Abstract
Item response theory (IRT) tries to construct a statistical model of measurement in psychology.
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Notes
- 1.
In the first edition of the present book, the place of the present chapter was used to discuss the independence of parameters in the context of form invariance with a non-Abelian symmetry group. Such symmetries make it difficult to integrate the posterior distribution over incidental parameters. Here, we solve the problem with the help of a Gaussian approximation to the posterior; see Chap. 10. Every posterior can be approximated by a Gaussian if the number N of events is large enough. Marginal distributions of a Gaussian are obtained easily; see Sect. B.2. The Gaussian approximation requires, however, that the ML estimators of all parameters exist for every event. This is one of the premises here. The considerations in the former chapter twelve are no longer needed.
- 2.
Georg Rasch, 1901–1980, Danish mathematician, professor at the University of Copenhagen.
- 3.
PISA means a “Programme for International Student Assessment” set up by the OECD, the Organisation for Economic Co-Operation and Development in Paris. Since the year 2000 the competence of high school participants has been measured and compared in most member states of the OECD as well as a number of partner states.
- 4.
The author is indebted to Prof. Andreas Müller at the Université de Genève for his proof that probability amplitudes allow a geometric representation of the symmetry of a statistical model. This is described in Sects. 4.13 and 4.14 of Ref. [14]. Here in Chap. 9, we have seen that the amplitudes provide the geometric measure.
- 5.
The Monte Carlo simulation has been executed in the framework of EXCEL 2003. This system provides the function RAND to generate random numbers. It is described under http://support.microsoft.com/kb/828795/en-us. The author is endebted to Dr. Henrik Bernshausen, University of Siegen (Germany), Fachbereich Didaktik der Physik, for carrying out the Monte Carlo simulation.
- 6.
The solution of the ML equations has been obtained with the help of the “Euler Math Toolbox”. It provides the same routines as “R” or “STATA” to deal with matrices. Details are given in Sections B and C of the Ph.D. thesis [14]. The author is indebted to Dr. Christoph Fuhrmann, Bergische Universität Wuppertal (Germany), Institut für Bildungsforschung, for the numerical calculations.
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Harney, H.L. (2016). Item Response Theory. In: Bayesian Inference. Springer, Cham. https://doi.org/10.1007/978-3-319-41644-1_12
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