Modification Indices
While fitting structural models, you may want to modify the
specified model in order to
- reduce the value significantly
- reduce the number of parameters to estimate without
increasing the value too much
If you specify the MODIFICATION or MOD option,
PROC CALIS computes and displays a default set of modification indices:
- Univariate Lagrange multiplier test indices
for most elements in the
model matrices that are constrained to equal constants.
These are second-order approximations of
the decrease in the value that would result from
allowing the constant matrix element to vary. Besides the value
of the Lagrange multiplier, the
corresponding probability (df=1) and the approximate change
of the parameter value (should the constant be changed
to a parameter) are displayed. If allowing the constant to be a
free estimated parameter would result in a singular information
matrix, the string 'sing' is displayed instead of the Lagrange
multiplier index. Not all elements in the model matrices
should be allowed to vary; the diagonal elements of the
inverse matrices in the RAM or LINEQS model must be constant ones.
The univariate Lagrange multipliers are displayed at the constant
locations of the model matrices.
- Univariate Wald test indices for those matrix
elements that correspond to
parameter estimates in the model.
These are second-order approximations of the increase in the
value that would result from constraining the parameter
to a 0 constant. The univariate Wald test indices are the
same as the t values that are displayed together with the parameter
estimates and standard errors. The univariate Wald test indices
are displayed at the parameter locations of the model matrices.
- Univariate Lagrange multiplier test indices
that are second-order
approximations of the decrease in the value that
would result from the release of equality constraints.
Multiple equality constraints containing n > 2 parameters
are tested successively in n steps, each assuming the
release of one of the equality-constrained parameters.
The expected change of the parameter values of the separated
parameter and the remaining parameter cluster are displayed, too.
- Univariate Lagrange multiplier test indices
for releasing active boundary constraints specified
by the BOUNDS statement
- Stepwise multivariate Wald test indices
for constraining
estimated parameters to 0 are computed and displayed.
In each step, the parameter
that would lead to the smallest increase in the multivariate
value is set to 0. Besides the multivariate value and its
probability, the univariate increments are also displayed.
The process stops when the univariate probability is smaller
than the specified value in the SLMW= option.
All of the preceding tests are approximations. You can often
get more accurate tests by actually fitting different models and
computing likelihood
ratio tests.
For more details about the Wald and the
Lagrange multiplier test,
refer to MacCallum (1986), Buse (1982),
Bentler (1986), or Lee (1985).
Note that, for large model matrices,
the computation time for the default modification indices can
considerably exceed the time needed for the minimization process.
The modification indices
are not computed for unweighted least-squares
or diagonally weighted least-squares
estimation.
Caution: Modification indices
are not computed if the model matrix is an identity matrix (IDE or ZID),
a selection matrix (PER), or the first matrix J in the LINEQS
model.
If you want to display the modification indices for such a matrix,
you should specify the matrix as another type;
for example, specify an identity matrix used in the COSAN statement
as a diagonal matrix with constant diagonal elements of 1.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.