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Convergence in mean square of factor predictors.

Wim P Krijnen1

  • 1University of Amsterdam, 1018 WB Amsterdam, The Netherlands. wim.krijnen@hetnet.nl

The British Journal of Mathematical and Statistical Psychology
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Summary

This study provides new sufficient conditions for factor predictor convergence in confirmatory factor analysis, addressing limitations of prior work, including Heywood cases. These conditions ensure unique true factor existence and aid in understanding factor indeterminacy.

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Area of Science:

  • Statistics
  • Psychometrics
  • Multivariate Analysis

Background:

  • Existing sufficient conditions for mean square convergence of factor predictors in common factor analysis (CFA) are insufficient for confirmatory factor analysis (CFA) and Heywood cases (zero error variance).
  • This gap limits the reliable estimation of true factors in specific, yet common, CFA models.

Purpose of the Study:

  • To establish new sufficient conditions for the mean square convergence of factor predictors in CFA.
  • To address the limitations of previous conditions, particularly in Heywood cases.
  • To ensure the existence of a unique true factor and provide a geometric interpretation of factor indeterminacy.

Main Methods:

  • Derivation of two novel sufficient conditions for the convergence of three basic factor predictors and a rotated principal components analysis predictor.
  • Analysis of these conditions for necessity under certain model specifications.
  • Development of a geometric interpretation for factor indeterminacy and mean square convergence.

Main Results:

  • Two new sufficient conditions are presented for factor predictor convergence in CFA, encompassing Heywood cases.
  • These conditions are shown to be necessary for specific model configurations.
  • The conditions guarantee the existence of a unique true factor.

Conclusions:

  • The established conditions enhance the theoretical foundation for factor prediction in CFA, particularly in challenging scenarios like Heywood cases.
  • The findings contribute to a deeper understanding of factor indeterminacy and the convergence properties of factor predictors.
  • This work provides a more robust framework for factor analysis model evaluation and interpretation.