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Influential Observations in Principal Factor Analysis

Published online by Cambridge University Press:  01 January 2025

Yutaka Tanaka  and
Yoshimasa Odaka
Yutaka Tanaka*
Affiliation:
Department of Statistics, Okayama University
Yoshimasa Odaka
Affiliation:
Information Processing Center, Okayama University of Science
*
Request for reprints should be sent to Yutaka Tanaka, Department of Statistics, Okayama University, 2-1-1 Tsushima-naka, Okayama 700. JAPAN.
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Abstract

We propose a method for detecting influential observations in iterative principal factor analysis. For this purpose we derive the influence functions I(x; LLT) and I(x; Δ) for the common variance matrix T =LLT and the unique variance matrix Δ, respectively, in the common factor decomposition Σ =LLT + Δ. A numerical example is given for illustration.

Keywords

influential observationsinfluence functioninfluence on the subspace spanned by eigenvectorsprincipal factor analysisperturbation theory of eigenvalue problems

Information

Type
Original Paper
Information
Psychometrika , Volume 54 , Issue 3 , September 1989 , pp. 475 - 485
Copyright
Copyright © 1989 The Psychometric Society

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Footnotes

The authors are grateful to Tomoyuki Tarumi and Atsuhiro Hayashi for their kind permission to use their software Seto/B for drawing Figures 1 and 2 and to anonymous reviewers for comments on the paper.

References

Anderson, T. W., & Rubin, H. (1956). Statistical inference in factor analysis. In Neyman, J. (Ed.), Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 5, 111–150.Google Scholar
Cook, R. D., & Weisberg, S. (1982). Residuals and influence in regression, New York: Chapman and Hall.Google Scholar
Critchley, F. (1985). Influence in principal component analysis. Biometrika, 72, 627–36.CrossRefGoogle Scholar
Hampel, F. R. (1974). The influence curve and its role in robust estimation. Journal of the American Statistical Association, 69, 383–93.CrossRefGoogle Scholar
Jolliffe, I. T. (1986). Principal component analysis, New York: Springer-Verlag.CrossRefGoogle Scholar
Kodake, K., & Tanaka, Y. (1985). Sensitivity analysis in factor analysis. Proceedings of the Second Japan-China Symposium on Statistics, 137–140.Google Scholar
Mardia, K. V., Kent, J. T., & Bibby, J. M. (1979). Multivariate analysis, New York: Academic Press.Google Scholar
Okamoto, M. (1986). Inshibunseki no kiso, Tokyo: JUSE Publishing (in Japanese)Google Scholar
Pack, P., Jolliffe, I. T., & Morgan, B. J. T. (1988). Influential observations in principal component analysis: A case study. Journal of Applied Statistics, 15, 37–50.CrossRefGoogle Scholar
Radhakrishnan, R., & Kshirsagar, A. M. (1981). Influence functions for certain parameters in multivariate analysis. Communications in Statistics, Series A, 10, 515–529.CrossRefGoogle Scholar
Rellich, F. (1969). Perturbation theory of eigenvalue problems, New York: Gordon and Breach.Google Scholar
Saaty, T. L., & Bram, J. (1964). Nonlinear mathematics, New York: McGraw-Hill.Google Scholar
Tanaka, Y. (1988). Sensitivity analysis in principal component analysis: Influence on the subspace spanned by principal components. Communications in Statistics, Series A, 17(9), 3157–3175.CrossRefGoogle Scholar
Yamane, Y. (1987). The influence of the factor analysis. Proceedings of the First Annual Conference of Japanese Society of Computational Statistics, 32–35. (in Japanese)Google Scholar