### Example 27.2 Estimating Covariances and Means Simultaneously

This example uses the same data set that is used in Example 27.1 and estimates the means and covariances. Use the MSTRUCT model specification as shown in the following statements:

```proc calis data=sales meanstr nostand;
mstruct var=q1-q4;
run;
```

In the PROC CALIS statement, specify the MEANSTR option to request the mean structure analysis in addition to the default covariance structure analysis. If you are not interested in the standardized solution, specify the NOSTAND option in the PROC CALIS statement to suppress computation of the standardized estimates. Without further model specification (such as the MATRIX statement), PROC CALIS assumes a saturated structural model with all means, variances, and covariances as model parameters.

Output 27.2.1 shows the modeling information. With the MEANSTR option specified in the PROC CALIS statement, the current analysis type is Means and Covariances, instead of the default Covariances in Example 27.1.

Output 27.2.1: Modeling Information of the Saturated Mean and Covariance Structure Model for the `Sales` Data

 Saturated Means and Covariance Structures Using MSTRUCT

The CALIS Procedure
Mean and Covariance Structures: Model and Initial Values

Modeling Information
Maximum Likelihood Estimation
Data Set WORK.SALES
N Records Used 14
N Obs 14
Model Type MSTRUCT
Analysis Means and Covariances

Variables in the Model
q1 q2 q3 q4

Output 27.2.2 shows the fit summary of the current model. Again, this is a perfect model fit with 0 chi-square value and 0 degrees of freedom.

Output 27.2.2: Fit Summary of the Saturated Mean and Covariance Structure Model for the `Sales` Data

Fit Summary
Chi-Square 0.0000
Chi-Square DF 0
Pr > Chi-Square .

Output 27.2.3 shows the estimates of the means, together with the standard error estimates and the t values. These estimated means are exactly the same as the sample means, which are not shown here.

Output 27.2.3: Mean Estimates for the `Sales` Data

MSTRUCT _Mean_ Vector
Variable Estimate Standard
Error
t Value
q1 1.36714 0.16132 8.47491
q2 2.07429 0.13146 15.77902
q3 2.20286 0.21596 10.20008
q4 3.65500 0.45281 8.07176

Output 27.2.4 shows the variance and covariance estimates. These estimates are exactly the same as the elements in the sample covariance matrix. In addition, these estimates match the estimates in Output 27.1.5 of Example 27.1, where only the covariance structures are analyzed.

Output 27.2.4: Variance and Covariance Estimates for the `Sales` Data

MSTRUCT _COV_ Matrix: Estimate/StdErr/t-value
q1 q2 q3 q4
q1
 0.3383 0.1327 2.5495
 0.000198 0.0765 0.002587
 0.0361 0.126 0.2865
 0.2214 0.2704 0.8186
q2
 0.000198 0.0765 0.002587
 0.2247 0.0881 2.5495
 0.1265 0.1082 1.1693
 0.2443 0.2251 1.0853
q3
 0.0361 0.126 0.2865
 0.1265 0.1082 1.1693
 0.6063 0.2378 2.5495
 0.6301 0.3935 1.6012
q4
 0.2214 0.2704 0.8186
 0.2443 0.2251 1.0853
 0.6301 0.3935 1.6012
 2.6655 1.0455 2.5495

These estimates are essentially the same as the sample means, variances, and covariances. This kind of analysis is much easier using PROC CORR with the NOMISS option. However, the main purpose of Example 27.1 and Example 27.2 is to introduce the MSTRUCT modeling language and some basic but important options in PROC CALIS. You can apply the MSTRUCT modeling language to more sophisticated situations that are beyond the saturated mean and covariance structure models. Example 27.3 and Example 27.4 fit some patterned covariance models that are nonsaturated. Also, options such as NOSE, NOSTAND, and MEANSTR are useful for all modeling languages in PROC CALIS.