(3.62)

or the

(3.63)

to hold. Corrections to the wordings of page 103 were posted on this website.

Your suggestion to allow confounding arcs beween *Xi* and *Xj*, is valid. However, allowing a confounding arc between *Li* and *Xj* (with *i* < *j*) is too permissive, as can be seen by the non-identified models of Figure 3.9 (b), (c), (d) and (g) in *Causality*.

In general, condition (3.62) is both over-restrictive and lacks intuitive basis. A more general and intuitive condition leading to (3.63) is formulated in (4.5) (*Causality*, p 122), which reads as follows:

**(3.62*) General condition for g-estimation **

__Comment 1:__ The new definition leads to improvements over (3,62), namely, there are cases where the *g*-formula (3.63) is valid with a subset *Lk* of the past but not with the entire past.

__Example 1__:

Assuming *U*1 and *U*2 are unobserved, and temporal order: *U*1,*Z, X*1, *U*2,*Y* we see that (3.62*), hence (3.63), are satisfied with *L*1 = 0, while taking the whole past *L*1 = *Z* would violate both.

(3.62) is also satisfied with the choice *L*1=0, but not with *L*1=*Z*.

__Comment 2:__ Defining *Lk* as the set of "nondescendants" of *Xk* (as opposed to temporal predecessors of *Xk*) also broadens (3.62).

__Example 2:__

with temporal order: *U*1,*X*1, *S,Y*

Both (3.62) and (3.62*) are satisfied with *L*1 = *S*, but not with *L*1 = 0.

__Comment 3:__ There are cases where (3.62) will not be satisfied even with the new interpretation of *Lk*, but the graphical condition (3.62*) is.

__Example 3:__(constructed by Ilya Shpitser)

It is easy to see that (3.62*) is satisfied; all back-door action-avoiding paths from *X*1 to *Y* are blocked by *X*0, *Z, Z'*.

At the same time, it is possible to show, though by a rather intricate method (see the Twin Network Method, page 213) that *Y*{*x*1, *x*2} is not independent of *X*1, given *Z, Z'* and *X*0.

(In the twin network model there is a *d*-connected path from *X*1 to *Yx*, as follows: *X*1 <–> *Z* <–> *Z** –> *Z'** –> *Y**) Therefore, (3.62) is not satisfied for *Y*{*x*1,*x*2} and *X*1.)

This example demonstrates one weakness of the Potential Response approach initially taken by Robins in deriving (3.63). The counterfactual condition (3.62) that legitimizes the use of the *g*-estimation formula is void of intuitive support, hence, epidemiologists who apply this formula are doing so under no guidance of substantive medical knowledge. Fortunately, graphical methods are slowly making their way into epidemiological practice, and more and more people begin to understand the assumptions behind *g*-estimation.

(Warning: Those who currently reign causal analysis in statistics are incurably graph-o-phobic and ruthlessly resist attempts to enlighten their students, readers and co-workers with graphical methods. This slows down progress in statistical research, but will eventually be overrun by commonsense.)

Best wishes,

========Judea Pearl