### Specifying the Full Measurement Model (H4) by the FACTOR Modeling Language: Lord Data

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 17.29:

Figure 17.29: Fit Summary of the Full Confirmatory Factor Model for Lord Data

Fit Summary
Chi-Square 0.7030
Chi-Square DF 1
Pr > Chi-Square 0.4018
Standardized RMR (SRMR) 0.0030
RMSEA Estimate 0.0000
Bentler Comparative Fit Index 1.0000

This is exactly the same fit summary as shown in Figure 17.18, 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 17.30.

Figure 17.30: Estimation Results of Full Confirmatory Factor Model for Lord Data

F1 F2
W
 7.5007 0.3234 23.1939 [_Parm1]
 0
X
 7.7027 0.3206 24.0235 [_Parm2]
 0
Y
 0
 8.5095 0.3269 26.0273 [_Parm3]
Z
 0
 8.6751 0.3256 26.6430 [_Parm4]

Factor Covariance Matrix: Estimate/StdErr/t-value
F1 F2
F1
 1.0000
 0.8986 0.0186 48.1800 [_Parm9]
F2
 0.8986 0.0186 48.1800 [_Parm9]
 1.0000

Error Variances
Variable Parameter Estimate Standard
Error
t Value
W _Parm5 30.13796 2.47037 12.19979
X _Parm6 26.93217 2.43065 11.08021
Y _Parm7 24.87396 2.35986 10.54044
Z _Parm8 22.56264 2.35028 9.60000

Again, these are the same estimates as those shown in Figure 17.19, which is for the PATH model specification. The FACTOR results displayed in Figure 17.30 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.