Loglinear Multidimensional Item Response Models for Polytomously Scored Items

  • Henk Kelderman
Chapter

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.

Keywords

Item Response Theory Item Parameter Category Weight Item Response Theory Model Loglinear Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 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.Google Scholar
  2. Agresti, A. (1984). Analysis of Ordinal Categorical Data. New York: Wiley.MATHGoogle Scholar
  3. 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.MathSciNetMATHGoogle Scholar
  4. Akaike, H. (1977). On entropy maximization principle. In P.R. Krisschnaiah (Ed), Applications of Statistics (pp. 27–41 ). Amsterdam: North Holland.Google Scholar
  5. Andersen, E.B. (1970). Asymptotic properties of conditional maximum likelihood estimators. Journal of the Royal Statistical Society B 32, 283301.Google Scholar
  6. Andersen, E.B. (1973). Conditional inference and multiple choice questionnaires British Journal of Mathematical and Statistical Psychology 26, 31–44.MathSciNetCrossRefGoogle Scholar
  7. Andersen, E.B. (1980). Discrete Statistical Models with Social Science Applications. Amsterdam: North Holland.MATHGoogle Scholar
  8. Andrich, D. (1978). A rating scale formulation for ordered response categories. Psychometrika 43, 561–573.MATHCrossRefGoogle Scholar
  9. Baker, R.J. and Neider, J.A. (1978). The GLIM System: Generalized Linear Interactive Modeling. Oxford: The Numerical Algorithms Group.Google Scholar
  10. Bishop, Y.M.M., Fienberg, S.E., and Holland, P.W. (1975). Discrete Multivariate Analysis. Cambridge, MA: MIT Press.MATHGoogle Scholar
  11. Bock, R.D. (1975). Multivariate Statistical Methods in Behavioral Research. New York: McGraw Hill.MATHGoogle Scholar
  12. Cox, M.A.A. and Hinkley, D.V. (1974). Theoretical Statistics. London: Chapman and Hall.MATHGoogle Scholar
  13. Cox, M.A.A. and Placket, R.L. (1980) Small samples in contingency tables. Biometrika 67, 1–13.MathSciNetMATHCrossRefGoogle Scholar
  14. Cressie, N. and Holland, P.W. (1983). Characterizing the manifest probabilities of latent trait models. Psychometrika 48, 129–142.MathSciNetMATHCrossRefGoogle Scholar
  15. de Leeuw, J. and Verhelst, N.D. (1986). Maximum likelihood estimation in generalized Rasch models. Journal of Educational Statistics 11, 183196.Google Scholar
  16. 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.Google Scholar
  17. Duncan, O.D. and Stenbeck, M. (1987). Are Likert scales unidimensional? Social Science Research 16, 245–259.CrossRefGoogle Scholar
  18. Fienberg, S.E. (1980). The Analysis of Cross-Classified Categorical Data. Cambridge, MA: MIT Press.MATHGoogle Scholar
  19. Fischer, G.H. (1974). Einführung in die Theorie psychologischer Tests [Introduction to the Theory of Psychological Tests]. Bern: Huber. (In German.)Google Scholar
  20. Fischer, G.H. (1987). Applying the principles of specific objectivity and generalizability to the measurement of change. Psychometrika 52, 565–587.MathSciNetMATHCrossRefGoogle Scholar
  21. Follmann, D.A. (1988). Consistent estimation in the Rasch model based on nonparametric margins. Psychometrika 53, 553–562.MathSciNetMATHCrossRefGoogle Scholar
  22. Goodman, L.A. (1970). Multivariate analysis of qualitative data. Journal of the American Statistical Association 65, 226–256.CrossRefGoogle Scholar
  23. Goodman, L.A. and Fay, R. (1974). ECTA Program, Description for Users. Chicago: Department of Statistics University of Chicago.Google Scholar
  24. Haberman, S.J. (1977). Log-linear models and frequency tables with small cell counts, Annals of Statistics 5, 1124–1147.MathSciNetMATHCrossRefGoogle Scholar
  25. Haberman, S.J. (1979). Analysis of Qualitative Data: New Developments (Vol. 2 ). New York: Academic Press.Google Scholar
  26. Hout, M., Duncan, O.D., and Sobel, M.E. (1987). Association and heterogeneity: Structural models of similarities and differences. Sociological Methodology 17, 145–184.CrossRefGoogle Scholar
  27. Kelderman, H. (1984). Loglinear Rasch model tests. Psychometrika 49, 223–245.MATHCrossRefGoogle Scholar
  28. Kelderman, H. (1989). Item bias detection using loglinear IRT. Psychometrika 54, 681–697.MathSciNetCrossRefGoogle Scholar
  29. Kelderman, H. (1992). Computing maximum likelihood estimates of log- linear IRT models from marginal sums Psychometrika 57, 437–450.MATHCrossRefGoogle Scholar
  30. Kelderman, H. and Rijkes, C.P.M. (1994). Loglinear multidimensional IRT models for polytomously scored items. Psychometrika 59, 147–177.CrossRefGoogle Scholar
  31. Kelderman, H. and Steen, R. (1993). LOGIMO: Loglinear Item Response Modeling [computer manual]. Groningen, The Netherlands: iec ProGAMMA.Google Scholar
  32. Koehler, K.J. (1977). Goodness-of-Fit Statistics for Large Sparse Multinomials. Unpublished doctoral dissertation, School of Statistics, University of Minnesota.Google Scholar
  33. Lancaster, H.O. (1961). Significance tests in discrete distributions. Journal of the American Statistical Association 56, 223–234.MathSciNetMATHCrossRefGoogle Scholar
  34. Larnz, K. (1978). Small-sample comparisons of exact levels for chi-square statistics. Journal of the American Statistical Association 73, 412–419.Google Scholar
  35. Lord, F.M. and Novick, M.R. (1968). Statistical Theories of Mental Test Scores. Reading, MA: Addison Wesley.MATHGoogle Scholar
  36. 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.MathSciNetMATHCrossRefGoogle Scholar
  37. Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika 47, 149–174.MATHCrossRefGoogle Scholar
  38. Neyman, J. and Scott, E.L. (1948). Consistent estimates based on partially consistent observations. Econometrica 16, 1–32.MathSciNetCrossRefGoogle Scholar
  39. Rao, C.R. (1973). Linear Statistical Inference and Its Applications ( 2nd ed. ). New York: Wiley.MATHCrossRefGoogle Scholar
  40. Rasch, G. (1960/1980). Probabilistic Models for Some Intelligence and Attainment Tests. Chicago: The University of Chicago Press.Google Scholar
  41. SPSS (1988). SPSS User’s Guide (2 ed.). Chicago, IL: Author.Google Scholar
  42. Stegelmann, W. (1983). Expanding the Rasch model to a general model having more than one dimension. Psychometrika 48, 257–267.CrossRefGoogle Scholar
  43. Thissen, D. and Mooney, J.A. (1989). Loglinear item response theory, with applications to data from social surveys. Sociological Methodology 19, 299–330.CrossRefGoogle Scholar
  44. Tjur, T. (1982). A connection between Rasch’s item analysis model and a multiplicative Poisson model. Scandinavian Journal of Statistics 9, 23–30.MathSciNetMATHGoogle Scholar
  45. van den Wollenberg, A.L. (1979). The Rasch Model and Time Limit Tests. Unpublished doctoral dissertation, Katholieke Universiteit Nijmegen, The Netherlands.Google Scholar
  46. van den Wollenberg, A.L. (1982). Two new test statistics for the Rasch model. Psychometrika 47, 123–140.MATHCrossRefGoogle Scholar
  47. 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.MathSciNetGoogle Scholar
  48. Verhelst, N.D., Glas, C.A.W., and Verstralen, H.H.F.M. (1993). OPLM: Computer Program and Manual. Arnhem, The Netherlands: Cito.Google Scholar
  49. Wilson, M. and Adams, R.J. (1993). Marginal maximum likelihood estimation for the ordered partition model. Journal of Educational Statistics 18, 69–90.CrossRefGoogle Scholar
  50. Wilson, M. and Masters, G.N. (1993). The partial credit model and null categories. Psychometrika 58, 87–99.CrossRefGoogle Scholar

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© Springer Science+Business Media New York 1997

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  • Henk Kelderman

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