3.3 Avoiding recantation
3.4.2 Interventional identification 1.0
Since cross-world independence (ii) thus enables expressing the cross-world counterfactual distribution P(Y(a, M(a0)) = y) in terms of interventional
distributions, as in expression (3.4), Pearl (2001) complemented (ii) with a second and third condition for identification. In particular, P(Y(a, M(a0)) =
y)can be estimated from observed data if, in addition to (ii),
P(M(a0) =m|C =c)is identifiable by some means, and (v)
P(Y(a, m) =y|C =c)is identifiable by some means. (vi) In accordance with Pearl (2014), we explicitly add ‘identifiable by some means’, since these last two conditions have often been interpreted too strictly in the literature in terms of identifiability by means of adjustment for C. Specifically, (v) has typically been replaced by
M(a0)
⊥⊥
A|C (v’)and (vi) by requiring that
Y(a, m)
⊥⊥
A|C, (vi’)3
both hold, implying once more Pearl’s well-known mediation formula (expression (3.5)).
The adjustment criterion for natural effects
A complete graphical criterion for identification of P(Y(a, M(a0)) = y)
under NPSEMs by the mediation formula was developed by Shpitser and VanderWeele (2011). They termed expression (3.5) the adjustment formula for natural direct and indirect effects in order to emphasize the restrictiveness of the identification strategy. This criterion generalizes the adjustment criterion (Shpitser et al., 2010), a complete graphical criterion for identification of total treatment effects P(Y(a) = y) by the adjustment formula ∑cP(Y = y|A =a, C=c)P(C =c), as discussed in section 2.4.2 of chapter 2.
Specifically, in order for P(Y(a, M(a0)) =y)to be identified by expres-
sion (3.5) under NPSEMs, this generalized adjustment criterion demands that both P(M(a) = m) and P(Y(a, m) = y)are identifiable by means of adjustment for a common set of measured baseline confounders C. That is, P(M(a) = m) is identified by ∑cP(M = m|A = a, C = c)P(C = c) and P(Y(a, m) = y) by ∑cP(Y = y|A = a, M = m, C = c)P(C = c), implying that P(M(a) = m|C = c) and P(Y(a, m) = y|C = c) in condi- tions (iii) and (iv) are readily identified as P(M = m|A = a, C = c) and P(Y =y|A = a, M =m, C = c), respectively, without needing additional auxiliary covariates for identification.
Intuitively, the adjustment criterion for natural effects can be thought of aiming to establish both cross-world independence (ii) and conditions (v) and (vi) solely by means of adjustment for a common measured covariate set C. First, it demands no unmeasured mediator-outcome confounding, as in Figure 3.2A, which would violate cross-world independence (ii) and, moreover, hamper identification of P(Y(a, m) = y)by means of covariate adjustment. Second, it demands the absence of treatment-induced mediator- outcome confounders, such as L in Figure 3.3A, since the presence of such intermediate confounders would both violate cross-world independence (ii) and hinder the availability of a common set C that enables identification
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of both P(M(a) = m)and P(Y(a, m) = y)by means of adjustment for C.4 Establishing cross-world independence (ii) and conditions (iii) and (iv) by means of the generalized adjustment criterion thus appear to go hand in hand.
Semi-parametric estimators
In section 2.4.3 of chapter 2, we illustrated that rewriting the adjustment formula for treatment effects (in point treatment studies) leads to two semi- parametric estimators. Similar estimators for natural direct and indirect effects have been developed based on the generalized adjustment formula for mediation analyis (or Pearl’s mediation formula) (expression (3.5)).
Ratio-of-mediator-probability-weighting estimator One such estimator
(Hong, 2010; Lange et al., 2012) requires a working model for the mediator distribution P(M|A, C)in order to calculate weights that are based on the ratio of mediator probabilities (under different treatment assignments). It can be seen to arise by rewriting expression (3.5) as follows
E{Y(a, M(a0))} =
∑
c,mE(Y|A= a, M=m, C=c)P(M=m|A= a 0, C=c)P(C=c) =∑
y,c,my·P(Y =y|A= a, M=m, C=c) ×P(M=m|A= a0, C=c)P(C=c, A =a) P(A =a|C =c) =∑
y,c,my·P(Y =y, M=m|A= a, C=c) × P(MP(M==mm|A=a0, C=c) |A =a, C=c) P(C =c|A= a)P(A= a) P(A=a|C =c) =∑
y,c,my·P(Y =y, M=m, C=c|A=a) P(A= a) P(A=a|C =c)4This can be seen upon noting that identification of P(M(a) =m)by covariate adjust-
ment insists L not to be included in C since doing so would amount to adjusting away part of the effect of interest. On the other hand, even though identification of P(Y(a, m) =y)
cannot be obtained by the adjustment criterion, a more general identifying functional for P(Y(a, m) =y)can be shown to require some form of adjustment for L.
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× P(MP(M==mm|A=a0, C=c) |A =a, C=c) =E YI(A = a) P(A=a|C) P(M|A =a0, C) P(M|A=a, C) .Just as for the inverse probability weighted estimator, discussed in sec- tion 2.4.3, one needs to (additionally) weight by the inverse of the probabil- ity of being assigned to treatment arm a, to account for the possibly selective nature of subjects with treatment assignment A =a. One thus additionally needs to fit a propensity score model P(A|C).
Imputation estimator Another estimator (Albert, 2012; Vansteelandt et al.,
2012b) requires an imputation model for the mean outcome given treatment, mediator and covariate set C, to impute counterfactual outcomes under a (possibly) counterfactual treatment assignment A = a. It can be seen to arise by rewriting expression (3.5) as follows
E{Y(a, M(a0))} =
∑
c,mE(Y|A=a, M =m, C =c)P(M=m|A=a 0, C=c)P(C=c) =∑
c,mE(Y|A=a, M =m, C =c) P(M =m, C=c, A =a0) P(A=a0|C =c) =∑
c,mE(Y|A=a, M =m, C =c) P(M =m, C=c|A =a0)P(A =a0) P(A= a0|C =c) =E I(A =a0) P(A =a0|C)E(Y|A= a, M, C) .Similarly, the resulting estimator additionally requires to weight by the inverse of the probability of being assigned to treatment level a0, to account
for the possibly selective nature of subjects with treatment assignment A =a0and hence also requires fitting a propensity score model P(A|C).5
Implementation of these estimators (as well as stratum-specific analogs) 5Technically, such a propensity score model could be avoided by a two-stage imputation
approach. This can be seen upon noting that expression (3.5) can also be rewritten as E{E[E(Y|A=a, M, C)|A=a0, C]|A=a0}.
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will be discussed in more detail in chapter 4. The unbiasedness of these estimators logically depends on whether the used covariate set C satisfies the adjustment criterion relative to both(A, M)and({A, M}, Y).
Increased identification power in observational studies
At least from a theoretical point of view, it can be argued that the adjustment criterion seriously limits the ability to identify P(Y(a, M(a0)) =y). Indeed,
as recently indicated by Pearl (2014), the interventional distributions in (v) and (vi) can be identified under a much wider range of research settings by Shpitser and Pearl (2006a)’s IDC algorithm for conditional treatment effects, which may involve
(a) additional adjustment for separate (but possibly overlapping) covari- ate sets or
(b) mediating instruments6 that enable application of the front-door esti- mator (as discussed in section 2.5.1 in chapter 2).
Pearl (2014) referred to the first identification strategy (a) as piecemeal deconfounding, because it can be regarded as a compromise between iden- tification by the adjustment criterion (which requires identification by ad- justment for a common set of covariates) and more general identification strategies such as (b). More specifically, this ‘divide and conquer’ strategy requires finding a set C that both satisfies (ii) and enables identification of P(M(a0) = m|C =c)and P(Y(a, m) = y|C= c)by means of adjusting for
sufficient adjustment sets Cm and Cy, respectively.7
The increased identification power of these additional strategies is, however, only relevant for observational studies (e.g. Imai et al., 2014), as both (v’) and (vi’) are satisfied by design if treatment is randomized, whereas (vi”) follows from combining either of these two assumptions 6This terminology was used in Pearl (2014) to refer to strong intermediate variables that
fully mediate certain effects whose identification can thus be obtained by the front-door criterion.
7This thus implies that P(M(a0) =m|C=c)is identified by ∑
cmP(M=m|A=a0, C=
c, Cm = cm)P(Cm =cm)and P(Y(a, m) =y|C =c)by ∑cyP(Y = y|A= a, M = m, C =
3
with (ii). This indeed implies that, in experimental studies in which treat- ment is randomized, only cross-world independence (ii) is required.