Introduction to Structural Equation Modeling with Latent Variables

H2: Two-Factor Model with Parallel Tests for Lord Data

The path diagram for the two-factor model with parallel tests is shown in Figure 17.26.

Figure 17.26: H2: Two-Factor Model with Parallel Tests for Lord Data

The hypothesis $H_{2}$ requires that variables or tests under each factor are "interchangeable." In terms of the measurement model, several pairs of parameters must be constrained to have equal estimates. That is, under the parallel-test model W and X should have the same effect or path coefficient $\beta _1$ from their common factor F1, and they should also have the same measurement error variance $\theta _1$ . Similarly, Y and Z should have the same effect or path coefficient $\beta _2$ from their common factor F2, and they should also have the same measurement error variance $\theta _2$ . These constraints are labeled in Figure 17.26.

You can impose each of these equality constraints by giving the same name for the parameters involved in the PATH model specification. The following statements specify the path diagram in Figure 17.26:

proc calis data=lord;
   path
      W <=== F1   = beta1,
      X <=== F1   = beta1,
      Y <=== F2   = beta2,
      Z <=== F2   = beta2;
   pvar
      F1  = 1.0,
      F2  = 1.0,
      W X = 2 * theta1,
      Y Z = 2 * theta2;
   pcov
      F1 F2;
run;

Note that the specification 2*theta1 in the PVAR statement means that theta1 is specified twice for the error variances of the two variables W and X. Similarly for the specification 2*theta2. An annotated fit summary is displayed in Figure 17.27.

Figure 17.27: Fit Summary, H2: Two-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
Adjusted GFI (AGFI)	0.9970
RMSEA Estimate	0.0000
Bentler Comparative Fit Index	1.0000

The chi-square value is 1.9335 (df=5, p=0.8583). This indicates that you cannot reject the hypothesized model H2. The standardized root mean square error (SRMSR) is 0.0076, which indicates a very good fit. Bentler’s comparative fit index is 1.0000. The adjusted GFI (AGFI) is 0.9970, and the RMSEA is close to zero. All results indicate that this is a good model for the data.

The estimation results are displayed in Figure 17.28.

Figure 17.28: Estimation Results, H2: Two-Factor Model with Parallel Tests for Lord Data

PATH List
Path			Parameter	Estimate	Standard Error	t Value	Pr > \|t\|
W	<===	F1	beta1	7.60099	0.26844	28.3158	<.0001
X	<===	F1	beta1	7.60099	0.26844	28.3158	<.0001
Y	<===	F2	beta2	8.59186	0.27967	30.7215	<.0001
Z	<===	F2	beta2	8.59186	0.27967	30.7215	<.0001

Variance Parameters
Variance Type	Variable	Parameter	Estimate	Standard Error	t Value	Pr > \|t\|
Exogenous	F1		1.00000
	F2		1.00000
Error	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

Covariances Among Exogenous Variables
Var1	Var2	Parameter	Estimate	Standard Error	t Value	Pr > \|t\|
F1	F2	_Parm1	0.89864	0.01865	48.1801	<.0001

Notice that because you explicitly specify the parameter names for the path coefficients (that is, beta1 and beta2), they are used in the output shown in Figure 17.28. The correlation between F1 and F2 is 0.8987, which is a very high correlation that suggests F1 and F2 might have been the same factor in the population. The next section sets this value to one so that the current model becomes a one-factor model with parallel tests.