Relationships among Estimation Criteria

The CALIS Procedure

Relationships among Estimation Criteria

The five estimation functions, F_ULS, F_GLS, F_ML, F_WLS, and F_DWLS, belong to the following two groups:

The functions F_ULS, F_GLS, and F_ML take into account all n² elements of the symmetric residual matrix S-C. This means that the off-diagonal residuals contribute twice to F, as lower and as upper triangle elements.
The functions F_WLS and F_DWLS take into account only the n(n+1)/2 lower triangular elements of the symmetric residual matrix S-C. This means that the off-diagonal residuals contribute to F only once.

The F_DWLS function used in PROC CALIS differs from that used by the LISREL 7 program. Formula (1.25) of the LISREL 7 manual (J $\ddot{o}$ reskog and S $\ddot{o}$ rbom 1988, p. 23) shows that LISREL groups the F_DWLS function in the first group by taking into account all n² elements of the symmetric residual matrix S-C.

Relationship between DWLS and WLS:
PROC CALIS: The F_DWLS and F_WLS estimation functions deliver the same results for the special case that the weight matrix W used by WLS estimation is a diagonal matrix.
LISREL 7: This is not the case.
Relationship between DWLS and ULS:
LISREL 7: The F_DWLS and F_ULS estimation functions deliver the same results for the special case that the diagonal weight matrix W used by DWLS estimation is an identity matrix (contains only 1s).
PROC CALIS: To obtain the same results with F_DWLS and F_ULS estimation, set the diagonal weight matrix W used in DWLS estimation to

$w_{ii,kk} = \{ 1. & {if i = k} \ 0.5 & {otherwise} .$

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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