The details of EM algorithms for maximum likelihood factor analysis are presented for both the exploratory and confirmatory models. The algorithm is essentially the same for both cases and involves only simple least squares regression operations; the largest matrix inversion required is for a q × q symmetric matrix where q is the matrix of factors. The example that is used demonstrates that the likelihood for the factor analysis model may have multiple modes that are not simply rotations of each other; such behavior should concern users of maximum likelihood factor analysis and certainly should cast doubt on the general utility of second derivatives of the log likelihood as measures of precision of estimation.

Rubin and Thayer recently presented equations to implement maximum likelihood (ML) estimation in factor analysis via the EM algorithm. They present an example to demonstrate the efficacy of the algorithm, and propose that their recovery of multiple local maxima of the ML function “certainly should cast doubt on the general utility of second derivatives of the log likelihood as measures of precision of estimation.” It is shown here, in contrast, that these second derivatives verify that Rubin and Thayer did not find multiple local maxima as claimed. The only known maximum remains the one found by Jöreskog over a decade earlier. The standard errors obtained from the second derivatives and the Fisher information matrix thus remain appropriate where ML assumptions are met. The advantages of the EM algorithm over other algorithms for ML factor analysis remain to be demonstrated.