Psychometrika

, Volume 60, Issue 2, pp 281–304

Isotonic ordinal probabilistic models (ISOP)

  • Hartmann Scheiblechner
Article

Abstract

The concept of an ordinal instrumental probabilistic comparison is introduced. It relies on an ordinal scale given a priori and on the concept of stochastic dominance. It is used to define a weakly independently ordered system, or isotonic ordinal probabilistic (ISOP) model, which allows the construction of separate “sample-free” ordinal scales on a set of “subjects” and a set of “items”. The ISOP-model is a common nonparametric theoretical structure for unidimensional models for quantitative, ordinal and dichotomous variables.

Fundamental theorems on dichotomous and polytomous weakly independently ordered systems are derived. It is shown that the raw score system has the same formal properties as the latent system, and therefore the latter can be tested at the observed empirical level.

Key words

sample-independence isotonic regression probabilistic pair comparison systems ordinal instrumental independence raw score systems scoring functions 

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References

  1. Andersen, E. B. (1973).Conditional inference and models for measuring. Copenhagen: Mentalhygiejnisk Forlag.Google Scholar
  2. Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In F. M. Lord & M. R. Novick (Eds.),Statistical theories of mental test scores (pp. 397–545). Reading: Addison-Wesley.Google Scholar
  3. Cliff, N., & Donoghue, J. R. (1992). Ordinal test fidelity estimated by an item sampling model.Psychometrika, 57, 217–236.CrossRefGoogle Scholar
  4. Fischer, G. H. (1974).Einführung in die Theorie psychologischer Tests [Introduction into the theory of psychological tests]. Bern: Huber.Google Scholar
  5. Fischer, G. H. (1987). Applying the principles of specific objectivity and of generalizability to the measurement of change.Psychometrika, 52, 565–587.CrossRefGoogle Scholar
  6. Fishburn, P. C. (1973). Binary choice probabilities: On the varieties of stochastic transitivity.Journal of Mathematical Psychology, 10, 327–352.Google Scholar
  7. Holland, P. W., & Rosenbaum, P. R. (1986). Conditional association and unidimensionality in monotone latent variable models.The Annals of Statistics, 14, 1523–1543.Google Scholar
  8. Irtel, H. (1987). On specific objectivity as a concept in measurement. In E. E. Roskam & R. Suck (Eds.),Progress in mathematical psychology-1, (pp. 35–45). Amsterdam North-Holland: Elsevier.Google Scholar
  9. Irtel, H. (1993). The uniqueness structure of simple latent trait models. In G. H. Fischer & D. Laming (Eds.),Contributions to Mathematical Psychology, Psychometrics, and Methodology (pp. 265–275). New York: Springer-Verlag.Google Scholar
  10. Irtel, H., & Schmalhofer, F. (1982). Psychodiadnostik auf Ordinalskalenniveau: Meßtheoretische Grundlagen, Modelltest und Parameterschätzung [Psychodiagnostics on ordinal scale level: Measurement theoretic foundations, model test and parameter estimation].Archiv für Psychologie, 134, 197–218.Google Scholar
  11. Junker, B. W. (1990).Progress in characterizing strictly unidimensional IRT representations. Pittsburgh: Carnegie Mellon University.Google Scholar
  12. Krantz, D. H., Luce, R. D., Suppes, P., & Tversky, A. (1971).Foundations of measurement, Vol. 1. New York: Academic Press.Google Scholar
  13. Meredith, W. (1965). Some results based on a general stochastic model for mental tests.Psychometrika, 30, 419–440.PubMedGoogle Scholar
  14. Mokken, R. J. (1971).A theory and procedure of scale analysis. Paris/Den Haag: Mouton.Google Scholar
  15. Mokken, R. J., & Lewis, C. (1982). A nonparametric approach to the analysis of dichotomous item responses.Applied Psychological Measurement, 6, 417–430.Google Scholar
  16. Rasch, G. (1960).Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.Google Scholar
  17. Rasch, G. (1977). On specific objectivity: An attempt at formalizing the request for generality and validity of scientific statements. In M. Blegvad (Ed.),The Danish yearbook of philosophy (Vol. 14, pp. 58–94). Copenhagen: Munksgaard.Google Scholar
  18. Robertson, T., Wright, F. T., & Dykstra, R. L. (1988).Order restricted statistical inference. New York: John Wiley.Google Scholar
  19. Rosenbaum, P. R. (1988). Item bundles.Psychometrika, 53, 349–359.CrossRefGoogle Scholar
  20. Scheiblechner, H. (1972). Personality and system influences on behavior in social contexts: Frequency models.Acta Psychologica, 36, 322–336.CrossRefPubMedGoogle Scholar
  21. Scheiblechner, H. (1979). Specifically objective stochastic latency mechanisms.Journal of Mathematical Psychology, 19, 18–38.CrossRefGoogle Scholar
  22. Scheiblechner, H. (1994). Estimation and testing procedures for isotonic probabilistic models (ISOP). In preparation.Google Scholar
  23. Stout, W. F. (1987). A nonparametric approach for assessing latent trait unidimensionality.Psychometrika, 52, 589–617.CrossRefGoogle Scholar
  24. Stout, W. F. (1990). A new item response theory modeling approach with applications to unidimensionality assessment and ability estimation.Psychometrica, 55, 293–325.Google Scholar
  25. Suppes, P., & Zinnes, J. (1963). Basic measurement theory. In R. D. Luce, R. R. Bush, & E. Galanter (Eds.),Handbook of mathematical psychology, Vol. 1. New York: Wiley.Google Scholar

Copyright information

© The Psychometric Society 1995

Authors and Affiliations

  • Hartmann Scheiblechner
    • 1
  1. 1.FB 04 Universität MarburgMarburgFRG

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