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Improving the convergence rate and speed of Fisher-scoring algorithm: ridge and anti-ridge methods in structural equation modeling


  • Ke-Hai Yuan

    () (University of Notre Dame)

  • Peter M. Bentler

    (University of California, Los Angeles)


Abstract In structural equation modeling (SEM), parameter estimates are typically computed by the Fisher-scoring algorithm, which often has difficulty in obtaining converged solutions. Even for simulated data with a correctly specified model, non-converged replications have been repeatedly reported in the literature. In particular, in Monte Carlo studies it has been found that larger factor loadings or smaller error variances in a confirmatory factor model correspond to a higher rate of convergence. However, studies of a ridge method in SEM indicate that adding a diagonal matrix to the sample covariance matrix also increases the rate of convergence for the Fisher-scoring algorithm. This article addresses these two seemingly contradictory phenomena. Using statistical and numerical analyses, the article clarifies why both approaches increase the rate of convergence in SEM. Monte Carlo results confirm the analytical results. Recommendations are provided on how to increase both the speed and rate of convergence in parameter estimation.

Suggested Citation

  • Ke-Hai Yuan & Peter M. Bentler, 2017. "Improving the convergence rate and speed of Fisher-scoring algorithm: ridge and anti-ridge methods in structural equation modeling," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(3), pages 571-597, June.
  • Handle: RePEc:spr:aistmt:v:69:y:2017:i:3:d:10.1007_s10463-016-0552-2
    DOI: 10.1007/s10463-016-0552-2

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    References listed on IDEAS

    1. P. Bentler & Jeffrey Tanaka, 1983. "Problems with EM algorithms for ML factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 48(2), pages 247-251, June.
    2. Masanori Ichikawa & Sadanori Konishi, 1995. "Application of the bootstrap methods in factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 60(1), pages 77-93, March.
    3. 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.
    4. Yuan, Ke-Hai & Chan, Wai, 2008. "Structural equation modeling with near singular covariance matrices," Computational Statistics & Data Analysis, Elsevier, vol. 52(10), pages 4842-4858, June.
    5. Anne Boomsma, 1985. "Nonconvergence, improper solutions, and starting values in lisrel maximum likelihood estimation," Psychometrika, Springer;The Psychometric Society, vol. 50(2), pages 229-242, June.
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