If only the covariance or correlation structures are considered, the five estimation functions, , , , , and , belong to the following two groups:

• The functions , , and take into account all elements of the symmetric residual matrix . This means that the off-diagonal residuals contribute twice to the discrepancy function , as lower and as upper triangle elements.

• The functions and take into account only the lower triangular elements of the symmetric residual matrix . This means that the off-diagonal residuals contribute to the discrepancy function only once.

The function used in PROC TCALIS 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 function in the first group by taking into account all elements of the symmetric residual matrix .

• Relationship between DWLS and WLS:
  PROC TCALIS: The and discrepancy functions deliver the same results for the special case that the weight matrix used by WLS estimation is a diagonal matrix.
  LISREL 7: This is not the case.

• Relationship between DWLS and ULS:
  LISREL 7: The and estimation functions deliver the same results for the special case that the diagonal weight matrix used by DWLS estimation is an identity matrix.
  PROC TCALIS: To obtain the same results with and estimation, set the diagonal weight matrix used in DWLS estimation to:

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