, Volume 64, Issue 3, pp 295–316 | Cite as

Additive conjoint isotonic probabilistic models (ADISOP)

  • Hartman Scheiblechner


The ISOP-model or model of twodimensional or bi-isotonicity (Scheiblechner, 1995) postulates that the probabilities of ordered response categories increase isotonically in the order of subject “ability” and item ”easiness”. Adding a conventional cancellation axiom for the factors of subjects and items gives the ADISOP model where the c.d.f.s of response categories are functions of an additive item and subject parameter and an ordinal category parameter. Extending cancellation to the interactions of subjects and categories as well as of items and categories (independence axiom of the category factor from the subject and item factor) gives the CADISOP model (completely additive model) in which the parallel c.d.f.s are functions of the sum of subject, item and category parameters. The CADISOP model is very close to the unidimensional version of the polytomous Rasch model with the logistic item/category characteristic(s) replaced by nonparametric axioms and statistics. The axioms, representation theorems and algorithms for model fitting of the additive models are presented.

Key words

ordered item response categories graded response models polytomous models isotonicity isotonic regression cancellation independence of conjoint measurement factors rating scale models nonparametric IRT models minimum ascending average algorithm MAA matrix partial order multidimensional coordinatewise partial order 


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  1. Abrahamowicz, M., & Ramsay, J.O. (1992). Multicategorial spline model for item response theory.Psychometrika, 57, 5–27.Google Scholar
  2. Andersen, E.B. (1995). Polytomous Rasch models and their estimation. In G. H. Fischer & I. W. Molenaar, (Eds.),Rasch models: Foundations, recent developments, and applications (pp. 271–292). New York: Springer.Google Scholar
  3. Andrich, D. (1978). A rating formulation for ordered response categories.Psychometrika, 43, 561–573.Google Scholar
  4. Bartholomew, D.J. (1983). Isotonic inference. In S. Kotz & N. L. Johnson (Eds.),Encyclopedia of Statistical Sciences, 4, 260–265.Google Scholar
  5. Cliff, N. (1994). Predicting ordinal relations.British Journal of Mathematical and Statistical Psychology, 47, 127–150.Google Scholar
  6. Cliff, N., & Donoghue, J.R. (1992). Ordinal test fidelity estimated by an item sampling model.Psychometrika, 57, 217–236.Google Scholar
  7. Dykstra, R.L. (1983). An algorithm for restricted least squares regression.Journal of American Statistical Association, 78, 384, 837–842.Google Scholar
  8. Ellis, J.L., & van den Wollenberg, A.L. (1993). Local homogeneity in latent trait models. A characterization of the homogeneous monotone IRT model.Psychometrika, 58, 417–429.Google Scholar
  9. Hemker, B.T. (1996).Unidimensional IRT models for polytomous items, with results for Mokken scale analysis. Unpublished Doctorial dissertation, Utrecht University.Google Scholar
  10. Hemker, B.T., Sijtsma, K., Molenaar, I.W., & Junker, B.W. (1996). Polytomous IRT models and monotone likelihood ratio of the total score.Psychometrika, 61, 679–693.Google Scholar
  11. Hemker, B.T., Sijtsma, K., Molenar, I.W., & Junker, B.W. (1997). Stochastic ordering using the latent trait and the sum score in polytomous IRT models.Psychometrika, 62, 331–347.Google Scholar
  12. 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
  13. Irtel, H. (1994). The uniqueness structure of simple latent trait models. In G. H. Fischer & D. Laming (Eds.),Contributions to mathematical psychology, psychometrics, and methodology (pp. 265–276). New York: Springer.Google Scholar
  14. Irtel, H., & Schmalhofer, F. (1982). Psychodiagnostik auf Ordinalskalenniveau: Meßtheoretische Grundlagen, Modelltests und Parameterschätzung [Psychodiagnostics on ordinal scale level: Measurement theoretic foundations, model test and parameter estimation].Archiev für Psychologie, 134, 197–218.Google Scholar
  15. Junker, B.W. (1998). Some remarks on Scheiblechner's treatment of ISOP models.Psychometrika, 63, 73–85.Google Scholar
  16. Krantz, D.H. (1974). Measurement theory and qualitative laws in psychophysics. In D. H. Krantz, R. D. Luce, R. C. Atkinson, & P. Suppes,Measurement, psychophysics, and neural information processing. Contemporary developments in mathematical psychology, Vol. 2 (pp. 160–199). San Francisco: Freeman and Company.Google Scholar
  17. Krantz, D.H., Luce, R.D., Suppes, P., & Tversky, A. (1971).Foundations of measurement. New York: Academic Press.Google Scholar
  18. Lehmann, E.L. (1986).Testing statistical hypothesis (2nd ed.). New York: J. Wiley.Google Scholar
  19. Luce, R.D., Krantz, D.H., Suppes, P., & Tversky, A. (1990).Foundations of measurement, Vol. 3. San Diego: Academic Press.Google Scholar
  20. Masters, G.N. (1982). A Rasch model for partial credit scoring.Psychometrika, 47, 149–174.Google Scholar
  21. Meredith, W. (1965). Some results based on a general stochastic model for mental tests.Psychometrika, 30, 419–440.Google Scholar
  22. Mokken, R.J. (1971).A theory and procedure for scale analysis. Paris/Den Haag: Mouton.Google Scholar
  23. Mokken, R.J., & Lewis, C. (1982). A nonparametric approach to the analysis of dichotomous item responses.Applied Psychological Measurement, 6, 417–430.Google Scholar
  24. Molenaar, I.W. (1991). A weighted Loevinger H-coefficient extending Mokken scaling to multicategory items.Kwantitatieve Methoden, 37, 97–117.Google Scholar
  25. Orth, B. (1974).Einführung in die Theorie des Messens [Introduction into the theory of measurement]. Stuttgart: Kohlhammer.Google Scholar
  26. Ramsay, J.O., & Abrahamowicz, M. (1989). Binomial regression with monotone splines: A psychometric application.Journal of the American Statistical Association, 84, 906–915.Google Scholar
  27. Rasch, G. (1961). On general laws and the meaning of measurement in psychology.Proceedings of the IV. Berkeley Symposium on mathematical statistics and probability, 4, 321–333.Google Scholar
  28. Robertson, T., Wright, F.T., & Dykstra, R.L. (1988).Order restricted statistical inference. New York: Wiley.Google Scholar
  29. Rosenbaum, P.R. (1988). Item bundles.Psychometrika, 53, 349–359.Google Scholar
  30. Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores.Psychometrika Monograph, No. 17.Google Scholar
  31. Scheiblechner, H. (1995). Isotonic ordinal probabilistic models (ISOP).Psychometrika, 60, 281–304.Google Scholar
  32. Scheiblechner, H. (1998). Corrections of theorems in Scheiblechner's treatment of ISOP models and comments on Junker's remarks.Psychometrika, 63, 87–91.Google Scholar
  33. Scheiblechner, H. (in press). Nonparametric IRT: Testing the bi-isotonicity of isotonic probabilistic models (ISOP).Psychometrika.Google Scholar
  34. Scott, D. (1964). Measurement models and linear inequalities.Journal of Mathematical Psychology, 1, 233–247.Google Scholar
  35. Schwarz, W. (1990). Experimental and theoretical results for some models of random dot pattern discrimination.Psychological Research, 52, 299–305.Google Scholar
  36. Sijtsma, K., & Junker, B.W. (1996). A survey of theory and methods of invariant item ordering.British Journal of Mathematical and Statistical Psychology, 49, 79–105.Google Scholar
  37. Sijtsma, K., & Meijer, R.R. (1992). A method for investigating the intersection of item response functions in Mokken's nonparametric IRT model.Applied Psychological Measurement, 16, 149–157.Google Scholar
  38. Stout, W.F. (1987). A nonparametric approach for assessing latent trait unidimensionality.Psychometrika, 52, 589–617.Google Scholar
  39. Stout, W.F. (1990). A new item response theory modeling approach with applications to unidimensionality assessment and ability estimation.Psychometrika, 55, 293–325.Google Scholar
  40. Suppes, P., & Zinnes, J.L. (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 1999

Authors and Affiliations

  • Hartman Scheiblechner
    • 1
  1. 1.Philipps UniversitätGermany

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