Psychometrika

, Volume 53, Issue 4, pp 553–562 | Cite as

Consistent estimation in the rasch model based on nonparametric margins

  • Dean Follmann
Article

Abstract

Consider the class of two parameter marginal logistic (Rasch) models, for a test ofm True-False items, where the latent ability is assumed to be bounded. Using results of Karlin and Studen, we show that this class of nonparametric marginal logistic (NML) models is equivalent to the class of marginal logistic models where the latent ability assumes at most (m + 2)/2 values. This equivalence has two implications. First, estimation for the NML model is accomplished by estimating the parameters of a discrete marginal logistic model. Second, consistency for the maximum likelihood estimates of the NML model can be shown (whenm is odd) using the results of Kiefer and Wolfowitz. An example is presented which demonstrates the estimation strategy and contrasts the NML model with a normal marginal logistic model.

Key words

nonparametric EM algorithm consistency identifiability marginal logistic model latent ability item analysis Rasch model 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andersen, E. B. (1973). Conditional inference for multiple-choice questionnaires.British Journal of Mathematical and Statistical Psychology, 26, 31–44.Google Scholar
  2. Andersen, E. B. (1980a). Comparing latent distributions,Psychometrika, 45, 121–134.Google Scholar
  3. Andersen, E. B. (1980b).Discrete statistical models with social science applications. Amsterdam: North-Holland.Google Scholar
  4. Andersen, E. B., & Madsen, M. (1977). Estimating the parameters of the latent population distribution.Psychometrika, 42, 357–374.Google Scholar
  5. Birnbaum, A. (1958).On the estimation of mental ability (series Report No. 15). Randolph Air Force Base, USAF School of Aviation Medicine.Google Scholar
  6. Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm.Psychometrika, 46, 443–459.Google Scholar
  7. Cressie, N., & Holland, P. W. (1983). Characterizing the manifest probabilities of latent trait models.Psychometrika, 48, 129–141.Google Scholar
  8. de Leeuw, J., & Verhelst, N. (1986). Maximum likelihood estimation in generalized Rasch models.Journal of Educational Statistics, 11, 183–196.Google Scholar
  9. Dempster, A. P., Laird, N., & Rubin, D. B. (1977). Maximum likelihood estimation with incomplete data via the EM algorithm.Journal of the Royal Statistical Society, Series B, 39, 1–38.Google Scholar
  10. Duncan, O. D. (1984). Rasch Measurement: Further Examples and Discussion, In C. F. Turner & E. Martin (eds.),Surveying subjective phenomena (vol. 2) New York: Russell Sage Foundation.Google Scholar
  11. Fienberg, S. E. (1986). The Rasch model, In S. Katz & N. L. Johnson (Eds.),Encyclopedia of statistical science (vol. 7). New York: John Wiley & Sons.Google Scholar
  12. Fienberg, S. E., & Meyer, M. M. (1983). Loglinear models and categorical data analysis with psychometric and econometric applications.Journal of Econometrics, 22, 191–214.Google Scholar
  13. Follmann, D. A. (1985).Nonparametric mixtures of logistic regression models. Unpublished doctoral dissertation, Carnegie-Mellon University, Pittsburgh, PA.Google Scholar
  14. Heckman, J. J., & Singer, B. (1984). A method for minimizing the impact of distributional assumptions in econometric models for duration data.Econometrica, 52, 271–320.Google Scholar
  15. Holland, P. W. (1981). When are item response models consistent with observed data?Psychometrika, 46, 79–92.Google Scholar
  16. Karlin, S., & Studden, W. J. (1966).Tchebycheff systems: With applications in analysis and statistics. New York: John Wiley & Sons.Google Scholar
  17. Kiefer, J., & Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters.The Annals of Statistics, 27, 805–811.Google Scholar
  18. Mislevy, R. J. (1984). Estimating latent distributions.Psychometrika, 49, 359–381.Google Scholar
  19. Rasch, G. (1960).Probabilistic models for some intelligence and attainment tests. Copenhagen: Danmarks Paedagogiske Institut.Google Scholar
  20. Redner, R. A., & Walker, H. F. (1984). Mixture Densities, Maximum Likelihood and the EM Algorithm.SIAM Review, 26 (2), 195–239.Google Scholar
  21. Sanathanan, L., & Blumenthal, S. (1978). The Logistic model and estimation of latent structure.Journal of the American Statistical Association, 73, 794–799.Google Scholar
  22. Simar, L. (1976). Maximum likelihood estimation of a compound Poisson process.The Annals of Statistics, 4, 1200–1209.Google Scholar
  23. Stouffer, S. A., & Toby, J. (1951). Role conflict and personality.American Journal of Sociology, 56, 395–406.Google Scholar
  24. Teicher, H. (1963). Identifiability of finite mixtures.Annals of Mathematical Statistics, 34, 1265–1269.Google Scholar
  25. Thissen, D. (1982). Marginal maximum likelihood for the one parameter logistic model.Psychometrika, 47, 201–214.Google Scholar
  26. Tjur, T. (1982). A connection between Rasch's item analysis model and a multiplicative Poisson model.Scandanavian Journal of Statistics, 9, 23–30.Google Scholar
  27. Wu, C. F. Jeff (1983). On the convergence properties of the EM algorithm.The Annals of Statistics, 11, 95–13.Google Scholar

Copyright information

© The Psychometric Society 1988

Authors and Affiliations

  • Dean Follmann
    • 1
  1. 1.National Heart, Lung, and Blood Institute Biostatistics Research BranchUSA

Personalised recommendations