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Determining the Number of Components from the Matrix of Partial Correlations

Published online by Cambridge University Press:  01 January 2025

Wayne F. Velicer
Wayne F. Velicer*
Affiliation:
University of Rhode Island
*
Requests for reprints should be sent to Wayne F. Velicer, Department of Psychology, University of Rhode Island, Kingston, R. I. 02881.
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Abstract

A common problem for both principal component analysis and image component analysis is determining how many components to retain. A number of solutions have been proposed, none of which is totally satisfactory. An alternative solution which employs a matrix of partial correlations is considered. No components are extracted after the average squared partial correlation reaches a minimum. This approach gives an exact stopping point, has a direct operational interpretation, and can be applied to any type of component analysis. The method is most appropriate when component analysis is employed as an alternative to, or a first-stage solution for, factor analysis.

Keywords

common variancefactor analysisimage component analysis
Type
Original Paper
Information
Psychometrika , Volume 41 , Issue 3 , September 1976 , pp. 321 - 327
Copyright
Copyright © 1976 The Psychometric Society

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References

Reference Note

Howe, W. G. Some contributions to factor analysis (Report No. ONRL-1919), 1955, Oak Ridge, Tenn.: Oak Ridge National Laboratory.CrossRefGoogle Scholar

References

Bartlett, M. S. Tests of significance in factor analysis. The British Journal of Psychology, 1950, 3, 77–85.Google Scholar
Bartlett, M. S. A further note on tests of significance in factor analysis. The British Journal of Psychology, 1951, 4, 1–2.Google Scholar
Bechtoldt, H. P. An empirical study of the factor analysis stability hypothesis. Psychometrika, 1961, 26, 405–432.CrossRefGoogle Scholar
Cattell, R. B. The scree test for the number of factors. Multivariate Behavioral Research, 1966, 1, 245–276.CrossRefGoogle ScholarPubMed
Emmett, W. C. Factor analysis by Lawley's method of maximum likelihood. British Journal of Psychology, Statistical Section, 1949, 2, 90–97.CrossRefGoogle Scholar
Gorsuch, R. L. Using Bartlett's significance test to determine the number of factors to extract. Educational and Psychological Measurement, 1973, 33, 361–364.CrossRefGoogle Scholar
Harman, H. H. Modern factor analysis, 1960, Chicago: University of Chicago Press.Google Scholar
Harman, H. H., & Jones, W. H. Factor analysis by minimizing residuals (minres). Psychometrika, 1966, 31, 351–368.CrossRefGoogle ScholarPubMed
Horst, P. Factor analysis of data matrices, 1965, New York: Holt, Rinehart, and Winston.Google Scholar
Jöreskog, K. G. Some contributions to maximum likelihood factor analysis. Psychometrika, 1967, 32, 443–482.CrossRefGoogle Scholar
Kaiser, H. F. The application of electronic computers to factor analysis. Educational and Psychological Measurement, 1960, 20, 141–151.CrossRefGoogle Scholar
Lord, F. M. A study of speed factors in tests and academic grades. Psychometrika, 1956, 21, 31–50.CrossRefGoogle Scholar
Maxwell, E. A. Recent trends in factor analysis. Journal of the Royal Statistical Society, 1961, 124, 49–59.CrossRefGoogle Scholar
Schönemann, P. H., & Wang, M. Some new results on factor indeterminacy. Psychometrika, 1972, 37, 61–91.CrossRefGoogle Scholar
Thurstone, L. L. The dimensions of temperament. Psychometrika, 1951, 16, 11–20.CrossRefGoogle Scholar
Van de Geer, J. P. Introduction to multivariate analysis for the social sciences, 1971, San Francisco: Freeman.Google Scholar
Velicer, W. F. An empirical comparison of the stability of factor analysis, principal component analysis, and image analysis. Educational and Psychological Measurement, 1974, 34, 563–572.CrossRefGoogle Scholar
Velicer, W. F. The relation between factor score estimates, image scores, and principal component scores. Educational and Psychological Measurement, 1976, 36, 149–159.CrossRefGoogle Scholar
Velicer, W. F. An empirical comparison of the similarity of principal component, image, and factor patterns. Multivariate Behavioral Research, in press.Google Scholar