Abstract
Over the last decade, there has been increasing interest in analyzing mental test data with loglinear models. Several authors have shown that the Rasch model for dichotomously scored items can be formulated as a loglinear model (Cressie and Holland, 1983; de Leeuw and Verhelst, 1986; Kelderman, 1984; Thissen and Mooney, 1989; Tjur, 1982). Because there are good procedures for estimating and testing Rasch models, this result was initially only of theoretical interest. However, the flexibility of loglinear models facilitates the specification of many other types of item response models. In fact, they give the test analyst the opportunity to specify a unique model tailored to a specific test. Kelderman (1989) formulated loglinear Rasch models for the analysis of item bias and presented a gamut of statistical tests sensitive to different types of item bias. Duncan and Stenbeck (1987) formulated a loglinear model specifying a multidimensional model for Likert type items. Agresti (1993) and Kelderman and Rijkes (1994) formulated a loglinear model specifying a general multidimensional response model for polytomously scored items.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adams, R.J. and Wilson, M. (1991). The random coefficients multinomial logit model: A general approach to fitting Rasch models. Paper presented at the Annual Meeting of the American Educational Research Association, Chicago, April.
Agresti, A. (1984). Analysis of Ordinal Categorical Data. New York: Wiley.
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.
Akaike, H. (1977). On entropy maximization principle. In P.R. Krisschnaiah (Ed), Applications of Statistics (pp. 27–41 ). Amsterdam: North Holland.
Andersen, E.B. (1970). Asymptotic properties of conditional maximum likelihood estimators. Journal of the Royal Statistical Society B 32, 283301.
Andersen, E.B. (1973). Conditional inference and multiple choice questionnaires British Journal of Mathematical and Statistical Psychology 26, 31–44.
Andersen, E.B. (1980). Discrete Statistical Models with Social Science Applications. Amsterdam: North Holland.
Andrich, D. (1978). A rating scale formulation for ordered response categories. Psychometrika 43, 561–573.
Baker, R.J. and Neider, J.A. (1978). The GLIM System: Generalized Linear Interactive Modeling. Oxford: The Numerical Algorithms Group.
Bishop, Y.M.M., Fienberg, S.E., and Holland, P.W. (1975). Discrete Multivariate Analysis. Cambridge, MA: MIT Press.
Bock, R.D. (1975). Multivariate Statistical Methods in Behavioral Research. New York: McGraw Hill.
Cox, M.A.A. and Hinkley, D.V. (1974). Theoretical Statistics. London: Chapman and Hall.
Cox, M.A.A. and Placket, R.L. (1980) Small samples in contingency tables. Biometrika 67, 1–13.
Cressie, N. and Holland, P.W. (1983). Characterizing the manifest probabilities of latent trait models. Psychometrika 48, 129–142.
de Leeuw, J. and Verhelst, N.D. (1986). Maximum likelihood estimation in generalized Rasch models. Journal of Educational Statistics 11, 183196.
Duncan, O.D. (1984). Rasch measurement: Further examples and discussion. In C.F. Turner and E. Martin (Eds), Surveying Subjective Phenomena (Vol. 2, pp. 367–403 ). New York: Russell Sage Foundation.
Duncan, O.D. and Stenbeck, M. (1987). Are Likert scales unidimensional? Social Science Research 16, 245–259.
Fienberg, S.E. (1980). The Analysis of Cross-Classified Categorical Data. Cambridge, MA: MIT Press.
Fischer, G.H. (1974). Einführung in die Theorie psychologischer Tests [Introduction to the Theory of Psychological Tests]. Bern: Huber. (In German.)
Fischer, G.H. (1987). Applying the principles of specific objectivity and generalizability to the measurement of change. Psychometrika 52, 565–587.
Follmann, D.A. (1988). Consistent estimation in the Rasch model based on nonparametric margins. Psychometrika 53, 553–562.
Goodman, L.A. (1970). Multivariate analysis of qualitative data. Journal of the American Statistical Association 65, 226–256.
Goodman, L.A. and Fay, R. (1974). ECTA Program, Description for Users. Chicago: Department of Statistics University of Chicago.
Haberman, S.J. (1977). Log-linear models and frequency tables with small cell counts, Annals of Statistics 5, 1124–1147.
Haberman, S.J. (1979). Analysis of Qualitative Data: New Developments (Vol. 2 ). New York: Academic Press.
Hout, M., Duncan, O.D., and Sobel, M.E. (1987). Association and heterogeneity: Structural models of similarities and differences. Sociological Methodology 17, 145–184.
Kelderman, H. (1984). Loglinear Rasch model tests. Psychometrika 49, 223–245.
Kelderman, H. (1989). Item bias detection using loglinear IRT. Psychometrika 54, 681–697.
Kelderman, H. (1992). Computing maximum likelihood estimates of log- linear IRT models from marginal sums Psychometrika 57, 437–450.
Kelderman, H. and Rijkes, C.P.M. (1994). Loglinear multidimensional IRT models for polytomously scored items. Psychometrika 59, 147–177.
Kelderman, H. and Steen, R. (1993). LOGIMO: Loglinear Item Response Modeling [computer manual]. Groningen, The Netherlands: iec ProGAMMA.
Koehler, K.J. (1977). Goodness-of-Fit Statistics for Large Sparse Multinomials. Unpublished doctoral dissertation, School of Statistics, University of Minnesota.
Lancaster, H.O. (1961). Significance tests in discrete distributions. Journal of the American Statistical Association 56, 223–234.
Larnz, K. (1978). Small-sample comparisons of exact levels for chi-square statistics. Journal of the American Statistical Association 73, 412–419.
Lord, F.M. and Novick, M.R. (1968). Statistical Theories of Mental Test Scores. Reading, MA: Addison Wesley.
Lindsay, B., Clogg, C.C., and 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.
Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika 47, 149–174.
Neyman, J. and Scott, E.L. (1948). Consistent estimates based on partially consistent observations. Econometrica 16, 1–32.
Rao, C.R. (1973). Linear Statistical Inference and Its Applications ( 2nd ed. ). New York: Wiley.
Rasch, G. (1960/1980). Probabilistic Models for Some Intelligence and Attainment Tests. Chicago: The University of Chicago Press.
SPSS (1988). SPSS User’s Guide (2 ed.). Chicago, IL: Author.
Stegelmann, W. (1983). Expanding the Rasch model to a general model having more than one dimension. Psychometrika 48, 257–267.
Thissen, D. and Mooney, J.A. (1989). Loglinear item response theory, with applications to data from social surveys. Sociological Methodology 19, 299–330.
Tjur, T. (1982). A connection between Rasch’s item analysis model and a multiplicative Poisson model. Scandinavian Journal of Statistics 9, 23–30.
van den Wollenberg, A.L. (1979). The Rasch Model and Time Limit Tests. Unpublished doctoral dissertation, Katholieke Universiteit Nijmegen, The Netherlands.
van den Wollenberg, A.L. (1982). Two new test statistics for the Rasch model. Psychometrika 47, 123–140.
Verhelst, N.D., Glas, C.A.W., and van der Sluis, A. (1984). Estimation problems in the Rasch model. Computational Statistics Quarterly 1, 245–262.
Verhelst, N.D., Glas, C.A.W., and Verstralen, H.H.F.M. (1993). OPLM: Computer Program and Manual. Arnhem, The Netherlands: Cito.
Wilson, M. and Adams, R.J. (1993). Marginal maximum likelihood estimation for the ordered partition model. Journal of Educational Statistics 18, 69–90.
Wilson, M. and Masters, G.N. (1993). The partial credit model and null categories. Psychometrika 58, 87–99.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media New York
About this chapter
Cite this chapter
Kelderman, H. (1997). Loglinear Multidimensional Item Response Models for Polytomously Scored Items. In: van der Linden, W.J., Hambleton, R.K. (eds) Handbook of Modern Item Response Theory. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2691-6_17
Download citation
DOI: https://doi.org/10.1007/978-1-4757-2691-6_17
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-2849-8
Online ISBN: 978-1-4757-2691-6
eBook Packages: Springer Book Archive