, Volume 47, Issue 2, pp 149–174 | Cite as

A rasch model for partial credit scoring

  • Geoff N. Masters


A unidimensional latent trait model for responses scored in two or more ordered categories is developed. This “Partial Credit” model is a member of the family of latent trait models which share the property of parameter separability and so permit “specifically objective” comparisons of persons and items. The model can be viewed as an extension of Andrich's Rating Scale model to situations in which ordered response alternatives are free to vary in number and structure from item to item. The difference between the parameters in this model and the “category boundaries” in Samejima's Graded Response model is demonstrated. An unconditional maximum likelihood procedure for estimating the model parameters is developed.

Key words

latent trait Rasch model ordered categories partial credit 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference notes

  1. Rasch, G. Objektivitet i samfundsvidenskaberne et metodeproblem. Paper presented at the University of Copenhagen, 1972 (mimeo).Google Scholar
  2. Masters, G. N. A Rasch model for rating scales. Doctoral dissertation, University of Chicago, 1980.Google Scholar
  3. Wright, B. D. & Masters, G. N. The measurement of knowledge and attitude.Research Memorandum No. 30, MESA Psychometric Laboratory, Department of Education, University of Chicago, 1981.Google Scholar
  4. Masters, G. N. & Wright, B. D. A model for partial credit scoring.Research Memorandum No. 31, MESA Psychometric Laboratory, Department of Education, University of Chicago, 1981.Google Scholar


  1. Andersen, E. B.Conditional inference and models for measuring, Copenhagen: Mentalhygiejnisk Forlag, 1973a.Google Scholar
  2. Andersen, E. B. Conditional inference and multiple choice questionnaires.British Journal of Mathematical and Statistical Psychology, 1973b,26, 31–44.Google Scholar
  3. Andersen, E. B. Sufficient statistics and latent trait models.Psychometrika, 1977,42, 69–81.Google Scholar
  4. Andersen, E. B.Discrete statistical models with social science applications. Amsterdam: North-Holland Publishing Company, 1980.Google Scholar
  5. Andrich, D. A binomial latent trait model for the study of Likert-style attitude questionnaires.British Journal of Mathematical and Statistical Psychology, 1978a,31, 84–98.Google Scholar
  6. Andrich, D. A rating formulation for ordered response categories.Psychometrika, 1978b,43, 561–573.CrossRefGoogle Scholar
  7. Andrich, D. Scaling attitude items constructed and scored in the Likert tradition.Educational and Psychological Measurement, 1978c,38, 665–680.Google Scholar
  8. Andrich, D. Application of a psychometric rating model to ordered categories which are scored with successive integers.Applied Psychological Measurement, 1978d,2, 581–594.Google Scholar
  9. Andrich, D. A model for contingency tables having an ordered response classification.Biometrics, 1979,35, 403–415.Google Scholar
  10. Edwards, A. L. & Thurstone, L. L. An internal consistency check for scale values determined by the method of successive intervals.Psychometrika, 1952,17, 169–180.CrossRefGoogle Scholar
  11. Fischer, G. H. A measurement model for the effect of mass-media.Acta Psychologica, 1972,36, 207–220.CrossRefGoogle Scholar
  12. Fischer, G. H. Some probabilistic models for the description of attitudinal and behavioral changes under the influence of mass communication. In W. F. Kempf and B. H. Repp,Mathematical models for social psychology. Vienna: Hans Huber, 1977.Google Scholar
  13. Mardell, C. & Goldenberg, D. S.DIAL: Developmental Indicators for the assessment of learning. Highland Park, Ill.: DIAL Inc., 1972.Google Scholar
  14. Mardell, C. & Goldenberg, D. S. For prekindergarten screening information: DIAL.Journal of Learning Disabilities, 1975,8, 140–147.Google Scholar
  15. Rasch, G.Probabilistic Models for Some Intelligence and Attainment Tests. Copenhagen, Denmark: Danmarks Paedogogiske Institut, 1960 (Chicago: University of Chicago Press, 1980).Google Scholar
  16. Rasch, G. On general laws and the meaning of measurement in psychology.Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 1961, 321–333.Google Scholar
  17. Rasch, G. On specific objectivity: An attempt at formalizing the request for generality and validity of scientific statements.Danish Yearbook of Philosophy, 1977,14, 58–94.Google Scholar
  18. Samejima, F. Estimation of latent ability using a response pattern of graded scores.Psychometrika, Monograph Supplement No. 17, 1969.Google Scholar
  19. Thurstone, L. L. The measurement of opinion.Journal of Abnormal and Social Psychology, 1928,22, 415–430.Google Scholar
  20. Wright, B. D. & Douglas, G. A. Conditional versus unconditional procedures for sample-free item analysis.Educational and Psychological Measurement, 1977,37, 47–60.Google Scholar
  21. Wright, B. D. & Masters, G. N.Rating scale analysis. Chicago: MESA Press, 1982.Google Scholar
  22. Wright, B. D. & Panchapakesan, N. A procedure for sample-free item analysis.Educational and Psychological Measurement, 1969,29, 23–48.Google Scholar

Copyright information

© The Psychometric Society 1982

Authors and Affiliations

  • Geoff N. Masters
    • 1
  1. 1.Department of EducationUniversity of ChicagoChicago

Personalised recommendations