UCLA

Estimation methods employed in Structural Equation Modeling (SEM) depend on asymptotic theory. When assumptions are violated (e.g., sample sizes are not especially large relative to the number of variables), methods break down, and conclusions are dubious. It has been suggested that ill-conditioned matrices contribute to poor performance (Huang & Bentler, 2015; Yuan & Bentler, 2017). In the present investigation, a Maximum a Posteriori (MAP) estimator was proposed and implemented for two matrices common to SEM to address conditioning: the sample covariance matrix and the asymptotic covariance matrix based on fourth order moments. This MAP estimator improves the condition of matrices by pushing down (pulling up) the over (under) estimated sample eigenvalues of poorly conditioned matrices, and better-conditioned matrices were expected to improve solution propriety as well as global model fit. Three differing implementations were proposed for Generalized Least Squares estimation methods (GLS and ADF) as well as correction methods to Maximum Likelihood solutions. Potential advantages of the approaches were evaluated using three Monte Carlo simulation studies across a wide range of sample sizes and estimation methods. The results reveal overall solution propriety is improved, and regularization when applied directly to weight matrices is more effective than indirect application (i.e., by modifying an input matrix or using correction methods). Moreover, results were dramatically improved for normal theory GLS even at samples sizes as small as N = 60 and greatly improved for ADF/RES methods at samples as small as N = 150. Generally, advantages did not carry over to non-normal conditions. Potential reasons for this result are given as well as prospective solutions. An illustrative example demonstrates the use of regularized GLS with real-world data.