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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Akaike, H. (1978). A New Look At The Bayes Procedure.Biometrika, 65(1), 53–59.
Anderson, J. A., & Richardson, S. C. (1979). Logistic discrimination and bias correction in maximum likelihood estimation.Technometrics, 21(1), 71–78.
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.
Baker, F. B. (1984). Ability metric transformations involved in vertical equating under item response theory.Applied Psychological Measurement, 8(3), 261–271.
Barndorff-Nielsen, O. (1983). On a formula for the distribution of the maximum likelihood estimator.Biometrika, 70(2), 343–365.
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.
Bock, R. D. (1983). The discrete Bayesian. In H. Wainer & S. Messick (Eds.),Principals of modern psychological measurement. A festschrift for Frederick M. Lord (pp. 103–115). NJ: Lawrence Erlbaum.
Bock, R. D., & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm.Psychometrika, 46, 443–459.
Bock, R. D., & Mislevy, R. J. (1981).Biweight estimates of latent ability. Unpublished manuscript.
Bradley, R. A., & Gart, J. J. (1962). The asymptotic properties of ML estimators when sampling from associated populations.Biometrika, 49, 205–214.
Cox, D. R., & Hinkley, D. V. (1974).Theoretical statistics. New York: Chapman & Hall. (Distributed by Halsted Press, New York)
Efron, B., & Hinkley, D. V. (1978). Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information.Biometrika, 65(3), 457–487.
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.
Hoadley, B. (1971). Asymptotic properties of maximum likelihood estimators for the independent not identically distributed case.Annals of Mathematical Statistics, 42(6), 1977–1991.
Jeffreys, H. (1961). Theory of Probability (3rd ed.). Oxford: Clarendon.
Jones, D. H. (1982).Redescending M-type estimators of latent ability (Program Statistics Tech. Rep. No. 82-30). Princeton, NJ: Educational Testing Service.
Kendall, M. G., & Stuart, A. (1973).The advanced theory of statistics (Vol. 2). New York: Hafner.
Kendall, M. G., & Stuart, A. (1977).The advanced theory of statistics (Vol. 1). London: Charles Griffin.
Lord, F. M. (1980).Applications of item response theory to practical testing problems. Hillsdale, NJ: Lawrence Erlbaum.
Lord, F. M. (1983a). Unbiased estimators of ability parameters, of their variance, and of their parallel-forms reliability.Psychometrika, 48, 233–245.
Lord, F. M. (1983b).Memorandum for: Ms. Stocking, Ms. M. Wang, Ms. Wingersky. Subject: Sampling variance and bias for MLE and Bayesian estimation of α. August 26, 1983 (Internal Memorandum) Princeton, NJ: Educational Testing Service.
Lord, F. M. (1984).Maximum likelihood and Bayesian parameter estimation in item response theory (Research Rep. No. RR-84-30-ONR). Princeton, NJ: Educational Testing Service.
Lord, F. M., & Novick, M. (1967).Statistical theories of mental test scores. Reading, MA: Addison-Wesley.
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.
Quenouille, M. H. (1956). Notes on Bias in Estimation.Biometrika, 43, 353–360.
Samejima, F. (1980).Is Bayesian estimation proper for estimating the individual's ability (Research Rep. 80-3). Knoxville, TN: University of Tennessee, Department of Psychology.
Schaefer, R. L. (1983). Bias correction in maximum likelihood logistic regression.Statistics in Medicine, 2, 71–78.
Wainer, H., & Wright, B. D. (1980). Robust estimation of ability in the Rasch model.Psychometrika, 45, 373–391.
Weiss, D. J., & McBride, J. R. (1984). Bias and information of Bayesian adaptive testing.Applied Psychological Measurement, 8(3), 273–285.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Warm, T.A. Weighted likelihood estimation of ability in item response theory. Psychometrika 54, 427–450 (1989). https://doi.org/10.1007/BF02294627
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02294627