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Descriptive Axioms for Common Factor Theory, Image Theory and Component Theory

Published online by Cambridge University Press:  01 January 2025

Roderick P. McDonald
Roderick P. McDonald*
Affiliation:
The Ontario Institute for Studies in Education
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Abstract

Through an extension of work by Guttman, common factor theory, image theory, and component theory are derived from distinct minimum subsets of assumptions chosen out of a set of five possible assumptions. It is thence shown that the problem of indeterminacy of factor scores in the common factor model is precisely reflected in the problem of the non-orthogonality of anti-images. Indeed, image scores are determinate for the same reason that the usual estimates of factor scores are determinate, and image scores cannot be used as though they were factor scores for the same reason that factor score estimates cannot be used as though they were factor scores.

Type
Original Paper
Information
Psychometrika , Volume 40 , Issue 2 , June 1975 , pp. 137 - 152
Copyright
Copyright © 1975 The Psychometric Society

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References

Bartlett, M. S.. The statistical conception of mental factors. British Journal of Psychology, 1937, 28, 97–104.Google Scholar
Guttman, L.. Multiple rectilinear prediction and the resolution into components. Psychometrika, 1940, 5, 75–99.CrossRefGoogle Scholar
Guttman, L.. General theory and methods for matric factoring. Psychometrika, 1944, 9, 1–16.CrossRefGoogle Scholar
Guttman, L.. Some necessary conditions for common factor analysis. Psychometrika, 1954, 19, 149–161.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
Guttman, L.. “Best possible” systematic estimates of communalities. Psychometrika, 1956, 21, 273–285.CrossRefGoogle Scholar
Guttman, L.. A necessary and sufficient formula for matric factoring. Psychometrika, 1957, 22, 79–82.CrossRefGoogle Scholar
Guttman, L.. Simple proofs of relations between the communality problem and multiple correlation. Psychometrika, 1957, 22, 147–158.CrossRefGoogle Scholar
Harris, C. W.. Some Rao-Guttman relationships. Psychometrika, 1962, 27, 247–264.CrossRefGoogle Scholar
Jöreskog, K. G.. On the statistical treatment of residuals in factor analysis. Psychometrika, 1962, 27, 335–354.CrossRefGoogle Scholar
Jöreskog, K. G.. Some contributions to maximum likelihood factor analysis. Psychometrika, 1967, 32, 443–482.CrossRefGoogle Scholar
Jöreskog, K. G.. Efficient estimation in image factor analysis. Psychometrika, 1969, 34, 51–75.CrossRefGoogle Scholar
Kaiser, H. F.. A second generation little jiffy. Psychometrika, 1970, 35, 401–415.CrossRefGoogle Scholar
Kaiser, H. F. and Rice, J., Little, Jiffy, Mark, I.V.. Educational and Psychological Measurement, 1974, 34, 111–117.CrossRefGoogle Scholar
McDonald, R. P.. The measurement of factor indeterminacy. Psychometrika, 1974, 39, 203–222.CrossRefGoogle Scholar
McDonald, R. P. and Swaminathan, H.. A simple matrix calculus with applications to multivariate analysis. General Systems, 1973, 18, 37–54.Google Scholar
Mulaik, S. A.. The foundations of factor analysis, 1972, N. Y.: McGraw-Hill.Google Scholar
Rao, G. R., and Mitra, S. J.. Generalized inverse of matrices and its applications, 1971, N. Y.: Wiley.Google Scholar
Roff, M.. Some properties of the communality in multiple factor theory. Psychometrika, 1936, 1, 1–6.CrossRefGoogle Scholar