, Volume 43, Issue 4, pp 561–573 | Cite as

A rating formulation for ordered response categories

  • David Andrich


A rating response mechanism for ordered categories, which is related to the traditional threshold formulation but distinctively different from it, is formulated. In addition to the subject and item parameters two other sets of parameters, which can be interpreted in terms of thresholds on a latent continuum and discriminations at the thresholds, are obtained. These parameters are identified with the category coefficients and the scoring function of the Rasch model for polychotomous responses in which the latent trait is assumed uni-dimensional. In the case where the threshold discriminations are equal, the scoring of successive categories by the familiar assignment of successive integers is justified. In the case where distances between thresholds are also equal, a simple pattern of category coefficients is shown to follow.

Key words

logistic latent trait Rasch model thresholds ordered categories 


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Reference notes

  1. Rasch, G.A mathematical theory of objectivity and its consequences for model construction. Paper presented at the European meeting on statistics, econometrics and management science. Amsterdam, 1968.Google Scholar
  2. Andrich, D.Applications of a psychometric model for ordered categories scored with successive integers. Paper presented at the A.E.R.A. Conference, New York, April, 1977.Google Scholar
  3. Kolakowski, D. & Bock, R. D.A Fortran IV program for maximum likelihood item analysis and test scoring: Logistic model for multiple item responses (Res. Mem. 13). Chicago, Ill.: Statistical Laboratory, Department of Education, University of Chicago, 1972.Google Scholar
  4. Waller, M.Estimating parameters in the Rasch model: Removing the effects of random guessing (ETS RB-76-8). Princeton, NJ: Educational Testing Service, 1976.Google Scholar


  1. Andersen, E. B. Conditional inference for multiple-choice questionnaires.British Journal of Mathematical and Statistical Psychology, 1973,26, 31–44.Google Scholar
  2. Andersen, E. B. Sufficient statistics and latent trait models.Psychometrika, 1977,42, 69–81.Google Scholar
  3. Birnbaum, A. Some latent trait models and their use in inferring an examinee's ability. In F. Lord & M. Novick,Statistical theories of mental test scores. Reading: Mass. Addison-Wesley, 1968.Google Scholar
  4. Lord, F. A theory of test scores.Psychometric Monograph,7, 1952.Google Scholar
  5. Lumsden, J. Person reliability.Applied Psychological Measurement, 1977,4, 477–482.Google Scholar
  6. Rasch, G. On general laws and the meaning of measurement in psychology. InProceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability. Berkeley: University of California Press, 1961, 321–34.Google Scholar
  7. Samejima, F. Estimation of latent ability using a response pattern of graded scores.Psychometric Monograph, 1969,34 (2, Whole No. 17).Google Scholar

Copyright information

© Psychometric Society 1978

Authors and Affiliations

  • David Andrich
    • 1
  1. 1.Department of EducationThe University of Western AustraliaNedlands

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