Fitting the factor analysis model

Abstract

When the covariance matrix Σ(p×P) does not satisfy the formal factor analysis model for m factors, there will be no factor matrix Λ(p×m) such that γ=(Σ-ΛΛ′) is diagonal. The factor analysis model may then be replaced by a tautology where γ is regarded as the covariance matrix of a set of “residual variates.” These residual variates are linear combinations of “discarded” common factors and unique factors and are correlated. Maximum likelihood, alpha and iterated principal factor analysis are compared in terms of the manner in which γ is defined, a “maximum determinant” derivation for alpha factor analysis being given. Weighted least squares solutions using residual variances and common variances as weights are derived for comparison with the maximum likelihood and alpha solutions. It is shown that the covariance matrix γ defined by maximum likelihood factor analysis is Gramian, provided that all diagonal elements are nonnegative. Other methods can define a γ which is nonGramian even when all diagonal elements are nonnegative.