## Overview of Structural Equation Modeling with Latent Variables

Structural equation modeling includes analysis of covariance structures and mean structures, fitting systems of linear structural equations, factor analysis, and path analysis. In terms of the mathematical and statistical techniques involved, these various types of analyses are more or less interchangeable because the underlying methodology is based on analyzing the mean and covariance structures. However, the different analysis types emphasize different aspects of the analysis.

The analysis of covariance structures refers to the formulation of a model for the observed variances and covariances among a set of variables. The model expresses the variances and covariances as functions of some basic parameters. Similarly, the analysis of mean structures refers to the formulation of a model for the observed means. The model expresses the means as functions of some basic parameters. Usually, the covariance structures are of primary interest. However, sometimes the mean structures are analyzed simultaneously with the covariance structures in a model.

Corresponding to this kind of abstract formulation of mean and covariance structure analysis, PROC CALIS offers you two matrix-based modeling languages for specifying your model:

• MSTRUCT: a matrix-based model specification language that enables you to directly specify the parameters in the covariance and mean model matrices

• COSAN: a general matrix-based model specification language that enables you to specify a very wide class of mean and covariance structure models in terms of matrix expressions

Instead of focusing directly on the mean and covariance structures, other generic types of structural equation modeling emphasize more about the functional relationships among variables. Mean and covariance structures are still the means of these analyses, but they are usually implied from the structural relationships, rather than being directly specified as in the COSAN or MSTRUCT modeling languages.

In linear structural equations, the model is formulated as a system of equations that relates several random variables with assumptions about the variances and covariances of the random variables. The variables involved in the system of linear structural equations could be observed (manifest) or latent. Causal relationships between variables are hypothesized in the model.

When all observed variables in the model are hypothesized as indicator measures of underlying latent factors and the main interest is about studying the structural relations among the latent factors, it is a modeling scenario for factor-analysis or LISREL (Keesling, 1972; Wiley, 1973; Jöreskog, 1973). PROC CALIS provides you two modeling languages that are closely related to this type of modeling scenario:

• FACTOR: a non-matrix-based model specification language that supports both exploratory and confirmatory factor analysis, including orthogonal and oblique factor rotations

• LISMOD: a matrix-based model specification language that enables you to specify the parameters in the LISREL model matrices

When causal relationships among observed and latent variables are freely hypothesized so that the observed variables are not limited to the roles of being measured indicators of latent factors, it is a modeling scenario for general path modeling (path analysis). In general path modeling, the model is formulated as a path diagram, in which arrows that connect variables represent variances, covariances, and path coefficients (effects). Depending on the way you represent the path diagram, you can use any of the following three different modeling languages in PROC CALIS:

• PATH: a non-matrix-based language that enables you to specify path-like relationships among variables

• RAM: a matrix-based language that enables you to specify the paths, variances, and covariance parameters in terms of the RAM model matrices (McArdle and McDonald, 1984)

• LINEQS: an equation-based language that uses linear equations to specify functional or path relationships among variables (for example, the EQS model by Bentler 1995)

Although various types of analyses are put into distinct classes (with distinct modeling languages), with careful parameterization and model specification, it is possible to apply any of these modeling languages to the same analysis. For example, you can use the PATH modeling language to specify a confirmatory factor-analysis model, or you can use the LISMOD modeling language to specify a general path model. However, for some situations some modeling languages are easier to use than others. See the section Which Modeling Language? of Chapter 27: The CALIS Procedure, for a detailed discussion of the modeling languages supported in PROC CALIS.

Loehlin (1987) provides an excellent introduction to latent variable models by using path diagrams and structural equations. A more advanced treatment of structural equation models with latent variables is given by Bollen (1989). Fuller (1987) provides a highly technical statistical treatment of measurement-error models.

This chapter illustrates applications of PROC CALIS, describes some of the main modeling features of PROC CALIS, and compares the CALIS procedure with the FACTOR and the SYSLIN procedures.