**From: Les Hayduk, University of Alberta
Date: January 1, 2001
Subject: On the Causal Interpretation of Path Coefficients
**

Les Hayduk asked whether the operational formula for path coefficient;

Like all models in science, a structural equation model (SEM) is interpreted as a mapping between physical operations in the real world (observations, interventions, etc.) and their representative mathematical operations on the model. The physical operation, denoted by

**Les Hayduk further asks:**
...the formula for b has no covariances on
the right hand side. Can you tell us how you think about connecting

and the estimand of

or

Thus, Les' question is legitimate: How do we get the estimands from the definition?

We can do it in two ways, the first is fairly familiar to SEM researchers, the second is more general and more instructive (demonstrated in

Once we prove that the equation:

"The differenceSimilar interpretation applies to IV-estimand or to any other estimand that one can find by algebraic methods, the only difference would be the "if" part, namely, different modeling assumptions should be cited, those that permit the derivation to go through. This is fairly standard in the literature, with the exception of two ingredients; the "if" part is often left implicit, and the interpretational part is rarely made explicit.b=E(Y|do(x,z+1)) -E(Y|do(x,z)) can be estimated consistently by the estimandb=cov(Z,Y) /var(Z) ifeis uncorrelated withXandZ."

The second method of analysis is non-algebraic; we derive the equality

directly from the definition of

Let us demonstrate this derivation in our example. To compute the expression

In addition to (1), there is another relationship between

In words, if we intervene and set the values of

Using Eqs. (1) and (2), we proceed to compute the controlled
expectation *E*(*Y*|*do*(*x*,*z*)) as follows:

We are done, because, in linear systems, E(Y|x,y) is given by:

From this we readily get:

This derivation is more general, because it can be applied to nonlinear systems, and because it applies to ANY expression involving

this method yields the standard expression of the total effect in terms of sums of products of path coefficients. In other words, the total effect is not

Another important feature of this derivation is that it maintains clear separation between the meaning of structural parameters and the methods used in their estimation - it lets the meaning dictate the estimation.

Next Discussion: (Battistin:
*Intuition for tight bounds under noncompliance)*