Introduction to structural equation modeling

Abstract

This article presents a short and non-technical introduction to Structural Equation Modeling or SEM. SEM is a powerful technique that can combine complex path models with latent variables (factors). Using SEM, researchers can specify confirmatory factor analysis models, regression models, and complex path models. We present the basic elements of a structural equation model, introduce the estimation technique, which is most often maximum Likelihood (ML), and discuss some problems concerning the assessment and improvement of the model fit, and model extensions to multigroup problems including factor means. Finally, we discuss some of the software, and list useful handbooks and Internet sites. What is Structural Equation Modeling? Structural Equation Modeling, or SEM, is a very general statistical modeling technique, which is widely used in the behavioral sciences. It can be viewed as a combination of factor analysis and regression or path analysis. The interest in SEM is often on theoretical constructs, which are represented by the latent factors. The relationships between the theoretical constructs are represented by regression or path coefficients between the factors. The structural equation model implies a structure for the covariances between the observed variables, which provides the alternative name covariance structure modeling. However, the model can be extended to include means of observed variables or factors in the model, which makes covariance structure modeling a less accurate name. Many researchers will simply think of these models as ‘Lisrel-models,’ which is also less accurate. LISREL is an abbreviation of LInear Structural RELations, and the name used by Joreskog for one of the first and most popular SEM programs. Nowadays structural equation models need not be linear, and the possibilities of SEM extend well beyond the original Lisrel program. Browne (1993), for instance, discusses the possibility to fit nonlinear curves. Structural equation modeling provides a very general and convenient framework for statistical analysis that includes several traditional multivariate procedures, for example factor analysis, regression analysis, discriminant analysis, and canonical correlation, as special cases. Structural equation models are often visualized by a graphical path diagram. The statistical model is usually represented in a set of matrix equations. In the early seventies, when this technique was first introduced in social and behavioral research, the software usually required setups that specify the model in terms of these matrices. Thus, researchers had to distill the matrix representation from the path diagram, and provide the software with a series of matrices for the different sets of 1 Note: The authors thank Alexander Vazsonyi and three anonymous reviewers for their comments on a previous version. We thank Annemarie Meijer for her permission to use the quality of sleep data. Introduction Structural Equation Modeling 2 parameters, such as factor loadings and regression coefficients. A recent development is software that allows the researchers to specify the model directly as a path diagram. This works well with simple problems, but may get tedious with more complicated models. For that reason, current SEM software still supports the commandor matrix-style model specifications too. This review provides a brief and non-technical review of the basic issues involved in SEM, including issues of estimation, model fit, and statistical assumptions. We include a list of available software, introductory books, and useful Internet resources. Examples of SEM-Models In this section, we set the stage by discussing examples of a confirmatory factor analysis, regression analysis, and a general structural equation model with latent variables. Structural equation modeling has its roots in path analysis, which was invented by the geneticist Sewall Wright (Wright, 1921). It is still customary to start a SEM analysis by drawing a path diagram. A path diagram consists of boxes and circles, which are connected by arrows. In Wright’s notation, observed (or measured) variables are represented by a rectangle or square box, and latent (or unmeasured) factors by a circle or ellipse. Single headed arrows or ‘paths’ are used to define causal relationships in the model, with the variable at the tail of the arrow causing the variable at the point. Double headed arrows indicate covariances or correlations, without a causal interpretation. Statistically, the single headed arrows or paths represent regression coefficients, and double-headed arrows covariances. Extensions of this notation have been developed to represent variances and means (cf. McArdle, 1996). The first example in Figure 1 is a representation of a confirmatory factor analysis model, with six observed variables and