Hostname: page-component-5cf477f64f-r2nwp Total loading time: 0 Render date: 2025-04-06T11:51:51.063Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Existence and Uniqueness of Maximum-Likelihood Estimates in the Rasch Model

Published online by Cambridge University Press:  01 January 2025

Gerhard H. Fischer
Gerhard H. Fischer*
Affiliation:
University of Vienna
*
Requests for reprints should be addressed to Gerhard H. Fischer, Institut für Psychologie, Universität Wien, Liebiggasse 5, A-1010 Wien, Austria.
Get access Rights & Permissions [Opens in a new window]

Abstract

Necessary and sufficient conditions for the existence and uniqueness of a solution of the so-called “unconditional” (UML) and the “conditional” (CML) maximum-likelihood estimation equations in the dichotomous Rasch model are given. The basic critical condition is essentially the same for UML and CML estimation. For complete data matrices A, it is formulated both as a structural property of A and in terms of the sufficient marginal sums. In case of incomplete data, the condition is equivalent to complete connectedness of a certain directed graph. It is shown how to apply the results in practical uses of the Rasch model.

Keywords

latent trait theoryRasch modelmaximum-likelihood estimation
Type
Original Paper
Information
Psychometrika , Volume 46 , Issue 1 , March 1981 , pp. 59 - 77
Copyright
Copyright © 1981 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Paper read at the European Meeting of the Psychometric Society, Groningen, June 19-21, 1980.

Part of the research reported herein was done while the author was staying at the Pulmologisches Zentrum der Stadt Wien; he is indebted to Professor Dr. F. Muhar and Dr. R. Mutschlechner for providing excellent working conditions.

References

Reference Notes

Andersen, E. B. The numerical solution of a set of conditional estimation equations (Rep. No. 5), 1970, Copenhagen: Copenhagen School of Economics and Business Administration.Google Scholar
Andersen, E. B. & Madsen, M. A computer program for parameter estimation and model check for a Rasch-model, 1975, Copenhagen: Universitetets Statistike Institut.Google Scholar
Gustafsson, J. E. & Lindblad, T. The Rasch model for dichotomous items: A solution of the conditional estimation problem for long tests and some thoughts about item screening procedures. Paper presented at the European Conference on Psychometrics and Mathematical Psychology, Uppsala, Sweden, June 15–17, 1978.Google Scholar
Rasch, G. An informal report on a theory of objectivity in comparisons. In Van der Kamp, L. J. Th. & Vlek, C. A. J. (Eds.), Measurement theory. Proceedings of the NUFFIC international summer session in science at “Het Oude Hof”, The Hague, July 14–28, 1966, 1967, Leiden: University of Leiden.Google Scholar
Rasch, G. A mathematical theory of objectivity and its consequences for model construction. Paper presented at the European Meeting on Statistics, Econometrics, and Management Science, Amsterdam, The Netherlands, September 2–7, 1968.Google Scholar
Rentz, R. R. & Rentz, C. C. Does the Rasch model really work? A synthesis of the literature for practitioners, 1978, Princeton: ERIC Clearinghouse on Tests, Measurement and Evaluation, Educational Testing Service.Google Scholar
Scheiblechner, H. A simple algorithm for CML-parameter-estimation in Rasch's probabilistic measurement model with two or more categories of answers (Res. Bull. No. 5), 1971, Vienna: Institute of Psychology, University of Vienna.Google Scholar

References

Andersen, E. B. Asymptotic properties of conditional likelihood ratio tests. Journal of the American Statistical Association, 1971, 66, 630–633.CrossRefGoogle Scholar
Andersen, E. B. The numerical solution of a set of conditional estimation equations. Journal of the Royal Statistical Society, Series B, 1972, 34, 42–54.CrossRefGoogle Scholar
Andersen, E. B. Conditional inference and models for measuring, 1973, Copenhagen: Mentalhygiejnisk Forlag.Google Scholar
Andersen, E. B. A goodness of fit test for the Rasch model. Psychometrika, 1973, 38, 123–140.CrossRefGoogle Scholar
Andersen, E. B. Conditional inference for multiple-choice questionnaires. The British Journal of Mathematical and Statistical Psychology, 1973, 26, 31–44.CrossRefGoogle Scholar
Andersen, E. B. Discrete statistical models with social science applications, 1980, Amsterdam: North-Holland Publishing Company.Google Scholar
Barndorff-Nielsen, O. Information and exponential families in statistical theory, 1978, New York: Wiley.Google Scholar
Beckenbach, E. F. & Bellman, R. Inequalities, 3rd ed., New York: Springer, 1971.Google Scholar
David, H. A. The method of paired comparisons, 1963, London: Griffin.Google Scholar
Fischer, G. H. Einführung in die Theorie psychologischer Tests, 1974, Bern: Huber (In German)Google Scholar
Fischer, G. H. Probabilistic test models and their applications. The German Journal of Psychology, 1978, 2, 298–319.Google Scholar
Fischer, G. H. & Allerup, P. Rechentechnische Fragen zu Raschs eindimensionalem Modell. In Fischer, G. H. (Eds.), Psychologische Testtheorie. Bern: Huber 1968, 269–280. (In German)Google Scholar
Fischer, G. H. & Scheiblechner, H. Algorithmen und Programme für das probabilistische Testmodell von Rasch. Psychologische Beiträge, 1970, 12, 23–51. (In German)Google Scholar
Ford, L. R. Jr. Solution of a ranking problem from binary comparisons. American Mathematical Monthly, 1957, 64, 28–33.CrossRefGoogle Scholar
Gustafsson, J. E. A solution of the conditional estimation problem for long tests in the Rasch model for dichotomous items. Educational and Psychological Measurement, 1980, 40, 377–385.CrossRefGoogle Scholar
Haberman, S. J. Maximum likelihood estimates in exponential response models. The Annals of Statistics, 1977, 5, 815–841.CrossRefGoogle Scholar
Hambleton, R. K., Swaminathan, H., Cook, L. L., Eignor, D. R. & Gifford, J. A. Developments in latent trait theory: Models, technical issues, and applications. Review of Educational Research, 1978, 48, 467–510.CrossRefGoogle Scholar
Harary, F., Norman, R. Z. & Cartwright, D. Structural models: An introduction to the theory of directed graphs, 1965, New York: Wiley.Google Scholar
Lord, F. M. Some how and which for practical tailored testing. In Van der Kamp, L. J. Th., Langerak, W. F. & De Gruijter, D. N. H. (Eds.), Psychometrics for educational debates. New York: Wiley. 1980, 189–205.Google Scholar
Rasch, G. Probabilistic models for some intelligence and attainment tests, 1960, Copenhagen: Paedagogiske Institut.Google Scholar
Rasch, G. On general laws and the meaning of measurement in psychology. In Neyman, J. (Eds.), Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability (Vol. 4). Berkeley: University of California Press. 1961, 321–333.Google Scholar
Rasch, G. An item analysis which takes individual differences into account. The British Journal of Mathematical and Statistical Psychology, 1966, 19, 49–57.CrossRefGoogle ScholarPubMed
Rasch, G. An individualistic approach to item analysis. In Lazarsfeld, P. F. & Henry, N. W. (Eds.), Readings in mathematical social science. Cambridge, Mass.: M.I.T. Press. 1966, 89–107.Google Scholar
Rasch, G. Objektivitet i samfundsvidenskaberne et metodeproblem. Nationaløkonomisk Tidsskrift, 1972, 110, 161–196. (In Danish)Google Scholar
Rasch, G. 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, 1977, Copenhagen: Munksgaard.Google Scholar
Ree, M. J. Estimating item characteristic curves. Applied Psychological Measurement, 1979, 3, 371–385.CrossRefGoogle Scholar
Samejima, F. A comment on Birnbaum's three-parameter logistic model in the latent trait theory. Psychometrika, 1973, 38, 221–233.CrossRefGoogle Scholar
Sixtl, F. Skalierungsverfahren: Grundzüge und ausgewählte Methoden sozialwissenschaftlichen Messens. In Holm, K. (Eds.), Die Befragung [The interview] (Vol. 4). München: Francke, 1976, 9–95. (In German)Google Scholar
Wood, R. Trait measurement and item banks. In De Gruijter, D. N. M. & Van der Kamp, L. J. Th. (Eds.), Advances in psychological and educational measurement. New York: Wiley. 1976, 247–263.Google Scholar
Wright, B. D. & Douglas, G. A. Best procedures for sample-free item analysis. Applied Psychological Measurement, 1977, 1, 281–294.CrossRefGoogle Scholar
Wright, B. D. & Douglas, G. A. Conditional versus unconditional procedures for sample-free item analysis. Educational and Psychological Measurement, 1977, 37, 573–586.CrossRefGoogle Scholar
Wright, B. D. & Stone, M. H. Best test design, 1979, Chicago: Mesa Press.Google Scholar
Zermelo, E. Die Berechnung der Turnierergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung [The computation of tournament results as a maximum-probability problem]. Mathematische Zeitschrift, 1929, 29, 436–460. (In German)CrossRefGoogle Scholar