Structural Equation Modeling as a Statistical Method: An Overview

Important attributes in many disciplines cannot be observed directly, and have to be measured by multiple indicators that are subject to errors. Structural equation modeling (SEM) has been a major tool for understanding the relationships between latent attributes and their observed indicators as well as those among latent attributes. This article provides a critical review of the statistical development of SEM methodology. Pros and cons of each method are noted, and the need for further development is pointed out. In particular, there is no effective procedure for SEM with small N and/ or large p, together with incomplete data from an unknown population distribution. ABBREVIATION AOSEM: An Overview of Structural Equation Modeling. INTRODUCTION In social, behavioral, education and health sciences, important attributes are often latent variables that cannot be observed directly, and have to be inferred from fallible measures. Their analyses are most effectively done by structural equation modeling (SEM). In contrast to statistical methods in traditional multivariate analysis, SEM has the mechanism of modeling manifest variables, latent variables, as well as measurement errors simultaneously. Although not familiar to many statisticians, SEM has become one of the most important methodology in many disciplines when analyzing survey or nonand quasi-experimental data. The journal, Structural Equation Modeling, launched in 1994, is consistently ranked as having the highest impact among all statistical and mathematicalmethod journals (http://en.wikipedia.org/wiki/Comparison_of_ statistics_journals). A SEM model can be equivalently represented as a mean and covariance structure model, and many commonly used statistical models are its special cases. The confirmatory factor model is among most widely used SEM models [1-2]. Other popular SEM models include the latent growth curve model [3-4], multitraitmultimethod model [2], and multiple-indicator and multiplecause model [2,5]. In particular, regression, MANOVA, and path analysis are also special cases of SEM models [6-7]. Commercial software for SEM includes AMOS, EQS, LISREL, MPLUS, and SAS CALIS; and free software includes Mx, sem and lavaan in R, and WebSEM. The purpose of this article is to provide an overview of SEM methodology, include estimation methods, test statistics, and different formulations of standard errors of parameter estimates. Pros and cons of each method are noted, and the need for further research is pointed out. ESTIMATION METHODS Like in any other disciplines of statistics, estimation methods in SEM were mostly developed for obtaining more efficient parameter estimates, catering for different types of data and/ or distributions in practice. Among all the methods of SEM, the normal-distribution-based maximum likelihood (NML) [8] is most widely used. Let x and S be the sample means and sample covariance matrix, and the corresponding mean and covariance structure model be ( ) μ θ and ( ) θ Σ . The NML method obtains parameter estimate θ̂ by minimizing the discrepancy function