Robust estimation of constrained covariance matrices for confirmatory factor analysis

Confirmatory factor analysis (CFA) is a data analysis procedure that is widely used in social and behavioral sciences in general and other applied sciences that deal with large quantities of data (variables). The classical estimator (and inference) procedures are based either on the maximum likelihood (ML) or generalized least squares (GLS) approaches which are known to be nonrobust to departures from the multivariate normal assumption underlying CFA. A natural robust estimator is obtained by first estimating the (mean and) covariance matrix of the manifest variables and then "plug-in" this statistic into the ML or GLS estimating equations. This two-stage method however does not fully take into account the covariance structure implied by the CFA model. An S-estimator for the parameters of the CFA model that is computed directly from the data is proposed instead and the corresponding estimating equations and an iterative procedure are derived. It is also shown that the two estimators have different asymptotic properties. A simulation study compares the finite sample properties of both estimators showing that the proposed direct estimator is more stable (smaller MSE) than the two-stage estimator.