Although coefficient alpha is the most widely used measure of internal consistency, it does not optimally describe the unidimensional internal consistency of a composite. Coefficients based on a one-factor model have been suggested as improved estimators of internal consistency reliability. When the 1-factor model does not fit the data, however, the meaning of such a coefficient is unclear. A new identification condition for factor analytic models is proposed that assures the composite can be modeled with only one common factor. The associated 1-factor reliability is the maximal internal consistency coefficient for a unit-weighted composite. The coefficient also describes k-factor reliability, the greatest lower bound to reliability, and reliability for any composite from a latent variable model with additive errors.

Model close fit is one key issue in the mean and covariance structure analysis. In this article, we utilize the latest results on the general distribution of likelihood ratio statistic in this methodology and propose several distribution free root mean square error of approximation (RMSEA) tests for evaluating the hypothesis of close fit of misspecified models. Simulation studies show that three of these tests have robust and desirable performance in spite of severe nonnormality across the examples when sample size is as large as 300. A new two-stage procedure which combines model exact fit tests and the proposed RMSEA tests for model close fit is futher propose for overall model fit evaluation.

This dissertation extends Dijkstra's (2011) consistent partial least squares estimator for structural equation models by deriving new estimators that are efficient. The new methods allow formal testing of models via chi-square statistics and evaluation of parameter estimates by deriving standard error estimates, which are previously not directly available. Two approaches are developed: (1) PLSe1: a one-step improvement methodology based on PLSc-estimated factor loadings and TSLS-estimated structural parameters; (2) PLSe2: an optimal generalized least squares methodology using PLSc-implied covariance matrix. The performances of the proposed methods are evaluated by Monte Carlo simulations. We generated data under a non-recursive structural equation model. We investigated the performances of the proposed estimators relative to the classical Maximum Likelihood estimator under a variety of sample sizes for both normal data (with PLSe1 and PLSe2) and non-normal data (with PLSe1 only). The results indicate that the proposed estimators provide estimates that are almost as good as the theoretically optimal ML estimator under normality. We also demonstrate that the standard error estimates closely correspond to the empirical Monte Carlo variation. Under non-normality, PLSe1 performs favorably with non-normal (i.e. robust) adjustments to model fit statistics and standard errors. The standard error estimates are consistent with corresponding sampling variance and the robust model fit chi-squares statistics are well calibrated. In particular, Satorra-Bentler's (1994) scaled chi-square statistic stands out clearly.