McArdle's approach is based on the following covariance structure model, which he has termed the RAM model:

Let **v** be a (*p* + *n*) � 1 random vector of *p*
manifest variables and *n* latent variables in the path model, possibly
partitioned into manifest and latent variables subsets in **m **and
**l**, respectively, in which case

(9)

(This partitioning is somewhat convenient,
but not necessary.) For simplicity assume all variables have zero means.
Let **F** be a matrix of multiple regression weights for predicting
each variable in **v** from the *p *+* n* -
1 other variables in **v**. **F** will have all diagonal elements
equal to zero. In general, some elements of **F** may be constrained
by hypothesis to be equal to each other, or to specified numerical values
(often zero). Let **r** be a vector of latent exogenous variables,
including residuals. The path model may then be written

**v** = **Fv**
+ **r** (10)

In path models, all endogenous variables are
perfectly predicted through the arrows leading to them. Since endogenous
variables are dependent variables in one or more linear equations, their
variances and covariances can be determined from the variances and covariances
of the variables with arrows pointing to them. Ultimately, the variances
and covariances of all endogenous variables are explained by a knowledge
of the linear equation set up and the variances and covariances of exogenous
variables in the system.) Consequently, elements of **r** corresponding
to endogenous variables in **v** will be null. The matrix **F**
contains the regression coefficients normally placed along the arrows
in a path diagram. **F***ij* is the path coefficient from *vj*
to *vi*. If a variable *vi* is exogenous, i.e., has no arrow
pointing to it, then row* i* of **F **will be null, and *ri*
= *vi*. Hence, the non-null elements of the variance covariance matrix
of **r** will be the coefficients in the "undirected" relationships
in the path diagram.

Define **P **=** ***E*(**rr� **). Furthermore,
let **W **=** ***E*(**vv� **), and **S
**=** ***E*(**mm�
**). The implications of Equation 10 for the structure of **S**,
the variance-covariance matrix of the manifest variables, can now be derived.
Regardless of whether the manifest and latent variables were partitioned
into distinct subsets in **v**, it is easy to construct a "filter
matrix" **J** which carries **v** into **m**. If the variables
in **v** are partitioned into manifest and latent variables, one obtains

(11)

**m **= **Jv** (12)

and consequently

**S
**=** ***E*(**mm�
**) = **J ***E*(**vv�
**) **J� **=
**JWJ � ** (13)

Since (assuming **I** -
**F** is nonsingular) Equation 13 may be rewritten in the form

**v** = (**I**
- **F**)-1**r** (14)

one obtains

**W **=(**I**
- **F**)-1**P**(**I**
- **F**)-1�
(15)

Equations 13 and 15 imply

**S
**= **J**(**I** -
**F**)-1**P**(**I**
- **F**)-1�
**J � ** (16)

This shows that any path model may be written in the form

**S**
= **F**1**F**2**PF**2� **F**1� (17)

as a COSAN model of order 2, where

(18)

and

**F**2
= (**F** - **I**)-1= **B**-1 (19)

McArdle's formulation may thus be characterized as follows:

1. For convenience, order the manifest variables
in the vector **m**, and the latent variables in the vector **l**.
The path model is then tested as a COSAN model of order 2, in which

2. , where **I** is of order *p � p* and **0 **is *p � n*.

3. **F**2**
**is the *inverse* of a square matrix **B**
of "directed relationships." **B**
is constructed from the path diagram as follows. Set all diagonal entries
of **B** to -1.
Examine the path diagram for arrows. For each arrow pointing from *vj *to *vi*,
record its path coefficient in position *bij*
matrix **B**.

4. **P** contains coefficients for "undirected"
paths between variable *vi
*and *vj* recorded in
positions *pij* and *pji*.

Obviously, *SEPATH* could have been written
around the elegant and straightforward RAM model. The approach would require
simply creating a list of manifest and latent variables, ordering them,
and filling the matrices **B** and **P** with coefficients obtained
by parsing PATH1 model statements.