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Factor Analysis Procedures Revisited from the Comprehensive Model with Unique Factors Decomposed into Specific Factors and Errors

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  • Kohei Adachi

    (Osaka University)

Abstract

Factor analysis (FA) procedures can be classified into three types (Adachi in WIREs Comput Stat https://onlinelibrary.wiley.com/doi/abs/10.1002/wics.1458 , 2019): latent variable FA (LVFA), matrix decomposition FA (MDFA), and its variant (Stegeman in Comput Stat Data Anal 99: 189–203, 2016) named completely decomposed FA (CDFA) through the theorems proved in this paper. We revisit those procedures from the Comprehensive FA (CompFA) model, in which a multivariate observation is decomposed into common factor, specific factor, and error parts. These three parts are separated in MDFA and CDFA, while the specific factor and error parts are not separated, but their sum, called a unique factor, is considered in LVFA. We show that the assumptions in the CompFA model are satisfied by the CDFA solution, but not completely by the MDFA one. Then, how the CompFA model parameters are estimated in the FA procedures is examined. The study shows that all parameters can be recovered well in CDFA, while the sum of the parameters for the specific factor and error parts is approximated by the LVFA estimate of the unique factor parameter and by the MDFA estimate of the specific factor parameter. More detailed results are given through our subdivision of the CompFA model according to whether the error part is uncorrelated among variables or not.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:psycho:v:87:y:2022:i:3:d:10.1007_s11336-021-09824-8
    DOI: 10.1007/s11336-021-09824-8
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    References listed on IDEAS

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    1. Henry Kaiser, 1958. "The varimax criterion for analytic rotation in factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 23(3), pages 187-200, September.
    2. Donald Rubin & Dorothy Thayer, 1982. "EM algorithms for ML factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 47(1), pages 69-76, March.
    3. 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.
    4. Jos Berge & Henk Kiers, 1991. "A numerical approach to the approximate and the exact minimum rank of a covariance matrix," Psychometrika, Springer;The Psychometric Society, vol. 56(2), pages 309-315, June.
    5. Harry Harman & Wayne Jones, 1966. "Factor analysis by minimizing residuals (minres)," Psychometrika, Springer;The Psychometric Society, vol. 31(3), pages 351-368, September.
    6. Carl Eckart & Gale Young, 1936. "The approximation of one matrix by another of lower rank," Psychometrika, Springer;The Psychometric Society, vol. 1(3), pages 211-218, September.
    7. repec:ucp:bkecon:9780226316529 is not listed on IDEAS
    8. 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.
    9. Kohei Adachi, 2013. "Factor Analysis with EM Algorithm Never Gives Improper Solutions when Sample Covariance and Initial Parameter Matrices Are Proper," Psychometrika, Springer;The Psychometric Society, vol. 78(2), pages 380-394, April.
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