Hostname: page-component-745bb68f8f-5r2nc Total loading time: 0 Render date: 2025-01-08T21:23:57.157Z Has data issue: false hasContentIssue false
  • English
  • Français

A Unified Nonparametric IRT Model for d-Dimensional Psychological Test Data (d-ISOP)

Published online by Cambridge University Press:  01 January 2025

Hartmann Scheiblechner
Hartmann Scheiblechner*
Affiliation:
FB Psychologie der Philipps-Universität
*
Requests for reprints should be sent to Hartmann Scheiblechner, FB Psychologie der Philipps-Universität, Gutenbergstraße 18, D-35032 Marburg, Germany. E-mail: scheible@staff.uni-marburg.de.
Get access Rights & Permissions [Opens in a new window]

Abstract

The (univariate) isotonic psychometric (ISOP) model (Scheiblechner, 1995) is a nonparametric IRT model for dichotomous and polytomous (rating scale) psychological test data. A weak subject independence axiom W1 postulates that the subjects are ordered in the same way except for ties (i.e., similarly or isotonically) by all items of a psychological test. A weak item independence axiom W2 postulates that the order of the items is similar for all subjects. Local independence (LI or W3) is assumed in all models. With these axioms, sample-free unidimensional ordinal measurements of items and subjects become feasible. A cancellation axiom (Co) gives, as a result, the additive isotonic psychometric (ADISOP) model and interval scales for subjects and items, and an independence axiom (W4) gives the completely additive isotonic psychometric (CADISOP) model with an interval scale for the response variable (Scheiblechner, 1999). The d-ISOP, d-ADISOP, and d-CADISOP models are generalizations to d-dimensional dependent variables (e.g., speed and accuracy of response).

Keywords

Weak subject independenceweak item independenceisotonic ordinal probabilistic modelssample-free measurementspeed-accuracy trade-offcoordinatewise partial order(modified) percentile score
Type
Original Paper
Information
Psychometrika , Volume 72 , Issue 1 , March 2007 , pp. 43 - 67
Copyright
Copyright © 2007 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The author would like to thank an Associate Editor and two anonymous referees and also Professor H.H. Schulze for their very valuable suggestions and corrections.

References

Antonovsky, A. (1987). Unraveling the mystery of health. How people manage stress and stay well, San Francisco: Jossey-Bass.Google Scholar
Falmagne, J.C. (1976). Random conjoint measurement and loudness summation. Psychological Review, 83(1), 65–79.CrossRefGoogle Scholar
Falmagne, J.C. (1979). On a class of probabilistic conjoint measurement models: Some diagnostic properties. Journal of Mathematical Psychology, 19, 73–88.CrossRefGoogle Scholar
Goodman, L.A., Kruskal, W.H. (1954). Measures of association for cross classifications, Part I. Journal of the American Statistical Association, 49, 732–764.Google Scholar
Goodman, L.A., Kruskal, W.H. (1959). Measures of association for cross classifications, Part II. Journal of the American Statistical Association, 54, 123–163.CrossRefGoogle Scholar
Goodman, L.A., Kruskal, W.H. (1963). Measures of association for cross classifications, Part III. Journal of the American Statistical Association, 58, 310–364.CrossRefGoogle Scholar
Goodman, L.A., Kruskal, W.H. (1972). Measures of association for cross classifications, Part IV. Journal of the American Statistical Association, 67, 415–421.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Hemker, B.T., Sijtsma, K., Molenaar, I.W., Junker, B.W. (1997). Stochastic ordering using the latent trait and the sum score in polytomous IRT models. Psychometrika, 62, 331–347.CrossRefGoogle Scholar
Irtel, H. (1987). On specific objectivity as a concept in measurement. In Roskam, E.E., Suck, R. (Eds.), Progress in mathematical psychology Vol. 1 (pp. 35–45). Amsterdam: North-Holland, Elsevier.Google Scholar
Irtel, H. (1995). An extension of the concept of specific objectivity. Psychometrika, 60, 115–118.CrossRefGoogle Scholar
Irtel, H., Schmalhofer, F. (1982). Psychodiagnostik auf Ordinalskalenniveau: Meßtheoretische Grundlagen, Modelltest und Parameterschätzung [Psychodiagnostics at the level of ordinal scales: Measurement theortical foundation, model controls and parameter estimation]. Archiv für Psychologie, 134, 197–218.Google Scholar
Joe, H. (2001). Majorization and stochastic orders. In Marley, A.A. (Eds.), International encyclopedia of the social & behavioral sciences: Mathematics and computer sciences, Oxford, UK: Elsevier Science.Google Scholar
Karabatsos, G. (2001). The Rasch model, additive conjoint measurement, and new models of probabilistic measurement theory. Journal of Applied Measurement, 2(4), 389–423.Google ScholarPubMed
Karabatsos, G. (2005). The exchangeable multinomial model as an approach to testing deterministic axioms of choice and measurement. Journal of Mathematical Psychology, 49, 51–69.CrossRefGoogle Scholar
Keeney, R.L., Raiffa, H. (1976). Decisions with multiple objectives: Preferences and value tradeoffs, New York: Wiley.Google Scholar
Krantz, D.H., Luce, R.D., Suppes, P., Tversky, A. (1971). Foundations of measurement. Vol. I, Additive and polynomial representations, New York: Academic Press.Google Scholar
Lutz, R. (1991). Vorhersagbarkeit der Interkorrelation pschodiagnostischer Skalen. [Predictability of correlations between psychodiagnostic scales]. Psychologische Beträge, 33, 47–61.Google Scholar
Lutz, R. (2001). Marburger Untersuchungs-Instrumentarium (MUM). [Marburg psychological diagnostics instruments (MUM)]. Unpublished manuscript, Department of Psychology, Philipps-Universität Marburg.Google Scholar
Michell, J. (1988). Some problems in testing the double cancellation condition in conjoint measurement. Journal of Mathematical Psychology, 32, 466–473.CrossRefGoogle Scholar
Mokken, R.J. (1971). A theory and procedure of scale analysis, Paris/Den Haag: Mouton.CrossRefGoogle Scholar
Molenaar, I.W., Sijtsma, K. (2000). User's manual MSP5 for windows, Groningen: ProGamma.Google Scholar
Narens, L. (1985). Abstract measurement theory, Cambridge, MA: MIT Press.Google Scholar
Narens, L. (2002). A meaningful justification for the representational theory of measurement. Journal of Mathematical Psychology, 46(6), 746–768.CrossRefGoogle 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 Vol. 14 (pp. 58–94). Copenhagen: Munksgaard.Google Scholar
Roberts, F.S. (1979). Measurement theory with applications to decision making, utility, and the social sciences, Reading, MA: Addison–Wesley.Google Scholar
Robertson, T., Wright, F.T., Dykstra, R.L. (1988). Order restricted statistical inference, New York: Wiley.Google Scholar
Roskam, E.E. (1997). Models for speed and time-limit tests. In van der Linden, W.J., Hambleton, R.K. (Eds.), Handbook of modern item response theory, New York: Springer-Verlag.Google Scholar
Scheiblechner, H. (1995). Isotonic ordinal probabilistic models (ISOP). Psychometrika, 60, 281–304.CrossRefGoogle Scholar
Scheiblechner, H. (1999). Additive conjoint isotonic probabilistic models (ADISOP). Psychometrika, 64, 295–316.CrossRefGoogle Scholar
Scheiblechner, H. (2002). Note on nonparametric IRT: Scoring functions, subject parameter estimation and further model controls of isotonic probabilistic models (ISOP). Preliminary draught. URL: http://www.staff.uni-marburg.de/~scheible/isoscore2.pdf.Google Scholar
Scheiblechner, H. (2003). Nonparametric IRT: Testing the bi-isotonicity of isotonic probabilistic models (ISOP). Psychometrika, 68, 79–96.CrossRefGoogle Scholar
Scheiblechner, H., & Lutz, R. (2006). Die Konstruktion eines optimalen eindimensionalen Tests mittels nichtparametrischer Testtheorie (NIRT) am Beispiel des MR-SOC. [The construction of an optimal one-dimensional test via nonparametric test theory (NIRT) by example of the MR-SOC]. Diagnostica, submitted.Google Scholar
Sijtsma, K., Molenaar, I.W. (2002). Introduction to nonparametric item response theory, London: Sage.CrossRefGoogle Scholar
Simon, H. (1981). The sciences of the artificial 2nd ed.,, Cambridge, MA: MIT Press.Google 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