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To What Extent can Communalities Reduce Rank?

Published online by Cambridge University Press:  01 January 2025

Louis Guttman
Louis Guttman*
Affiliation:
The Israel Institute of Applied Social Research and The Hebrew University in Jerusalem
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Abstract

The question is raised as to whether the null hypothesis concerning the number of common factors underlying a given set of correlations should be that this number is small. Psychological and algebraic evidence indicate that a more appropriate null hypothesis is that the number is relatively large, and that smallness should be but an alternative hypothesis. The question is also raised as to why approximation procedures should be aimed primarily at the observed correlation matrix R and not at, say, R−1. What may be best for R may be worst for R−1, and conversely, yet R−1 is directly involved in problems of multiple and partial regressions. It is shown that a widely accepted inequality for the possible rank to which R can be reduced, when modified by communalities, is indeed false.

Type
Original Paper
Information
Psychometrika , Volume 23 , Issue 4 , December 1958 , pp. 297 - 308
Copyright
Copyright © 1958 The Psychometric Society

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Footnotes

*

This research was facilitated by a noncommitted grant-in-aid to the writer from the Ford Foundation.

References

Burt, C. Bipolar factors as a cause of cyclic overlap. Brit. J. Psychol., Statist. Sect., 1952, 5, 197–202.Google Scholar
Cattell, R. B. Index universel pour les facteurs psychologiques. L'analyse factorielle et ses application, Paris: Centre National de la Recherche Scientifique, 1956.Google Scholar
Doob, J. W. Stochastic processes, New York: Wiley, 1954.Google Scholar
Guttman, L. Image theory for the structure of quantitative variates. Psychometrika, 1953, 18, 277–296.CrossRefGoogle Scholar
Guttman, L. Some necessary conditions for common-factor analysis. Psychometrika, 1954, 19, 149–161.CrossRefGoogle Scholar
Guttman, L. A new approach to factor analysis: the radex. In Lazarsfeld, P. F. (Eds.), Mathematical thinking in the social sciences, Glencoe, Ill.: Free Press, 1954.Google Scholar
Guttman, L. A generalized simplex for factor analysis. Psychometrika, 1955, 20, 173–192.Google Scholar
Guttman, L. The determinacy of factor score matrices with implications for five other basic problems of common-factor theory. Brit. J. statist. Psychol., 1955, 8, 65–81.Google Scholar
Guttman, L. Une solution au problème des communautés. Bulletin du Centre d'Études et Recherches Psychotechniques, 1956, 6, 123–128.Google Scholar
Guttman, L. “Best possible” systematic estimates of communalities. Psychometrika, 1956, 21, 273–285.Google Scholar
Guttman, L. Empirical verification of the radex structure of mental abilities and personality traits. Educ. psychol. Measmt, 1957, 17, 391–407.CrossRefGoogle Scholar
Guttman, L. What lies ahead for factor analysis? Educ. psychol. Measmt, in press.Google Scholar
Lawley, D. N. The estimation of factor loadings by the method of maximum likelihood. Proc. Roy. Soc. Edin., 1940, 60, 64–82.CrossRefGoogle Scholar
Ledermann, W. On the rank of the reduced correlation matrix in multiple-factor analysis. Psychometrika, 1937, 2, 85–93.CrossRefGoogle Scholar
Lord, F. M. A study of speed factors in tests and academic grades. Psychometrika, 1956, 21, 31–50.Google Scholar
Rao, C. R. Estimation and tests of significance in factor analysis. Psychometrika, 1955, 20, 93–111.Google Scholar
Thomson, G. H. The factorial analysis of human ability 5th ed., London: Univ. London Press, 1951.Google Scholar
Thurstone, L. L. Multiple-factor analysis, Chicago: Univ. Chicago Press, 1947.Google Scholar
Wrigley, C. The distinction between common and specific variance in factor theory. Brit. J. statist. Psychol., 1957, 10, 81–98.Google Scholar