Hostname: page-component-7dd5485656-j7khd Total loading time: 0 Render date: 2025-10-29T16:55:26.184Z Has data issue: false hasContentIssue false
  • English
  • Français

An Irt-Based Model for Dichotomous Longitudinal Data

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 G. H. Fischer, Institut für Psychologic, Universitfit Wien, Liebiggasse 5, A-1010 Vienna, AUSTRIA
Get access Rights & Permissions [Opens in a new window]

Abstract

The LLRA (linear logistic model with relaxed assumptions; Fischer, 1974, 1977a, 1977b, 1983a) was developed, within the framework of generalized Rasch models, for assessing change in dichotomous item score matrices between two points in time; it allows to quantify change on latent trait dimensions and to explain change in terms of treatment effects, treatment interactions, and a trend effect. A remarkable feature of the model is that unidimensionality of the item set is not required. The present paper extends this model to designs with any number of time points and even with different sets of items presented on different occasions, provided that one unidimensional subscale is available per latent trait. Thus unidimensionality assumptions within subscales are combined with multidimensionality of the item set. Conditional maximum likelihood methods for parameter estimation and hypothesis testing are developed, and a necessary and sufficient condition for unique identification of the model, given the data, is derived. Finally, a sample application is presented.

Keywords

Rasch modelslogistic modelsmeasurement of change

Information

Type
Original Paper
Information
Psychometrika , Volume 54 , Issue 4 , December 1989 , pp. 599 - 624
Copyright
Copyright © 1989 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.)

Article purchase

Temporarily unavailable

Footnotes

To my friend Josef Roppert who has taught me how to apply statistical reasoning to substantive problems.

This research was supported in part by Österreichische Forschungsgemeinschaft under grant No. 01/0054. The author wishes to thank B. Wild for the numerical computation of the sample application in section 5.

References

Andersen, E. B. (1973). A goodness of fit test for the Rasch model. Psychometrika, 38, 123–140.CrossRefGoogle Scholar
Andersen, E. B. (1980). Discrete statistical models with social science applications, Amsterdam: North-Holland.Google Scholar
Bock, R. D. (1979). Univariate and multivariate analysis for time-structured data. In Nesselroade, J. R. & Baltes, P. B. (Eds.), Longitudinal research in the study of behavior and development (pp. 199–231). New York: Academic Press.Google Scholar
Christofides, N. (1975). Graph theory. An algorithmic approach, New York: Academic Press.Google Scholar
Fischer, G. H. (1974). Einführung in die Theorie psychologischer Tests [Introduction to mental test theory], Berne: Huber.Google Scholar
Fischer, G. H. (1976). Some probabilistic models for measuring change. In de Gruijter, D. N. N. & van der Kamp, L. J. Th (Eds.), Advances in psychological and educational measurement (pp. 97–110). New York: J. Wiley.Google Scholar
Fischer, G. H. (1977). 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 (pp. 102–151). Berne: Huber, and New York: Wiley.Google Scholar
Fischer, G. H. (1977). Linear logistic latent trait models: Theory and application. In Spada, H. & Kempf, W. F. (Eds.), Structural models of thinking and learning (pp. 203–225). Berne: Huber.Google Scholar
Fischer, G. H. (1981). On the existence and uniqueness of maximum-likelihood estimates in the Rasch model. Psychometrika, 46, 53–77.CrossRefGoogle Scholar
Fischer, G. H. (1983). Logistic latent trait models with linear constraints. Psychometrika, 48, 3–26.CrossRefGoogle Scholar
Fischer, G. H. (1983). Some latent trait models for measuring change in qualitative observations. In Weiss, D. J. (Eds.), New horizons in testing (pp. 309–329). New York: Academic Press.Google Scholar
Fischer, G. H. (1986). Ein probabilistischer Ansatz der Veränderungsmessung [A probabilistic approach to the measurement of change]. In Petillon, H. & Wagner, J., Wolf, B. (Eds.), Schülergerechte Diagnose (pp. 159–185). Weinheim: Beltz.Google Scholar
Fischer, G. H. (1987). Applying the principles of specific objectivity and generalizability to the measurement of change. Psychometrika, 52, 565–587.CrossRefGoogle Scholar
Fischer, G. H., & Formann, A. K. (1982). Veränderungsmessung mittels linear-logistischer Modelle [Measuring change by means of linear logistic models]. Zeitschrift für Differentielle und Diagnostische Psychologie, 3, 75–99.Google Scholar
Formann, A. K. (1986). A note on the computation of the second-order derivatives of the elementary symmetric functions in the Rasch model. Psychometrika, 51, 335–339.CrossRefGoogle Scholar
Gustafsson, J.-E. (1980). A solution of the conditional estimation problem for long tests in the Rasch model for dichotomous items. Educational and Psychological Measurement, 40, 377–385.CrossRefGoogle Scholar
Haberman, S. J. (1974). The analysis of frequency data, Chicago: The University of Chicago Press.Google Scholar
Haberman, S. J. (1978). Analysis of qualitative data (Vol. 1), New York: Academic Press.Google Scholar
Heckl, U. (1976). Therapieerfolge bei der Behandlung sprachgestörter Kinder [Effects of therapy in speech-handicapped children], Vienna: University of Vienna. Unpublished doctoral dissertationGoogle Scholar
Iby, M. (1987). Die Effektivität von Kommunikationsseminaren in der innerbetrieblichen Ausbildung der Zentralsparkasse [The efficacy of communication seminars for employees of the “Zentralsparkasse”], Vienna: University of Vienna.Google Scholar
Jansen, P. G. W. (1984). Computing the second-order derivatives of the symmetric functions in the Rasch model. Kwantitatieve Methoden, 5, 131–147.Google Scholar
Krauth, J. (1983). Bewertung der Änderungssensitivität von Items [Assessing item sensitivity to change]. Zeitschrift für Differentielle und Diagnostische Psychologie, 4, 7–28.Google Scholar
Lehmann, E. L. (1986). Testing statistical hypotheses 2nd ed.,, New York: J. Wiley.CrossRefGoogle Scholar
Mutschlechner, R. (1987). Der Patient im Krankenhaus—Ein Versuch, die Wirksamkeit einer psychologischen Betreuung nachzuweisen [The patient in hospital—An attempt to assess the effects of psychological treatment], Vienna: University of Vienna.Google Scholar
Pendl, P. (1976). Effektivität des Sprachlabors an Höheren Schulen [The efficacy of a language laboratory in high schools], Vienna: University of Vienna.Google Scholar
Plewis, I. (1985). Analysing change, Chichester: J. Wiley.Google Scholar
Rasch, G. (1967). An informal report on a theory of objectivity in comparisons. In van der Kamp, L. T. Th. & Vlek, C. A. J. (Eds.), Measurement theory (pp. 1–19). Leyden: University of Leyden.Google Scholar
Rasch, G. (1968). A mathematical theory of objectivity and its consequences for model construction (pp. GR1–GR31). Amsterdam, The Netherlands: Econometrics and Management Science.Google Scholar
Rasch, G. (1972). Objectivitet i samfundsvidenskaberne et metodeproblem [Objectivity in the social sciences as a methodological problem]. Nationaløkonomisk Tidsskrift, 110, 161–196.Google Scholar
Rasch, G. (1977). 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 (pp. 58–94). Copenhagen: Munksgaard.Google Scholar
Rop, I. (1977). 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 (pp. 281–293). Berne: Huber.Google Scholar
Rost, J. & Spada, H. (1983). Die Quantifizierung von Lerneffekten anhand von Testdaten [Quantifying practice effects in test data]. Zeitschrift für Differentielle und Diagnostische Psychologie, 4, 29–49.Google Scholar
Schmied, Ch. (1987). Die Effektivität von Managementseminaren—durchgeführt bei den Österreichischen Bundesbahnen [The efficacy of management seminars for employees of the Austrian Federal Railways], Vienna: University of Vienna.Google Scholar
Schumacher, M. (1980). Point estimation is quantal response models. Biometrical Journal, 22, 315–334.CrossRefGoogle Scholar
Seber, G. A. F. (1979). The linear hypothesis: A general theory, London: Charles Griffin.Google Scholar
Spada, H. (1983). Die Analyse von Veränderungen im Rahmen unterschiedlicher testtheoretischer Modelle [Analyzing change within the framework of various models in test theory]. In Minsel, W. R. & Scheller, R. (Eds.), Brennpunkte der klinischen Psychologie (Vol. 5), München: Kösel.Google Scholar
Spiel, Ch., Formann, A. K. (1988). Veränderungsmessung mit paarweise homogenen Items [The measurement of change based on pairs of homogeneous items]. In Kubinger, K. D. (Eds.), Moderne Testtheorie (pp. 296–310). Weinheim and München: Psychologie Verlags Union.Google Scholar
Tanzer, N. (1984). On the existence of unique joint maximum-likelihood estimates in linear logistic latent trait models for incomplete dichotomous data, Vienna: University of Vienna, Department of Psychology.Google Scholar
Unkelbach, H. D., & Wolf, T. (1985). Qualitative Dosis-Wirkungs-Analysen [Qualitative analysis of doseresponse-functions], Stuttgart: Gustav Fischer.Google Scholar
Verhelst, N. D., & Glas, C. A. W. (1984). Using the Rasch-model in the analysis of multifactorial (quasi-)experiments and interrupted time series, Utrecht: State University of Utrecht, Vakgroep Psychometrie, Statistiek en Modelvorming.Google Scholar
Verhelst, N. D., Glas, C. A. W., & van der Sluis, A. (1984). Estimation problems in the Rasch model: The basic symmetric functions. Computational Statistics Quarterly, 1, 245–262.Google Scholar
Widowitz, E. (1987). Der Effekt autogenen Trainings bei funktionellen Erkrankungen [The effect of “autogenous training” on the functional syndrome], Vienna: University of Vienna.Google Scholar
Zimprich, H. (1980). Behandlungskonzepte und—resultate bei psychosomatischen Erkrankungen im Kindesalter [Designs and results of treatments of psychosomatic diseases in children]. In Zimprich, H. (Eds.), Kind und Umwelt (pp. 131–198). Vienna: Springer.CrossRefGoogle Scholar