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Isotonic Ordinal Probabilistic Models (ISOP)

Published online by Cambridge University Press:  01 January 2025

Hartmann Scheiblechner
Hartmann Scheiblechner*
Affiliation:
Philipps Universität Marburg
*
Requests for reprints should be sent to Hartmann Scheiblechner, FB 04 Universität Marburg, Gutenbergstrage 18, D-35032 Marburg, FRG.
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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.

Keywords

sample-independenceisotonic regressionprobabilistic pair comparison systemsordinal instrumental independenceraw score systemsscoring functions

Information

Type
Original Paper
Information
Psychometrika , Volume 60 , Issue 2 , June 1995 , pp. 281 - 304
Copyright
Copyright © 1995 The Psychometric Society

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Footnotes

I wish to thank 3 reviewers and 2 editors who contributed a lot to the readability and precision of the article.

References

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