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Some Mathematical Properties of the Matrix Decomposition Solution in Factor Analysis

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  • Kohei Adachi

    (Osaka University)

  • Nickolay T. Trendafilov

    (Open University)

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.

Suggested Citation

  • Kohei Adachi & Nickolay T. Trendafilov, 2018. "Some Mathematical Properties of the Matrix Decomposition Solution in Factor Analysis," Psychometrika, Springer;The Psychometric Society, vol. 83(2), pages 407-424, June.
    • Handle: RePEc:spr:psycho:v:83:y:2018:i:2:d:10.1007_s11336-017-9600-y
    DOI: 10.1007/s11336-017-9600-y
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    References listed on IDEAS

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    1. Stanley Mulaik, 1976. "Comments on “the measurement of factorial indeterminacy”," Psychometrika, Springer;The Psychometric Society, vol. 41(2), pages 249-262, June.
    2. Masamori Ihara & Yutaka Kano, 1986. "A new estimator of the uniqueness in factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 51(4), pages 563-566, December.
    3. Donald Rubin & Dorothy Thayer, 1982. "EM algorithms for ML factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 47(1), pages 69-76, March.
    4. Harry Harman & Wayne Jones, 1966. "Factor analysis by minimizing residuals (minres)," Psychometrika, Springer;The Psychometric Society, vol. 31(3), pages 351-368, September.
    5. Jos Berge, 1983. "A generalization of Kristof's theorem on the trace of certain matrix products," Psychometrika, Springer;The Psychometric Society, vol. 48(4), pages 519-523, December.
    6. repec:ucp:bkecon:9780226316529 is not listed on IDEAS
    7. Stegeman, Alwin, 2016. "A new method for simultaneous estimation of the factor model parameters, factor scores, and unique parts," Computational Statistics & Data Analysis, Elsevier, vol. 99(C), pages 189-203.
    8. Naomichi Makino, 2015. "Generalized data-fitting factor analysis with multiple quantification of categorical variables," Computational Statistics, Springer, vol. 30(1), pages 279-292, March.
    9. Unkel, S. & Trendafilov, N.T., 2010. "A majorization algorithm for simultaneous parameter estimation in robust exploratory factor analysis," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 3348-3358, December.
    10. Steffen Unkel & Nickolay T. Trendafilov, 2010. "Simultaneous Parameter Estimation in Exploratory Factor Analysis: An Expository Review," International Statistical Review, International Statistical Institute, vol. 78(3), pages 363-382, December.
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    Cited by:

    1. Kohei Adachi, 2022. "Factor Analysis Procedures Revisited from the Comprehensive Model with Unique Factors Decomposed into Specific Factors and Errors," Psychometrika, Springer;The Psychometric Society, vol. 87(3), pages 967-991, September.
    2. Kohei Uno & Kohei Adachi & Nickolay T. Trendafilov, 2019. "Clustered Common Factor Exploration in Factor Analysis," Psychometrika, Springer;The Psychometric Society, vol. 84(4), pages 1048-1067, December.

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