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The CALIS Procedure |

- The functions
*F*_{ULS},*F*_{GLS}, and*F*_{ML}take into account all*n*elements of the symmetric residual matrix^{2}**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 (Jreskog and Srbom 1988, p. 23) shows
that LISREL groups the *F*_{DWLS} function in the first group
by taking into account all *n ^{2}* elements of the symmetric residual
matrix

- 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

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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