PROC CALIS: Relationships among Estimation Criteria :: SAS/STAT(R) 9.2 User's Guide, Second Edition

The CALIS Procedure

Relationships among Estimation Criteria

The five estimation functions, $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ , belong to the following two groups:

The functions $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ take into account all $\text{[math]}$ elements of the symmetric residual matrix $\text{[math]}$ . This means that the off-diagonal residuals contribute twice to $\text{[math]}$ , as lower and as upper triangular elements.
The functions $\text{[math]}$ and $\text{[math]}$ take into account only the $\text{[math]}$ lower triangular elements of the symmetric residual matrix $\text{[math]}$ . This means that the off-diagonal residuals contribute to $\text{[math]}$ only once.

The $\text{[math]}$ function used in PROC CALIS differs from that used by the LISREL 7 program. Formula (1.25) of the LISREL 7 manual (Jöreskog and Sörbom 1988, p. 23) shows that LISREL groups the $\text{[math]}$ function in the first group by taking into account all $\text{[math]}$ elements of the symmetric residual matrix $\text{[math]}$ .

Relationship between DWLS and WLS:
PROC CALIS: The $\text{[math]}$ and $\text{[math]}$ estimation functions deliver the same results for the special case that the weight matrix $\text{[math]}$ used by WLS estimation is a diagonal matrix.
LISREL 7: This is not the case.
Relationship between DWLS and ULS:
LISREL 7: The $\text{[math]}$ and $\text{[math]}$ estimation functions deliver the same results for the special case that the diagonal weight matrix $\text{[math]}$ used by DWLS estimation is an identity matrix (contains only 1s).
PROC CALIS: To obtain the same results with $\text{[math]}$ and $\text{[math]}$ estimation, set the diagonal weight matrix $\text{[math]}$ used in DWLS estimation to

$\text{[math]}$

Because the reciprocal elements of the weight matrix are used in the goodness of fit function, the off-diagonal residuals are weighted by a factor of 2.

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