Hostname: page-component-65f69f4695-6knnn Total loading time: 0 Render date: 2025-06-29T14:49:15.684Z Has data issue: false hasContentIssue false
  • English
  • Français

A Comparison of Four Methods of Constructing Factor Scores

Published online by Cambridge University Press:  01 January 2025

Roderick P. McDonald  and
E. J. Burr
Roderick P. McDonald
Affiliation:
University of New England, N.S.W., Australia
E. J. Burr
Affiliation:
University of New England, N.S.W., Australia
Get access Rights & Permissions [Opens in a new window]

Abstract

Four least-squares methods for constructing factor scores have been described in the literature. The formal properties of these scores are developed, and they are compared in terms of four generally desirable properties of constructed factor scores. In particular, it is shown that two of the methods yield scores that are conditionally unbiassed, and univocal in the sense of Guilford and Michael, though not orthogonal, while one of the other methods yields orthogonal scores.

It is shown that constructed factor scores cannot be simultaneously univocal and orthogonal, unless we choose the special basis in factor space given by Canonical Factor Analysis.

The general problem of choosing between the methods is discussed, on the basis of the theoretical relations obtained.

Information

Type
Original Paper
Information
Psychometrika , Volume 32 , Issue 4 , December 1967 , pp. 381 - 401
Copyright
Copyright © 1967 The 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.)

Article purchase

Temporarily unavailable

Footnotes

*

EDITOR’S NOTE: The reader will quickly discover that this article develops several of the generalizations given in the second part of the preceding article, “On Factors and Factor Scores.” Independent development of the same generalizations is, of course, not a new phenomenon. Because the Presidential Address automatically is accepted for publication and given space in the December issue, it was decided that the only fair thing to do was to print this article in the same issue.

References

Anderson, T. W. and R., H. Statistical inference in factor analysis. Proceedings of the third Berkeley Symposium on mathematical statistics and probability Vol. V, 1956.Google Scholar
Bartlett, M. S. The statistical conception of mental factors. British Journal of Psychology, 1937, 28, 97–104.Google Scholar
Burkett, G. R. A study of reduced rank models for multiple prediction. Psychometric Monographs, No. 12, 1964.Google Scholar
Guilford, J. P. and Michael, W. B. Approaches to univocal factor scores. Psychometrika, 1948, 13, 1–22.CrossRefGoogle ScholarPubMed
Gulliksen, H. Theory of mental tests, New York: John Wiley, 1950.CrossRefGoogle Scholar
Guttman, L. The determinacy of factor score matrices with implications for five other basic problems of common-factor theory. British Journal of Statistical Psychology, 1955, 8, 65–82.CrossRefGoogle Scholar
Hadley, G. Linear algebra, Mass.: Addison-Wesley, 1965.Google Scholar
Heermann, E. F. Univocal or orthogonal estimators of orthogonal factors. Psychometrika, 1963, 28, 161–172.CrossRefGoogle Scholar
Heermann, E. F. The geometry of factorial indeterminacy. Psychometrika, 1964, 29, 371–382.CrossRefGoogle Scholar
Horst, P. Factor analysis of data matrices, New York: Holt, Rinehart & Winston, 1965.Google Scholar
Kendall, M. G. and Kendall, S. A. The advanced theory of statistics, London: Griffin, 1961.CrossRefGoogle Scholar
Kestelman, H. The fundamental equation of factor analysis. British Journal of Psychology, Statistical Section, 1952, 5, 1–6.CrossRefGoogle Scholar
Lawley, D. N. The estimation of factor loadings by the method of maxium likelihood. Proceedings of the Royal Society of Edinburgh, 1940, 60, 64–82.CrossRefGoogle Scholar
McDonald, R. P. A general approach to nonlinear factor analysis. Psychometrika, 1962, 27, 397–415.CrossRefGoogle Scholar
McDonald, R. P. Numerical methods for polynomical models in nonlinear factor analysis. Psychometrika, 1967, 32, 77–112.CrossRefGoogle Scholar
Rao, C. R. Estimation and tests of significance in factor analysis. Psychometrika, 1955, 20, 93–112.CrossRefGoogle Scholar
Thomson, G. H. The factorial analysis of human ability (2nd ed.), New York: Houghton Mifflin, 1946.Google Scholar
Thurstone, L. L. The vectors of mind, Chicago: Univ. of Chicago Press, 1935.Google Scholar