Hostname: page-component-5f745c7db-f9j5r Total loading time: 0 Render date: 2025-01-06T05:46:52.517Z Has data issue: true hasContentIssue false
  • English
  • Français

A Strong True-Score Theory, with Applications

Published online by Cambridge University Press:  01 January 2025

Frederic M. Lord
Frederic M. Lord*
Affiliation:
Educational Testing Service Princeton University
Get access Rights & Permissions [Opens in a new window]

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.

Type
Original Paper
Information
Psychometrika , Volume 30 , Issue 3 , September 1965 , pp. 239 - 270
Copyright
Copyright © 1965 Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

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.

References

Appell, P. and de Fériet, Kampé, J. Fonctions hypergeometriques et hyperspheriques. Polynomes d' Hermite. Gauthier-Villars, 1926.Google Scholar
Buckingham, R. A. Numerical methods, London: Pitman, 1957.Google Scholar
Bückner, H. F. Numerical methods for integral equations. In Todd, J. (Eds.), A survey of numerical analysis, New York: McGraw-Hill, 1962.Google Scholar
Cochran, W. G. Some methods for strengthening the common x2 tests. Biometrics, 1954, 10, 417–451.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Elderton, W. P. Frequency curves and correlation (3rd ed.), Cambridge, England: Cambridge Univ. Press, 1938.Google Scholar
Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. Higher transcendental functions. Vol. 1, New York: McGraw-Hill, 1953.Google Scholar
Fox, L. and Goodwin, E. T. The numerical solution of non-singular linear equations. Phil. Trans. roy. Soc. London (Series A), 1952, 245, 501–535.Google Scholar
Gulliksen, H. O. Theory of mental tests, New York: Wiley, 1950.CrossRefGoogle Scholar
Hirschman, I. I. and Widder, D. V. The convolution transform, Princeton: Princeton Univ. Press, 1955.Google Scholar
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, 1960.Google Scholar
Kantorovich, L. W. and Krylov, V. I. Approximate methods of higher analysis, New York: Interscience, 1958.Google Scholar
Keats, J. A. and Lord, F. M. A theoretical distribution for mental test scores. Psychometrika, 1962, 27, 59–72.CrossRefGoogle Scholar
Kendall, M. G. and Stuart, A. The advanced theory of statistics, New York: Hafner, 1958.Google Scholar
Kunz, K. S. Numerical analysis, New York: McGraw-Hill, 1957.Google Scholar
Lord, F. M. A theory of test scores, Richmond, Va.: William Byrd Press, 1952.Google Scholar
Lord, F. M. An approach to mental test theory. Psychometrika, 1959, 24, 283–302.CrossRefGoogle Scholar
Lord, F. M. Use of true-score theory to predict moments of univariate and bivariate observed-score distributions. Psychometrika, 1960, 25, 325–342.CrossRefGoogle Scholar
Lord, F. M. Estimating true measurements from fallible measurements (Binomial Case)—expansion in a series of beta distributions, Princeton, N. J.: Educ. Test. Serv., 1962.CrossRefGoogle Scholar
Lord, F. M. True-score theory—The four parameter beta model with binomial errors, Princeton, N. J.: Educ. Test. Serv., 1964.CrossRefGoogle Scholar
Medgyessy, P. Decomposition of superpositions of distribution functions, Budapest: Publishing House of the Hungarian Academy of Sciences, 1961.Google Scholar
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.CrossRefGoogle Scholar
Pollard, H. Distribution functions containing a Gaussian factor. Proc. Amer. math. Soc., 1953, 4, 578–582.CrossRefGoogle Scholar
Rasch, G. Probabilistic models for some intelligence and attainment tests, Copenhagen, Denmark: Nielsen and Lydiche, 1960.Google Scholar
Teicher, H. On the mixture of distributions. Ann. math. Statist., 1960, 31, 55–73.CrossRefGoogle Scholar
Tricomi, F. G. Integral equations, New York: Interscience, 1957.Google Scholar
Trumpler, R. J. and Weaver, H. F. Statistical astronomy, Berkeley: Univ. California Press, 1953.CrossRefGoogle Scholar
Tucker, L. R. A note on the estimation of test reliability by the Kuder-Richardson formula (20). Psychometrika, 1949, 14, 117–120.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Walsh, J. E. Corrections to two papers concerned with binomial events. Sankhyā, 1963, 25, 427–427.Google Scholar
Walther, A. and Dejon, B. General report on the numerical treatment of integral and integro-differential equations. Basel/Stuttgart: Birkhauser Verlag, 1960, 645–671.Google Scholar
Young, A. Approximate product-integration. Proc. roy. Soc., A, 1954, 224, 552–561.Google Scholar
Young, A. The application of product-integration to the numerical solution of integral equations. Proc. roy. Soc., A, 1954, 224, 561–573.Google Scholar