IDEAS home Printed from https://ideas.repec.org/a/spr/psycho/v83y2018i2d10.1007_s11336-017-9600-y.html
   My bibliography  Save this article

Some Mathematical Properties of the Matrix Decomposition Solution in Factor Analysis

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11336-017-9600-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11336-017-9600-y?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Uno, Kohei & Satomura, Hironori & Adachi, Kohei, 2016. "Fixed factor analysis with clustered factor score constraint," Computational Statistics & Data Analysis, Elsevier, vol. 94(C), pages 265-274.
    3. Kohei Adachi & Nickolay T. Trendafilov, 2018. "Sparsest factor analysis for clustering variables: a matrix decomposition approach," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 12(3), pages 559-585, September.
    4. 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.
    5. 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.
    6. Sundberg, Rolf & Feldmann, Uwe, 2016. "Exploratory factor analysis—Parameter estimation and scores prediction with high-dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 148(C), pages 49-59.
    7. Jushan Bai & Serena Ng, 2020. "Simpler Proofs for Approximate Factor Models of Large Dimensions," Papers 2008.00254, arXiv.org.
    8. Liu, Litao & Cao, Zhi & Liu, Xiaojie & Shi, Lei & Cheng, Shengkui & Liu, Gang, 2020. "Oil security revisited: An assessment based on complex network analysis," Energy, Elsevier, vol. 194(C).
    9. Paolo Giordani & Roberto Rocci & Giuseppe Bove, 2020. "Factor Uniqueness of the Structural Parafac Model," Psychometrika, Springer;The Psychometric Society, vol. 85(3), pages 555-574, September.
    10. Lars Eldén & Nickolay Trendafilov, 2019. "Semi-sparse PCA," Psychometrika, Springer;The Psychometric Society, vol. 84(1), pages 164-185, March.
    11. Matteo Barigozzi & Matteo Luciani, 2019. "Quasi Maximum Likelihood Estimation and Inference of Large Approximate Dynamic Factor Models via the EM algorithm," Papers 1910.03821, arXiv.org, revised Sep 2024.
    12. Zirogiannis, Nikolaos & Tripodis, Yorghos, 2013. "A Generalized Dynamic Factor Model for Panel Data: Estimation with a Two-Cycle Conditional Expectation-Maximization Algorithm," Working Paper Series 142752, University of Massachusetts, Amherst, Department of Resource Economics.
    13. Dorota Toczydlowska & Gareth W. Peters & Man Chung Fung & Pavel V. Shevchenko, 2017. "Stochastic Period and Cohort Effect State-Space Mortality Models Incorporating Demographic Factors via Probabilistic Robust Principal Components," Risks, MDPI, vol. 5(3), pages 1-77, July.
    14. Aßmann, Christian & Boysen-Hogrefe, Jens & Pape, Markus, 2012. "The directional identification problem in Bayesian factor analysis: An ex-post approach," Kiel Working Papers 1799, Kiel Institute for the World Economy (IfW Kiel).
    15. Jos Berge & Henk Kiers, 1993. "An alternating least squares method for the weighted approximation of a symmetric matrix," Psychometrika, Springer;The Psychometric Society, vol. 58(1), pages 115-118, March.
    16. Chen, Derek H. C. & Gawande, Kishore, 2007. "Underlying dimensions of knowledge assessment : factor analysis of the knowledge assessment methodology data," Policy Research Working Paper Series 4216, The World Bank.
    17. Kenneth Bollen & Stanislav Kolenikov & Shawn Bauldry, 2014. "Model-Implied Instrumental Variable—Generalized Method of Moments (MIIV-GMM) Estimators for Latent Variable Models," Psychometrika, Springer;The Psychometric Society, vol. 79(1), pages 20-50, January.
    18. Zhou, Lin & Tang, Yayong, 2021. "Linearly preconditioned nonlinear conjugate gradient acceleration of the PX-EM algorithm," Computational Statistics & Data Analysis, Elsevier, vol. 155(C).
    19. Naomichi Makino, 2015. "Generalized data-fitting factor analysis with multiple quantification of categorical variables," Computational Statistics, Springer, vol. 30(1), pages 279-292, March.
    20. Kim, Jiwhan & Nam, Changi & Ryu, Min Ho, 2020. "IPTV vs. emerging video services: Dilemma of telcos to upgrade the broadband," Telecommunications Policy, Elsevier, vol. 44(4).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:psycho:v:83:y:2018:i:2:d:10.1007_s11336-017-9600-y. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.