For interpreting linear structural relationships,
it is often desirable to have structural parameters *standardized*,
i.e., constrained so that all latent variables have unit variance. It
is rather easy, in traditional computational methods of analysis of covariance
structures, to constrain the variances of *exogenous* latent variables
to unity, since these variances appear as parameters in the standard model
specification. One simply sets these parameters equal to a fixed value
of 1. This approach was not available with *endogenous* latent variables,
because their variances could not be specified directly. Consequently,
"standardized" solutions were generated in EzPATH 1.0 (as in,
say, EQS 3.0, LISREL VI, and CALIS) by first computing the unstandardized
solution, then computing (non-iteratively) the values of standardized
coefficients after the fact, using standard regression algebra. There
are, in practice, some problems with such solutions. First, standard errors
are not available. Second some equality constraints specified in the model
coefficients, which are satisfied in the unstandardized solution, may
not be achieved in the standardized version.

*SEPATH* offers
an option (the *New* option in the *Standardization* group in
the *Analysis Parameters* window), which produces a standardized
solution by constraining the variances of endogenous latent variables
during iteration. This method, described by Browne and DuToit (1987),
and Mels (1989), is a constrained Fisher Scoring algorithm. The algorithm
works as follows. Describe the *r* constraints on the endogenous
latent variable variances in the form

**c**(**q**) = **0** (131)

where **c**(**q**)
is a differentiable, continuous function of the parameter vector **q**. Let **L** be the Jacobian
matrix of **c**(**q**),
i.e.,

(132)

During minimization, approximate the constraint function with its first-order Taylor expansion, i.e.,

(133)

The nonlinear constraints required to establish unit variances for the endogenous latent variables (i.e., those of Equation 131) can thus be approximated by the linear constraints

On each iteration, the increment vector dk is calculated by solving the linear equation system

(134)

(where gk *r *Lagrange multipliers
corresponding to the *r *constraints. If there are *t *free
parameters in the model, the degrees of freedom for the *Chi-square*
statistic is

(135)

The above approach can be used as a general
method to minimize a discrepancy function subject to constraints on the
parameters. Here some very specific constraints are of interest. Specifically,
suppose there are *r* endogenous latent variables in a model. The
last *r* diagonal elements of the matrix **Y**
(see Equation 42) must be constrained to be equal to unity. Hence, in
this special case, a typical element of is given by

(136)

where *p *is the number of manifest variables.
Mels (1989, page 35) shows how to calculate a typical element of the Jacobian
matrix Lk.**
**

During iteration, progress is monitored so that the augmented discrepancy functions satisfy the inequality

(137)

Once the algorithm has converged, an estimate
of the covariance matrix of the elements of **q
**may be obtained by dividing the first txt
principal submatrix of the *inverse *of the augmented Hessian

(138)

by *N *-* 1. *Further details, including detailed formulae
for calculating the necessary derivatives for implementing the constrained
estimation procedure, are provided in clear and compact form by Mels (1989).