Example 27.1 Estimating Covariances and Correlations
This example shows how you can use PROC CALIS to estimate the covariances and correlations of the variables in your data set.
Estimating the covariances introduces you to the most basic form of covariance structures—a saturated model with all variances
and covariances as parameters in the model. To fit such a saturated model when there is no need to specify the functional
relationships among the variables, you can use the MSTRUCT modeling language of PROC CALIS.
The following data set contains four variables q1
–q4
for the quarterly sales (in millions) of a company. The 14 observations represent 14 retail locations in the country. The
input data set is shown in the following DATA step:
data sales;
input q1 q2 q3 q4;
datalines;
1.03 1.54 1.11 2.22
1.23 1.43 1.65 2.12
3.24 2.21 2.31 5.15
1.23 2.35 2.21 7.17
.98 2.13 1.76 2.38
1.02 2.05 3.15 4.28
1.54 1.99 1.77 2.00
1.76 1.79 2.28 3.18
1.11 3.41 2.20 3.21
1.32 2.32 4.32 4.78
1.22 1.81 1.51 3.15
1.11 2.15 2.45 6.17
1.01 2.12 1.96 2.08
1.34 1.74 2.16 3.28
;
Use the following PROC CALIS specification to estimate a saturated covariance structure model with all variances and covariances
as parameters:
proc calis data=sales pcorr;
mstruct var=q1-q4;
run;
In the PROC CALIS statement, specify the data set with the DATA= option. Use the PCORR option to display the observed and
predicted covariance matrix. Next, use the MSTRUCT statement to fit a covariance matrix of the variables that are provided
in the VAR= option. Without further specifications such as the MATRIX statement, PROC CALIS assumes all elements in the covariance
matrix are model parameters. Hence, this is a saturated model.
Output 27.1.1 shows the modeling information. Information about the model is displayed: the name and location of the data set, the number
of data records read and used, and the number of observations in the analysis. The number of data records read is the actual
number of records (or observations) that PROC CALIS processes from the data set. The number of data records used might or
might not be the same as the actual number of records read from the data set. For example, records with missing values are
read but not used in the analysis for the default maximum likelihood (ML) method. The number of observations refers to the
N used for testing statistical significance and model fit. This number might or might not be the same as the number of records
used for at least two reasons. First, if you use a frequency variable in the FREQ statement, the number of observations used is a weighted sum of the number of records, with the frequency variable being
the weight. Second, if you use the NOBS= option in the PROC CALIS statement, you can override the number of observations that
are used in the analysis. Because the current data set does not have any missing data and there are no frequency variables
or an NOBS= option specified, these three numbers are all 14.
The model type is MSTRUCT because you use the MSTRUCT statement to define your model. The analysis type is covariances, which
is the default. Output 27.1.1 then shows the four variables in the covariance structure model.
Output 27.1.1: Modeling Information of the Saturated Covariance Structure Model for the Sales
Data
The CALIS Procedure
Covariance Structure Analysis: Model and Initial Values
WORK.SALES |
14 |
14 |
14 |
MSTRUCT |
Covariances |
Output 27.1.2 shows the initial covariance structure model for these four variables. All lower triangular elements (including the diagonal
elements) of the covariance matrix are parameters in the model. PROC CALIS generates the names for these parameters: _Add01
–_Add10
. Because the covariance matrix is symmetric, all upper triangular elements of the matrix are redundant. The initial estimates
for covariance are denoted by missing values no initial values were specified.
Output 27.1.2: Initial Saturated Covariance Structure Model for the Sales
Data
The PCORR option in the PROC CALIS statement displays the sample covariance matrix in Output 27.1.3. By default, PROC CALIS computes the unbiased sample covariance matrix (with variance divisor equal to N – 1) and uses it for the covariance structure analysis.
Output 27.1.3: Sample Covariance Matrix for the Sales
Data
0.33830 |
0.00020 |
0.03610 |
0.22137 |
0.00020 |
0.22466 |
0.12653 |
0.24425 |
0.03610 |
0.12653 |
0.60633 |
0.63012 |
0.22137 |
0.24425 |
0.63012 |
2.66552 |
The fit summary and the fitted covariance matrix are shown in Output 27.1.4 and Output 27.1.5, respectively.
Output 27.1.4: Fit Summary of the Saturated Covariance Structure Model for the Sales
Data
Output 27.1.5: Fitted Covariance Matrix for the Sales
Data
In Output 27.1.4, the model fit chi-square is 0 (df = 0). The p-value cannot be computed because the degrees of freedom is zero. This fit is perfect because the model is saturated.
Output 27.1.5 shows the fitted covariance matrix, along with standard error estimates and t values in each cell. The variance and covariance estimates match exactly those of the sample covariance matrix shown in Output 27.1.3.
A common practice for determining statistical significance for estimates in structural equation modeling is to require the
absolute value of t to be greater than 1.96, which is the critical value of a standard normal variate at =0.05. While all diagonal elements in Output 27.1.5 show statistical significance, all off-diagonal elements are not significantly different from zero. The t values for these elements range from 0.002 to 1.601.
Output 27.1.6 shows the standardized estimates of the variance and covariance elements. This is also the correlation matrix under the MSTRUCT
model. Standard error estimates and t values are computed with the correlation estimates. Note that because the diagonal element values are fixed at 1, no standard
errors or t values are shown.
Output 27.1.6: Standardized Covariance Matrix for the Sales
Data
Sometimes researchers do not need to estimate the standard errors that are in their models. You can suppress the standard
error and t value computations by using the NOSE option in the PROC CALIS statement:
proc calis data=sales nose;
mstruct var=q1-q4;
run;
Output 27.1.7 shows the fitted covariance matrix with the NOSE option. These values are exactly the same as in the sample covariance matrix
shown in Output 27.1.3.
Output 27.1.7: Fitted Covariance Matrix without Standard Error Estimates for the Sales
Data
0.3383 |
0.000198 |
0.0361 |
0.2214 |
0.000198 |
0.2247 |
0.1265 |
0.2443 |
0.0361 |
0.1265 |
0.6063 |
0.6301 |
0.2214 |
0.2443 |
0.6301 |
2.6655 |
This example shows a very simple application of PROC CALIS: estimating the covariance matrix with standard error estimates.
The covariance structure model is saturated. Several extensions of this very simple model are possible. To estimate the means
and covariances simultaneously, see Example 27.2. To fit nonsaturated covariance structure models with certain hypothesized patterns, see Example 27.3 and Example 27.4. To fit structural models with implied covariance structures that are based on specified functional relationships among variables,
see Example 27.6.