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Maximum Likelihood Estimation of the Joint Covariance Matrix for Sections of Tests given to Distinct Samples with Application to Test Equating

Published online by Cambridge University Press:  01 January 2025

Dorothy T. Thayer
Dorothy T. Thayer*
Affiliation:
Educational Testing Service
*
Reprint requests should be addressed to Dorothy T. Thayer, Educational Testing Service 21-T, Princeton, New Jersey, 08541.
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Abstract

Consider an old test X consisting of s sections and two new tests Y and Z similar to X consisting of p and q sections respectively. All subjects are given test X plus two variable sections from either test Y or Z. Different pairings of variable sections are given to each subsample of subjects. We present a method of estimating the covariance matrix of the combined test (X1, ..., Xs, Y1, ..., Yp, Z1, ..., Zq) and describe an application of these estimation techniques to linear, observed-score, test equating.

Keywords

correlationsincomplete datamissing datainestimable parametersmaximum likelihoodtest equatingEM algorithm
Type
Original Paper
Information
Psychometrika , Volume 48 , Issue 2 , June 1983 , pp. 293 - 297
Copyright
Copyright © 1983 The Psychometric Society

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Footnotes

The author is indebted to Paul W. Holland and Donald B. Rubin for their encouragement and many helpful comments and suggestions that contributed significantly to the development of this paper.

This research was supported by the Program Statistics Research Project of the ETS Research Statistics Group.

References

Angoff, W. H. Scales, norms, and equivalent scores. In Thorndike, R. (Eds.), Educational measurement second edition, (pp. 508–600). Washington, D.C.: American Council on Education, 1971.Google Scholar
Beaton, A. E. The use of special matrix operations in statistical calculus, Princeton, N.J.: Educational Testing Service, 1964.Google Scholar
Braun, H. I. & Holland, P. W. Observed score test equating: A mathematical analysis of some ETS equating procedures. In Holland, P. W. and Rubin, D. B. (Eds.), Test equating, New York: Academic Press, Inc., 1982.Google Scholar
Dempster, A. P., Laird, N. M., & Rubin, D. B. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society Series B, 1977, 39, 1–38.CrossRefGoogle Scholar
Holland, P. W. & Thayer, D. T. Section pre-equating the Graduate Record Examination, Princeton, N.J.: Educational Testing Service, 1981.CrossRefGoogle Scholar
Holland, P. W. & Wightman, L. E. Section pre-equating: A preliminary investigation. In Holland, P. W. and Rubin, D. B. (Eds.), Test equating, New York: Academic Press, Inc., 1982.Google Scholar
Rubin, D. B. Characterizing the estimation of parameters in incomplete-data problems. Journal of the American Statistical Association, 1974, 69, 467–474.CrossRefGoogle Scholar
Rubin, D. B. Inference and missing data. Biometrika, 1976, 63, 581–592.CrossRefGoogle Scholar
Rubin, D. B. & Thayer, D. T. Relating tests given to different samples. Psychometrika, 1978, 43, 3–10.CrossRefGoogle Scholar