Factor models: Evaluation and improvement

Factor analytic models use a variance-covariance matrix to estimate underlying factors through the equation Sigma = Lambda Phi Lambda' + Psi . In order to provide estimates for the matrices Lambda, Phi, and Psi, some of the parameters need to be restricted and at least k2 restrictions are required to provide a unique estimate (Joreskog, 1969). The most commonly specified models place restrictions on Phi or Lambda.;The estimation of the specified models, which make restrictions on Lambda or Phi, was compared and evaluated. Several different Lambda and Phi conditions were used to assess whether the specified models can reproduce these target Lambda and Phi conditions. The estimation results indicated that when the Lambda and Phi conditions matched the restrictions imposed by the specified models, the models performed well in reproducing the target values. However, when the conditions did not match the imposed restrictions the specified models performed poorly. In practice, a researcher must assume "a priori" knowledge of the restrictions to be imposed on Lambda and Phi, and hence, would not really know if the estimation results are correct or not.;Several measures of fit (fit statistics) are commonly used to assess factor analytic models and were evaluated using a simulation. The simulation results indicated in general that fit statistics have no consistent relationship with the estimation quality of the specified models. In particular, the fit statistics could not identify when the estimated correlation between the factors was incorrect.;In an attempt to provide better estimation of factor analytic models, a new model was developed which places no restrictions on Lambda or Phi. This was accomplished by expanding Lambda by a set of known constants which then could be appropriately restricted, thus satisfying Joreskog's requirement. The estimation based on this model performed better than the other methods considered and appears to be capable of estimating Lambda and Phi of any possible form.