On Item Parameter Estimation in Certain Latent Trait Models

  • J. Pfanzagl
Part of the Recent Research in Psychology book series (PSYCHOLOGY)


The paper deals with the estimation of item difficulty parameters (δ1,…, δm) if a subject with ability α produces the response vector (x 1,…, x m) ∈ {0, 1}m with probability \(\prod_{i=1}^{m}H{(\alpha-\delta_{i})}^{x_{i}}(1-H(\alpha-\delta_{i}))^{1-x_{i}}\).

If H = Ψ (the logistic distribution function), conditional maximum likelihood estimators are consistent under mild conditions on the ability parameters α j , j = 1, 2,…. If the ability parameters are a random sample from an unknown distribution, conditional maximum likelihood estimators are consistent and asymptotically efficient. Marginal maximum likelihood estimators share these properties, since they are asymptotically equivalent to conditional maximum likelihood estimators by a theorem of de Leeuw and Verhelst.

If H = Φ, estimating (δ1,…, δm) is impossible if the distribution of ability parameters is completely unknown, since the item difficulty parameters are not identifiable in this case.


Mixture Model Parametric Family Nuisance Parameter Unknown Distribution Ability Parameter 
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. Aczél, J. (1966). Lectures on functional equations and their applications. New York: Academic Press.Google Scholar
  2. Andersen, E.B. (1973a). Conditional inference and models for measuring. Copenhagen: Mentalhygiejnisk Forskningsforlag.Google Scholar
  3. Andersen, E.B. (1973b). A goodness of fit test for the Rasch model. Psychometrika, 38, 123–140.CrossRefGoogle Scholar
  4. Andersen, E.B. (1980). Discrete statistical models with social science applications. Amsterdam: North-Holland.Google Scholar
  5. Balakrishnan, N. (Ed.) (1992). Handbook of the logistic distribution. New York: Marcel Dekker.Google Scholar
  6. 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 (pp. 397–479). Reading, MA: Addison-Wesley.Google Scholar
  7. Bock, R.D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443–459.CrossRefGoogle Scholar
  8. Chambers, E.A., & Cox, D.R. (1967). Discrimination between alternative binary response models. Biometrika, 54, 573–578.PubMedGoogle Scholar
  9. de Leeuw, J., & Verhelst, N. (1986). Maximum likelihood estimation in generalized Rasch models. Journal of Educational Statistics, 11, 183–196.CrossRefGoogle Scholar
  10. Falmagne, J.C. (1985). Elements of psychophysical theory. Oxford Psychology Series Vol. 6. Oxford: Clarendon Press.Google Scholar
  11. Feller, W. (1939/41). On the logistic law of growth and its empirical verifications in biology. Acta Biotheoretica, A, 5, 51–66.CrossRefGoogle Scholar
  12. Fischer, G.H. (1968). Psychologische Testtheorie. Bern: Huber.Google Scholar
  13. Fischer, G.H. (1974). Einführung in die Theorie psychologischer Tests. Bern: Huber.Google Scholar
  14. Fischer, G.H. (1981). On the existence and uniqueness of maximum likelihood estimates in the Rasch model. Psychometrika, 46, 59–77.CrossRefGoogle Scholar
  15. Fischer, G.H. (1988). Spezifische Objektivität: Eine wissenschaftstheoretische Grundlage des Rasch-Modells. In K.D. Kubinger (Ed.), Moderne Testtheorie (pp. 87–111 ). Weinheim und München: Psychologie Verlags Union.Google Scholar
  16. Follman, D. (1988). Consistent estimation in the Rasch model based on non-parametric margins. Psychometrika, 53, 553–562.CrossRefGoogle Scholar
  17. Gigerenzer, G. (1981). Messung und Modellbildung in der Psychologie. München: Ernst Reinhardt.Google Scholar
  18. Hambleton, R.K., & Swaminathan, H. (1985). Item response theory. Boston: Kluwer and Nijhoff.Google Scholar
  19. Hillgruber, G. (1990). Schätzung von Parametern in psychologischen Testmodellen. Master’s thesis. Cologne: Department of Mathematics, University of Cologne.Google Scholar
  20. Lindsay, B., Clogg, C.C., & Grego, J. (1991). Semiparametric estimation in the Rasch model and related exponential response models, including a simple latent class model for item analysis. Journal of the American Statistical Association, 86, 96–107.CrossRefGoogle Scholar
  21. Pfanzagl, J. (1971). Theory of measurement (2nd ed.). New York: Wiley.Google Scholar
  22. Pfanzagl, J. (1990a). Estimation in semiparametric models. Some recent developments. Lecture Notes in Statistics, Vol. 63. New York: Springer-Verlag.Google Scholar
  23. Pfanzagl, J. (1990b). Incidental versus random nuisance parameters. Preprints in Statistics 128. Cologne: Department of Mathematics, University of Cologne. (To appear in: Annals of Statistics.)Google Scholar
  24. Pfanzagl, J. (1991a). On the identifiability of structural parameters in mixtures; applications to psychological tests. Preprints in Statistics 130. Cologne: Department of Mathematics, University of Cologne. (To appear in: Journal of Statistical Planning and Inference.)Google Scholar
  25. Pfanzagl, J. (1991b). A case of asymptotic equivalence between conditional and marginal maximum likelihood estimators. Preprints in Statistics 132. Cologne: Department of Mathematics, University of Cologne. (To appear in: Journal of Statistical Planning and Inference.)Google Scholar
  26. Pfanzagl, J. (1992). On the consistency of conditional maximum likelihood estimators. Preprints in Statistics 134. Cologne: Department of Mathematics, University of Cologne. (To appear in: Annals of the Institute of Statistical Mathematics.)Google Scholar
  27. Pfanzagl, J., & Wefelmeyer, W. (1982). Contributions to a general asymptotic statistical theory. Lecture Notes in Statistics, Vol. 13. New York: Springer-Verlag.Google Scholar
  28. Roskam, E.E., & Jansen, P.G.W. (1984). A new derivation of the Rasch model. In E. Degreef & J. van Buggenhaut, (Eds.), Trends in mathematical psychology (pp. 293–307 ). Amsterdam: North-Holland.CrossRefGoogle Scholar
  29. Thissen, D. (1982). Marginal maximum likelihood estimation for the one-parameter logistic model. Psychometrika, 47, 175–186.CrossRefGoogle Scholar
  30. Thurstone, L.L. (1927). A law of comparative judgement. Psychological Review, 34, 273–286.CrossRefGoogle Scholar
  31. Wright, B.D., & Douglas, G.A. (1977). Best procedures for sample-free item analysis. Applied Psychological Measurement, 1, 281–294.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • J. Pfanzagl
    • 1
  1. 1.Mathematical InstituteUniversity of CologneCologne 41Germany

Personalised recommendations