# Introduction to Structural Equation Modeling with Latent Variables

### H1: One-Factor Model with Parallel Tests for Lord Data

The path diagram for the one-factor model with parallel tests is shown in Figure 17.29.

Figure 17.29: H1: One-Factor Model with Parallel Tests for Lord Data

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 double-headed path that connects F1 and F2 in Figure 17.29. Again, you need only minimal modification of the preceding specification for to specify the path diagram in Figure 17.29, 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 17.30.

Figure 17.30: Fit Summary, H1: One-Factor Model with Parallel Tests for Lord Data

Fit Summary
Chi-Square 37.3337
Chi-Square DF 6
Pr > Chi-Square <.0001
Standardized RMR (SRMR) 0.0286
RMSEA Estimate 0.0898
Bentler Comparative Fit Index 0.9785

The chi-square 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 17.31.

Figure 17.31: Estimation Results, H1: One-Factor Model with Parallel Tests for Lord Data

PATH List
Path Parameter Estimate Standard
Error
t Value Pr > |t|
W <=== F1 beta1 7.18623 0.26598 27.0180 <.0001
X <=== F1 beta1 7.18623 0.26598 27.0180 <.0001
Y <=== F2 beta2 8.44198 0.28000 30.1494 <.0001
Z <=== F2 beta2 8.44198 0.28000 30.1494 <.0001

Variance Parameters
Variance
Type
Variable Parameter Estimate Standard
Error
t Value Pr > |t|
Exogenous F1   1.00000
F2   1.00000
Error W theta1 34.68865 1.64634 21.0701 <.0001
X theta1 34.68865 1.64634 21.0701 <.0001
Y theta2 26.28513 1.39955 18.7812 <.0001
Z theta2 26.28513 1.39955 18.7812 <.0001

Covariances Among Exogenous Variables
Var1 Var2 Estimate Standard
Error
t Value Pr > |t|
F1 F2 1.00000

The goodness-of-fit tests for the four hypotheses are summarized in the following table.

Number of

Degrees of

Hypothesis

Parameters

Freedom

p-value

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 one-factor models), both hypotheses were rejected. Hypotheses and (both two-factor 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 chi-square statistics and their degrees of freedom, yielding a chi-square of 1.23 with four degrees of freedom, which is obviously not significant. In a sense, the chi-square 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 two-factor model with parallel tests provides the best explanation of the data.