The path diagram for the onefactor model with parallel tests is shown in Figure 17.26.
The hypothesis differs from in that F1
and F2
have a perfect correlation in . This is indicated by the fixed value 1.0 for the doubleheaded path that connects F1
and F2
in Figure 17.26. Again, you need only minimal modification of the preceding specification for to specify the path diagram in Figure 17.26, as shown in the following statements:
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 = 1.0; run;
The only modification of the preceding specification is in the PCOV statement, where you put a constant 1 for the covariance
between F1
and F2
. An annotated fit summary is displayed in Figure 17.27.
Figure 17.27: Fit Summary, H1: OneFactor Model with Parallel Tests for Lord Data
Fit Summary  

ChiSquare  37.3337 
ChiSquare DF  6 
Pr > ChiSquare  <.0001 
Standardized RMR (SRMR)  0.0286 
Adjusted GFI (AGFI)  0.9509 
RMSEA Estimate  0.0898 
Bentler Comparative Fit Index  0.9785 
The chisquare value is 37.3337 (df=6, p<0.0001). This indicates that you can reject the hypothesized model H1 at the 0.01 level. The standardized root mean square error (SRMSR) is 0.0286, the adjusted GFI (AGFI) is 0.9509, and Bentler’s comparative fit index is 0.9785. All these indicate good model fit. However, the RMSEA is 0.0898, which does not support an acceptable model for the data.
The estimation results are displayed in Figure 17.28.
Figure 17.28: Estimation Results, H1: OneFactor Model with Parallel Tests for Lord Data
PATH List  

Path  Parameter  Estimate  Standard Error 
t Value  
W  <===  F1  beta1  7.18623  0.26598  27.01802 
X  <===  F1  beta1  7.18623  0.26598  27.01802 
Y  <===  F2  beta2  8.44198  0.28000  30.14943 
Z  <===  F2  beta2  8.44198  0.28000  30.14943 
Variance Parameters  

Variance Type 
Variable  Parameter  Estimate  Standard Error 
t Value 
Exogenous  F1  1.00000  
F2  1.00000  
Error  W  theta1  34.68865  1.64634  21.07010 
X  theta1  34.68865  1.64634  21.07010  
Y  theta2  26.28513  1.39955  18.78119  
Z  theta2  26.28513  1.39955  18.78119 
Covariances Among Exogenous Variables  

Var1  Var2  Estimate  Standard Error 
t Value 
F1  F2  1.00000 
The goodnessoffit tests for the four hypotheses are summarized in the following table.
Number of 
Degrees of 


Hypothesis 
Parameters 

Freedom 
pvalue 


4 
37.33 
6 
< .0001 
1.0 

5 
1.93 
5 
0.8583 
0.8986 

8 
36.21 
2 
< .0001 
1.0 

9 
0.70 
1 
0.4018 
0.8986 
Recall that the estimates of for and are almost identical, about 0.90, indicating that the speeded and unspeeded tests are measuring almost the same latent variable. However, when was set to 1 in and (both onefactor models), both hypotheses were rejected. Hypotheses and (both twofactor models) seem to be consistent with the data. Since is obtained by adding four constraints (for the requirement of parallel tests) to (the full model), you can test versus by computing the differences of the chisquare statistics and their degrees of freedom, yielding a chisquare of 1.23 with four degrees of freedom, which is obviously not significant. In a sense, the chisquare difference test means that representing the data by would not be significantly worse than representing the data by . In addition, because offers a more precise description of the data (with the assumption of parallel tests) than , it should be chosen because of its simplicity. In conclusion, the twofactor model with parallel tests provides the best explanation of the data.