In the previous chapter, we went over the concepts of path coefficients and covariance algebra. In reality, these terms, though used for exploratory factor analysis, come from the tradition of SEM. Exploratory factor analysis (EFA) simply attempted to model covariance structure based on identifying common sources of variance. Alternatively, SEM attempts to use covariance to model many, very explicit relationships between variables. Like EFA, SEM can incorporate both observed and unobserved variables, but unlike EFA, SEM does not necessarily need to have unobserved variables. In SEM, the relationships between variables can be represented as a series of paths, whether those variables are observed or latent. The correlation between any two variables is a path coefficient. Each observed variable will also have some residual correlation, and residual correlations may be correlated with one another, something that is not allowed in EFA.
The following is a list of the components of an SEM model:
- Observed variable: This is a variable for which we directly collect data that goes into model estimation.
- Latent variable: This is a variable that we do not observe but theorize to exist and behave in a certain way. We use observed data and theorized model constraints to estimate quantitative relationships between observed and unobserved variables, and possibly relationships among unobserved variables themselves.
- Path: This is a theorized relationship between two variables. Paths are thought of as going from a causal variable to a caused variable. In some cases, a path can go in both the directions.
- Residual: This is the portion of the data not explained by the causal paths in the model.
- Covariance algebra: These are a set of mathematical tricks describing how covariances behave algebraically. They are as follows:
In the previous chapter, we described the common factor that the model has relying on the relationship in the following figure:
In the preceding formula, F is a common underlying factor. This is a really special case. SEM allows for an extension of this basic idea to more complex models in which the observed variables can themselves be connected as we will see further in this chapter. This also calls for different estimation methods.
The most typical method to represent an SEM model is with a path diagram. These diagrams use boxes and arrows to represent proposed causal relationships and a set of standard graphical elements have been proposed to represent models.
- Observed variable: This is represented with a rectangular box.
- Latent variable: This is represented with an elliptical box.
- Causal path: This is represented with a straight arrow. The arrow points from causal to effect. A double-headed arrow is also allowed.
- Residual or correlation: This is represented with a two-headed curved arrow. If both arrow heads point to the same variable, it is a residual variance. If they point to two different variables, they show a correlated error term. See the figures related to the political economy of SEM as an example in the next sections.