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.

Following up and correcting prior work by Teel, Bearden, and Sharma (1986) in this journal, a general approach to variance explained in latent dependent variables of nonrecursive linear structural equation models is given. A new method of its estimation, easily implemented in EQS or LISREL, is described and illustrated.

Problems about whether a hypothesized covariance structure models is an appropriate representation of the population covariance structure of multiple variables can be addressed using goodness-of-fit testing in structural equation modeling. Many test statistics and their extensions have been developed for various specific conditions and some of them have been extensively used in practice. However, their expected performances might break down under violations of multivariate normality or sufficiently large sample sizes. This paper evaluates the robustness of four modified goodness-of-fit test statistics T_SB(new), T_MV, T_YB and T_F in SEM. Monte Carlo simulation demonstrates that the robustness of covariance structure statistics vary as a function of the correctness of the model as well as distributional characteristics of observed data. Suggestions for application of these modified test statistics are given after taking both the literature and current simulation result into account. A surprising result was the failure of T_MV, the Satorra-Bentler mean-scaled and variance-adjusted test statistic, to perform correctly even asymptotically in one condition.

Donoho and Gavish(2013) proposed a method of recovering a matrix by selecting singular values above a hard threshold. In their paper, 4/√3 is proved to be asymptotic MSE-optimal choice of hard threshold. We empirically test the applicability of Donoho and Gavish's method in factor analysis with simulated datasets and assess its asymptotic property when both the matrix and sample size grow while keeping the true number of factors fixed.

The factor analytic model is usually an approximation that may not represent well the latent structure of a set of variables. This paper studies the distortion to common factor analysis due to doublets, pairwise associations in variables over and above those postulated by the factor model. Three methodologies to estimate the parameters of the factor model while minimizing the influence of doublets on the solution, i.e., methodologies that are resistant to the effects of doublets and near doublets, proposed by Yates(1987), Mulaik(2010) and Bentler(2012), are reviewed and compared. Examples and a simulation verify that these methodologies achieve their goal.