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Weighted Likelihood Estimation of Ability in Item Response Theory

Published online by Cambridge University Press:  01 January 2025

Thomas A. Warm
Thomas A. Warm*
Affiliation:
FAA Academy
*
Requests for reprints should be sent to Thomas A. Warm, FAA Academy AAC-934, Mike Monroney Aeronautical Center, PO Box 25082, Oklahoma City, OK 73125.
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Abstract

Applications of item response theory, which depend upon its parameter invariance property, require that parameter estimates be unbiased. A new method, weighted likelihood estimation (WLE), is derived, and proved to be less biased than maximum likelihood estimation (MLE) with the same asymptotic variance and normal distribution. WLE removes the first order bias term from MLE. Two Monte Carlo studies compare WLE with MLE and Bayesian modal estimation (BME) of ability in conventional tests and tailored tests, assuming the item parameters are known constants. The Monte Carlo studies favor WLE over MLE and BME on several criteria over a wide range of the ability scale.

Keywords

maximum likelihood estimationunbiased estimationstatistical biasBayesian modal estimationitem response theorytailored testingadaptive testing

Information

Type
Original Paper
Information
Psychometrika , Volume 54 , Issue 3 , September 1989 , pp. 427 - 450
Copyright
Copyright © 1989 The Psychometric Society

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References

Akaike, H. (1978). A New Look At The Bayes Procedure. Biometrika, 65(1), 53–59.CrossRefGoogle Scholar
Anderson, J. A., & Richardson, S. C. (1979). Logistic discrimination and bias correction in maximum likelihood estimation. Technometrics, 21(1), 71–78.CrossRefGoogle Scholar
Andrews, D. F., Bickel, P. J., Hampel, F. R., Huber, P. J., Rogers, W. H., & Tukey, J. W. (1972). Robust estimates of location, Princeton, NJ: Princeton University Press.Google Scholar
Baker, F. B. (1984). Ability metric transformations involved in vertical equating under item response theory. Applied Psychological Measurement, 8(3), 261–271.CrossRefGoogle Scholar
Barndorff-Nielsen, O. (1983). On a formula for the distribution of the maximum likelihood estimator. Biometrika, 70(2), 343–365.CrossRefGoogle Scholar
Basu, A. P., & Ghosh, J. K. (1980). Asymptotic properties of a solution to the likelihood equation with life-testing applications. Journal of the American Statistical Association, 75(370), 410–414.Google Scholar
Bock, R. D. (1983). The discrete Bayesian. In Wainer, H. & Messick, S. (Eds.), Principals of modern psychological measurement. A festschrift for Frederick M. Lord (pp. 103–115). NJ: Lawrence Erlbaum.Google Scholar
Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika, 46, 443–459.CrossRefGoogle Scholar
Bock, R. D., & Mislevy, R. J. (1981). Biweight estimates of latent ability. Unpublished manuscript.Google Scholar
Bradley, R. A., & Gart, J. J. (1962). The asymptotic properties of ML estimators when sampling from associated populations. Biometrika, 49, 205–214.CrossRefGoogle Scholar
Cox, D. R., & Hinkley, D. V. (1974). Theoretical statistics, New York: Chapman & Hall.CrossRefGoogle Scholar
Efron, B., & Hinkley, D. V. (1978). Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information. Biometrika, 65(3), 457–487.CrossRefGoogle Scholar
Goldstein, H. (1980). Dimensionality, bias, independence and measurement scale problems in latent trait test score models. British Journal of Mathematical and Statistical Psychology, 33, 234–246.CrossRefGoogle Scholar
Hoadley, B. (1971). Asymptotic properties of maximum likelihood estimators for the independent not identically distributed case. Annals of Mathematical Statistics, 42(6), 1977–1991.CrossRefGoogle Scholar
Jeffreys, H. (1961). Theory of Probability 3rd ed.,, Oxford: Clarendon.Google Scholar
Jones, D. H. (1982). Redescending M-type estimators of latent ability, Princeton, NJ: Educational Testing Service.CrossRefGoogle Scholar
Kendall, M. G., & Stuart, A. (1973). The advanced theory of statistics (Vol. 2), New York: Hafner.Google Scholar
Kendall, M. G., & Stuart, A. (1977). The advanced theory of statistics (Vol. 1), London: Charles Griffin.Google Scholar
Lord, F. M. (1980). Applications of item response theory to practical testing problems, Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
Lord, F. M. (1983). Unbiased estimators of ability parameters, of their variance, and of their parallel-forms reliability. Psychometrika, 48, 233–245.CrossRefGoogle Scholar
Lord, F. M. (1983). Memorandum for: Ms. Stocking, Ms. M. Wang, Ms. Wingersky. Subject: Sampling variance and bias for MLE and Bayesian estimation of α., Princeton, NJ: Educational Testing Service.Google Scholar
Lord, F. M. (1984). Maximum likelihood and Bayesian parameter estimation in item response theory, Princeton, NJ: Educational Testing Service.CrossRefGoogle Scholar
Lord, F. M., & Novick, M. (1967). Statistical theories of mental test scores, Reading, MA: Addison-Wesley.Google Scholar
Owen, R. J. (1975). A Bayesian sequential procedure for quantal response in the context of adaptive mental testing. Journal of the American Statistical Association, 70, 351–356.CrossRefGoogle Scholar
Quenouille, M. H. (1956). Notes on Bias in Estimation. Biometrika, 43, 353–360.CrossRefGoogle Scholar
Samejima, F. (1980). Is Bayesian estimation proper for estimating the individual's ability, Knoxville, TN: University of Tennessee, Department of Psychology.Google Scholar
Schaefer, R. L. (1983). Bias correction in maximum likelihood logistic regression. Statistics in Medicine, 2, 71–78.CrossRefGoogle ScholarPubMed
Wainer, H., & Wright, B. D. (1980). Robust estimation of ability in the Rasch model. Psychometrika, 45, 373–391.CrossRefGoogle Scholar
Weiss, D. J., & McBride, J. R. (1984). Bias and information of Bayesian adaptive testing. Applied Psychological Measurement, 8(3), 273–285.CrossRefGoogle Scholar