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Interventional identification 2.0

3.3 Avoiding recantation

3.4.3 Interventional identification 2.0

We now show how the insights provided by the central notion of recantation can be incorporated within recent work on complete identification algo- rithms for interventional distributions (Huang and Valtorta, 2006; Shpitser and Pearl, 2006a,b, 2008a; Tian and Pearl, 2003) so as to arrive at identifi- cation strategies with complete identification power. Specifically, Shpitser (2013) recently proved that any π-specific effect of A on Y is identifiable under NPSEMs if, and only if the following two conditions hold:

there is no recanting district for the π-specific effect of A on Y (vii) P(Y(a) =y)is identifiable by some means. (viii) Since π may refer to a subset of paths that constitute either a natural direct or a natural indirect effect, this result also provides a complete identification criterion for P(Y(a, M(a0)) =y).

Below we demonstrate that this result opens avenues towards novel strategies for identifying natural effects, mainly by resorting to alterna- tive cross-world assumptions that may substitute for cross-world indepen- dence (ii). Before doing this, we give a more detailed review of Shpitser (2013)’s main results, followed by some examples.

From cross-world to interventional quantities

In the beginning of this section, we already mentioned that the recant- ing district criterion enables translating cross-world quantities – used to define path-specific effects – into interventional quantities. It does so by demanding there to be no recanting district for the π-specific effect of in- terest, such that conflicting treatment assignments only occur in different districts. Specifically, if condition (vii) holds, by Theorem 3.4.1, π-specific effects on some outcome Y can be expressed as a functional of interventional

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distributions8

d\y

i

P(Di|do(PA (Di) = pai (Di))), (3.6)

where PA (Di) = PA(Di)\ Di denotes the set of parents of all nodes in district Di (excluding nodes in Di itself). Here, the product runs across all districts Di in the subgraphGD, and the summation is made over all possible realisations of the nodes in these districts, except for the outcome. Note that this expression closely matches Tian and Pearl (2003)’s identifying functional for interventional distributions P(Y(a) =y)(expression (2.16) in chapter 2), which builds on c-component factorization and can be consid- ered a generalization of the truncated factorization formula to DAGs with hidden variables. In fact, expression (3.6) differs from expression (2.16) only to the extent that treatment assignments are allowed to be different across districts. This is indicated by the superscript i in pai (Di), which denotes the vector of value assignments to PA (Di). For instance, pai (Di)includes the treatment assignment a or a0depending on whether Diincludes children

of treatment A that transmit part of the natural direct or indirect effect, re- spectively. More generally, if interest lies in identifying a certain π-specific effect, then pai (D

i) includes e.g. the active treatment assignment a (or baseline assignment a0) depending on whether (or not) Diincludes children

of treatment A that transmit part of that π-specific effect, respectively.

From interventional to observed quantities

Further identification of π-specific effects in terms of observational distribu- tions thus depends on whether each factor P(Di|do(PA (Di) = pai (Di))) in expression (3.6) is identifiable from the observed data. However, be- cause expression (3.6) only differs from Tian and Pearl (2003)’s identifying functional in that the former allows for conflicting treatment assignments between districts, identifiability of P(Y(a) = y) by Tian’s ID algorithm logically implies each factor in expression (3.6) to be expressible in terms of 8For notational simplicity, we choose to display expression (3.6) in terms of interven-

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observational distributions. This is made explicit in condition (viii).

Some examples

To illustrate the above results, we give a number of examples in which the recanting district criterion mainly serves to establish cross-world in- dependence (ii); natural effects are therefore also identified under Pearl’s identifying conditions, as discussed in section 3.4.2. In the next section, we develop a novel complementary identification strategy that is inspired by the logic of the recanting district criterion and circumvents cross-world independence (ii).

A simple example In the subgraphGDof causal diagramGin Figure 3.2B,

there are two districts in D ={C, M, Y}, i.e. D1 = {M, C}and D2 = {Y}, such that, by Theorem 3.4.1, P(Y(a, M(a0)) =y)is expressible as

c,mP(Y=y|do(A=a, M =m, C=c))P(M =m, C=c|do(A= a

0)).

Since we know that P(Y(a) =y) is identifiable, each of the factors in this expression must also be identifiable. Indeed, by Tian’s ID algorithm, we find that this expression can be written as a functional of the observed data, i.e. expression (3.5).

If C were unmeasured, such as in Figure 3.2A, the identifying functional for P(Y(a) =y)would be

m P(Y =y, M=m|do(A=a)),

an expression that necessarily groups together M and Y into a joint interven- tional distribution, because they belong to a common district{M, Y}. This grouping into joint distributions once more illustrates the obstacle to enable conflicting treatment assignments for nodes in a common district, as was also exemplified by the inability to marginalise over U in expression (3.3).

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G A M Y C1 C2 C3 U1 U2 U3 GD M Y C1 C3

Figure 3.4: The somewhat more involved graphG from chapter 2 along with its subgraphGD.

A somewhat more involved example Consider again the causal diagram

G from Figure 2.7, which corresponds to Figure 5F in Pearl (2014) and is reproduced here in Figure 3.4, for convenience, along with the subgraph of interestGD.

In section 2.5.4 of chapter 2, we had demonstrated that P(Y(a) =y)is not identifiable in G via the adjustment criterion. Similarly, no common set of baseline covariates C can be found such that the adjustment criterion is satisfied relative to both (A, M) and ({A, M}, Y). That is, any subset of {C1, C2, C3} that includes C2 but not C3 can be shown to satisfy the adjustment criterion relative to (A, M). This can be seen by noting that, in any case, we must adjust for C2. Since C3is a collider, adjusting for it opens spurious pathways A U3 → C3 ← C1 → M and A ← U3 → C3 ← U1 → C2 ← U2 → M that cannot be blocked by additionally adjusting for C1. However, the adjustment criterion relative to({A, M}, Y) insists that C3be included in the adjustment set, because the spurious path A ←U3 →C3→Y can only be blocked by C3. The adjustment criterion for natural effects (Shpitser and VanderWeele, 2011) thus tells us that, logically, since P(Y(a) = y)is not identifiable via the adjustment criterion, neither is P(Y(a, M(a0)) =y).

Nonetheless, since there is no recanting district for the natural direct or indirect effect inGD, the identifying functional for P(Y(a, M(a0)) =y)can

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be expressed as

c1,c3,m

P(Y|do(A =a, M=m, C3 =c3))P(M=m|do(A= a0, C1 =c1)) ×P(C3=c3|do(C1=c1))P(C1 =c1). (3.7) Moreover, since – as illustrated in section 2.5.4 of chapter 2 – P(Y(a) = y) is identifiable by Tian’s ID algorithm, the above expression is likewise identified from the observed data, namely as:

c1,c2,c3,m

P(Y|A= a, M=m, C3 =c3)P(M=m|A =a0, C1 =c1, C2 =c2) ×P(C1 =c1)P(C2=c2)P(C3 =c3|C1=c1). (3.8) Note that, in contrast to the previous example, this functional cannot be reduced to an expression of the form of the adjustment formula.

Nonetheless, it can easily be verified that Pearl’s ‘piecemeal deconfound- ing’ approach would have yielded the same identification result. However, it would have required searching the space of candidate covariate sets C that not only satisfied cross-world indendence (ii) (i.e. {C1},{C3}or{C1, C3}) but also conditions (v) and (vi) (i.e. only {C1}). Shpitser’s identification approach is not only (more) complete, but arguably also more insightful as it clarifies that identification of P(Y(a) = y) is the main difficulty in identifying the natural (in)direct effect, which cannot be achieved solely by covariate adjustment.