# Introduction to Structural Equation Modeling with Latent Variables

### Specifying the Parallel Tests Model (H2) by the FACTOR Modeling Language: Lord Data

In the section H2: Two-Factor Model with Parallel Tests for Lord Data, you fit a two-factor model with parallel tests for the Lord data by the PATH modeling language in PROC CALIS. Some paths and error variance are constrained under the PATH model. You can also specify this parallel tests model by the FACTOR modeling language, as shown in the following statements:

```proc calis data=lord;
factor
F1  ===>  W  X    = 2 * beta1,
F2  ===>  Y  Z    = 2 * beta2;
pvar
F1  = 1.0,
F2  = 1.0,
W X = 2 * theta1,
Y Z = 2 * theta2;
cov
F1 F2;
run;
```

In this specification, you specify some parameters explicitly. You apply the parameter `beta1` to the loadings of both `W` and `X` on `F1`. This means that `F1` has the same amount of effect on `W` and `X`. Similarly, you apply the parameter `beta2` to the loadings of `Y` and `Z` on `F2`. The constraints on the error variances for `W`, `X`, `Y`, and `Z` in this FACTOR model specification are done in the same way as in the PATH model specification in the section H2: Two-Factor Model with Parallel Tests for Lord Data.

The fit summary table for this parallel tests model is shown in Figure 17.34.

Figure 17.34: Fit Summary of the Confirmatory Factor Model with Parallel Tests for Lord Data

Fit Summary
Chi-Square 1.9335
Chi-Square DF 5
Pr > Chi-Square 0.8583
Standardized RMR (SRMR) 0.0076
RMSEA Estimate 0.0000
Bentler Comparative Fit Index 1.0000

All the fit indices shown in Figure 17.34 for the FACTOR model match the corresponding PATH model results displayed in Figure 17.27. All the estimation results in Figure 17.35 for the FACTOR model are the same as those for the corresponding PATH model in Figure 17.28.

Figure 17.35: Estimation Results of the Confirmatory Factor Model with Parallel Tests for Lord Data

F1 F2
W
 7.6010 0.2684 28.3158 <.0001 [beta1]
 0
X
 7.6010 0.2684 28.3158 <.0001 [beta1]
 0
Y
 0
 8.5919 0.2797 30.7215 <.0001 [beta2]
Z
 0
 8.5919 0.2797 30.7215 <.0001 [beta2]

Factor Covariance Matrix: Estimate/StdErr/t-value/p-value
F1 F2
F1
 1.0000
 0.8986 0.0187 48.1801 <.0001 [_Parm1]
F2
 0.8986 0.0187 48.1801 <.0001 [_Parm1]
 1.0000

Error Variances
Variable Parameter Estimate Standard
Error
t Value Pr > |t|
W theta1 28.55545 1.58641 18.0000 <.0001
X theta1 28.55545 1.58641 18.0000 <.0001
Y theta2 23.73200 1.31844 18.0000 <.0001
Z theta2 23.73200 1.31844 18.0000 <.0001