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.
References
G. Rasch, On general laws and the meaning of measurement in psychology, in Fourth Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, 1961), pp. 321–333
G. Rasch, An item analysis which takes individual differences into account. Br. J. Math. Stat. Psychol. 19, 49–57 (1966)
G. Rasch, An informal report on a theory of objectivity, in comparisons, in Measurement Theory. Proceedings of the NUFFIC International Summer Session in Science at ‘Het Oude Hof’. The Hague, July 14–28, 1966, ed. by LJTh van der Kamp, C.A.J. Vlek (University of Leiden, Leiden, 1967), pp. 1–19
G. Rasch, A mathematical theory of objectivity and its consequences for model contribution, in European Meeting on Statistics, Econometrics, and Management Science (Amsterdam, 1968)
G. Rasch, On specific objectivity: An attempt at formalising the request for generality and validity of scientific statements. Dan. Yearb. Philos. 14, 58–94 (1977)
G. Rasch, Probabilistic Models for Some Intelligence and Attainment Tests (The University of Chicago Press, Chicago, 1980)
G.H. Fischer, Spezifische Objektivität: Eine wissenschaftstheoretische Grundlage des Rasch-Modells (Psychologie Verlagsunion, Weinheim, 1988), pp. 87–111
G.H. Fischer, Derivations of the rasch model. In Rasch-models: foundations, recent developments and applications, New York, 1995. Workshop held at the University of Vienna (1995), pp. 15–38. 25–27 Feb. 1993
G.H. Fischer, Rasch Models (Elsevier, Amsterdam, 2007), pp. 515–585. chapter 16
E.B. Anderson, Asymptotic properties of conditional maximum likelihood estimators. J. R. Stat. Soc. 32, 283–301 (1970)
E.B. Anderson, Conditional inference for multiple-choice questionaires. Br. J. Math. Stat. Psychol. 26, 31–44 (1973)
E.B. Anderson, Sufficient statistic and latent trait models. Psychometrika 42, 69–81 (1977)
J. Rost, Lehrbuch Testtheorie—Testkonstruktion, 2nd edn. (Hans Huber, Bern, 2004)
C. Fuhrmann, Eine trigonometrische Parametrisierung von Kompetenzen. Zur Methodologie der probabilistischen Bildungsforschung. Ph.D. thesis, Ruhr-Universität Bochum, Bochum, Germany. In print at Springer vs. To appear in 2017
OECD. PISA 2000. Zusammenfassung zentraler Befunde, http://www.oecd.org/germany/33684930.pdf. Accessed 25 May 2015
OECD. Pisa 2003, Technical report, http://www.oecd.org/edu/school/ programmeforinternationastudentassessmentpisa/35188570.pdf. Accessed 16 Sept 2013
OECD. The PISA 2003 assessment framework—mathematics, reading, science and problem knowledge and skills, http://www.oecd.org/edu/preschoolandschool/programmeforinternationastudentassessmentpisa/336694881.pdf. Accessed 30 Jan, 2 Feb 2013
OECD. Pisa 2003 data analysis manual, spss user 2 ed, http://browse.oecdbookshop.org/oecd/pdfs/free/9809031e.pdf. Accessed on 3 Sept 2008 and 28 Oct 2012
OECD. PISA 2012 Ergebnisse im Fokus, http://www.oecd.org/berlin/themen/PISA-2012-Zusammenfassung.pdf. Accessed 9 July 2014
OECD. PISA 2009 results: Overcoming social background, equity in learning opportunities and outcomes (volume II), http://dx.doi.org/10.1787/9789264091504-en in http://www.oecd-ilibrary.org/education. Accessed 25 April 2013
L.L. Thurstone, The Measurement of Values, 3rd edn. (University of Chicago Press, Chicago, 1963), p. 195
X. Liu, W. Boone, Introduction to Rasch Measurement in Science Education (JAM Press, Maple Grove, 2006), pp. 1–22
F. Samejima, Constant information model on dichotomous response level, in New Horizons in Testing (Academic Press, New York, 1983), pp. 63–79
K. Harney, C. Fuhrmann, H.L. Harney, Der schiefe Turm von PISA. die logistischen Parameter des Rasch-Modells sollten revidiert werden, in ZA-Information. Zentralarchiv für Empirische Sozialforschung an der Universität zu Köln 59 (2006), pp. 10–55
L. Guttman, The Basis of Scalogram Analysis (Gloucester, Smith, 1973)
I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 2015)
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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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