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

Published online by Cambridge University Press:  01 January 2025

Stefano Noventa ,
Luca Stefanutti  and
Giulio Vidotto
Stefano Noventa*
Affiliation:
Assessment Center, University of Verona
Luca Stefanutti
Affiliation:
FISSPA, University of Padova
Giulio Vidotto
Affiliation:
Department of General Psychology, University of Padova
*
Requests for reprints should be sent to Stefano Noventa, Assessment Center University of Verona, Verona, Italy. E-mail: stefano.noventa@univr.it
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Abstract

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.

Keywords

Rasch modelitem response theorymost probable distribution
Type
Original Paper
Information
Psychometrika , Volume 79 , Issue 3 , July 2014 , pp. 377 - 402
Copyright
Copyright © 2013 The Psychometric Society

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References

Adams, E.W. (1965). Elements of a theory of inexact measurement. Philosophy of Science, 32(3), 205228CrossRefGoogle Scholar
Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43, 561–573CrossRefGoogle Scholar
Andrich, D. (1982). An extension of the Rasch model for ratings providing both location and dispersion parameters. Psychometrika, 47(1), 105113CrossRefGoogle Scholar
Aczel, J. (1966). Lectures on functional equations and their applications, New York: Academic PressGoogle Scholar
Aczel, J., Dohmbres, J. (1989). Functional equations in several variables, Cambridge: Cambridge University PressCrossRefGoogle Scholar
Bradley, R.A., Terry, M.E. (1952). Rank analysis of incomplete block designs: I. The method of paired comparisons. Biometrika, 39(3/4), 324345Google Scholar
Bernardo, J.M., Smith, A.F.M. (1994). Bayesian theory, Chichester: WileyCrossRefGoogle Scholar
Barton, M.A., Lord, F.M. (1981). An upper asymptote for the three-parameter logistic item-response model, Princeton: Educational testing serviceCrossRefGoogle Scholar
Clinton, W.L., Massa, L.J. (1972). Derivation of a statistical mechanical distribution function by a method of inequalities. American Journal of Physics, 40, 608–610CrossRefGoogle Scholar
Davis-Stober, C.P. (2009). Analysis of multinomial models under inequalities constraints: applications to measurement theory. Journal of Mathematical Psychology, 53, 1–13CrossRefGoogle Scholar
Fischer, G.H. (1973). The linear logistic test model as an instrument in educational research. Acta Psychologica, 37, 359–374CrossRefGoogle Scholar
Fischer, G.H. (1995). Some neglected problems in IRT. Psychometrika, 60(4), 459487CrossRefGoogle Scholar
Fischer, G.H., Molenaar, I.W. (1995). Rasch models: foundations, recent developments, and applications, New York: SpringerCrossRefGoogle Scholar
Fishburn, P.C. (1981). Uniqueness properties in finite-continuous additive measurement. Mathematical Social Sciences, 1(2), 145153CrossRefGoogle Scholar
Gonzales, C. (2000). Two factor additive conjoint measurement with one solvable component. Journal of Mathematical Psychology, 44, 285–309CrossRefGoogle ScholarPubMed
Holland, P.W. (1990). On the sampling theory foundations of item response theory models. Psychometrika, 55(4), 577601CrossRefGoogle Scholar
Huang, K. (1987). Statistical mechanics, New York: WileyGoogle Scholar
Irtel, H. (1987). On specific objectivity as a concept in measurement. In Roskam, E.E., Suck, R. (Eds.), Progress in mathematical psychology-1, Amsterdam: ElsevierGoogle Scholar
Irtel, H. (1993). The uniqueness of simple latent trait models. In Fischer, G.H., Laming, D. (Eds.), Contributions to mathematical psychology, psychometrics, and methodology, New York: SpringerGoogle Scholar
Jaynes, E.T. (1957). Information theory and statistical mechanics. The Physical Review, 106(4), 620630CrossRefGoogle Scholar
Jaynes, E.T. (1968). Prior probabilities. IEEE Transactions on Systems Science and Cybernetics, 4(3), 227241CrossRefGoogle Scholar
Kagan, A.M., Linnik, V.Y., Rao, C.R. (1973). Characterization problems in mathematical statistics, New York: WileyGoogle Scholar
Karabatsos, G. (2001). The Rasch model, additive conjoint measurement, and new models of probabilistic measurement theory. Journal of Applied Measurement, 2(4), 389423Google ScholarPubMed
Kyngdon, A. (2008). The Rasch model from the perspective of the representational theory of measurement. Theory & Psychology, 18, 89–109CrossRefGoogle Scholar
Kyngdon, A. (2011). Plausible measurement analogies to some psychometric models of test performance. British Journal of Mathematical & Statistical Psychology, 64, 478–497CrossRefGoogle ScholarPubMed
Krantz, D.H., Luce, R.D., Suppes, P., Tversky, A. (1971). Foundations of measurement, San Diego: Academic PressGoogle Scholar
Landsberg, P.T. (1954). On most probable distributions. Proceedings of the National Academy of Sciences, 40, 149–154CrossRefGoogle ScholarPubMed
Lord, F.M., Novik, M.R. (1968). Statistical theories of mental test scores, London: Addison-WesleyGoogle Scholar
Luce, R.D. (1959). Individual choice behavior: a theoretical analysis, New York: WileyGoogle Scholar
Luce, R.D., Krantz, D.H., Suppes, S., Tversky, A. (1990). Foundations of measurement, San Diego: Academic PressGoogle Scholar
Luce, R.D., Narens, L. (1994). Fifteen problems concerning the representational theories of measurement. In Humpreys, P. (Eds.), Patrick suppes: scientific philosopher, Dordrecht: Kluwer AcademicGoogle Scholar
Luce, R.D., Tukey, J.W. (1964). Simultaneous conjoint measurement: a new scale type of fundamental measurement. Journal of Mathematical Psychology, 1, 1–27CrossRefGoogle Scholar
Masters, G.N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149174CrossRefGoogle Scholar
Michell, J. (1990). An introduction to the logic of psychological measurement, Hillsdale: ErlbaumGoogle Scholar
Michell, J. (2009). The psychometricians’ fallacy: too clever by half?. British Journal of Mathematical & Statistical Psychology, 62, 41–55CrossRefGoogle ScholarPubMed
Perline, R., Wright, B.D., Wainer, H. (1979). The Rasch model as additive conjoint measurement. Applied Psychological Measurement, 3, 237–255CrossRefGoogle Scholar
Pfanzagl, J. (1971). Theory of measurement, Wurzburg and Vienna: Physica-VerlagCrossRefGoogle Scholar
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests, Copenhagen: Nielsen & LydicheGoogle Scholar
Rasch, G. (1972). 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, Copenhagen: MunksgaardGoogle Scholar
Scott, D. (1964). Measurement structures and linear inequalities. Journal of Mathematical Psychology, 1, 233–247CrossRefGoogle Scholar
Suppes, P., Zinnes, J.L. (1963). Basic theory of measurement. In Luce, R.D., Bush, R.R., Galanter, E. (Eds.), Handbook Math. Psych., New York: WileyGoogle Scholar
Scheiblechner, H. (1972). Das Lernen und Lösen komplexer Denkaufgaben [The learning and solving of complex reasoning items]. Zeitschrift für Experimentelle und Angewandte Psychologie, 3, 456–506Google Scholar
Scheiblechner, H. (1995). Isotonic psychometrics models. Psychometrika, 60, 281–304CrossRefGoogle Scholar
Scheiblechner, H. (1999). Additive conjoint isotonic probabilistic models. Psychometrika, 64, 295–316CrossRefGoogle Scholar
Tversky, A. (1967). A general theory of polynomial conjoint measurement. Journal of Mathematical Psychology, 4, 1–20CrossRefGoogle Scholar