Question to author
The definition of the back-door condition (Causality,
page 79, Definition 3.3.1) seems to be contrived. The exclusion of
descendants of X (Condition (i)) seems to be introduced as an after
fact, just because we get into trouble if we dont.
Why cant we get it from first principles;
first define sufficiency of Z in terms of
the goal of removing bias and, then, show
that, to achieve this goal, you neither want nor need
descendants of X in Z.
Author answer:
The exclusion of descendants from the back-door criterion
is not a contrived "fix", but is based indeed on first principles.
The principles are as follows:
We wish to measure a certain quantity (causal effect) and,
instead, we measure a dependency P(y|x) that results
from all the paths in the diagram, some are spurious
(the back-door paths) and some are genuine (the directed paths).
Thus, we need to modify the measured dependency
and make it equal to the desired quantity. To do it
systematically, we condition on Z while ensuring that:
To see why, consider the path
The intermediate variables,
S_{}1, S_{2}..., (as well as Y)
are affected by noise factor e_{}1, e_{}2,...
which are not shown explicitly in the diagram.
However, under magnification, the chain unfolds into
the graph:
Now imagine that we condition on a descendant Z
of S_{1}
Since S_{1} is a collider, this creates dependency between
X and e_{1} which is equivalent to a back-door path
By principle 3, such spurious paths should not be created.
Note that a descendant Z of X that is not also a descendant of some S_{i} escapes this exclusion; it can safely be conditioned on without introducing bias. (Though it may decrease the efficiency of the associated estimator of the causal effect of X on Y.)
Further questions from Reader:
This is a reasonable explanation (for excluding
descendants of X,) but it has two shortcomings:
Author's Answer
Thus, if we wish to test the (null) hypothesis
that there is no causal effect of X on Y,
adjusting for Z = {C,F} is perfectly legitimate,
and the graph shows it (i.e., C and F are
non-descentant of X). However, it is not
a legitimate adjustment for assessing the causal
effect of X on Y, and the back-door criterion
tells us so, because the graph under this task is
F becomes a descendant of X, excluded by
the back-door criterion.
If they sound at variance with traditional epidemiology, it is only because traditional epidemiologists did not have a formal system of removing and adding dependencies. All they had was healthy intuition; graphs translates this intuition into a formal system of closing and opening paths,
We should realize that before 1985, causal analysis in epidemiology was in a state of chaos, or, as I put it politely: "a state of healthy intuition." Even the idea that confounding stands for a "difference between two dependencies, one that we want to measure, the other that we do measure" was resisted by many (see chapter 6 of Causality), because they could not express the former mathematically.
As to finding "something in graph language that is closest to traditional meaning," we can do much better. Graphs provide the actual meaning of what we call "tradition meaning".
In other words, graph theory is formal, traditional meaning is just intuition. Suppose we find a conflict between the two, who should we believe? tradition or formal mathematics?
Luckily, we will never find such conflict. Why? because tradition is slippery and healthy intuition is slippery and one can always say: what they really meant was different...
In short, what graphs gives us is the ONLY sensible interpretation of the "traditional meaning of epidemiological concepts."