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Factor analysis with EM algorithm never gives improper solutions when sample covariance and initial parameter

Kohei Adachi1

  • 1Graduate School of Human Sciences, Osaka University, 1-2 Yamadaoka, Suita, Osaka, 565-0871, Japan, adachi@hus.osaka-u.ac.jp.

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The EM algorithm for factor analysis guarantees proper solutions with positive unique variances and valid factor correlations. Modified formulas ensure convergence, even when standard methods fail due to numerical precision issues.

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Area of Science:

  • Psychometrics
  • Statistical analysis
  • Factor analysis

Background:

  • The Expectation-Maximization (EM) algorithm was introduced for maximum likelihood factor analysis.
  • Ensuring proper solutions in factor analysis is crucial for valid interpretation.

Purpose of the Study:

  • To mathematically prove the EM algorithm yields proper solutions in factor analysis.
  • To demonstrate the EM algorithm's superiority over gradient algorithms for problematic datasets.
  • To address convergence issues in confirmatory factor analysis using the EM algorithm.

Main Methods:

  • Mathematical proof of EM algorithm properties.
  • Numerical simulations comparing EM and gradient algorithms.
  • Analysis of formula modifications for improved EM algorithm stability.

Main Results:

  • The EM algorithm is proven to consistently produce proper solutions with positive unique variances and valid factor correlations.
  • The EM algorithm successfully analyzed data that caused gradient algorithms to yield improper solutions.
  • Modified EM algorithm formulas prevent asymmetry in factor correlation matrices, ensuring convergence.

Conclusions:

  • The EM algorithm provides a robust method for factor analysis, guaranteeing proper solutions.
  • The EM algorithm offers advantages over gradient methods, particularly with challenging datasets.
  • Adjustments to the EM algorithm enhance its reliability in confirmatory factor analysis under limited numerical precision.