Models and Methods - McArdle's RAM Model

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. Fij 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)-1r (14)

one obtains

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

Equations 13 and 15 imply

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

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

S = F1F2PF2 F1 (17)

as a COSAN model of order 2, where

(18)

and

F2 = (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. F2 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.