduct problems. Journal of Child and Family Studies, 10, 181–197.CrossRefGoogle Scholar
Tjur, T. (1982). A connection between Rasch’s item analysis model and a multiplicative Poisson model. Scandinavian Journal of Statistics, 9, 23–30.Google Scholar
Wirth, R.J., & Edwards, M.C. (2007). Item factor analysis: current approaches and future directions. Psychological Methods, 12, 58–79.CrossRefGoogle ScholarPubMed