This section begins with a brief description
of the McDonald's COSAN model. Let* ***S**
be a population variance-covariance matrix for a set of manifest variables.
The COSAN model (McDonald, 1978) holds if **S**
may be expressed as

**S**
= **F**1**F**2...**F**k**PF**k¢ ...**F**2¢ **F**1¢ (8)

where **P** is symmetric and Gramian, and
any of the elements of any **F** matrix or **P** may be constrained
under the model to be a function of the others, or to be specified numerical
values. As a powerful additional option, any square **F** matrix may
be specified to be the *inverse* of a patterned matrix. This "*patterned
inverse*" option is critical for applications to path analysis.
A COSAN model with *k* **F** matrices is referred to as "a
COSAN model of order *k*."

Obvious special cases are: Orthogonal and oblique common factor models, confirmatory factor models, and patterned covariance matrices.

McDonald's COSAN model is a powerful and original approach which offers many benefits to the prospective tester of covariance structure models. Testing and estimation for the model were implemented in a computer program called, aptly enough, COSAN (See Fraser and McDonald, 1988 for details on a recent version of this program, which has been available since 1978).

In 1978, J. J. McArdle proposed some simple rules for translating any path diagram directly to a structural model. In collaboration with McDonald, he proposed an approach which yielded a model directly testable with the COSAN computer program.