From: L.B.S., University of Arizona, and S.M., Georgia Institute of Technology

Subject: Can

**Question to author:**

L.B.S., from University of Arizona, questioned whether
the *do*(*x*) operator can represent realistic actions or
experiments:
"Even an otherwise perfectly executed randomized experiment may yield
perfectly misleading conclusions if, for example, the construct validity
of the treatment is zero, e.g., there is a serious confounding. A good
example is a study involving injected vitamin *E* as a treatment for
incubated children at risk for retrolental fibroplasia. The randomized
experiment indicated efficacy for the injections, but it was soon
discovered that the actual effective treatment was opening the
pressurized, oxygen-saturated incubators several times per day to give the
injections, thus lowering the barometric pressure and oxygen levels in the
blood of the infants (Leonard, Major Medical Mistakes). Any statistical
analysis would have been misleading in that case."

S.M., from Georgia Institute of Technology, adds:

"Your example of the misleading causal effect, shows the kind of thing
that troubles me about the *do*(*x*) concept. You *do*(*x*)
or don't *do*(*x*) but something else, and this seems correlated
with an effect. But it may be something else that is correlated with
*do*(*x*) that is the cause and not the *do*(*x*) per se."

**Author reply**

The *do*(*x*) operator was devised to help
prevent such troubles. It stands for doing *X=x* in an ideal
experiment, where *X*, and *X* alone is manipulated,
not any other variable in the model. Mathematics deals
with ideal situations, and it is the experimenter's job
to make sure that the experimental conditions approximate
the mathematical ideal as closely as possible.

In your example of the vitamin E injection (above), there is another
variable being manipulated together with *X*, namely the
incubator cover, *Z*, which turns the experiment into a *do*(*x,z*),
condition instead of *do*(*x*). Thus, the experiment was far from ideal,
and far even from the standard experimental protocol, which
requires the use of placebo. Had placebo been used (to approximate the
requirement of the *do*(*x*) operator), the result would
not have been biased.

Such sensitivity to deviations from the ideal is
not unique to manipulation. Consider the
notion of conditional expectation, *E*(*Y*|*x*), which stands for the
expected value of *Y*, given that we observed *X=x*, and
ONLY *X=x*. If, in practice, we actually observe some other
variable, say *Z=z*, which we fail to report, predictions
based of our report would be biased.
Yet we hardly find scholars complaining that this sensitivity
is a troublesome feature of the conditioning concept.
Quite the contrary; scholars find the conditioning operator
to be a powerful notational tool for distinguishing
among predictions that are based on different sets
of observations.

Returning to manipulations, the *do*(*x*) operator
is a mathematical device that helps us specify
explicitly and formally what should be held constant, and what
is free to vary in any given experiment. The *do*-calculus
then helps us predict the logical ramifications of
such specifications, assuming they are executed
faithfully, and assuming we have a valid causal model
of the environment.

Next Discussion (L.H./S.M.: *The
causal interpretation of structural coefficients*)