Models and Methods - The LISREL Model
In the following sections, all variables will be assumed to be in deviation score form (i.e., have zero means) unless explicitly stated otherwise.
The statistical model behind SEPATH is best understood in historical context, and so this section begins with a review of several important models for the analysis of covariance structures.
In his 1986 review of developments in structural modeling, Bentler described 3 general approaches to covariance structure representations. The first and most familiar involved integration of the psychometric factor analytic (FA) tradition with the econometric simultaneous equations model (SEM). This approach, originated by a number of authors including Keesling, Wiley, and Jöreskog was described by Bentler with the neutral acronym FASEM. The well-known LISREL model is of course the best known example of this approach.
The LISREL model can be written in three interlocking equations. Perhaps the key equation is the structural equation model, which relates latent variables.
h = Bh + Gx + z(1)
The endogenous, or "dependent" latent variables are collected in the vector h, while the exogenous, or "independent" latent variables are in x. B and G are coefficient matrices, while z is a random vector of residuals, sometimes called "errors in equations" or "disturbance terms." The elements of B and G would be path coefficients for directed relationships among latent variables. It is assumed in general that z and x are uncorrelated, and that I - B is of full rank.
Because usually h and x are not observed without error, there are also factor model (or "measurement model") equations to account for measurement of these latent variables through manifest variables. The measurement models for the two sets of latent variables are
y = Lyh + e(2)
and
x = Lx x + d(3)
With
(4)
the LISREL model is that
Syy= Ly(I - B)-1(GFG¢+ Y)(I - B¢)-1Ly¢ + Qe(5)
Sxx = LxFLx¢ + Qd(6)
Sxy = LxFG¢(I - B¢)-1Ly¢(7)
F, Y, Qe,and Qd are the covariance matrices for x, z, e, and d respectively. There seems to be considerable confusion in the literature about the precise assumptions required for Equations 5 through 7 to hold. Jöreskog and Sörbom (1989) state the assumptions that (1) z is uncorrelated with x, (2) e is uncorrelated with h, (3) d is uncorrelated with x, (4) z, e, and d are mutually uncorrelated, and (5) I - B is of full rank. However, it appears that Equation 7 also requires an assumption not stated by Jöreskog and Sörbom (1989), i.e., that e and x are uncorrelated.
This model reduces to a number of well-known special cases. For example, if there are no y-variables, then the model reduces to the common factor model, as can be seen from Equation 6.
An important aspect of the LISREL approach is that, in using it, variables must be arranged according to type. Manifest and latent, "exogenous" and "endogenous" variables are used in different places in different equations. Moreover, LISREL's typology for manifest variables is somewhat different from that used by other models. Specifically, in LISREL a manifest variable is designated as x or y on the basis of the type (exogenous or endogenous) of latent variable it loads on.
It is, of course, possible to translate models from a path diagram representation of a model to a LISREL model. However, this is not always easy. In some well known cases special strategies must be used to "trick" the LISREL model into analyzing a path diagram representation. For example, LISREL does not allow direct representation of a path in which an arrow goes from a manifest exogenous variable to a latent endogenous variable. Consequently a dummy latent exogenous variable (identical to the manifest variable) must be created in such cases.
In his review, Bentler (1986) referred to the models of McArdle (1978) and Bentler and Weeks (1979) as "generic" approaches, in that their emphasis was on the distinction between independent (exogenous) and dependent (endogenous) variables, rather than manifest and latent variables.
McArdle (1978) proposed an approach that was considerably simpler than the LISREL model. This approach, in essence, did not require any partitioning of variables into types. One could represent all paths in only two matrices, one representing directed relationships among variables, the other undirected relationships. McArdle's approach, which he called the RAM model, could be tested easily as a special case of McDonald's COSAN model.
McArdle's specification was innovative, and offered substantial benefits. It allowed path models to be grasped and fully specified in their simplest form — as linear equations among manifest and latent variables. Instead of 18 model matrices, and a plethora of different variable types, one only needed 3 matrices! After reading some of McArdle's early papers, the present author was motivated to seek an automated approach to structural modeling. So SEPATH, and its precursor EzPATH, owe a special debt to McArdle.
Ironically, it took some time for McArdle's work to receive the attention it deserved. The work initially met with a lukewarm reception from journal editors and rather harsh opposition from some reviewers. It took 4 years for a detailed algebraic treatment (McArdle & McDonald, 1984) to pass through the review process and achieve publication. By then, unfortunately, the full credit due to McArdle had been diluted.