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Sequential Models for Ordered Responses

  • Gerhard Tutz
Chapter

Abstract

The model considered in this chapter is suited for a special type of response. First, the response should be from ordered categories, i.e., a graded response. Second, the categories or levels should be recorded successively in a stepwise manner. An example illustrates this type of item, which is often found in practice:

Wright and Masters (1982) consider the item \(\sqrt {9.0/0.3 - 5} = ?\) Three levels of performance may be distinguished: No subproblem solved (Level 0), 9.0/0.3 = 30 solved (Level 1), 30 − 5 = 25 solved (Level 2), \(\sqrt {25} = 5\) (Level 3). The important feature is that each level in a solution to the problem can be reached only if the previous level is reached.

Keywords

Item Response Theory Item Difficulty Item Parameter Sequential Model Partial Credit 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. Agresti, A. (1984). Analysis of Ordinal Categorical Data. New York: Wiley.zbMATHGoogle Scholar
  2. Andersen, E.B. (1973). A goodness of fit test for the Rasch model. Psychometrika 38, 123–139.MathSciNetzbMATHCrossRefGoogle Scholar
  3. Anderson, D.A. and Aitkin, M. (1985). Variance component models with binary responses: Interviewer variability. Journal of the Royal Statistical Society B 47, 203–210.MathSciNetGoogle Scholar
  4. Anderson, D.A. and Hinde, J. (1988). Random effects in generalized linear models and the EM algorithm. Communications in Statistical Theory and Methodology 17, 3847–3856.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Andrich, D.A. (1978). A rating formulation for ordered response categories. Psychometrika 43, 561–573.zbMATHCrossRefGoogle Scholar
  6. Bauer, P., Hommel, G., and Sonnemann, E. (1988). Multiple Hypotheses Testing. Heidelberg: Springer-Verlag.CrossRefGoogle Scholar
  7. Bock, R.D. and Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika 46, 443–459.MathSciNetCrossRefGoogle Scholar
  8. Fahrmeir, L., Frost, H., Hennevogl, W., Kaufmann, H., Kranert, T., and Tutz, G. (1990). GLAMOUR: User and Example Guide. Regensburg, Germany: University of Regensburg.Google Scholar
  9. Fahrmeir, L. and Tutz, G. (1994). Multivariate Statistical Modelling Based on Generalized Linear Models. New York: Springer-Verlag.zbMATHGoogle Scholar
  10. Fischer, G.H. (1981). On the existence and uniqueness of maximum-likelihood estimates in the Rasch-model. Psychometrika 46, 59–77.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Hambleton, R.K. and Swaminathan, H. (1985). Item Response Theory: Principles and Applications. Boston: Kluwer Academic Publishers.Google Scholar
  12. Kalbfleisch, J. and Prentice, R. (1980). The Statistical Analysis of Failure Time Data. New York: Wiley.zbMATHGoogle Scholar
  13. Kelderman, H. (1984). Loglinear Rasch model tests. Psychometrika 49, 223–245.zbMATHCrossRefGoogle Scholar
  14. Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika 47, 149–174.zbMATHCrossRefGoogle Scholar
  15. McCullagh, P. (1980). Regression models for ordinal data. Journal of the Royal Statistical Society A 135, 370–384.Google Scholar
  16. Molenaar, I.W. (1983). Item Steps (Heymans Bulletin 83–630-EX). Groningen, The Netherlands: Psychologische Instituten, Rijksuniversiteit Groningen.Google Scholar
  17. Rost, J. (1991). A logistic mixture distribution model for polychotomous item responses. British Journal of Mathematical and Statistical Psychology 44, 75–92.CrossRefGoogle Scholar
  18. Rost, J. and Daviér, M.v. (1992). MIRA: A PC Program for the Mixed Rasch Model (User Manual). Kiel, Germany. Institute of Science Eduation (IPN).Google Scholar
  19. Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika, Monograph No. 17.Google Scholar
  20. Stiratelli, R., Laird, N., and Ware, J.H. (1984). Random-effects models for serial observation with binary response. Biometrics 40, 961–971.CrossRefGoogle Scholar
  21. Thissen, D.M. (1982). Marginal maximum likelihood estimation for the one-parameter logistic model. Psychometrika 47, 175–186.zbMATHCrossRefGoogle Scholar
  22. Tutz, G. (1990). Sequential item response models with an ordered response. British Journal of Mathematical and Statistical Psychology 43, 39–55.MathSciNetzbMATHCrossRefGoogle Scholar
  23. Tutz, G. (1991). Sequential models in categorical regression. Computational Statistics Data Analysis 11, 275–295.MathSciNetzbMATHCrossRefGoogle Scholar
  24. Waclawiw, M. and Liang, K.Y. (1993). Prediction of random effects in the generalized linear model. Journal of the American Statistical Association 88, 171–178.zbMATHGoogle Scholar
  25. Wright, B.D. and Masters, G.N. (1982). Rating Scale Analysis. Chicago: MESA Press.Google Scholar
  26. Zeger, S.L., Liang, K.Y., and Albert, P.S. (1988). Models for longitudinal data: A generalized estimating equation approach. Biometrics 44, 1049–1060.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

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  • Gerhard Tutz

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