Abstract
It has long been part of the item response theory (IRT) folklore that under the usual empirical Bayes unidimensional IRT modeling approach, the posterior distribution of examinee ability given test response is approximately normal for a long test. Under very general and nonrestrictive nonparametric assumptions, we make this claim rigorous for a broad class of latent models.
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References
Bock, R. D., & Mislevy, R. J. (1982). Adaptive EAP estimation of ability in a microcomputer environment.Applied Psychological Measurement, 6, 431–444.
Chang, H. (1992).Some theoretical and applied results concerning item response theory model estimation. Unpublished doctoral dissertation. University of Illinois at Urbana-Champaign, Department of Statistics.
Chang, H., & Stout, W. F. (1991).The asymptotic posterior normality of the latent trait in an IRT model (ONR Research Report 91-4). Urbana-Champaign: University of Illinois, Department of Statistics.
Clarke, B. S., & Junker, B. W. (1991).Inference from the product of marginals of a dependent likelihood (Technical Report No. 508). Pittsburgh, PA: Carnegie Mellon University, Department of Statistics.
Drasgow, F. (1987). A study of measurement bias of two standard psychological tests.Journal of Applied Psychology, 72, 19–30.
Holland, P. W. (1990a). The Dutch identity: A new tool for the study of item response theory models.Psychometrika, 55, 5–18.
Holland, P. W. (1990b). On the sampling theory foundations of item response theory models.Psychometrika, 55, 577–601.
Lehmann, E. L. (1983).Theory of point estimation. New York: John Wiley & Sons.
Lindley, D. V. (1965).Introduction to probability and statistics. Part 2: Inference. London: Cambridge University Press.
Lord, F. M. (1980).Applications of item response theory to practical testing problems. Hillsdale, NJ: Lawrence Erlbaum.
Stout, W. F. (1974).Almost sure convergence. New York: Academic Press.
Stout, W. F. (1990). A new item response theory modeling approach with applications to unidimensionality assessment and ability estimation.Psychometrika, 55, 293–325.
Wald, A. (1949). Note on the consistency of the maximum likelihood estimate.Annals of Mathematical Statistics, 20, 595–601.
Walker, A. M. (1969). On the asymptotic behaviour of posterior distributions.Journal of the Royal Statistical Society, Series B, 31, 80–88.
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This research was partially supported by Office of Naval Research Cognitive and Neural Sciences Grant N0014-J-90-1940, 442-1548, National Science Foundation Mathematics Grant NSF-DMS-91-01436, and the National Center for Supercomputing Applications. We wish to thank Kumar Joag-dev and Zhiliang Ying for enlightening suggestions concerning the proof of the basic result.
The authors wish to thank Kumar Joag-Dev, Brian Junker, Bert Green, Paul Holland, Robert Mislevy, and especially Zhiliang Ying for their useful comments and discussions.
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Chang, HH., Stout, W. The asymptotic posterior normality of the latent trait in an IRT model. Psychometrika 58, 37–52 (1993). https://doi.org/10.1007/BF02294469
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DOI: https://doi.org/10.1007/BF02294469