Approximate Standard Errors

The CALIS Procedure

Approximate Standard Errors

Except for unweighted and diagonally weighted least-squares estimation, approximate standard errors can be computed as the diagonal elements of the matrix

${c \over NM} H^{-1} , {where}$

$NM = \{ (N - 1) & {if the CORR or COV matrix is analyzed} \ & {or the intercept... ...atrix is analyzed} \ & {and the intercept variable is not used in the model} \ .$

The matrix H is the approximate Hessian matrix of F evaluated at the final estimates, c=1 for the WLS estimation method, c=2 for the GLS and ML method, and N is the sample size. If a given correlation or covariance matrix is singular, PROC CALIS offers two ways to compute a generalized inverse of the information matrix and, therefore, two ways to compute approximate standard errors of implicitly constrained parameter estimates, t values, and modification indices. Depending on the G4= specification, either a Moore-Penrose inverse or a G2 inverse is computed. The expensive Moore-Penrose inverse computes an estimate of the null space using an eigenvalue decomposition. The cheaper G2 inverse is produced by sweeping the linearly independent rows and columns and zeroing out the dependent ones. The information matrix, the approximate covariance matrix of the parameter estimates, and the approximate standard errors are not computed in the cases of unweighted or diagonally weighted least-squares estimation.

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