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Applying the Principles of Specific Objectivity and of Generalizability to the Measurement of Change

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 Psychologie, Universität Wien, Liebiggasse 5, A-1010 Vienna, AUSTRIA.
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Abstract

A formal framework for measuring change in sets of dichotomous data is developed and implications of the principle of specific objectivity of results within this framework are investigated. Building upon the concept of specific objectivity as introduced by G. Rasch, three equivalent formal definitions of that postulate are given, and it is shown that they lead to latent additivity of the parametric structure. If, in addition, the observations are assumed to be locally independent realizations of Bernoulli variables, a family of models follows necessarily which are isomorphic to a logistic model with additive parameters, determining an interval scale for latent trait measurement and a ratio scale for quantifying change. Adding the further assumption of generalizability over subsets of items from a given universe yields a logistic model which allows a multidimensional description of individual differences and a quantitative assessment of treatment effects; as a special case, a unidimensional parameterization is introduced also and a unidimensional latent trait model for change is derived. As a side result, the relationship between specific objectivity and additive conjoint measurement is clarified.

Keywords

latent trait theoryitem response theorymeasurement of changelogistic latent trait modelsspecific objectivityconjoint measurementadditivity
Type
Original Paper
Information
Psychometrika , Volume 52 , Issue 4 , December 1987 , pp. 565 - 587
Copyright
Copyright © 1987 The Psychometric Society

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Footnotes

This research was supported in part by Österreichische Forschungsgemeinschaft under grant No. 01/0054. The author is indebted to A. Kriegl, E. E. Roskam, I. W. Molenaar, J. O. Ramsey, and J. Irtel for valuable critical comments on a previous draft of this paper.

References

Abel, N. H. (1826). Untersuchung der Functionen zweier unabhängig veränderlichen Grössenx undy, wief(x, y), welche die Eigenschaft haben, dassf (z, f(x, y)) eine symmetrische Function vonz, x undy ist [Investigation of the functions of two independent variables x und y, like f(x, y), with the property that f(z, f(x, y)) is a symmetric function of z, x, and y]. Journal für die reine und angewandte Mathematik, 1, 11–15.Google Scholar
Aczél, J. (1949). Über einparametrige Transformationen [On one-parametric transformations]. Publicationes Mathematicae, 1, 243–247.CrossRefGoogle Scholar
Aczél, J. (1966). Lectures on functional equations and their applications, New York: Academic Press.Google Scholar
Aczél, J., Belousov, V. D., Hosszú, M. (1960). Generalized associativity and bisymmetry on quasigroups. Acta Mathematica Academiae Scientiarum Hungaricae, 11, 127–136.CrossRefGoogle Scholar
Albert, D. (1985). Über Beziehungen zwischen Rasch-Modell und Prozeßmodellen. Ein Beitrag zur allgemeinen differentiellen Psychologie [On the relationship between Rasch models and process models. A contribution to general differential psychology]. Psychologische Beiträge, 27, 297–317.Google Scholar
Andersen, E. B. (1977). Sufficient statistics and latent trait models. Psychometrika, 42, 69–81.CrossRefGoogle Scholar
Andersen, E. B. (1980). Discrete statistical models with social science applications, Amsterdam: North-Holland.Google Scholar
Andersen, E. B. (1980). Comparing latent distributions. Psychometrika, 45, 121–134.CrossRefGoogle Scholar
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (Eds.), Statistical theories of mental test scores (pp. 395–479). Reading, MA: Addison-Wesley.Google Scholar
Borchsenius, K. (1974). A group theoretical formalization of the concept of objectivity. Unpublished manuscript, Copenhagen.Google Scholar
Colonius, H. (1979). Zur Eindeutigkeit der Parameter im Rasch-Modell [On the uniqueness of the parameters in the Rasch model]. Psychologische Beiträge, 21, 414–416.Google Scholar
Colonius, H. (1980). Representation and uniqueness of the Bradley-Terry-Luce model for pair comparisons. British Journal of Mathematical and Statistical Psychology, 33, 99–103.CrossRefGoogle Scholar
Fischer, G. H. (1968). Neue Entwicklungen in der psychologischen Testtheorie [Recent developments in psychological test theory]. In Fischer, G. H. (Eds.), Psychologische Testtheorie (pp. 15–158). Berne: Huber.Google Scholar
Fischer, G. H. (1972). A measurement model for the effect of mass-media. Acta Psychologica, 36, 207–220.CrossRefGoogle Scholar
Fischer, G. H. (1976). Some probabilistic models for measuring change. In de Gruijter, D. N. M., van der Kamp, L. J. Th. (Eds.), Advances in educational and psychological measurement (pp. 97–110). New York: John 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: John Wiley.Google Scholar
Fischer, G. H. (1977). Linear logistic test models. Theory and application. In Spada, H., Kempf, W. F. (Eds.), Structural models of thinking and learning (pp. 203–225). Berne: Hans Huber.Google 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, New York: Academic Press.Google Scholar
Fischer, G. H. (1985). Some consequences of specific objectivity for the measurement of change. In Roskam, E. E. (Eds.), Measurement and personality assessment (pp. 39–55). Amsterdam: North-Holland.Google Scholar
Fischer, G. H., Formann, A. K. (1982). Some applications of logistic latent trait models with linear constraints on the parameters. Applied Psychological Measurement, 6, 397–416.CrossRefGoogle Scholar
Hamerle, A. (1979). Über die messtheoretischen Grundlagen von Latent-Trait-Modellen [Foundations of measurement in latent trait models]. Archiv für Psychologie, 132, 19–39.Google Scholar
Hosszú, M. (1953). On the functional equation of transitivity. Acta Scientiarum Mathematicarum (Szegedina), 15, 203–208.Google Scholar
Krantz, D. H., Luce, R. D., Suppes, P., Tversky, A. (1971). Foundations of measurement, New York: Academic Press.Google Scholar
Lazarsfeld, P. F., Henry, N. W. (1968). Latent structure analysis, Boston: Houghton Mifflin.Google Scholar
Pexider, H. W. (1903). Une application d'une formule de Cauchy [An application of a formula of Cauchy]. R. C. Circulo matematico Palermo, 17, 236–240.CrossRefGoogle Scholar
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests, Copenhagen: Paedagogiske Institut.Google Scholar
Rasch, G. (1967). 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 Leiden.Google Scholar
Rasch, G. (1968). A mathematical theory of objectivity and its consequences for model construction, Amsterdam: 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, Copenhagen: Munksgaard.Google Scholar
Roskam, E. E. (1983). Allgemeine Datentheorie [General data theory]. In Feger, H., Bredenkamp, J. (Eds.), Messen und Testen, Göttingen: Verlag für Psychologie Hogrefe.Google Scholar
Roskam, E. E., Jansen, P. G. W. (1984). A new derivation of the Rasch model. In Degreef, E., van Buggenhaut, J. (Eds.), Trends in mathematical psychology, Amsterdam: Elsevier.Google Scholar
Schweitzer, A. R. (1911). On a functional equation. Bulletin of the American Mathematical Society, 18, 299–302.CrossRefGoogle Scholar
Stäckel, P. (1897). Über eine von Abel untersuchte Funktionalgleichung [On a functional equation investigated by Abel]. Zeitschrift für Mathematik und Physik, 42, 323–326.Google Scholar