### The causal interpretation of structural coefficients

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

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."

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 InterpretationThe 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}+efor the effects of variables

X_{1},X_{2}, andX_{3}on variableX_{0}. We can interpret the estimate ofb_{1}as the magnitude of the change inX_{0}that would be predicted to accompany a unit change INCREASE inX_{1}withX_{2}andX_{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 forb_{2}andb_{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

X_{0}=a + b_{1}X_{1}+b_{2}X_{2}+b_{3}X_{3}+e (1)If Eq. (1) is a regression equation, then

b_{1}stands for the magnitude of the change inX_{0}that would be predicted to accompany a unit change inX_{1}if we find thatX_{2}andX_{3}remain constant, at their original values.We formally express this interpretation using conditional expectations:

b_{1}=E(X_{0}|x_{1}+1,x_{2},x_{3}) –E(X_{0}|x_{1},x_{2},x_{3}) (2)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 inX_{1}" must be qualified to mean "a unit interventional change inX_{1}" , thus ruling out changes inX_{1}that are produced by other variables in the model. Second, the phrase "if we find thatX_{2}andX_{3}remain constant" must be abandoned and replaced by the phrase "if we holdX_{1}andX_{2}constant", thus ensuring constancy even contrary to model predictions.Formally, these two modifications are expressed as:

b_{1}=E(X_{0}|do(x_{1}+1,x_{2},x_{3})) –E(X_{0}|do(x_{1},x_{2},x_{3})) (3)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

X_{0}would correspond to the total effectE(X_{0}|do(x_{1}+1)) –E(X_{0}|do(x_{1})) (4)or to the marginal conditional expectation

E(X_{0}|x_{1}+1) –E(X_{0}|x_{1}), (5)depending on whether the change in

X_{1}is interventional or statistical. None of (4) or (5) matches the meaning ofb_{1}in Eq. (1), regardless of whether we treat (1) as structural or regression equation.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}andX_{3}are affected byX_{1}. In contrast, expression (2) coincides withb_{1}only under very special circumstances. It is for this reason that we consider (3), not (2), to be an "interpretation" ofb_{1}; (2) interprets the "regression estimate" ofb_{1}(which might well be biased), while (3) interpretsb_{1}itself.Best wishes,

========Judea Pearl

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