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In search of operational definitions

3.6 From mediating instruments to conceptual clarity

3.6.1 In search of operational definitions

When it comes to the interpretation of natural direct effects, critics adhering to the slogan ‘no causation without manipulation’ have repeatedly empha- sised the operational question of how exactly one may go about blocking the treatment’s effect on the mediator, in order to recover M(0)in treated

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subjects, without affecting the direct path from treatment to outcome (e.g. Didelez et al., 2006). Inevitably, any answer to this question invokes a me- diating instrument, such as L in Figure 3.5A, that can be intervened on in order to prevent treatment from exerting its effect on the mediator.

In order to avoid interpretational ambiguities or strong, untestable cross- world assumptions, critics have therefore proposed to, instead, study ‘ma- nipulable’ causal quantities, such as controlled direct effects (e.g. Naimi et al., 2014a), which express what the treatment effect would have been if the mediator were kept fixed at some predetermined level m uniformly in the population. As opposed to natural effects’ descriptive formulation, their prescriptive interpretation has been claimed to be more directly informative as to potential policy implications of certain interventions. However, apart from the fact that, in many settings, the mediator may often be difficult to manipulate, when aiming to decompose the treatment effect into a mediated and unmediated component, general attempts to define the controlled direct effect’s counterpart, a so-called ‘controlled indirect effect’, have stumbled upon similar operational difficulties. Whereas some would simply define such an effect as the difference between treatment effect and the controlled direct effect, such a definition is not entirely satisfactory, especially in the presence of treatment-mediator interactions, in which case, their interpre- tation may be highly ambiguous. Without the introduction of a mediating instrument, such as Z in Figure 3.5B, it is indeed difficult to conceive of an intervention that would block only the direct path from treatment to outcome (also see VanderWeele, 2011b).

3.6.2 Deterministic expanded graphs

It thus seems that mediating instruments provide some sort of necessary extension to the original causal diagram12 that allows for interventionist – some may say, empirically meaningful – interpretations of natural ef- fects. Moreover, they are key for a clear operational definition of controlled 12Note that the expanded graphs with mediating instruments, depicted in Figure 3.5, can

be marginalized over L and/or Z to result in the original causal diagram in Figure 3.2A, only if the above exclusion restrictions pertaining to L and/or Z hold, such that neither L nor Z is a common cause of any two variables on the original graph in Figure 3.2A.

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indirect effects. The conceptual notion of an expanded graph with two mediating instruments, as depicted in Figure 3.5C, corresponds very closely to what has been described in Robins and Richardson (2010).

In this – possibly unrealistic – setting, Shpitser (2013)’s result tells us that identification of P(Y(a, M(a0)) = y)may be obtained – provided (viii)

holds and L and Z are truly mediating instruments – if L and Z are in separate districts. If not, their district would be recanting and identification would fail. The associated cross-world assumption – i.e. Z(a)

⊥⊥

L(a0)

– indeed formalizes the need for no unmeasured confounding between the two instruments. However, this can never be guaranteed unless, as postulated by Robins and Richardson (2010), both L and Z are deterministic functions of (a randomized) treatment A. In that case, both Z(a)and L(a0)

are constant, and hence trivially independent.13

Ironically, this required determinism seems to leave us incapable of pulling apart the causal pathways that we meant to assess in the first place, which brings us back to square one. However, progress can be made if one can conceive of separate interventions on L and Z that would enable to break their perfect correlation. From this perspective, the deterministic characterization of an expanded graph such as Figure 3.5C, gives rise to a specific type of experimental design that requires one to think of L and Z as inherent, but distinct, properties of the treatment, which may be intervened on separately, but when combined, fully capture all of its active ingredients; see section 3.6.3 for an example. The feasibility of such designs thus pri- marily mirrors the extent to which different active components of treatment or exposure can be conceived of being manipulated in isolation (Didelez, 2013b).14 Moreover, when combined with the aforementioned exclusion 13In addition, as in Robins and Richardson (2010), independence of Z(a)and L(a0)can

be shown to lead to cross-world assumption (ii).

14One may (justly) claim that, in theory, only one of the mediating instruments would

need to be a deterministic function of treatment. However, it seems difficult to conceptually conceive of any scenario where only one of the mediating instruments would be fully determined by treatment. Try, for instance, to imagine manipulating A, while L is fully determined by A. Nonetheless, one needs to be able to manipulate L, which captures an inherent aspect of A, while leaving untouched all other aspects of A. This seems to necessarily imply that all other inherent aspects of A need to be captured by another deterministic variable, which would then naturally lead to Z.

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restrictions, such designs thus entail separate manipulations of L and Z, which capture distinct but exhaustive features of the treatment to which, respectively, solely M and Y are (directly) responsive. Importantly, this characterization enables to interpret natural effects as specific interventional contrasts.

Consider, for instance, the causal diagram in Figure 3.2A and its expan- sion in Figure 3.5C, where Z and L are deterministic functions of treatment which can be conceived as two complementary components that fully char- acterise treatment such that A = {L, Z}, a = {la, za} and a0 = {la0, za0}.

Then identification of P(Y(a, M(a0)) = y) under the NPSEM associated

with the causal diagram in Figure 3.5C is tantamount to identification of the interventional distribution P(Y(za, la0) = y)since

P(Y(a, M(a0)) = y) =

l,z,m

P(Y =y, M=m|do(L=l, Z =z))

×P(L =l|do(A=a0))P(Z=z|do(A= a)) =

l,z,m P(Y =y|L =l, Z=z, M=m)P(M =m|L =l) ×P(L =l|A= a0)P(Z =z|A= a) (3.9) =

l,z

P(Y =y|L=l, Z=z)I(L=la0)I(Z =za)

=P(Y=y|L=la0, Z=za)

=P(Y(za, la0) =y).

The first equality holds by expression (3.6), the second by Tian’s ID al- gorithm and the third by the conditional independence M

⊥⊥

Z|L and determinism.

The mere feasibility of such an experimental design and the plausibility of the two aforementioned exclusion restrictions thus provide the necessary context for interpreting natural effects as manipulable and hence – as critics may claim – policy-relevant parameters (Robins and Richardson, 2010). In other words, instead of actually conducting such an experiment, one could estimate natural effects based on available (experimental or observational) data, provided (vii) and (viii) hold, while the construction of a scientifically plausible story – encoded in a deterministic extended graph – then serves

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to license their interventional interpretation. Moreover, the problem of identification of natural effects from available data on observable variables represented in a deterministic extended graph is thus effectively reduced to one of identification of the effect of a joint intervention on {L, Z}, to which the usual calculus for joint treatment effects (Tian and Pearl, 2003) is applicable, which typically avoids reliance on cross-world assumptions.15