, Volume 79, Issue 3, pp 377–402 | Cite as

An Analysis of Item Response Theory and Rasch Models Based on the Most Probable Distribution Method

  • Stefano NoventaEmail author
  • Luca Stefanutti
  • Giulio Vidotto


The most probable distribution method is applied to derive the logistic model as the distribution accounting for the maximum number of possible outcomes in a dichotomous test while introducing latent traits and item characteristics as constraints to the system. The item response theory logistic models, with a particular focus on the one-parameter logistic model, or Rasch model, and their properties and assumptions, are discussed for both infinite and finite populations.

Key words

Rasch model item response theory most probable distribution 



We wish to thank the two anonymous reviewers of the journal for their insight into the work and their helpful comments and suggestions.


  1. Adams, E.W. (1965). Elements of a theory of inexact measurement. Philosophy of Science, 32(3), 205–228. CrossRefGoogle Scholar
  2. Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 561–573. CrossRefGoogle Scholar
  3. Andrich, D. (1982). An extension of the Rasch model for ratings providing both location and dispersion parameters. Psychometrika, 47(1), 105–113. CrossRefGoogle Scholar
  4. Aczel, J. (1966). Lectures on functional equations and their applications. New York: Academic Press. Google Scholar
  5. Aczel, J., & Dohmbres, J. (1989). Functional equations in several variables. Cambridge: Cambridge University Press. CrossRefGoogle Scholar
  6. Bradley, R.A., & Terry, M.E. (1952). Rank analysis of incomplete block designs: I. The method of paired comparisons. Biometrika, 39(3/4), 324–345. CrossRefGoogle Scholar
  7. Bernardo, J.M., & Smith, A.F.M. (1994). Bayesian theory. Chichester: Wiley. CrossRefGoogle Scholar
  8. Barton, M.A., & Lord, F.M. (1981). An upper asymptote for the three-parameter logistic item-response model. Princeton: Educational testing service. Google Scholar
  9. Clinton, W.L., & Massa, L.J. (1972). Derivation of a statistical mechanical distribution function by a method of inequalities. American Journal of Physics, 40, 608–610. CrossRefGoogle Scholar
  10. Davis-Stober, C.P. (2009). Analysis of multinomial models under inequalities constraints: applications to measurement theory. Journal of Mathematical Psychology, 53, 1–13. CrossRefGoogle Scholar
  11. Fischer, G.H. (1973). The linear logistic test model as an instrument in educational research. Acta Psychologica, 37, 359–374. CrossRefGoogle Scholar
  12. Fischer, G.H. (1995). Some neglected problems in IRT. Psychometrika, 60(4), 459–487. CrossRefGoogle Scholar
  13. Fischer, G.H., & Molenaar, I.W. (1995). Rasch models: foundations, recent developments, and applications. New York: Springer. CrossRefGoogle Scholar
  14. Fishburn, P.C. (1981). Uniqueness properties in finite-continuous additive measurement. Mathematical Social Sciences, 1(2), 145–153. CrossRefGoogle Scholar
  15. Gonzales, C. (2000). Two factor additive conjoint measurement with one solvable component. Journal of Mathematical Psychology, 44, 285–309. PubMedCrossRefGoogle Scholar
  16. Holland, P.W. (1990). On the sampling theory foundations of item response theory models. Psychometrika, 55(4), 577–601. CrossRefGoogle Scholar
  17. Huang, K. (1987). Statistical mechanics. New York: Wiley. Google Scholar
  18. Irtel, H. (1987). On specific objectivity as a concept in measurement. In E.E. Roskam & R. Suck (Eds.), Progress in mathematical psychology-1. Amsterdam: Elsevier. Google Scholar
  19. Irtel, H. (1993). The uniqueness of simple latent trait models. In G.H. Fischer & D. Laming (Eds.), Contributions to mathematical psychology, psychometrics, and methodology. New York: Springer. Google Scholar
  20. Jaynes, E.T. (1957). Information theory and statistical mechanics. The Physical Review, 106(4), 620–630. CrossRefGoogle Scholar
  21. Jaynes, E.T. (1968). Prior probabilities. IEEE Transactions on Systems Science and Cybernetics, 4(3), 227–241. CrossRefGoogle Scholar
  22. Kagan, A.M., Linnik, V.Y., & Rao, C.R. (1973). Characterization problems in mathematical statistics. New York: Wiley. Google Scholar
  23. Karabatsos, G. (2001). The Rasch model, additive conjoint measurement, and new models of probabilistic measurement theory. Journal of Applied Measurement, 2(4), 389–423. PubMedGoogle Scholar
  24. Kyngdon, A. (2008). The Rasch model from the perspective of the representational theory of measurement. Theory & Psychology, 18, 89–109. CrossRefGoogle Scholar
  25. Kyngdon, A. (2011). Plausible measurement analogies to some psychometric models of test performance. British Journal of Mathematical & Statistical Psychology, 64, 478–497. CrossRefGoogle Scholar
  26. Krantz, D.H., Luce, R.D., Suppes, P., & Tversky, A. (1971). Foundations of measurement. Vol. 1: Additive and polynomial representations. San Diego: Academic Press. Google Scholar
  27. Landsberg, P.T. (1954). On most probable distributions. Proceedings of the National Academy of Sciences, 40, 149–154. CrossRefGoogle Scholar
  28. Lord, F.M., & Novik, M.R. (1968). Statistical theories of mental test scores. London: Addison-Wesley. Google Scholar
  29. Luce, R.D. (1959). Individual choice behavior: a theoretical analysis. New York: Wiley. Google Scholar
  30. Luce, R.D., Krantz, D.H., Suppes, S., & Tversky, A. (1990). Foundations of measurement. Vol. 3: Representation, axiomatization and invariance. San Diego: Academic Press. Google Scholar
  31. Luce, R.D., & Narens, L. (1994). Fifteen problems concerning the representational theories of measurement. In P. Humpreys (Ed.), Patrick suppes: scientific philosopher, (Vol. 2). Dordrecht: Kluwer Academic. Google Scholar
  32. Luce, R.D., & Tukey, J.W. (1964). Simultaneous conjoint measurement: a new scale type of fundamental measurement. Journal of Mathematical Psychology, 1, 1–27. CrossRefGoogle Scholar
  33. Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149–174. CrossRefGoogle Scholar
  34. Michell, J. (1990). An introduction to the logic of psychological measurement. Hillsdale: Erlbaum. Google Scholar
  35. Michell, J. (2009). The psychometricians’ fallacy: too clever by half? British Journal of Mathematical & Statistical Psychology, 62, 41–55. CrossRefGoogle Scholar
  36. Perline, R., Wright, B.D., & Wainer, H. (1979). The Rasch model as additive conjoint measurement. Applied Psychological Measurement, 3, 237–255. CrossRefGoogle Scholar
  37. Pfanzagl, J. (1971). Theory of measurement. Wurzburg and Vienna: Physica-Verlag. CrossRefGoogle Scholar
  38. Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Nielsen & Lydiche. Google Scholar
  39. Rasch, G. (1972). 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. Copenhagen: Munksgaard. Google Scholar
  40. Scott, D. (1964). Measurement structures and linear inequalities. Journal of Mathematical Psychology, 1, 233–247. CrossRefGoogle Scholar
  41. Suppes, P., & Zinnes, J.L. (1963). Basic theory of measurement. In R.D. Luce, R.R. Bush, & E. Galanter (Eds.), Handbook Math. Psych.: Vol. 1. New York: Wiley. Google Scholar
  42. Scheiblechner, H. (1972). Das Lernen und Lösen komplexer Denkaufgaben [The learning and solving of complex reasoning items]. Zeitschrift für Experimentelle und Angewandte Psychologie, 3, 456–506. Google Scholar
  43. Scheiblechner, H. (1995). Isotonic psychometrics models. Psychometrika, 60, 281–304. CrossRefGoogle Scholar
  44. Scheiblechner, H. (1999). Additive conjoint isotonic probabilistic models. Psychometrika, 64, 295–316. CrossRefGoogle Scholar
  45. Tversky, A. (1967). A general theory of polynomial conjoint measurement. Journal of Mathematical Psychology, 4, 1–20. CrossRefGoogle Scholar

Copyright information

© The Psychometric Society 2013

Authors and Affiliations

  • Stefano Noventa
    • 1
    Email author
  • Luca Stefanutti
    • 2
  • Giulio Vidotto
    • 3
  1. 1.Assessment CenterUniversity of VeronaVeronaItaly
  2. 2.FISSPAUniversity of PadovaPadovaItaly
  3. 3.Department of General PsychologyUniversity of PadovaPadovaItaly

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