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Essential Independence and Likelihood-Based Ability Estimation for Polytomous Items

Published online by Cambridge University Press:  01 January 2025

Brian W. Junker
Brian W. Junker*
Affiliation:
Department of Statistics, University of Illinois at Urbana-Champaign
*
Requests for reprints should be sent to Brian W. Junker, Department of Statistics, 232 Baker Hall, Carnegie Mellon University, Pittsburgh, PA 15213.
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Abstract

A definition of essential independence is proposed for sequences of polytomous items. For items satisfying the reasonable assumption that the expected amount of credit awarded increases with examinee ability, we develop a theory of essential unidimensionality which closely parallels that of Stout. Essentially unidimensional item sequences can be shown to have a unique (up to change-of-scale) dominant underlying trait, which can be consistently estimated by a monotone transformation of the sum of the item scores. In more general polytomous-response latent trait models (with or without ordered responses), an M-estimator based upon maximum likelihood may be shown to be consistent for θ under essentially unidimensional violations of local independence and a variety of monotonicity/identifiability conditions. A rigorous proof of this fact is given, and the standard error of the estimator is explored. These results suggest that ability estimation methods that rely on the summation form of the log likelihood under local independence should generally be robust under essential independence, but standard errors may vary greatly from what is usually expected, depending on the degree of departure from local independence. An index of departure from local independence is also proposed.

Keywords

item response theory (IRT)polytomous item responsesessential independenceunidimensionalitylatent trait identifiabilitylikelihood-based trait estimationasymptotic standard errorsstructural robustnesslocal dependence
Type
Original Paper
Information
Psychometrika , Volume 56 , Issue 2 , June 1991 , pp. 255 - 278
Copyright
Copyright © 1991 The Psychometric Society

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Footnotes

This work was supported in part by Office of Naval Research Grant N00014-87-K-0277 and National Science Foundation Grant NSF-DMS-88-02556. The author is grateful to William F. Stout for many helpful comments, and to an anonymous reviewer for raising the questions addressed in section 2. A preliminary version of section 6 appeared in the author's Ph.D. thesis.

References

Anastasi, A. (1988). Psychological testing 6th ed.,, New York: Macmillan.Google Scholar
Ash, R. B. (1972). Real analysis and probability, New York: Academic Press.Google Scholar
Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M. & Novick, M. R. (Eds.), Statistical theories of mental test scores, Reading, MA: Addison-Wesley.Google Scholar
Bock, R. D. (1972). Estimating item parameters and latent ability when the responses are scored in two or more nominal categories. Psychometrika, 37, 29–51.CrossRefGoogle Scholar
Bock, R. D.,& Mislevy, R. J. (1982). Adaptive EAP estimation of ability in a microcomputer environment. Applied Psychological Measurement, 6, 431–444.CrossRefGoogle Scholar
Bradley, R. C. (1985). Counterexamples to the central limit theorem under strong mixing conditions. In Revesz, P. (Eds.), Colloquia mathematica societitas Janos Bolyai 36. Limit theorems in probability and statistics, Veszprem (Hungary), 1982. Volume I (pp. 153–172). Amsterdam: North-Holland.Google Scholar
Bradley, R. C. (1989). A stationary, pairwise independent, absolutely regular sequence for which the central limit theorem fails. Probability Theory and Related Fields, 81, 1–10.CrossRefGoogle Scholar
Drasgow, F., & Parsons, C. K. (1983). Application of unidimensional item response theory models to multidimensional data. Applied Psychological Measurement, 7, 189–199.CrossRefGoogle Scholar
Dvoretzky, A. (1972). Asymptotic normality for sums of dependent random variables. Proceedings of the sixth Berkeley symposium on mathematical probability and statistics, 2, 513–535.Google Scholar
Foutz, R. V. (1977). On the unique consistent solution to the likelihood equations. Journal of the American Statistical Association, 72, 147–148.CrossRefGoogle Scholar
Gibbons, R. D., Bock, R. D., & Hedeker, D. R. (1989). Conditional dependence, Chicago, IL: University of Illinois at Chicago, Office of Naval Research and Illinois State Psychiatric Institute.CrossRefGoogle Scholar
Holland, P. W., & Rosenbaum, P. R. (1986). Conditional association and unidimensionality in monotone latent trait models. Annals of Statistics, 14, 1523–1543.CrossRefGoogle Scholar
Jannarone, R. J. (1986). Conjunctive item response theory kernels. Psychometrika, 51, 357–373.CrossRefGoogle Scholar
Jogdeo, K. (1978). On a probability bound of Marshall and Olkin. Annals of Statistics, 6, 232–234.CrossRefGoogle Scholar
Junker, B. W. (1988). Statistical aspects of a new latent trait model. Unpublished doctoral dissertation, University of Illinois at Urbana-Champaign, Department of Statistics.Google Scholar
Lehmann, E. L. (1983). Theory of point estimation, New York: Wiley.CrossRefGoogle Scholar
Levine, M. V. (1989). Ability distributions, pattern probabilities, and quasidensities, Champaign, IL: University of Illinois at Urbana-Champaign, Office of Naval Research and Model-Based Measurement Laboratory.Google Scholar
Lord, F. M. (1980). Applications of item response theory to practical testing problems, Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
Lord, F. M. (1986). Maximum likelihood and Bayesian parameter estimation in item response theory. Journal of Educational Measurement, 23, 157–162.CrossRefGoogle Scholar
Nandakumar, R. (1990). Traditional dimensionality versus essential dimensionality. Journal of Educational Measurement, 28, 99–117.CrossRefGoogle Scholar
Rosenbaum, P. R. (1988). Item bundles. Psychometrika, 53, 349–359.CrossRefGoogle Scholar
Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph No. 17, 24(4, Pt. 2).Google Scholar
Samejima, F. (1972). A general model for free-response data. Psychometrika Monograph No. 18, 37(4, Pt. 2).Google Scholar
Samejima, F. (1973). A comment of Birnbaum's three parameter logistic model in the latent trait theory. Psychometrika, 38, 221–233.CrossRefGoogle Scholar
Serfling, R. J. (1980). Approximation theorems in mathematical statistics, New York: Wiley.CrossRefGoogle Scholar
Spray, J. A., & Ackerman, T. A. (1987). The effect of item response dependence on trait or ability dimensionality, Iowa City, IA: ACT.Google Scholar
Stout, W. F. (1987). A nonparametric approach for assessing latent trait unidimensionality. Psychometrika, 52, 589–617.CrossRefGoogle Scholar
Stout, W. F. (1990). A new item response theory modeling approach with applications to unidimensionality assessment and ability estimation. Psychometrika, 55, 293–325.CrossRefGoogle Scholar
Thissen, D., & Steinberg, L. (1986). A taxonomy of item response models. Psychometrika, 51, 567–577.CrossRefGoogle Scholar
Tsutakawa, R. K., & Soltys, M. J. (1988). Approximation for Bayesian ability estimation. Journal of Educational Statistics, 13, 117–130.CrossRefGoogle Scholar
Wainer, H., & Wright, B. D. (1980). Robust estimation of ability in the Rasch model. Psychometrika, 45, 373–391.CrossRefGoogle Scholar
Walker, A. M. (1969). On the asymptotic behaviour of posterior distributions. Journal of the Royal Statistical Society, Series B, 31, 80–88.CrossRefGoogle Scholar