Abstract
The joint maximum likelihood estimation of the parameters of the Rasch model is hampered by several drawbacks, the most relevant of which are that: (1) the estimates are not available for item or person with perfect scores; (2) the item parameter estimates are severely biased, especially for short tests. To overcome both these problems, in this paper a new method is proposed, based on a fuzzy extension of the empirical probability function and the minimum Kullback–Leibler divergence estimation approach. The new method warrants the existence of finite estimates for both person and item parameters and results very effective in reducing the bias of joint maximum likelihood estimates.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Andersen, E. B. (1980). Discrete statistical models with social sciences applications. Amsterdam: North-Holland.
Andrich, D., Lyne, A., Sheridan, B., & Luo, G. (2003). RUMM 2020 [Computer Software]. Perth, Australia: RUMM Laboratory.
Bartolucci, F., Bellio, R., Salvan, A., & Sartori, N. (2012). Modified profile likelihood for panel data models. Available at SSRN repository: http://ssrn.com/abstract=2000666.
Bertoli-Barsotti, L. (2005). On the existence and uniqueness of JML estimates for the partial credit model. Psychometrika, 70, 517–531.
Bertoli-Barsotti, L., & Punzo, A. (2012). Comparison of two bias reduction techniques for the Rasch model. Electronic Journal of Applied Statistical Analysis, 5(3), 360–366.
Bertoli-Barsotti, L., & Bacci, S. (2014). Identifying Guttman structures in incomplete Rasch datasets. Communications in Statistics – Theory and Methods, 43(3), 470–497. doi:10.1080/03610926.2012.66555.
Cohen, J., Chan, T., Jiang, T., & Seburn, M. (2008). Consistent estimation of Rasch item parameters and their standard errors under complex sample designs. Applied Psychological Measurement, 32, 289–310.
Fischer, G. H. (1981). On the existence and uniqueness of maximum-likelihood estimates in the Rasch model. Psychometrika, 46, 59–77.
Firth, D. (1993). Bias reduction of maximum likelihood estimates. Biometrika, 80, 27–38.
Haldane, J. B. S. (1956). The estimation and significance of the logarithm of a ratio of frequencies. Annals of Human Genetics, 20, 309–311.
Jeffreys, H. (1939). Theory of probability. Oxford: Oxford University Press.
Jeffreys, H. (1946). An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society of London. Series A: Mathematics and Physical Sciences, 186, 453–461.
Linacre, J. M. (2009). WINSTEPS®. Rasch measurement computer program. Beaverton, OR: Winsteps.com.
Molenaar, I. W. (1995). Estimation of item parameters. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models: Foundations, recent developments, and applications (pp. 39–51). New York: Springer.
R Development Core Team. (2012). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing.
Warm, T. A. (1989). Weighted likelihood estimation of in item response theory. Psychometrika, 54, 427–450.
Wright, B. D. (1988). The efficacy of unconditional maximum likelihood bias correction: Comment on Jansen, van den Wollenberg, and Wierda. Applied Psychological Measurement, 12, 315–318.
Wu, M. L., Adams, R. J., Wilson, M. R., & Haldane, S. A. (2007). ACER ConQuest: Generalised item response modeling software - version 2.0. ACER Press edition
Acknowledgments
This research was partially funded in the framework of the project “Opportunity for young researchers”, reg. no. CZ.1.07/2.3.00/30.0016, supported by Operational Programme Education for Competitiveness and co-financed by the European Social Fund and the state budget of the Czech Republic.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Bertoli-Barsotti, L., Lando, T., Punzo, A. (2014). Estimating a Rasch Model via Fuzzy Empirical Probability Functions. In: Vicari, D., Okada, A., Ragozini, G., Weihs, C. (eds) Analysis and Modeling of Complex Data in Behavioral and Social Sciences. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Cham. https://doi.org/10.1007/978-3-319-06692-9_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-06692-9_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06691-2
Online ISBN: 978-3-319-06692-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)