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Factor Analysis with EM Algorithm Never Gives Improper Solutions when Sample Covariance and Initial Parameter Matrices Are Proper

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  • Kohei Adachi

Abstract

Rubin and Thayer (Psychometrika, 47:69–76, 1982 ) proposed the EM algorithm for exploratory and confirmatory maximum likelihood factor analysis. In this paper, we prove the following fact: the EM algorithm always gives a proper solution with positive unique variances and factor correlations with absolute values that do not exceed one, when the covariance matrix to be analyzed and the initial matrices including unique variances and inter-factor correlations are positive definite. We further numerically demonstrate that the EM algorithm yields proper solutions for the data which lead the prevailing gradient algorithms for factor analysis to produce improper solutions. The numerical studies also show that, in real computations with limited numerical precision, Rubin and Thayer’s (Psychometrika, 47:69–76, 1982 ) original formulas for confirmatory factor analysis can make factor correlation matrices asymmetric, so that the EM algorithm fails to converge. However, this problem can be overcome by using an EM algorithm in which the original formulas are replaced by those guaranteeing the symmetry of factor correlation matrices, or by formulas used to prove the above fact. Copyright The Psychometric Society 2013

Suggested Citation

  • 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.
    • Handle: RePEc:spr:psycho:v:78:y:2013:i:2:p:380-394
    DOI: 10.1007/s11336-012-9299-8
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    References listed on IDEAS

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    1. Donald Rubin & Dorothy Thayer, 1982. "EM algorithms for ML factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 47(1), pages 69-76, March.
    2. James Anderson & David Gerbing, 1984. "The effect of sampling error on convergence, improper solutions, and goodness-of-fit indices for maximum likelihood confirmatory factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 49(2), pages 155-173, June.
    3. David Gerbing & James Anderson, 1987. "Improper solutions in the analysis of covariance structures: Their interpretability and a comparison of alternate respecifications," Psychometrika, Springer;The Psychometric Society, vol. 52(1), pages 99-111, March.
    4. Otto Driel, 1978. "On various causes of improper solutions in maximum likelihood factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 43(2), pages 225-243, June.
    5. Kuroda, Masahiro & Sakakihara, Michio, 2006. "Accelerating the convergence of the EM algorithm using the vector [epsilon] algorithm," Computational Statistics & Data Analysis, Elsevier, vol. 51(3), pages 1549-1561, December.
    6. Robert Jennrich & Stephen Robinson, 1969. "A Newton-Raphson algorithm for maximum likelihood factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 34(1), pages 111-123, March.
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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. Lorenzo Finesso & Peter Spreij, 2016. "Factor analysis models via I-divergence optimization," Psychometrika, Springer;The Psychometric Society, vol. 81(3), pages 702-726, September.

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