Analyzing Invariance Properties - Types of Invariance

Invariance under a constant scaling factor
In this case, A is in the group G1 where

(146)

In essence, a model is invariant under a constant factor if multiplying S by a non-zero constant will not change the fact that a model fits S. Most covariance structures of any practical interest are at least invariant under a constant scaling factor. Some of the non-centrality based fit indices developed by Steiger (1989, 1990) require that the model fitted be invariant under a constant scaling factor (ICSF).

Invariance under changes of scale
In this case A is in the group G2, where

(147)

where Diag (t) is a diagonal matrix with diagonal elements given by the p components of the vector t.