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Logistic Latent Trait Models with Linear Constraints

Published online by Cambridge University Press:  01 January 2025

Gerhard H. Fischer
Gerhard H. Fischer*
Affiliation:
University of Vienna
*
Requests for reprints should be sent to Gerhard H. Fischer, Institut für Psychologic, Universität Wen, Liebiggasse 5, A-1010 Wien, Austria.
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Abstract

Two linearly constrained logistic models which are based on the well-known dichotomous Rasch model, the ‘linear logistic test model’ (LLTM) and the ‘linear logistic model with relaxed assumptions’ (LLRA), are discussed. Necessary and sufficient conditions for the existence of unique conditional maximum likelihood estimates of the structural model parameters are derived. Methods for testing composite hypotheses within the framework of these models and a number of typical applications to real data are mentioned.

Keywords

latent trait theoryRasch modellog-linear modelsconditional maximum likelihood estimation

Information

Type
Original Paper
Information
Psychometrika , Volume 48 , Issue 1 , March 1983 , pp. 3 - 26
Copyright
Copyright © 1983 The Psychometric Society

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Footnotes

This research was supported in part by the Österreichische Forschungsgemeinschaft under grant No.01/0054. The author is indebted to Norbert Tanzer for many valuable comments.

Paper read at the Meeting of the Psychometric Society at Chapel Hill, N.C., May 27-29, 1981.

References

References Notes

Piswanger, K. Interkulturelle Vergleiche mit dem Matrizentest von Formann (Cross-cultural comparisons with Formann's Matrices Test). Vienna: University of Vienna, Philosophical Dissertation, 1975.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, Leiden: University of Leyden, 1967.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 Nederlands, September 2–7, 1968.Google Scholar
Fischer, G. H. & Formann, A. K. An algorithm and a FORTRAN program for estimating the item parameters of the linear logistic test model, Vienna: Institute of Psychology, University of Vienna, 1972.Google Scholar
Formann, A. K. Die Konstruktion eines neuen Matrizentests und die Untersuchung des Lösungsverhaltens mit Hilfe des linearen logistischen Testmodells (Constructing a new matrices test and analyzing the test behavior by means of the linear logistic test model). Vienna: University of Vienna, Philosophical Dissertation, 1973.Google Scholar
Nährer, W. Modellkontrollen bei Anwendung des linearen logistischen Testmodells in der Psychologie (Tests of fit in applications of the linear logistic test model in psychology). Vienna: University of Vienna, Philosophical Dissertation, 1977.Google Scholar
Heckl, U. Therapieerfolge bei der Behandlung sprachgestörter Kinder (Therapeutic effects in speech-handicapped children). Vienna: University of Vienna, Philosophical Dissertation, 1976.Google Scholar
Glatz, E. M. Die Wirksamkeit eines verhaltenstherapeutischen Eßtrainings bei geistig retardierten Kindern (The effects of behavior therapy on eating behavior in mentally retarded children). Vienna: University of Vienna, Philosophical Dissertation, 1977.Google Scholar

References

Andersen, E. B. Discrete statistical models with social science applications, Amsterdam: North-Holland Publishing Company, 1980.Google Scholar
Andersen, E. B. Comparing latent distributions. Psychometrika, 1980, 45, 121–134.CrossRefGoogle Scholar
Andrich, D. & Kline, P. Within and among population item fit with the simple logistic model. Educational and Psychological Measurement, 1981, 41, 35–48.CrossRefGoogle Scholar
Barndorff-Nielsen, O. Information and exponential families in statistical theory, New York: Wiley, 1978.Google Scholar
Bock, R. D. & Aitkin, M. Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 1981, 46, 443–459.CrossRefGoogle Scholar
Breslow, N. Regression analysis of the log odds ratio: A method for retrospective studies. Biometrics, 1976, 32, 409–416.CrossRefGoogle ScholarPubMed
Christofides, N. Graph theory. An algorithmic approach, New York: Academic Press, 1975.Google Scholar
Cliff, N. Complete orders from incomplete data: Interactive ordering and tailored testing. Psychological Bulletin, 1975, 82, 289–302.CrossRefGoogle Scholar
Cox, D. R. The analysis of binary data, London: Methuen, 1970.Google Scholar
Dantzig, G. B. Linear programming and extensions, Princeton: Princeton University Press, 1963.Google Scholar
Fischer, G. H. A measurement model for the effect of mass-media. Acta Psychologica, 1972, 36, 207–220.CrossRefGoogle Scholar
Fischer, G. H. The linear logistic test model as an instrument in educational research. Acta Psychologica, 1973, 37, 359–374.CrossRefGoogle Scholar
Fischer, G. H. Einführung in die Theorie psychologischer Tests (Introduction to the theory of psychological tests). Bern: Huber, 1974.Google Scholar
Fischer, G. H. Some probabilistic models for measuring change. In de Gruijter, D. N. M. & van der Kamp, J. L. Th. (Eds.), Advances in psychological and educational measurement . New York: Wiley. 1976, 97–110.Google Scholar
Fischer, G. H. Some probabilistic models for the description of attitudinal and behavioral changes under the influence of mass communication. In Kempf, W. F. & Repp, B. (Eds.), Mathematical models for social psychology . Bern: Huber. 1977, 102–151.Google Scholar
Fischer, G. H. Linear logistic test models: Theory and application. In Spada, H. & Kempf, W. F. (Eds.), Structural models of thinking and learning . Bern: Huber. 1977, 203–225.Google Scholar
Fischer, G. H. On the existence and uniqueness of maximum-likelihood estimates in the Rasch model. Psychometrika, 1981, 46, 59–77.CrossRefGoogle Scholar
Fischer, G. H. & Formann, A. K. Veränderungsmessung mittels linear-logistischer Modelle (Measuring change by means of linear logistic models). Zeitschrift für Differentielle und Diagnostische Psychologie, 1982, 3, 75–99.Google Scholar
Fischer, G. H. & Pendl, P. Individualized testing on the basis of the dichotomous Rasch model. In van der Kamp, L. J. Th., Langerak, W. F. & de Gruijter, D. N. M. (Eds.), Psychometrics for educational debates . New York: Wiley. 1980, 171–188.Google Scholar
Formann, A. K. & Piswanger, K. Wiener Matrizen-Test (The Viennese Matrices Test), Weinheim: Beltz, 1979.Google 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. The analysis of frequency data, Chicago: The University of Chicago Press, 1974.Google Scholar
Harary, F., Norman, R. Z. & Cartwright, D. Structural models: An introduction to the theory of directed graphs, New York: Wiley, 1965.Google Scholar
Koch, G. G., Landis, J. R., Freeman, J. L., Freeman, D. H. Jr. & Lehnen, R. G. A general methodology for the analysis of experiments with repeated measurement of categorical data. Biometrics, 1977, 33, 133–158.CrossRefGoogle ScholarPubMed
Marascuilo, L. A. & Serlin, R. C. Tests and contrasts for comparing change parameters for a multiple sample McNemar data model. The British Journal of Mathematical and Statistical Psychology, 1979, 32, 105–112.CrossRefGoogle Scholar
Mellenbergh, G. J. Applicability of the Rasch model in two cultures. In Cronbach, L. J. & Drenth, P. J. D. (Eds.), Mental tests and cultural adaptation . The Hague: Mouton. 1972, 453–457.Google Scholar
Mellenbergh, G. J. & Vijn, P. The Rasch model as a loglinear model. Applied Psychological Measurement, 1981, 5, 369–376.CrossRefGoogle Scholar
Nährer, W. Zur Analyse von Matrizenaufgaben mit dem linearen logistischen Testmodell (On the analysis of matrices items with the linear logistic test model). Zeitschrift für Experimentelle und Angewandte Psychologie, 1980, 27, 553–564.Google Scholar
Nenty, H. J. & Dinero, Th.E. A cross-cultural analysis of the fairness of the Cattell Culture Fair Intelligence Test using the Rasch model. Applied Psychological Measurement, 1981, 5, 355–368.CrossRefGoogle Scholar
Plewis, I. A comparison of the approaches to the analysis of longitudinal categoric data. The British Journal of Mathematical and Statistical Psychology, 1981, 34, 118–123.CrossRefGoogle Scholar
Rasch, G. Probabilistic models for some intelligence and attainment tests, Copenhagen: Paedagogiske Institut, 1960.Google Scholar
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 (Objectivity in the social sciences as a methodological problem). Nationaløkonomisk Tidsskrift, 1972, 110, 161–196.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 . Copenhagen: Munksgaard. 1977, 58–94.Google Scholar
Rop, I. The application of a linear logistic model describing the effects of pre-school curricula on cognitive growth. In Spada, H. & Kempf, W. F. (Eds.), Structural models of thinking and learning . Bern: Huber. 1977, 281–293.Google Scholar
Scheiblechner, H. Das Lernen und Lösen komplexer Denkaufgaben (The learning and solving of complex reasoning tasks). Zeitschrift für Experimentelle und Angewandte Psychologie, 1972, 19, 476–506.Google Scholar
Scheuneman, J. Latent-trait theory and item bias. In van der Kamp, L. J.Th., Langerak, W. F. & de Gruijter, D. N. M. (Eds.), Psychometrics for educational debates . New York: Wiley. 1980, 139–151.Google Scholar
Spada, H. Intelligenztheorie und Intelligenzmessung (Theory and measurement of intelligence). Psychologische Beiträge, 1970, 12, 83–96.Google Scholar
Spada, H. Modelle des Denkens und Lernens (Models of thinking and learning), Bern: Huber, 1976.Google Scholar
Wainer, H., Morgan, A. & Gustafsson, J.-E. A review of estimation procedures for the Rasch model with an eye toward longish tests. Journal of Educational Statistics, 1980, 5, 35–64.CrossRefGoogle Scholar
Whitely, S. E. & Dawis, R. V. The nature of objectivity with the Rasch model. Journal of Educational Measurement, 1974, 11, 163–178.CrossRefGoogle Scholar
Wright, B. D. & Stone, M. H. Best test design, Chicago: Mesa Press, 1979.Google Scholar