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

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.