Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-09T21:58:44.011Z Has data issue: false hasContentIssue false
  • English
  • Français

Loglinear Multidimensional IRT Models for Polytomously Scored Items

Published online by Cambridge University Press:  01 January 2025

Henk Kelderman  and
Carl P. M. Rijkes
Henk Kelderman*
Affiliation:
University of Twente
Carl P. M. Rijkes
Affiliation:
University of Twente
*
Requests for reprints should be sent to H. Kelderman, Department of Work and Organizational Psychology, Faculty of Psychology and Pedagogics, Vrije Universiteit, De Boelelaan 1081c, NL 1081 HV Amsterdam, THE NETHERLANDS.
Get access Rights & Permissions [Opens in a new window]

Abstract

A loglinear IRT model is proposed that relates polytomously scored item responses to a multidimensional latent space. The analyst may specify a response function for each response, indicating which latent abilities are necessary to arrive at that response. Each item may have a different number of response categories, so that free response items are more easily analyzed. Conditional maximum likelihood estimates are derived and the models may be tested generally or against alternative loglinear IRT models.

Keywords

multidimensional item response theoryloglinear modelRasch modelmultidimensional Rasch modelpolytomous responsespartial credit modelgoodness-of-fit testing

Information

Type
Original Paper
Information
Psychometrika , Volume 59 , Issue 2 , June 1994 , pp. 149 - 176
Copyright
Copyright © 1994 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

1

Hank Kelderman is currently affiliated with Vrije Universiteit, Amsterdam.

We thank Linda Vodegel-Matzen of the Division of Developmental Psychology of the University of Amsterdam for making available the data used in the example in this article.

References

Akaike, H. (1977). On entropy maximization principle. In Krisschnaiah, P. R. (Eds.), Applications of statistics (pp. 27–41). Amsterdam: North Holland.Google Scholar
Andersen, E. B. (1973). Conditional inference and multiple choice questionnaires. British Journal of Mathematical and Statistical Psychology, 26, 31–44.CrossRefGoogle Scholar
Andersen, E. B. (1983). A general latent structure model for contingency table data. In Wainer, H., Messick, S. (Eds.), Principals of modern psychological measurement (pp. 117–138). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
Andrich, D. (1978). A rating scale formulation for ordered response categories. Psychometrika, 43, 561–573.CrossRefGoogle Scholar
Andrich, D. (1982). An extension of the Rasch model for ratings providing both location and dispersion parameters. Psychometrika, 47, 105–113.CrossRefGoogle Scholar
Baglivo, J., Olivier, D., Pagano, M. (1992). Methods for exact goodness-of-fit tests. Journal of the American Statistical Association, 87, 464–469.CrossRefGoogle Scholar
Bishop, Y. M. M., Fienberg, S. E., Holland, P. W. (1975). Discrete multivariate analysis, Cambridge, MA: MIT Press.Google Scholar
Bock, R. D. (1972). Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika, 37, 29–51.CrossRefGoogle Scholar
Carpenter, P. A., Just, M. A., Shell, P. (1990). What one intelligence test measures: A theoretical account of the processing in the Raven Progressive Matrices Test. Psychological Review, 97, 404–431.CrossRefGoogle Scholar
Cox, M. A. A., Placket, R. L. (1980). Small samples in contingency tables. Biometrika, 67, 1–13.CrossRefGoogle Scholar
Cressie, N., Holland, P. W. (1983). Characterizing the manifest probabilities of latent trait models. Psychometrika, 48, 129–142.CrossRefGoogle Scholar
de Leeuw, J., Verhelst, N. D. (1986). Maximum likelihood estimation in generalized Rasch models. Journal of Educational Statistics, 11, 183–196.CrossRefGoogle Scholar
Duncan, O. D. (1984). Rasch measurement: Further examples and discussion. In Turner, C. F., Martin, E. (Eds.), Surveying subjective phenomena, Vol. 2 (pp. 367–403). New York: Russell Sage Foundation.Google Scholar
Duncan, O. D., Stenbeck, M. (1987). Are Likert scales unidimensional?. Social Science Research, 16, 245–259.CrossRefGoogle Scholar
Embretson, S. E. (1984). A general latent trait model for response processes. Psychometrika, 49, 175–186.CrossRefGoogle Scholar
Embretson, S. E. (1985). Multicomponent latent trait models for test design. In Embretson, S. E. (Eds.), Test design: Developments in psychology and psychometrics (pp. 195–218). Orlando, FL: Academic Press.CrossRefGoogle Scholar
Embretson, S. E. (1991). Measuring and validating the cognitive modifiability construct. Poster Presentation at the Annual Meeting of the American Educational Research Association.Google Scholar
Fischer, G. H. (1972). A measurement model for the effect of mass-media. Acta Psychologica, 36, 207–220.CrossRefGoogle Scholar
Fischer, G. H. (1973). The linear logistic test model as an instrument in educational research. Acta Psychologica, 36, 359–374.CrossRefGoogle Scholar
Fischer, G. H. (1974). Einführung in die Theorie psychologischer Tests [Introduction to the theory of psychological test], Bern: Huber.Google Scholar
Fischer, G. H. (1976). Some probabilistic models for measuring change. In de Gruyter, D. N. M., van der Kamp, L. J. Th. (Eds.), Advances in psychological and educational measurement (pp. 97–110). New York: Wiley.Google Scholar
Fischer, G. H., Forman, A. K. (1982). Some applications of logistic latent trait models with linear constraints on the parameters. Applied Psychological Measurement, 6, 397–416.CrossRefGoogle Scholar
Forbes, A. R. (1964). An item analysis of the advanced matrices. British Journal of Educational Psychology, 34, 1–14.CrossRefGoogle Scholar
Frederiksen, J. R. (1982). A componential theory of reading skills and their interactions. In Sternberg, R. J. (Eds.), Advances in the psychology of human intelligence Vol. 1 (pp. 125–180). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
Glas, CA. W., Verhelst, N. D. (1989). Extensions of the Partial Credit Model. Psychometrika, 54, 635–660.CrossRefGoogle Scholar
Haberman, S. J. (1977). Log-linear models and frequency tables with small cell counts. Annals of Statistics, 5, 1124–1147.CrossRefGoogle Scholar
Haberman, S. J. (1979). Analysis of qualitative data: New developments, Vol. 2, New York: Academic Press.Google Scholar
Hunt, E. B. (1974). Quote the Raven? Nevermore!. In Greg, L. W. (Eds.), Knowledge and cognition (pp. 129–158). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
Imrey, P. B., Koch, G. C., Stokes, M. E. (1981). Categorical data analysis: Some reflections on the loglinear model and logistic regression. Part I: Historical and methodological overview. International Statistical Overview, 49, 265–283.Google Scholar
Kelderman, H. (1984). Loglinear Rasch model tests. Psychometrika, 49, 223–245.CrossRefGoogle Scholar
Kelderman, H. (1989). Item bias detection using loglinear IRT. Psychometrika, 54, 681–697.CrossRefGoogle Scholar
Kelderman, H. (1992). Computing maximum likelihood estimates of loglinear IRT models from marginal sums. Psychometrika, 57, 437–450.CrossRefGoogle Scholar
Kelderman, H., Steen, R. (1988). Logimo: Loglinear IRT modeling, Enschede, The Netherlands: University of Twente.Google Scholar
Koehler, K. J. (1977). Goodness-of-fit statistics for large sparse multinomials. Unpublished doctoral dissertation, University of Minnesota, School of Statistics.Google Scholar
Koehler, K. J. (1986). Goodness-of-fit tests for log-linear models in sparse contingency tables. Journal of the American Statistical Association, 81, 483–493.CrossRefGoogle Scholar
Lancaster, H. O. (1961). Significance tests in discrete distributions. Journal of the American Statistical Association, 56, 223–234.CrossRefGoogle Scholar
Lehmann, E. L. (1983). The theory of point estimation, New York: John Wiley.CrossRefGoogle Scholar
Marshalek, B., Lohman, D. F., Snow, R. E. (1983). The complexity continuum in the radex and hierarchical models of intelligence. Intelligence, 7, 107–127.CrossRefGoogle Scholar
Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.CrossRefGoogle Scholar
Muraki, E. (1990). Fitting a polytomous item response model to Likert type data. Applied Psychological Measurement, 14, 59–71.CrossRefGoogle Scholar
Newell, A. (1977). On the analysis of human problem solving protocols. In Johnson-Laird, P. N., Wason, P. C. (Eds.), Thinking (pp. 46–61). London: Cambridge University Press.Google Scholar
Neyman, J., Scott, E. L. (1948). Consistent estimates based on partially consistent observations. Econometrica, 16, 1–32.CrossRefGoogle Scholar
Rao, C. R. (1973). Linear statistical inference and its applications, New York: Wiley.CrossRefGoogle Scholar
Rasch, G. (1961). On general laws and the meaning of measurement in psychology. Proceedings of the fourth Berkeley symposium on mathematical statistics and probability (pp. 321–333). Berkeley, CA: University of California Press.Google Scholar
Rasch, G. (1980). Probabilistic models for some intelligence and attainment tests, Chicago: The University of Chicago Press.Google Scholar
Raven, J., Raven, J. C., Court, J. H. (1991). Manual for Raven's progressive matrices and vocabulary scales (section 1): General overview, Oxford: Oxford Psychologists Press.Google Scholar
Read, T. R. C., Cressie, N. (1988). Goodness-of-fit statistics for discrete multivariate data, New York: Springer-Verlag.CrossRefGoogle Scholar
Rost, J. (1988). Measuring attitudes with a threshold model drawing on a traditional scaling concept. Applied Psychological Measurement, 12, 397–409.CrossRefGoogle Scholar
Samejima, F. (1972). A general model for free-response data. Psychometrika Monograph No. 18, 37 (4, Pt. 2).Google Scholar
Scheiblechner, H. (1972). Das lernen und lòsen komplexer Denkaufgaben [Learning and solving complex cognitive problems]. Zeitschrift für experimentelle und Angewandte Psychologie, 19, 476–506.Google Scholar
Spada, H. (1976). Modelle des Denkens und Lernens [Models of thinking and learning], Bern: Huber.Google Scholar
Stenner, A. J., Smith, M. III, Burdick, D. (1983). Toward a theory of construct definition. Journal of Educational Measurement, 20, 303–316.CrossRefGoogle Scholar
Sternberg, R. J. (1982). Advances in the psychology of human intelligence, Vol. 1, Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
Thissen, D., Steinberg, L. (1984). A response model for multiple choice items. Psychometrika, 49, 501–519.CrossRefGoogle Scholar
Theunissen, T. J. J. M. (1985). Binary programming and test design. Psychometrika, 50, 411–420.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
van den Wollenberg, A. L. (1982). Two new test statistics for the Rasch model. Psychometrika, 47, 123–140.CrossRefGoogle Scholar
van der Linden, W. J., Boekkooi-Timminga, E. (1989). A maximum model for test design with practical constraints. Psychometrika, 54, 237–248.CrossRefGoogle Scholar
Wilson, M. (1989). The partial order model. Paper presented at the Fifth International Objective Measurement Workshop, Berkeley, CA.Google Scholar
Wilson, M. (1990). An extension of the partial credit model to incorporate diagnostic information, Berkeley, CA: University of California, Graduate School of Education.Google Scholar
Wright, B. D., Masters, G. N. (1982). Rating scale analysis, Chicago: MESA Press.Google Scholar