Example 9: Testing for Stability of a Correlation Matrix over Time
Suppose you measure a set of variables twice, and want to test the hypothesis
that the correlation coefficient has not changed from time 1 to time 2.
For example, suppose 120 individuals are measured twice on verbal, quantitative,
and analytical ability. In this case, the covariance matrix would be 6x6.
A 6x6 correlation matrix is given in the Twocorr.sta
data file.

Select Structural
Equation Modeling from the Statistics
- Advanced Linear\Nonlinear Models menu.
In the resulting Structural
Equation Modeling dialog click on the Advanced
tab.
Here, we modify the general approach to setting up a pattern hypothesis
described above in order to compare correlations measured at two times.
(1) Click the Set parameters
button and select Correlations
under Data to Analyze in the
resulting of the Analysis
Parameters dialog and click OK..

(2) Click the Path Tool button
to display the Path
Construction Tool dialog. This is used to create all the paths
for a correlation matrix for the relevant variables. Select Correl.
under Path Type, then highlight
all 6 of the manifest variables in the To:
column.

(3) Next, click the Add>> button.
This action creates all the paths for the full correlation matrix for
the 6 variables.
(4) You want to constrain the correlations between the variables at
time two to be the same as those at time 1. Notice first that there are
3 correlations between the 3 variables at time 1, and they are assigned
parameter numbers 1-3. Highlight the corresponding 3 correlations at time
2 by clicking on them while holding down the CTRL key.
(5) Set the Renumber box to
renumber these 3 correlations starting at 1.

(6) Next, highlight all the paths and renumber them, starting at 1.
(7) Click OK.
Your final model should look like this.

The commands for performing the hypothesis test are in a file called
Twocorr.cmd (/Examples/Datasets/SEPATH
directory of STATISTICA). Part
of this file is shown below in report
format. (You can run this model by clicking the Open
model button on the Advanced
tab of the Structural
Equation Modeling dialog, choosing this file, and clicking
the OK (Run model) button.) The
estimated correlations among the alias latent variables are maximum likelihood
estimates of the correlations among the manifest variables, under the
null hypothesis.

A note of caution should be added here. The tests performed by SEPATH
on correlation matrices are valid asymptotic statistics. However, the
very limited evidence available at this time suggests that (especially
at smaller sample sizes) tests especially designed for correlational hypotheses
(Steiger 1980a, 1980b; Steiger & Browne, 1984; Steiger & Hakstian,
1982; Wilson & Martin, 1983), which use the Fisher transform, may
be somewhat more accurate and more powerful than tests performed with
SEPATH.