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On the Relationships between Sum Score Based Estimation and Joint Maximum Likelihood Estimation
Published online by Cambridge University Press: 01 January 2025
Abstract
This paper analyzes the sum score based (SSB) formulation of the Rasch model, where items and sum scores of persons are considered as factors in a logit model. After reviewing the evolution leading to the equality between their maximum likelihood estimates, the SSB model is then discussed from the point of view of pseudo-likelihood and of misspecified models. This is then employed to provide new insights into the origin of the known inconsistency of the difficulty parameter estimates in the Rasch model. The main results consist of exact relationships between the estimated standard errors for both models; and, for the ability parameters, an upper bound for the estimated standard errors of the Rasch model in terms of those for the SSB model, which are more easily available.
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- Theory and Methods
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- Copyright © 2007 The Psychometric Society
Footnotes
The authors acknowledge partial financial support from the FONDECYT Project No. 1060722 from the Chilean Government, and the BIL05/03 grant to P. De Boeck, E. Lesaffre and G. Molenberghs (Flanders) for a collaboration with G. del Pino, E. San Martín, F. Quintana and J. Manzi (Chile).
References
Andersen, E.B. (1980). Discrete statistical models with social science applications, Amsterdam: North-Holland.Google Scholar
Besag, J. (1975). Statistical analysis of non-lattice data. The Statistician, 24, 179–195.10.2307/2987782CrossRefGoogle Scholar
Blackwood, L.G., & Bradley, E.L. (1989). The equivalence of two methods of parameter estimation for the Rasch model. Psychometrika, 54, 751–754.CrossRefGoogle Scholar
Boomsma, A., van Duijn, M.A.J., & Snijders, T.A.B. (2001). Essays on item response theory, New York: Springer.10.1007/978-1-4613-0169-1CrossRefGoogle Scholar
Fischer, G.H., & Molenaar, I.W. (1995). Rasch models. Foundations, recent developments and applications, New York: Springer.Google Scholar
Ghosh, M. (1995). Inconsistent maximum likelihood estimators for the Rasch model. Statistical and Probability Letters, 23, 165–170.10.1016/0167-7152(94)00109-LCrossRefGoogle Scholar
Haberman, S.J. (2004). Joint and conditional maximum likelihood estimation for the Rasch model with binary responses. ETS Research Report RR-04-20, May 2004.CrossRefGoogle Scholar
Holland, P. (1990). On the sampling theory foundations of item response theory models. Psychometrika, 55, 577–601.10.1007/BF02294609CrossRefGoogle Scholar
McCullagh, P., & Nelder, J.A. (1989). Generalized linear models, (2nd ed.). New York: Chapman & Hall/CRC.10.1007/978-1-4899-3242-6CrossRefGoogle Scholar
Mellenbergh, G.J., & Vijn, P. (1981). The Rasch model as a loglinear model. Applied Psychological Measurement, 5, 369–376.10.1177/014662168100500310CrossRefGoogle Scholar
Molenaar, I.W. (1995). Estimation of item parameters. In Fischer, G.H. & Molenaar, I.W. (Eds.), Rasch models. Foundations, recent developments and applications (Chap. 3), New York: Springer.Google Scholar
Perline, R., Wright, B.D., & Wainer, H. (1979). The Rasch model as additive conjoint measurement. Applied Psychological Measurement, 3, 237–255.10.1177/014662167900300213CrossRefGoogle Scholar
Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests, Copenhagen: The Danish Institute for Educational Research. (Expanded edition, 1980, Chicago: The University of Chicago Press.)Google Scholar
Van der Linden, W., & Hambleton, R.K. (1997). Handbook of modern item response theory, New York: Springer.10.1007/978-1-4757-2691-6CrossRefGoogle Scholar
Verhelst, N., & Molenaar, I. (1988). Logit-based parameter estimation in the Rasch model. Statistica Neerlandica, 42, 273–295.10.1111/j.1467-9574.1988.tb01240.xCrossRefGoogle Scholar
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