### Journals Information

Mathematics and Statistics Vol. 10(3), pp. 690 - 700
DOI: 10.13189/ms.2022.100326
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## Form Invariance - An Alternative Answer to the Measurement Problem of Item Response Theory

Henrik Bernshausen 1, Christoph Fuhrmann 2, Hanns-Ludwig Harney 3, Klaus Harney 4, Andreas Muller 5,*
1 Faculty of Science and Technology, Physics Education Group, University of Siegen, D-57068 Siegen, Germany
2 School of Education, Institute for educational research, University of Wuppertal, D-42097 Wuppertal, Germany
3 Max-Planck-Institute for Nuclear Physics, Postfach 103980, D-69029 Heidelberg, Germany
4 Faculty of Philosophy and Educational Science, Institute for educational research, Ruhr-Universität, D-44801 Bochum, Germany
5 Faculty of Sciences/Physics Department and Institut Universitaire de Formation des Enseignants, University of Geneva, CH - 1211 Geneva 4, Switzerland

ABSTRACT

The measurement problem of item response theory is the question of how to assign ability parameters to persons and difficulty parameters to items such that the comparison of abilities is independent of the specific set of difficulties . Correspondingly, the comparison of difficulties should be independent of the specific set of abilities . These requirements are called specific objectivity. They are the basis of the Rasch model. It measures and on one and the same scale. The present paper asks the different question of how to assign ability parameters to persons in a way that the comparison of abilities is independent of the position on the scale where the measurement takes place. Correspondingly, the comparison of difficulties should also be independent of the position on the scale where the calibration of difficulties takes place. Again, and measured on one and the same scale. These requirements are called form invariance. They lead to an item response function (IRF) different from that of the Rasch model. It integrates information from and beyond the mere score dependence and also shows specific objectivity (in a generalized mathematical form). The properties of the form invariant item response function are compared to that of the Rasch model, and related to previous work by Warm, Jaynes and Samejima. Moreover, several numerical examples for the use of it are provided.

KEYWORDS
Item Response Theory, Form Invariance, Invariant Measure, Individual Answer Patterns

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Henrik Bernshausen , Christoph Fuhrmann , Hanns-Ludwig Harney , Klaus Harney , Andreas Muller , "Form Invariance - An Alternative Answer to the Measurement Problem of Item Response Theory," Mathematics and Statistics, Vol. 10, No. 3, pp. 690 - 700, 2022. DOI: 10.13189/ms.2022.100326.

(b). APA Format:
Henrik Bernshausen , Christoph Fuhrmann , Hanns-Ludwig Harney , Klaus Harney , Andreas Muller (2022). Form Invariance - An Alternative Answer to the Measurement Problem of Item Response Theory. Mathematics and Statistics, 10(3), 690 - 700. DOI: 10.13189/ms.2022.100326.