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 The CALIS Procedure

## 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.

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