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A strong true-score theory, with applications

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Abstract

A “strong” mathematical model for the relation between observed scores and true scores is developed. This model can be used

  1. 1.

    To estimate the frequency distribution of observed scores that will result when a given test is lengthened.

  2. 2.

    To equate true scores on two tests by the equipercentile method.

  3. 3.

    To estimate the frequencies in the scatterplot between two parallel (nonparallel) tests of the same psychological trait, using only the information in a (the) marginal distribution(s).

  4. 4.

    To estimate the frequency distribution of a test for a group that has taken only a short form of the test (this is useful for obtaining norms).

  5. 5.

    To estimate the effects of selecting individuals on a fallible measure.

  6. 6.

    To effect matching of groups with respect to true score when only a fallible measure is available.

  7. 7.

    To investigate whether two tests really measure the same psychological function when they have a nonlinear relationship.

  8. 8.

    To describe and evaluate the properties of a specific test considered as a measuring instrument.

The model has been tested empirically, using it to estimate bivariate distributions from univariate distributions, with good results, as checked by chi-square tests.

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References

  1. Appell, P. and Kampé de Fériet, J. Fonctions hypergeometriques et hyperspheriques. Polynomes d' Hermite. Gauthier-Villars, 1926.

  2. Buckingham, R. A.Numerical methods. London: Pitman, 1957.

    Google Scholar 

  3. Bückner, H. F. Numerical methods for integral equations. In J. Todd (Ed.),A survey of numerical analysis. New York: McGraw-Hill, 1962.

    Google Scholar 

  4. Cochran, W. G. Some methods for strengthening the common x2 tests.Biometrics, 1954,10, 417–451.

    Google Scholar 

  5. Eisenberg, H. B., Geoghagen, R. R. M., and Walsh, J. E. A general probability model for binomial events with application to surgical mortality.Biometrics, 1963,19, 152–157.

    Google Scholar 

  6. Elderton, Sir W. P.Frequency curves and correlation. (3rd ed.). Cambridge, England: Cambridge Univ. Press, 1938.

    Google Scholar 

  7. Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G.Higher transcendental functions. Vol. 1. New York: McGraw-Hill, 1953.

    Google Scholar 

  8. Fox, L. and Goodwin, E. T. The numerical solution of non-singular linear equations.Phil. Trans. roy. Soc. London (Series A), 1952–53,245, 501–535.

    Google Scholar 

  9. Gulliksen, H. O.Theory of mental tests. New York: Wiley, 1950.

    Google Scholar 

  10. Hirschman, I. I. and Widder, D. V.The convolution transform. Princeton: Princeton Univ. Press, 1955.

    Google Scholar 

  11. Jeffries, J. and Orrall, F. The numerical solution of Fredholm integral equations of the first kind. (Unclassified report reproduced by the Armed Services Technical Information Agency.) Arlington, Va.: Arlington Hall Station. November 1960. AD 250–862.

    Google Scholar 

  12. Kantorovich, L. W. and Krylov, V. I.Approximate methods of higher analysis. New York: Interscience, 1958.

    Google Scholar 

  13. Keats, J. A. and Lord, F. M. A theoretical distribution for mental test scores.Psychometrika, 1962,27, 59–72.

    Google Scholar 

  14. Kendall, M. G. and Stuart, A.The advanced theory of statistics. New York: Hafner, 1958.

    Google Scholar 

  15. Kunz, K. S.Numerical analysis. New York: McGraw-Hill, 1957.

    Google Scholar 

  16. Lord, F. M. A theory of test scores.Psychometric Monogr., No. 7. Richmond, Va.: William Byrd Press, 1952.

    Google Scholar 

  17. Lord, F. M. An approach to mental test theory.Psychometrika, 1959,24, 283–302.

    Google Scholar 

  18. Lord, F. M. Use of true-score theory to predict moments of univariate and bivariate observed-score distributions.Psychometrika, 1960,25, 325–342.

    Google Scholar 

  19. Lord, F. M. Estimating true measurements from fallible measurements (Binomial Case)—expansion in a series of beta distributions. Res. Bull. 62-23. Princeton, N. J.: Educ. Test. Serv., 1962.

    Google Scholar 

  20. Lord, F. M. True-score theory—The four parameter beta model with binomial errors. Res. Bull. 64-6. Princeton, N. J.: Educ. Test. Serv., 1964.

    Google Scholar 

  21. Medgyessy, P.Decomposition of superpositions of distribution functions. Budapest: Publishing House of the Hungarian Academy of Sciences, 1961.

    Google Scholar 

  22. Phillips, D. L. A technique for the numerical solution of certain integral equations of the first kind.J. Ass. comput. Machinery, 1962,9, 84–97.

    Google Scholar 

  23. Pollard, H. Distribution functions containing a Gaussian factor.Proc. Amer. math. Soc., 1953,4, 578–582.

    Google Scholar 

  24. Rasch, G.Probabilistic models for some intelligence and attainment tests. Copenhagen, Denmark: Nielsen and Lydiche (M. Simmelkiaer), 1960.

    Google Scholar 

  25. Teicher, H. On the mixture of distributions.Ann. math. Statist., 1960,31, 55–73.

    Google Scholar 

  26. Tricomi, F. G.Integral equations. New York: Interscience, 1957.

    Google Scholar 

  27. Trumpler, R. J. and Weaver, H. F.Statistical astronomy. Berkeley: Univ. California Press, 1953.

    Google Scholar 

  28. Tucker, L. R. A note on the estimation of test reliability by the Kuder-Richardson formula (20).Psychometrika, 1949,14, 117–120.

    Google Scholar 

  29. Twomey, S. On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature.J. Ass. comput. Machinery, 1963,10, 97–101.

    Google Scholar 

  30. Walsh, J. E. Corrections to two papers concerned with binomial events.Sankhyā, Ser. A., 1963,25, 427.

    Google Scholar 

  31. Walther, A. and Dejon, B. General report on the numerical treatment of integral and integro-differential equations. InSymposium on the Numerical Treatment of Ordinary Differential Equations, Integral and Integro-Differential Equations—Proceedings of the Rowe Symposium (20–24 September 1960). Basel/Stuttgart: Birkhauser Verlag, 1960. Pp. 645–671.

    Google Scholar 

  32. Young, A. Approximate product-integration.Proc. roy. Soc., A, 1954,224, 552–561.

    Google Scholar 

  33. Young, A. The application of product-integration to the numerical solution of integral equations.Proc. roy. Soc., A, 1954,224, 561–573.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Educational Testing Service, USA

    Frederic M. Lord

  2. Princeton University, USA

    Frederic M. Lord

Authors
  1. Frederic M. Lord
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Additional information

This work was supported in part by contract Nonr-2752(00) between the Office of Naval Research and Educational Testing Service. Reproduction in whole or in part for any purpose of the United States Government is permitted. Many of the extensive computations were done on Princeton University computer facilities, supported in part by National Science Foundation Grant NSF-GP579. The last portion of the work was carried out while the writer was Brittingham Visiting Professor at the University of Wisconsin. The writer is indebted to Diana Lees, who wrote many of the computer programs, checked most of the mathematical derivations, and gave other invaluable assistance throughout the project.

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Lord, F.M. A strong true-score theory, with applications. Psychometrika 30, 239–270 (1965). https://doi.org/10.1007/BF02289490

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  • DOI: https://doi.org/10.1007/BF02289490

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