Some important recent papers (Browne & Shapiro, 1989, 1991; Dijkstra, 1990) have investigated invariance properties of covariance structures. This topic is important for several reasons. First, some of the statistics calculated (i.e., the G1 and G2 coefficients) depend on a certain scale invariance property. Second, some analyses discussed in the preceding section depend on invariance properties of the covariance structure. Finally, and perhaps most important, to fully understand the relationship between data and model, it is crucial to understand what aspects of the model are affected by rescalings of the data, and what aspects (if any) are unaffected.

Shapiro and Browne (1989) established a number
of key invariance properties for covariance structures. Their paper concerned
situations where a model fitting a covariance matrix **S**
of a random vector X would continue
to fit if the manifest variables in **x **were linearly transformed,
i.e.,

(145)

Shapiro and Browne (1989) studied how covariance
structures remained invariant when **S**
is allowed to be transformed by **A **matrices of various types. They
used the following definition. Consider a multiplicative group *G*
of nonsingular p x p matrices.
That is, if AÎG and BÎG, then AB-1ÎG. Under certain side
conditions met by the groups under consideration, *G* constitutes
a Lie group with matrix multiplication as the group operation. Associated
with *G* is a corresponding Lie group *G** of transformations
defined on the set of symmetric positive definite matrices by S ® ASA' . A *covariance structure* is
a symmetric matrix valued function **S**(**q**) which relates a parameter
vector **q** from a subset
of Âq
to **S**.

**Definition. **A covariance structure **S**(**q**) is said to be invariant
under the group *G** if for every and AÎG there exists a such
that **S**(**q***) = **S**(**q*******)A'.

This means that for any AÎG, the set of positive definite matrices corresponding to the given model remains invariant under the transformation S ® ASA'.

Browne and Shapiro (1989) studied several types
of **A** matrices corresponding to different kinds of invariance.

These included two kinds of invariance which
are of particular interest to *SEPATH*
users (see Types
of Invariance).