# Introduction to Structural Equation Modeling with Latent Variables

### Testing Built-In Covariance Patterns in PROC CALIS

Some covariance patterns are well-known in multivariate statistics. For example, testing the diagonal pattern for a covariance matrix in the preceding section is a test of uncorrelatedness between the observed variables. Under the multivariate normal assumption, this test is also a test of independence between the observed variables. This test of independence is routinely applied in maximum likelihood factor analysis for testing the zero common factor hypothesis for the observed variables. For testing such a well-known covariance pattern, PROC CALIS provides an efficient way of specifying a model. With the COVPATTERN= option, you can invoke the built-in covariance patterns in PROC CALIS without the MSTRUCT model specifications, which could become laborious when the number of variables are large.

For example, to test the diagonal pattern (uncorrelatedness) of the motor skills, you can simply use the following specification:

```proc calis data=motor covpattern=uncorr;
run;
```

The COVPATTERN=UNCORR option in the PROC CALIS statement invokes the diagonally patterned covariance matrix for the motor skills. PROC CALIS then generates the appropriate free parameters for this built-in covariance pattern. As a result, the MATRIX statement is not needed for specifying the free parameters, as it is if you use explicit MSTRUCT model specifications. Some of the output for using the COVPATTERN= option is shown in Figure 17.3.

Figure 17.3: Fit Summary: Testing Uncorrelatedness with the COVPATTERN= Option

Fit Summary
Chi-Square 8.8071
Chi-Square DF 3
Pr > Chi-Square 0.0320

MSTRUCT _COV_ Matrix: Estimate/StdErr/t-value/p-value
x1 x2 x3
x1
 3.5660 0.8524 4.1833 <.0001 [_varparm_1]
 0
 0
x2
 0
 4.0120 0.9591 4.1833 <.0001 [_varparm_2]
 0
x3
 0
 0
 3.7760 0.9026 4.1833 <.0001 [_varparm_3]

In the second table of Figure 17.3, the estimates of variances and their standard errors are the same as those shown in Figure 17.2. The only difference is that the parameter names (for example, `_varparm_1`) for the variances in Figure 17.3 are generated by PROC CALIS, instead of being specified as those in Figure 17.2.

However, the current chi-square test for the model fit is 8.8071 (df=3, p=0.0320), which is different from that in Figure 17.2 for testing the same covariance pattern. The reason is that the chi-square correction due to Bartlett (1950) has been applied automatically to the current built-in covariance pattern testing. Theoretically, this corrected chi-square value is more accurate. Therefore, in addition to its efficiency in specification, the built-in covariance pattern with the COVPATTERN= option offers an extra advantage in the automatic chi-square correction.

The COVPATTERN= option supports many other built-in covariance patterns. For details, see the COVPATTERN= option. See also the MEANPATTERN= option for testing built-in mean patterns.