Consistent estimation in the rasch model based on nonparametric margins Article Received: 13 August 1986 Revised: 27 August 1987 DOI:
10.1007/BF02294407 Cite this article as: Follmann, D. Psychometrika (1988) 53: 553. doi:10.1007/BF02294407 Abstract
Consider the class of two parameter marginal logistic (Rasch) models, for a test of
m 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 (when m 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
This research was supported by NIMH traning grant, 2 T32 MH 15758-06 and by ONR contract N00014-84-K-0588. The author would like to thank Diane Lambert, John Rolph, and Stephen Fienberg for their assistance. Also, the comments of the referees helped to substantially improve the final version of this paper.
Andersen, E. B. (1973). Conditional inference for multiple-choice questionnaires.
British Journal of Mathematical and Statistical Psychology, 26
Andersen, E. B. (1980a). Comparing latent distributions,
Andersen, E. B. (1980b).
Discrete statistical models with social science applications
. Amsterdam: North-Holland.
Andersen, E. B., & Madsen, M. (1977). Estimating the parameters of the latent population distribution.
Birnbaum, A. (1958).
On the estimation of mental ability (series Report No. 15). Randolph Air Force Base, USAF School of Aviation Medicine.
Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm.
Cressie, N., & Holland, P. W. (1983). Characterizing the manifest probabilities of latent trait models.
de Leeuw, J., & Verhelst, N. (1986). Maximum likelihood estimation in generalized Rasch models.
Journal of Educational Statistics, 11
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
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.
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.
Fienberg, S. E., & Meyer, M. M. (1983). Loglinear models and categorical data analysis with psychometric and econometric applications.
Journal of Econometrics, 22
Follmann, D. A. (1985).
Nonparametric mixtures of logistic regression models
. Unpublished doctoral dissertation, Carnegie-Mellon University, Pittsburgh, PA.
Heckman, J. J., & Singer, B. (1984). A method for minimizing the impact of distributional assumptions in econometric models for duration data.
Holland, P. W. (1981). When are item response models consistent with observed data?
Karlin, S., & Studden, W. J. (1966).
Tchebycheff systems: With applications in analysis and statistics
. New York: John Wiley & Sons.
Kiefer, J., & Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters.
The Annals of Statistics, 27
Mislevy, R. J. (1984). Estimating latent distributions.
Rasch, G. (1960).
Probabilistic models for some intelligence and attainment tests
. Copenhagen: Danmarks Paedagogiske Institut.
Redner, R. A., & Walker, H. F. (1984). Mixture Densities, Maximum Likelihood and the EM Algorithm.
SIAM Review, 26
Sanathanan, L., & Blumenthal, S. (1978). The Logistic model and estimation of latent structure.
Journal of the American Statistical Association, 73
Simar, L. (1976). Maximum likelihood estimation of a compound Poisson process.
The Annals of Statistics, 4
Stouffer, S. A., & Toby, J. (1951). Role conflict and personality.
American Journal of Sociology, 56
Teicher, H. (1963). Identifiability of finite mixtures.
Annals of Mathematical Statistics, 34
Thissen, D. (1982). Marginal maximum likelihood for the one parameter logistic model.
Tjur, T. (1982). A connection between Rasch's item analysis model and a multiplicative Poisson model.
Scandanavian Journal of Statistics, 9
Wu, C. F. Jeff (1983). On the convergence properties of the EM algorithm.
The Annals of Statistics, 11
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