Testing Rank Deficiency in the Approximate Covariance
The inverse of the information matrix (or approximate
Hessian matrix) is used for the covariance matrix of
the parameter estimates, which is needed for the
computation of approximate standard errors and
modification indices. The numerical condition
of the information matrix (computed as the crossproduct
J'J of the Jacobian matrix J) can be very poor in
many practical applications, especially for the analysis
of unscaled covariance data.
The following four-step strategy is used for the
inversion of the information matrix.
- The inversion (usually of a normalized matrix
D-1HD-1) is tried using a modified form
of the Bunch and Kaufman (1977) algorithm, which
allows the specification of a different singularity criterion
for each pivot. The following three criteria for the
rank loss in the information matrix are used to specify
If no rank loss is detected, the inverse of the
information matrix is used for the covariance matrix
of parameter estimates, and the next two steps
- ASING specifies absolute singularity.
- MSING specifies relative singularity
depending on the whole matrix norm.
- VSING specifies relative singularity
depending on the column matrix norm.
- The linear dependencies among the parameter subsets
are displayed based on the singularity criteria.
- If the number of parameters t is smaller than the
value specified by the G4= option (the default value is 60),
the Moore-Penrose inverse is computed based on the
eigenvalue decomposition of the information matrix.
If you do not specify the NOPRINT option, the distribution of
eigenvalues is displayed, and those eigenvalues that
are set to zero in the Moore-Penrose inverse are
indicated. You should inspect this eigenvalue
- If PROC CALIS did not set the right
subset of eigenvalues to zero, you can specify the COVSING=
option to set a larger or smaller subset of eigenvalues
to zero in a further run of PROC CALIS.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.