The Uniqueness Structure of Simple Latent Trait Models

  • Hans Irtel
Part of the Recent Research in Psychology book series (PSYCHOLOGY)


A latent trait system is a set of subjects A, a set of items X, and a response function r, mapping A × X into the real numbers. Numerical representations of such a system map A and X into the reals, such that r is represented by a numerical operation. It is shown for an additive latent trait system that its internal structure may be characterized by its automorphism group and that homogeneity and uniqueness of this group make the system ratio scalable. A non-additive case is also considered. Here the two factors are combined in a non-additive way, but the system’s internal structure induces an independent system on one of the two factors which is interval-scalable.


Automorphism Group Latent Trait Response Probability Mathematical Psychology Latent Trait Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adams, E.W., Fagot, R.F., & Robinson, R. (1970). On the empirical status of axioms in fundamental theories of measurement. Journal of Mathathematical Psychology, 7, 379–409.CrossRefGoogle Scholar
  2. Andersen, E.B. (1973). Conditional inference for multiple-choice questionnaires. British Journal of Mathematical and Statistical Psychology, 26, 31–44.CrossRefGoogle Scholar
  3. Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee’s ability. In F.M. Lord, & M.R. Novick, Statistical theories of mental test scores. Reading/Mass.: Addison-Wesley.Google Scholar
  4. Colonius, H. (1979). Zur Eindeutigkeit der Parameter im Rasch-Modell. Psychologische Beiträge, 21, 414–416.Google Scholar
  5. Fischer, G.H. (1974). Einführung in die Theorie psychologischer Tests. Bern: Huber.Google Scholar
  6. Fischer, G.H. (1981). On the existence and uniqueness of maximum-likelihood estimates in the Rasch model. Psychometrika, 46, 59–77.CrossRefGoogle Scholar
  7. Fischer, G.H. (1988). Spezifische Objektivität: Eine wissenschaftstheoretische Grundlage des Rasch-Modells. In K.D. Kubinger (Ed.), Moderne Testtheorie - Ein Abriß samt neuesten Beiträgen. Weinheim: Psychologie Verlags Union.Google Scholar
  8. Hamerle, A. (1979). Über die meßtheoretischen Grundlagen von Latent-Trait-Modellen. Archiv für Psychologie, 21, 153–167.Google Scholar
  9. Hamerle, A., & Tutz, G. (1980). Goodness of fit tests for probabilistic measurement models. Journal of Mathematical Psychology, 21, 153–167.CrossRefGoogle Scholar
  10. Irtel, H. (1987). On specific objectivity as a concept in measurement. In E.E. Roskam & R. Suck (Eds.), Progress in mathematical psychology-I. Amsterdam: North-Holland.Google Scholar
  11. Krantz, D.H., Luce, R.D., Suppes, P., & Tversky, A. (1971). Foundations of measurement (Vol. 1). New York: Academic Press.Google Scholar
  12. Luce, R.D., & Cohen, M. (1983). Factorizable automorphisms in solvable conjoint structures I. Journal of Pure and Applied Algebra, 27, 225–261.CrossRefGoogle Scholar
  13. Luce, R.D., & Tukey, J.W. (1964). Simultaneous conjoint measurement: A new type of fundamental measurement. Journal of Mathematical Psychology, 1, 1–27.CrossRefGoogle Scholar
  14. Narens, L. (1985). Abstract measurement theory. Cambridge, MA: MIT Press.Google Scholar
  15. Pfanzagl, J. (1971). Theory of measurement. Würzburg: Physica-Verlag.Google Scholar
  16. Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Paedagogiske Institut.Google Scholar
  17. Roberts, F.S., & Rosenbaum, Z. (1988/89). Tight and loose value automorphisms. Discrete Applied Mathematics, 22, 169–179.CrossRefGoogle Scholar
  18. Roskam, E.E. (1983). Allgemeine Datentheorie. In H. Feger & J. Bredenkamp (Eds.), Messen und Testen, Enzyklopädie der Psychologie, Forschungsmethoden, Vol. 3. Göttingen: Hogrefe.Google Scholar
  19. 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
  20. Wottawa, H. (1980). Grundriß der Testtheorie. München: Juventa Verlag.Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • Hans Irtel
    • 1
  1. 1.Department of PsychologyUniversity RegensburgRegensburgGermany

Personalised recommendations