**5.2 Effect decomposition into path-specific effects**

**5.2.2 Decomposition in a setting with two sequential medi-**

In most mediation analyses, even when interest lies in a single mediator, one cannot ignore the possible presence of multiple mediators, as the following motivating example illustrates.

**Motivating example**

For illustrative purposes, we revisit previous analyses (VanderWeele and Vansteelandt, 2010; Vansteelandt et al., 2012b) on survey data from 5,882 adult respondents from the Large Analysis and Review of European Hous- ing and Health Status (LARES) project conducted by the World Health Organization (Shenassa et al., 2007). These analyses focused on the effect

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A M1 M2 Y C**Figure 5.2:**Causal diagram with two sequential mediators M1and M2.

of living in damp and moldy conditions (binary exposure A) on the risk
of depression (binary outcome Y) and put forward perceived control over
one’s home as a putative mediating mechanism (M). Corresponding natu-
ral direct and indirect effects (via perceived control) were estimated under
the assumption that available individual and housing characteristics (C)
were sufficient to control for confounding so that conditions (i)-(iii) were
met (as reflected by the DAG in Figure 5.1). Kaufman (2010), however,
indicated that mold exposure is likely to also cause physical illness, which
may, in turn, compromise both one’s sense of control and mental health.
This hypothetical scenario (as reflected by the DAG in Figure 5.2) therefore
violates assumption (iv) and thus hinders identification of the targeted nat-
ural effects discussed earlier. It moreover implies that both physical illness
(M1) and perceived control (M2) act as sequential mediators, giving rise to
a finest possible decomposition that involves four distinct pathways from
exposure to outcome (i.e. pathways A_{→}Y, A_{→} M1 →Y, A→ M2 →Y
and A→ M1 → M2→Y).

In the remainder of this section, we first outline a sequential approach that bears close resemblance to VanderWeele and Vansteelandt (2013), start- ing from a coarse two-way decomposition which is next refined into a three-way decomposition. We then demonstrate how natural effect models can be extended to parameterize component effects of the resulting and alternative decompositions and articulate required identification conditions.

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**A sequential approach**

Let Y(a, M1(a0), M2(a0, M1(a0)))be the counterfactual outcome that would
be observed if A were set to a and M1and M2were set to the natural value
they would have taken if A had been a0_{. The first stage then corresponds}

to a two-way decomposition with respect to the joint mediator{M1, M2}, separating pathway A →Y from the remaining pathways as follows:

E{Y(1)−Y(0)}

= E{Y(1, M1(1), M2(1, M1(1)))−Y(1, M1(0), M2(0, M1(0)))} (5.6)
+E_{{}Y(1, M1(0), M2(0, M1(0)))−Y(0, M1(0), M2(0, M1(0)))}. (5.7)
That is, the effect transmitted along either one or both mediators, or so-
called joint natural indirect effect (expression (5.6)), is separated from the
remaining effect through neither of the mediators, or the joint natural direct
effect (expression (5.7)), denoted E_{A}_{→}_{Y}(0, 0)(see Table 5.1).

In a second stage, a more fine-grained, three-way decomposition can be obtained by further partitioning expression (5.6) into the entire effect transmitted along M1and the effect transmitted along M2only, respectively denoted EA→M1Y(1, 1)and EA→M2→Y(1, 0)(see Table 5.1):

E{Y(1, M1(1), M2(1, M1(1)))−Y(1, M1(0), M2(0, M1(0)))}

= E_{{}Y(1, M1(1), M2(1, M1(1)))−Y(1, M1(0), M2(1, M1(0)))} (5.8)
+E{Y(1, M1(0), M2(1, M1(0)))−Y(1, M1(0), M2(0, M1(0)))}. (5.9)
The first contrast (expression (5.8)) captures the notion of activating all
paths along M1that feed into Y, either directly or indirectly via M2, while
blocking all other pathways. It corresponds to the natural indirect effect as
defined with respect to M1(i.e. along the combined pathways A → M1 →Y
and A _{→} M1 → M2 → Y), under the composition assumption that
Y(a, M1(a0), M2(a, M1(a0))) = Y(a, M1(a0)). The second contrast (expres-
sion (5.9)) expresses the so-called semi-natural indirect effect (Pearl, 2014) or
partial indirect effect (Huber, 2014) with respect to M2(i.e. A → M2→Y),
as it only captures part of the effect mediated by M2that bypasses M1.

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E_{A}

_{→}

_{Y}(a0

_{, a}00

_{) =}

_{g E}

_{{}

_{Y(1, M}1(a0), M2(a00, M1(a0)))} −g E{Y(0, M1(a0), M2(a00, M1(a0)))} =

*θ*1+

*θ*4a0+

*θ*5a00+

*θ*7a0a00 EA→M1Y(a, a00) = g E{Y(a, M1(1), M2(a00, M1(1)))} −g E{Y(a, M1(0), M2(a00, M1(0)))} =

*θ*2+

*θ*4a+

*θ*6a00+

*θ*7aa00 EA→M2→Y(a, a0) = g E{Y(a, M1(a0), M2(1, M1(a0)))} −g E{Y(a, M1(a0), M2(0, M1(a0)))} =

*θ*3+

*θ*5a+

*θ*6a0+

*θ*7aa0

**Table 5.1:**Shorthand notation for the component effects from a three-way decom-
position in the presence of two causally ordered mediators M1and M2and their

parameterization in model (5.10), for which the link function g(_{·)}is the identity
link.

Further decomposition will generally fail without imposing strong para- metric constraints, as in the linear SEM framework (Avin et al., 2005) (al- though see Daniel et al. (2015) for a sensitivity analysis approach). Likewise, alternative decompositions of expression (5.6) that involve the natural in- direct effect with respect to M2 (instead of M1; i.e. along the combined pathways A → M2 → Y and A → M1 → M2 → Y) cannot be recovered without making certain no-interaction assumptions (Huber, 2014; Imai and Yamamoto, 2013; Petersen et al., 2006; Robins, 2003; Tchetgen Tchetgen and VanderWeele, 2014). These decompositions are beyond the scope of this chapter (see Technical appendix 5.A.1 for a detailed overview and comparison of targeted decompositions).

**Natural effect models**

Natural effect models can be extended to characterize the three-way decom- position of the previous section. For instance, in the following saturated natural effect model for a binary exposure A

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+*θ*4aa0+*θ*5aa00+*θ*6a0a00+*θ*7aa0a00,
(5.10)
for a, a0 _{and a}00 _{equal to 0 or 1, the total effect, ∑}7

i=1*θ*i, can be partitioned
into the joint natural direct effect

E_{A}_{→}_{Y}(0, 0) = *θ*1,

and the joint natural indirect effect

EA→M1Y(1, 1) +EA→M2→Y(1, 0) =

7

### ∑

i=2
*θ*i.

The latter can be further partitioned into the natural indirect effect with respect to M1

EA→M1Y(1, 1) = *θ*2+*θ*4+*θ*6+*θ*7,

and the partial indirect effect with respect to M2(see Table 5.1)

EA→M2→Y(1, 0) = *θ*3+*θ*5.

Model (5.10) is a special case of the wider class of generalized linear natural effect models for three-way decomposition

E{Y(a, M1(a0), M2(a00, M1(a0)))|C∗} = g−1{*θ*>W(a, a0, a00, C∗)},
with W(a, a0_{, a}00_{, C}∗_{)}_{a known vector with components that may depend on}

a, a0_{, a}00 _{and (possibly) covariates C}∗_{.}

*Different ways of accounting for the interaction terms θ*4 *to θ*7 yield
*another five possible decompositions, listed in Table 5.2. For instance, θ*4
can be apportioned to either E_{A}_{→}_{Y}or E_{A}_{→}_{M}_{1}_{Y}*. Similarly, θ*5can be appor-
tioned to EA→Y or EA→M2→Y*, θ*6 to EA→M1Y or EA→M2→Y *and θ*7to either

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(1) E_{A}

_{→}

_{Y}(0, 0) +E

_{A}

_{→}

_{M}

_{1}

_{Y}(1, 1) +E

_{A}

_{→}

_{M}

_{2}

_{→}

_{Y}(1, 0) = (θ1) + (θ2+

*θ*4+

*θ*6+

*θ*7) + (θ3+

*θ*5) (2) EA→Y(1, 1) +EA→M1Y(0, 0) +EA→M2→Y(0, 1) = (θ1+

*θ*4+

*θ*5+

*θ*7) + (θ2) + (θ3+

*θ*6) (3) EA→Y(0, 0) +EA→M1Y(1, 0) +EA→M2→Y(1, 1) = (θ1) + (θ2+

*θ*4) + (θ3+

*θ*5+

*θ*6+

*θ*7) (4) E

_{A}

_{→}

_{Y}(1, 1) +E

_{A}

_{→}

_{M}

_{1}

_{Y}(0, 1) +E

_{A}

_{→}

_{M}

_{2}

_{→}

_{Y}(0, 0) = (θ1+

*θ*4+

*θ*5+

*θ*7) + (θ2+

*θ*6) + (θ3) (5) E

_{A}

_{→}

_{Y}(0, 1) +E

_{A}

_{→}

_{M}

_{1}

_{Y}(1, 1) +E

_{A}

_{→}

_{M}

_{2}

_{→}

_{Y}(0, 0) = (θ1+

*θ*5) + (θ2+

*θ*4+

*θ*6+

*θ*7) + (θ3) (6) EA→Y(1, 0) +EA→M1Y(0, 0) +EA→M2→Y(1, 1) = (θ1+

*θ*4) + (θ2) + (θ3+

*θ*5+

*θ*6+

*θ*7)

**Table 5.2:**All six possible three-way decompositions and their parameterization in
model (5.10). Each component on the lefthand side of the equation is represented by
a linear combination of parameters on the righthand side (grouped in parentheses).

on the first two decompositions in Table 5.2 as their sequential approach
builds on identification of E_{A}_{→}_{Y}(0, 0)and E_{A}_{→}_{M}_{1}_{Y}(1, 1), as outlined in the
previous section. The remaining four decompositions involve instances
of EA→Y(a0, a00)with a0 6= a00, and instances of EA→M1Y(a, a00)with a 6= a00,

which require slightly stronger identification assumptions, as discussed next.

**Identification**

Two-way decomposition into joint natural direct and indirect effects can be obtained if assumptions (5.1)-(5.4) hold with respect to the joint mediator {M1, M2}. We refer to the corresponding conditions in NPSEMs as (i’)-(iv’).

Such first-stage decomposition can be obtained for the DAG in Figure 5.2, but also for the DAGs in Figures 5.3A and 5.3B. This may come as a sur- prise since the effect of M1on M2is confounded either by an unmeasured

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confounder U (Figure 5.3A) or (measured) intermediate confounder L (Fig- ure 5.3B). However, this does not hinder identification of the joint natural direct and indirect effect because (i’)-(iv’) do not impose restrictions on the structural relation between the mediators. The other DAGs, however, do not enable such two-way decomposition. In Figures 5.3C and 5.3D, (ii’) and (iv’) are violated because of unmeasured confounding by U and intermediate confounding by L, respectively.

All six three-way decompositions in Table 5.2 can be recovered under NPSEMs if, in addition to (i’)-(iv’), (v’) the effect of M1 on M2 is uncon- founded within strata of{A, C}and (vi’) none of the M1−M2confounders are affected by exposure. In contrast to assumptions (i’)-(iv’), (v’) and (vi’) do not allow for unmeasured or intermediate confounding of the effect of M1on M2. Consequently, these assumptions are violated in all discussed DAGs (except the one in Figure 5.2). However, decomposition with respect to the three sequential mediators L, M1 and M2 becomes possible under

(A) A M1 M2 Y C U (C) A M1 M2 Y C U (B) A M1 M2 Y C L (D) A M1 M2 Y C L

**Figure 5.3:** Causal diagrams with two sequential mediators M1and M2and un-

measured confounder U (in panels A and C) or intermediate confounder L (in panels B and D).

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more general identification conditions for multiple mediators (see Technical appendix 5.A.2).

Finally, consistent with VanderWeele and Vansteelandt (2013), we show in Technical appendix 5.A.2.5 that the first two decompositions in Table 5.2 necessitate slightly weaker assumptions than (i’)-(vi’). In Technical ap- pendix 5.A.2, we also provide a more detailed and formal discussion of identification assumptions, as well as extensions to more than two medi- ators. Importantly, we generalize the adjustment criterion for two-way decomposition in a single mediator setting (Shpitser and VanderWeele, 2011) to (k+1)-way decompositions in settings with k causally ordered mediators.