Structural Equation Modeling With Unknown Population Distributions: Ridge Generalized Least Squares

Abstract

This article proposes 2 classes of ridge generalized least squares (GLS) procedures for structural equation modeling (SEM) with unknown population distributions. The weight matrix for the first class of ridge GLS is obtained by combining the sample fourth-order moment matrix with the identity matrix. The weight matrix for the second class is obtained by combining the sample fourth-order moment matrix with its diagonal matrix. Empirical results indicate that, with data from an unknown population distribution, parameter estimates by ridge GLS can be much more accurate than those by either GLS or normal-distribution-based maximum likelihood; and standard errors of the parameter estimates also become more accurate in predicting the empirical ones. Rescaled and adjusted statistics are proposed for overall model evaluation, and they also perform much better than the default statistic following from the GLS method. The use of the ridge GLS procedures is illustrated with a real data set.

Keywords:

• efficiency
• Monte Carlo
• rescaled statistics
• root mean square error
• sandwich-type covariance matrix
• type I errors

Notes

¹ There is a GLS procedure for SEM in which the weight matrix is estimated using S according to the normal distribution assumption. Like ML, this GLS method does not correctly account for the variability in S in general, and is not studied in this article.

² A random variable z following standardized can be obtained by with u being simulated from the distribution ; and can be obtained by with t being simulated from the distribution .

³ For all estimation methods, the criterion for convergence is defined as the maximum difference between consecutive values of all the parameters being less than 10⁻⁴. If convergence cannot be reached within 300 iterations of the Fisher-scoring algorithm, the replication is declared as failure to reach a convergence.

⁴ Four files (corresponding to the four distribution conditions) containing the RMSE, RD, and the average of the absolute bias across the 33 parameters, as well as the number of rejections by each statistic at each value of a can be downloaded at www2.nd.edu/~kyuan/ridgeGLS/. These files also contain the separate results of bias, RMSE, and RD averaged across the 15 factor loadings, three factor correlations, and 15 error variances, respectively.

Additional information

Funding

This research was supported by the National Science Foundation under Grant No. SES-1461355, and part of the work was conducted while Ke-Hai Yuan was visiting The Chinese University of Hong Kong.