Abstract
This paper discusses thecompatibility of the polychotomous Rasch model with dichotomization of the response continuum. It is argued that in the case of graded responses, the response categories presented to the subject are essentially an arbitrary polychotomization of the response continuum, ranging for example from total rejection or disagreement to total acceptance or agreement of an item or statement. Because of this arbitrariness, the measurement outcome should be independent of the specific polychotomization applied, for example, presenting a specific multicategory response format should not affect the measurement outcome. When such is the case, the original polychotomous model is called “compatible” with dichotomization.
A distinction is made between polychotomization or dichotomization “before the fact,” that is, in constructing the response format, and polycho- or dichotomization “after the fact,” for example in dichotomizing existing graded response data.
It is shown that, at least in case of dichotomization after-the-fact, the polychotomous Rasch model is not compatible with dichotomization, unless a rather special condition of the model parameters is met. Insofar as it may be argued that dichotomization before the fact is not essentially different from dichotomization after the fact, the value of the unidimensional polychotomous Rasch model is consequently questionable. The impact of our conclusion on related models is also discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aitchison, J., & Silvey, S. D. (1957). The generalization of profit analysis to the case of multiple responses.Biometrika, 44, 131–140.
Andersen, E. B. (1973). Conditional inference for multiple choice questionnaires.British Journal of Mathematical and Statistical Psychology, 26, 31–57.
Andersen, E. B. (1977). Sufficient statistics and latent trait models.Psychometrika, 42, 69–81.
Andrich, D. (1978). A rating formulation for ordered response categories.Psychometrika, 43, 561–573.
Andrich, D. (1979). A model for contingency tables having an ordered response classification.Biometrics, 35, 403–415.
Benzécri, J. P. (1973).Analyse des données. II: L'analyse des correspondances [Data analysis, II: Correspondence analysis]. Paris-Bruxelles: Dunot.
Berg, I. A., & Rapaport, G. M. (1954). Response bias in an unstructured questionnaire.The Journal of Psychology, 38, 475–481.
Cohen, J. (1983). The cost of dichotomization.Applied Psychological Measurement, 7, 249–253.
Coombs, C. H. (1964).Theory of Data. New York: Wiley. (Reprinted, 1976, Ann Arbor: Mathesis Press)
Drasgow, F., & Parsons, C. K. (1983). Application of unidimensional item response theory models to multidimensional data.Applied Psychological Measurement, 7, 189–199.
Fischer, G. H. (1974).Einführung in die Theorie psychologischer tests [Introduction to psychological test theory]. Bern: Huber.
Guttman, L. (1967). The development of nonmetric space analysis; a letter to John Ross.Multivariate Behavioral Research, 2, 71–82. (Reprinted in J. C. Lingoes, E. E. Roskam, & I. Borg, Eds., Geometric representations of relational data; readings in multidimensional scaling, pp. 237–248, Ann Arbor: Mathesis Press)
Hoogstraten, J. (1979).De machteloze onderzoeker [The helpless researcher]. Meppel: Boom.
Jansen, P. G. W. (1983). Rasch analysis of attitudinal data. Unpublished doctoral dissertation, University of Nijmegen, and Rijks Psychologische Dienst, Den Haag.
Koch, W. R. (1983). Likert scaling using the graded response latent trait model.Applied Psychological Measurement, 7, 15–32.
Likert, R. (1932). A technique for the measurement of attitudes.Archives of Psychology, 140, 1–59.
Lingoes, J. C. (1968). The rationale of the Guttman-Lingoes nometric series: A letter to Doctor Philip Runkel.Multivariate Behavioral Research, 3, 495–508. (Reprinted in J. C. Lingoes, E. E. Roskam, & I. Borg, Eds.,Geometric representations of relational data: Readings in multidimensional scaling, pp. 249–261, Ann Arbor: Mathesis Press)
Lord, F. M. (1980).Applications of item response theory to practical testing problems. Hillsdale: Lawrence Erlbaum.
Masters, G. N. (1982). A Rasch model for partial credit scoring.Psychometrika, 47, 149–174.
Masters, G. N. (1985, March). A latent trait analysis of computer-administered items with feedback. Paper presented at the annual meeting of the American Educational Research Association, Chicago.
Nishisato, S. (1980).Analysis of categorical data: Dual scaling and its applications. Toronto: University of Toronto Press.
Rasch, G. (1961). On general laws and the meaning of measurement in psychology.Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability (pp. 321–333). Berkeley and Los Angeles: University of California Press.
Roskam, E. E. (1983). Allgemeine Datentheorie [General Theory of Data]. In H. Feger & H. Bredenkamp (Eds.),Messen und Testen, Band 3 der Serie Forschungsmethoden der Psychologie der Enzyklopadie der Psychologie [Measurement and testing, Vol. 3 of the series psychological research methods of the encyclopedia of psychology], Gottingen: Hogrefe, 1–135.
Roskam, E. E., & Jansen, P. G. W. (1984). A new derivation of the Rasch model. In E. Degreef & J. van Buggenhaut (Eds.),Trends in mathematical psychology (pp. 293–307). Amsterdam: Elsevier-North-Holland.
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded responses [Monograph].Psychometrika,34(4, Pt. 2).
Thissen, D., & Steinberg, L. (1984). A response model for multiple choice items.Psychometrika, 49, 501–519.
Tzeng, O. C. S. (1983). A comparative evaluation of four response formats in personality ratings.Educational and Psychological Measurement, 43, 935–950.
Van den Wollenberg, A. L. (1982). Two new test statistics for the Rasch model.Psychometrika, 47, 123–140.
Van Heerden, J., & Hoogstraten, J. (1979). Response tendency in a questionnaire without questions.Applied Psychological Measurement, 3, 117–121.
Wright, B. D., & Stone, M. H. (1979).Best test design: Rasch measurement. Chicago: MESA Press.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jansen, P.G.W., Roskam, E.E. Latent trait models and dichotomization of graded responses. Psychometrika 51, 69–91 (1986). https://doi.org/10.1007/BF02294001
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02294001