st: Structural equations, latent variables and path analysis

[Dick Campbell <dcamp@uic.edu>]

A recent query on the list asked about software for structural equation
modeling. Among social scientists outside of economics,
the term "structural equation" has come to mean so many
things that it is virtually meaningless. Thus, when people ask
for structural equation modeling software it is necessary to
go through a sort of triage process to find out what they actually
want to do.

Here are several meanings of the term and  brief comments on
what is available in Stata.

1) A sequence of regression equations involving directly
observed variables (as opposed to latent) arranged in a simple recursive
system (loosely "one way causation") estimable by OLS,
sometimes referred to, again loosely, as "path analysis." (Actually, path
analysis is more general, but I will leave that aside).

As noted, for models involving continuous outcome variables,
Phil Ender's Pathreg is an easy way to specify such a set of equations.
Most people who do analyses of this kind will also want to get estimates of
direct and indirect effects along with their standard errors (or at least they
should), however Pathreg will not do that and, so far as I know, no
other Stata package will. There are routines for doing so in most latent
variable packages (see below), however for recursive models the computations
can be done by hand, although computations of standard errors can get a bit 
tedious.

2) A system of equations of a non-recursive nature often involving
simultaneity among endogenous variables and requiring instrumental
variables or some other means of identifying the model.

In this case, and in case 1, the term "structural equation" in the
econometric literature is used in sharp distinction to  "reduced form."
There are many Stata routines to deal with this situation, beginning with 
IVREG.
Findit "instrumental variables" is a good place to start. However, many people
who use the term "structural equation model" are not referring to this 
distinction.

3) Models,  with latent variables, including confirmatory factor analysis,.

Unfortunately, in some areas of social science, the term "structural equation"
has come to refer to latent variable models which can be dealt with by 
programs
such as LISREL, MPLUS, EQS, AMOS and several others, all of which also deal
quite nicely with directly observed variables and most of which do effect 
decompositions.
Latent variable models can be recursive or non-recursive and thus will
sometimes involve reduced form and structural equations. It is important
to realize however, that the term "structural equation" has a very distinct
meaning in the econometric literature which has nothing to do with latent 
variables.
A more correct term would be SEMUV or Structural Equation Models
with Unmeasured Variables, however that term, like this note perhaps, 
borders on the pedantic.

Within Stata, GLLAMM will deal with a very broad class of latent variable 
models as
well as cases 1 and 2 for both continuous and categorical outcomes. I don't 
believe
that it does the kind of effect decomposition referred to above, although
that is a relatively minor issue. A routine for doing so would be a nice 
addition to the Stata tool kit.

A complete and rigorous discussion of the relationship between structural
equation models and latent variable models can be found in Skrondal and 
Rabe-Hesketh's .
book "Generalized Latent Variable Modeling: Multilevel, Longitudinal and 
Structural
Equation Models."

Richard T. Campbell
University of Illinois at Chicago



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