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Psychometrika

, Volume 60, Issue 1, pp 7–26 | Cite as

Models for measurement, precision, and the nondichotomization of graded responses

  • David Andrich
Article

Abstract

It is common in educational, psychological, and social measurement in general, to collect data in the form of graded responses and then to combine adjacent categories. It has been argued that because the division of the continuum into categories is arbitrary, any model used for analyzing graded responses should accommodate such action. Specifically, Jansen and Roskam (1986) enunciate ajoining assumption which specifies that if two categoriesj andk are combined to form categoryh, then the probability of a response inh should equal the sum of the probabilities of responses inj andk. As a result, they question the use of the Rasch model for graded responses which explicitly prohibits the combining of categories after the data are collected except in more or less degenerate cases. However, the Rasch model is derived from requirements of invariance of comparisons of entities with respect to different instruments, which might include different partitions of the continuum, and is consistent with fundamental measurement. Therefore, there is a strong case that the mathematical implication of the Rasch model should be studied further in order to understand how and why it conflicts with the joining assumption. This paper pursues the mathematics of the Rasch model and establishes, through a special case when the sizes of the categories are equal and when the model is expressed in the multiplicative metric, that its probability distribution reflects the precision with which the data are collected, and that if a pair of categories is collapsed after the data are collected, it no longer reflects the original precision. As a consequence, and not because of a qualitative change in the variable, the joining assumption is destroyed when categories are combined. Implications of the choice between a model which satisfies the joining assumption or one which reflects on the precision of the data collection considered are discussed.

Key words

latent trait theory dichotomization rating data graded responses Rasch models 

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References

  1. Andersen, E. B. (1973). Conditional inference for multiple choice questionnaires.British Journal of Mathematical and Statistical Psychology, 26, 31–44.Google Scholar
  2. Andersen, E. B. (1977). Sufficient statistics and latent trait models.Psychometrika, 42, 69–81.Google Scholar
  3. Andrich, D. (1978). A rating formulation for ordered response categories.Psychometrika, 43, 357–374.Google Scholar
  4. Dawes, R. M. (1972).Fundamentals of attitude measurement. New York: John Wiley.Google Scholar
  5. Fischer, G. (1977). Some probabilistic models for the description of attitudinal and behavioral changes under the influence of mass communication. In W. F. & R. Repp (Eds.),Mathematical models for social psychology (pp. 102–151). Berne: Huber.Google Scholar
  6. Fischer, G. H. (1981). On the existence and uniqueness of maximum-likelihood estimates in the Rasch model.Psychometrika, 46, 1, 59–77.Google Scholar
  7. Jansen, P. G. W., & Roskam, E. E. (1984). The polychotomous Rasch model and dichotomization of graded responses. In E. Degreef & J. van Buggenhaut (Ed.),Trends in mathematical psychology. North-Holland: Elsevier Science Publishers B. V.Google Scholar
  8. Jansen, P. G. W., & Roskam, E. E. (1986). Latent trait models and dichotomization of graded responses.Psychometrika, 51(1), 69–91.Google Scholar
  9. Ramsay, J. O. (1975). Review of Foundations of measurement, Vol. I, by D. H. Krantz, R. D. Luce, P. Suppes, and A. Tverskey.Psychometrika, 40, 257–62.Google Scholar
  10. Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. (An expanded edition with a foreword and afterword by B. D. Wright was published in 1980 by the University of Chicago Press.) Copenhagen: Danish Institute for Educational Research.Google Scholar
  11. Rasch, G. (1961). On general laws and the meaning of measurement in psychology. In J. Neyman (Ed.),Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, IV (pp. 321–334). Berkeley CA: University of California Press.Google Scholar
  12. Rasch, G. (1966). An individualistic approach to item analysis. In P. F. Lazarsfeld and N. W. Henry, (Eds.),Readings in mathematical social science (pp. 89–108). Chicago: Science Research Associates.Google Scholar
  13. Rasch, G. (1977). On specific objectivity: An attempt at formalising the request for generality and validity of scientific statements.Danish Yearbook of Philosophy, 14, 58–94.Google Scholar
  14. Roskam, E. E., & Jansen, P. G. W. (1989). Conditions for Rasch dichotomizability of the unidimensional polytomous Rasch model.Psychometrika, 54, 317–332.Google Scholar
  15. Thurstone, L. L. (1927). A law of comparative judgement.Psychological Review, 34, 278–286.Google Scholar
  16. Vogt, D. K., & Wright, B. D. (Undated). Parameter Estimation for the Polychotomous Rasch Model. Unpublished manuscript. University of Chicago: School of Education.Google Scholar
  17. Wright, B. D. (1985). Additivity in psychological measurement. In E. E. Roskam (Ed.),Measurement and personality assessment. Selected papers, XXIII International Congress of Psychology, 8, 101–111.Google Scholar

Copyright information

© The Psychometric Society 1995

Authors and Affiliations

  • David Andrich
    • 1
  1. 1.School of EducationMurdoch University, Murdoch UniversityMurdochAustralia

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