Structural Equation Modeling

General Approach

A set of variables have an observed covariance matrix, which accounts for all the relationships among the variables. A structural regression model that estimates all covariance paths among the variables will reproduce the covariance matrix exactly, but is unlikely to resolve a theoretical understanding of how the variables relate. The aim of SEM is to define a parsimonious regression model that specifies theoretically consistent paths among variables. To the extent that model paths reflect established theory, the model will generally have good external validity.

Based on the specified regression model, a software algorithm will estimate a model-implied covariance matrix. This estimation can be done by hand following rules of path tracing for unstandardized or standardized parameter estimates. The estimation of model parameters includes correlation and regression coefficients for the specified paths, variances of exogenous variables, and residual variances of endogenous variables. Nonspecified paths are constrained, usually by default, to a value of 0, but the modeler may constrain paths to any value. With most real-world data sets, as the number of model constraints increases, there is increasing deviation of the implied covariance matrix from the observed covariance matrix. The residual covariance matrix shows this deviation. Thus, a further aim of SEM is to define a parsimonious model that reproduces the observed covariance matrix with minimal residuals. To the extent that residuals are minimized, the model will generally have statistical validity, also described as good model fit.

The results of SEM should thus be grounded solidly in theory and satisfy certain statistical requirements before they are interpreted. Assuming that study design, sampling, and data collection further abide rigorous standards, thoughtful interpretation of SEM results can contribute meaningfully to scholarship.