Models and Methods - The Bentler- Weeks Model

The RAM model is somewhat wasteful in terms of the size of some of its matrices. Bentler and Weeks (1979) produced an alternative model which is somewhat more efficient in the size of its matrices. Specifically, the F2 and P matrices are quite large in the RAM model, and have a large number of zero elements. Bentler and Weeks showed how, in situations where there are no manifest exogenous variables (i.e., all manifest variables have at least one arrow pointing to them), the McArdle-McDonald approach may be modified to reduce the size of the model matrices.

Partition v in the form

(20)

where the subscripts x and n refer to "exogenous" and "endogenous," respectively.

Then one may write v = Fv + r in a partitioned form as

(21)

Now define n as a vector containing all the endogenous, or "dependent" variables. We may partition m into manifest exogenous and endogenous variables, i.e.,

(22)

Then

(23)

One may then write

n = F0n + Glx (24)

where

(25)

The derivation now proceeds with an algebraic development similar to the RAM-COSAN equations. Rearranging Equation 24, one obtains

(I - F0)n = Glx (26)

n = (I - F0)-1Glx (27)

(28)

whence, letting

(29)

F2 = (I - F0)-1 (30)

F3 = G (31)

and

P = E(lxlx¢ ) (32)

we have

S = GF2F3PF3¢ F2¢ G¢ (33)

G is a filter matrix similar to J in the McArdle-McDonald specification. F2 = B2-1, where B2 is a matrix containing path coefficients for directed relationships among endogenous variables only, and having -1 as each diagonal element. F3 contains path coefficients from exogenous variables to endogenous variables only, and P contains coefficients for undirected relationships, i.e., the variance-covariance parameters for the latent exogenous variables.

This clever algebraic refinement allowed some of the virtues of the McArdle approach to be retained, while expressing the essential relationships in smaller matrices. (Notice how several of the null submatrices are eliminated.) However, this model also had some minor drawbacks. It required partitioning variables into exogenous and endogenous types, and it did not allow direct expression of manifest exogenous variables.

An alternative model allows us to treat manifest exogenous variables explicitly. If you add a vector of manifest variables to each of the two variable lists in the Bentler-Weeks (1979) model, and modify the regression coefficient matrices accordingly, you arrive at the model used in SEPATH. In this model, which is similar to one given by Bentler and Weeks (1980), variables are partitioned into two groups.