From: L. H., University of Alberta and S.M., Georgia Tech

Subject: The causal interpretation of structural coefficients

**Question to author:**

In response to my comments (e.g., *Causality,* Section 5.4) that
the causal interpretation of structural coefficients
is practically unknown among SEM researchers,
and my more recent comment that a correct causal interpretation
is conspicuously absent from *all* SEM books
and papers, including *all* 1970-1999 texts in economics,
two readers wrote that the "unit-change" interpretation
is common and well accepted in the SEM literature.

L.H. from the University of Alberta wrote:

"Page 245 of L. Hayduk, Structural Equation Modeling
with LISREL: Essentials and Advances, 1986, has a chapter headed
"Interpreting it All", whose first section is titled "The basics
of interpretation," whose first paragraph, has a second sentence
which says in italics (with notation changed to correspond to the
above) that a slope can be interpreted as: the magnitude of the
change in

*y* that would be predicted to accompany a unit change
in *x* with the other variables in the equation left
untouched at their original values." ... "Seems to me that
O.D. Duncan, Introduction to Structural
Equation Models 1975 pages 1 and 2 are pretty clear on *b* as causal.
"More precisely, it [*byx*] says that a change of one unit in *x* ...
produces a change of *b* units in *y*" (page 2). I suspect
that H. M. Blalock's book "Causal models in the social Sciences",
and D. Heise's book "Causal analysis." probably speak of *b* as causal."

S.M., from Georgia Tech concurs:

"I concur with L.H. that Heise, author of Causal Analysis
(1975) regarded the

*b* of causal equations to be how much
a unit change in a cause produced an effect in an effect
variable. This is a well-accepted idea."

**Author reply**

The "unit change" idea appears, sporadically, in
several SEM publications, yet, invariably, its appearance is
marred by evidence that essential elements of
this idea have not been properly conceptualized,
and that its technical ramifications have not been utilized.
It is not surprising, therefore, that SEM texts do not
go beyond parameter estimation, and do not show readers
how structural parameters, once estimated, can be put
into meaningful use -- such treatment would require
crisp and unambiguous understanding
of the unit-change idea.

The paragraph cited above (from Hayduk 1986) is an example of how the unit-change idea has been misunderstood even by authors who are open to causal interpretations. I will first cite the paragraph, then point out its two major errors, and finally, I will offer a concise revision that coheres with the modern conception of "unit-change".

********* Original paragraph from Hayduk's book 1986 p.245 *********

**8.1 The Basics of Interpretation**

The interpretation of structural coefficients as "effect coefficients"
originates with ordinary regression equations like

*X*_{0} = *a + b*_{1}*X*_{1} + b_{2}*X*_{2} + *b*_{3}*X*_{3} + *e*

for the effects of variables *X*_{1}, *X*_{2}, and *X*_{3} on variable *X*_{0}. We can
interpret the estimate of *b*_{1} as the magnitude of the change in *X*_{0} that
would be predicted to accompany a unit change INCREASE in *X*_{1} with *X*_{2}
and *X*_{3} left untouched at their original values. We avoid ending with the
phrase "held constant" because this phrase must be abandoned for models
containing multiple equations, as we shall later see. Parallel
interpretations are appropriate for *b*_{2} and *b*_{3}.

**********end of original paragraph ***********

This paragraph illustrates how two basic distinctions
have not been appreciated by SEM authors.
The first is the distinction between the meaning of
structural coefficients and the meaning
of the statistical estimates of those coefficients.
We rarely find the former defined
independently of the latter -- a confusion that
is rampant and especially embarrassing in econometric texts.
The second is the distinction between "held constant"
and "left untouched" or "found to remain constant",
for which the *do*(*x)* operator was devised.
(Related distinctions: "doing" versus "seeing" or
"interventional change" versus "natural change".)

To emphasize the centrality of these distinctions I will now offer a concise revision of Hayduk's paragraph,

****** revised paragraph ************

**8.1 The Basics of Interpretation (Revised)**

The interpretation of structural coefficients as "effect coefficients"
bears some resemblance to, but differs fundamentally from the
interpretation of coefficients in regression equations like

If Eq. (1) is a regression equation, then

We formally express this interpretation using conditional
expectations:

However, if Eq. (1) represents a structural equation,
the interpretation of *b*_{1} must be modified in two
fundamental ways. First, the phrase "a unit change in
*X*_{1}" must be qualified to mean "a unit interventional
change in *X*_{1}" , thus ruling out changes in
*X*_{1} that are produced by other variables in the model.
Second, the phrase "if we find that *X*_{2} and
*X*_{3} remain constant" must be abandoned and replaced by the
phrase "if we hold *X*_{1} and *X*_{2} constant",
thus ensuring constancy even contrary to model predictions.

Formally, these two modifications are expressed as:

Under no circumstance should one use the phrase "left untouched at their original values." Leaving variables untouched permits those variables to vary (in response to interventions or natural influences), in which case the change in

or to the marginal conditional expectation

depending on whether the change in

The interpretation expressed in (3) holds
in ALL models, including those containing multiple equations,
recursive and nonrecursive, regardless of whether
e is correlated with other variables in the model
and regardless of whether *X*_{2} and *X*_{3} are affected
by *X*_{1}. In contrast, expression (2) coincides with
*b*_{1} only under very special circumstances. It is for this
reason that we consider (3), not (2), to be an "interpretation"
of *b*_{1}; (2) interprets the "regression estimate" of *b*_{1}
(which might well be biased), while (3) interprets
*b*_{1} itself.

Next Discussion (Many readers: *Homework, or Gentle
Introduction to Causal Models*)