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Limits on Log Odds Ratios for Unidimensional Item Response Theory Models

Published online by Cambridge University Press:  01 January 2025

Shelby J. Haberman,
Paul W. Holland and
Sandip Sinharay
Shelby J. Haberman
Affiliation:
Educational Testing Service
Paul W. Holland
Affiliation:
Educational Testing Service
Sandip Sinharay*
Affiliation:
Educational Testing Service
*
Requests for reprints should be sent to Sandip Sinharay, Center for Statistical and Psychometrical Theory and Practice, Educational Testing Service, Rosedale Road, MS 12-T, Princeton, NJ 08541, USA. E-mail: ssinharay@ets.org
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Abstract

Bounds are established for log odds ratios (log cross-product ratios) involving pairs of items for item response models. First, expressions for bounds on log odds ratios are provided for one-dimensional item response models in general. Then, explicit bounds are obtained for the Rasch model and the two-parameter logistic (2PL) model. Results are also illustrated through an example from a study of model-checking procedures. The bounds obtained can provide an elementary basis for assessment of goodness of fit of these models.

Keywords

Rasch model2PL modelnormal ogive modelcumulant generating function

Information

Type
Theory and Methods
Information
Psychometrika , Volume 72 , Issue 4 , December 2007, pp. 551 - 561
Copyright
Copyright © 2007 The Psychometric Society

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Footnotes

Any opinions expressed in this publication are those of the authors and not necessarily those of the Educational Testing Service.

The authors thank Dan Eignor, Matthias von Davier, Lydia Gladkova, Brian Junker, and the three anonymous reviewers for their invaluable advice. The authors gratefully acknowledge the help of Kim Fryer with proofreading.

References

Berk, R.H. (1972). Consistency and asymptotic normality of MLE’s for exponential models. The Annals of Mathematical Statistics, 43, 193–204.CrossRefGoogle Scholar
Bishop, Y.M.M., Fienberg, S.E., Holland, P.W. (1975). Discrete multivariate analysis: Theory and practice, Cambridge: MIT Press.Google Scholar
Brascamp, H.J., Lieb, E.H. (1976). On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. Journal of Functional Analysis, 22, 366–389.CrossRefGoogle Scholar
Cramér, H. (1946). Mathematical methods of statistics, Princeton: Princeton University Press.Google Scholar
Feller, W. (1968). An introduction to probability theory and its applications (3rd ed., Vol. 1). New York: Wiley.Google Scholar
Fleming, W.H. (1965). Functions of several variables, Reading: Addison-Wesley.Google Scholar
Haberman, S.J. (1974). The analysis of frequency data, Chicago: University of Chicago Press.Google Scholar
Haberman, S.J. (1980). A note on the conditional moments of a multivariate normal distribution confined to a convex set. Journal of Multivariate Analysis, 10, 398–404.CrossRefGoogle Scholar
Holland, P.W. (1990). The Dutch identity: A new tool for the study of item response models. Psychometrika, 55, 5–18.CrossRefGoogle Scholar
Holland, P.W., Rosenbaum, P.R. (1986). Conditional association and unidimensionality in montone latent variable models. The Annals of Statistics, 14, 1523–1543.CrossRefGoogle Scholar
Lord, F.M., Novick, M.R. (1968). Statistical theories of mental test scores, Reading: Addison-Wesley.Google Scholar
Plackett, R.L. (1981). The analysis of categorical data, (2nd ed.). London: Charles Griffin.Google Scholar
Rao, C.R. (1973). Linear statistical inference and its applications, (2nd ed.). New York: Wiley.CrossRefGoogle Scholar
Rockafellar, R.T. (1970). Convex analysis, Princeton: Princeton University Press.CrossRefGoogle Scholar
Scheffé, H. (1947). A useful convergence theorem for probability distributions. The Annals of Mathematical Statistics, 18, 434–438.CrossRefGoogle Scholar
Sinharay, S., Johnson, M.S., Stern, H.S. (2006). Posterior predictive assessment of item response theory models. Applied Psychological Measurement, 30, 298–321.CrossRefGoogle Scholar