Hostname: page-component-745bb68f8f-b6zl4 Total loading time: 0 Render date: 2025-01-08T10:47:58.026Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Mathematical Properties of the Matrix Decomposition Solution in Factor Analysis

Published online by Cambridge University Press:  01 January 2025

Kohei Adachi  and
Nickolay T. Trendafilov
Kohei Adachi*
Affiliation:
Graduate School of Human Sciences, Osaka University
Nickolay T. Trendafilov
Affiliation:
Open University
*
Correspondence should be made to Kohei Adachi, Graduate School of Human Sciences, Osaka University, 1-2 Yamadaoka, Suita, Osaka565-0871, Japan. Email: adachi@hus.osaka-u.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

A new factor analysis (FA) procedure has recently been proposed which can be called matrix decomposition FA (MDFA). All FA model parameters (common and unique factors, loadings, and unique variances) are treated as fixed unknown matrices. Then, the MDFA model simply becomes a specific data matrix decomposition. The MDFA parameters are found by minimizing the discrepancy between the data and the MDFA model. Several algorithms have been developed and some properties have been discussed in the literature (notably by Stegeman in Comput Stat Data Anal 99:189–203, 2016), but, as a whole, MDFA has not been studied fully yet. A number of new properties are discovered in this paper, and some existing ones are derived more explicitly. The properties provided concern the uniqueness of results, covariances among common factors, unique factors, and residuals, and assessment of the degree of indeterminacy of common and unique factor scores. The properties are illustrated using a real data example.

Keywords

exploratory factor analysismodel identifiabilitycovariances between factors and residualshigher rank approximationfactor indeterminacy
Type
Original Paper
Information
Psychometrika , Volume 83 , Issue 2 , June 2018 , pp. 407 - 424
Copyright
Copyright © 2017 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.)

References

Adachi, K., (2012). Some contributions to data-fitting factor analysis with empirical comparisons to covariance-fitting factor analysis, Journal of the Japanese Society of Computational Statistics, 25, 25–38.CrossRefGoogle Scholar
Adachi, K., (2015). A matrix-intensive approach to factor analysis, Journal of the Japan Statistical Society, Japanese Issue, 44, 363–382(in Japanese).Google Scholar
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 (Vol. 5, pp. 111–150). Berkeley, CA: University of California Press.Google Scholar
Bartlett, M.S., (1950). Tests of significance in factor analysis, British Journal of Psychology (Statistics Section), 3, 77–85.CrossRefGoogle Scholar
de Leeuw, J., (2004). Least squares optimal scaling of partially observed linear systems. In van Montfort, K., Oud, J., & Satorra, A. (Eds.), Recent developments of structural equation models: Theory and applications (pp. 121–134). Dordrecht: Kluwer Academic Publishers.CrossRefGoogle Scholar
Gifi, A., (1990). Nonlinear multivariate analysis. Chichester:Wiley.Google Scholar
Guttman, L., (1955). The determinacy of factor score matrices with implications for five other basic problems of common factor theory, British Journal of Statistical Psychology, 8, 65–81.CrossRefGoogle Scholar
Harman, H.H., (1976). Modern factor analysis. 3 Chicago:The University of Chicago Press.Google Scholar
Harman, H.H., Jones, W.H., (1966). Factor analysis by minimizing residuals (Minres), Psychomerika, 31, 351–369.CrossRefGoogle ScholarPubMed
Ihara, M., Kano, Y., (1986). A new estimator of the uniqueness in factor analysis, Psychometrika, 51, 563–566.CrossRefGoogle Scholar
Makino, N., (2015). Generalized data-fitting factor analysis with multiple quantification of categorical variables, Computational Statistics, 30, 279–292.CrossRefGoogle Scholar
Mulaik, S.A., (1976). Comments on “The measurement of factorial indeterminacy”, Psychometrika, 41, 249–262.CrossRefGoogle Scholar
Mulaik, S.A., (2010). Foundations of factor analysis. 2 Boca Raton:CRC Press.Google Scholar
Rubin, D.B., Thayer, D.T., (1982). EM algorithms for ML factor analysis, Psychometrika, 47, 69–76.CrossRefGoogle Scholar
Sočan, G., (2003). The incremental value of minimum rank factor analysis. Ph.D. Thesis, Groningen: University of Groningen.Google Scholar
Spearman, C., (1904). “General Intelligence,” objectively determined and measured, American Journal of Psychology, 15, 201–293.CrossRefGoogle Scholar
Stegeman, A., (2016). A new method for simultaneous estimation of the factor model parameters, factor scores, and unique parts, Computational Statistics and Data Analysis, 99, 189–203.CrossRefGoogle Scholar
Tanaka, Y., Tarumi, T., (1995). Handbook for statistical analysis: Multivariate analysis (windows version). Tokyo:Kyoritsu-Shuppan(in Japanese).Google Scholar
ten Berge, J.M.F., (1983). A generalization of Kristof’s theorem on the trace of certain matrix products, Psychometrika, 48, 519–523.CrossRefGoogle Scholar
ten Berge, J.M.F., (1993). Least squares optimization in multivariate analysis. Leiden:DSWO Press.Google Scholar
Thurstone, L.L., (1935). The vectors of mind. Chicago:University if Chicago Press.Google Scholar
Unkel, S., Trendafilov, N.T., (2010). Simultaneous parameter estimation in exploratory factor analysis: An expository review, International Statistical Review, 78, 363–382.CrossRefGoogle Scholar