Introduction to Structural Equation Modeling with Latent Variables

The measurement models described in the section Some Measurement Models are also known as confirmatory factor models. PROC CALIS has a specific modeling language, called FACTOR, for confirmatory factor models. You can use this modeling language for both exploratory and confirmatory factor analysis.

For example, the full measurement model H4 in the section H4: Full Measurement Model for Lord Data can be specified equivalently by the FACTOR modeling language with the following statements:

proc calis data=lord;
   factor
      F1  ===>  W  X,
      F2  ===>  Y  Z;
   pvar
      F1 = 1.0,
      F2 = 1.0,
      W X Y Z;
   cov
      F1 F2;
run;

In the specification, you use the FACTOR statement to invoke the FACTOR modeling language. In the FACTOR statement, you specify the paths from the latent factors to the measurement indicators. For example, F1 has two paths to its indicators, W and X. Similarly, F2 has two paths to its indicators, Y and Z. Next, you use the PVAR statement to specify the variances, which is exactly the same way you use the PATH model specification in the section H4: Full Measurement Model for Lord Data. Lastly, you use the COV statement to specify the covariance among the factors, much like you use the PCOV statement to specify the same covariance in the PATH model specification.

Given the same confirmatory factor model, there is a major difference between the paths specified by the PATH statement and the paths specified by the FACTOR statement. In the FACTOR statement, each path must start with a latent factor followed by a right arrow and the variable list. In the PATH statement, each path can start or end with an observed or latent variable, and the direction of the arrow can be left or right.

The fit summary table for the FACTOR model is shown in Figure 32:

This is exactly the same fit summary as shown in Figure 21, which is for the PATH model specification. Therefore, this confirms that the same model is being fit by the FACTOR model specification.

The estimation results are shown in Figure 33.

Again, these are the same estimates as those shown in Figure 22, which is for the PATH model specification. The FACTOR results displayed in Figure 33 are arranged differently though. No paths are shown there. The relationships between the latent factors and its indicators are shown in matrix form. The factor variance and covariances are also shown in matrix form.