The notion of the total effect decomposition into the natural direct effect and the natural indirect effect can be facilitated by intuitively understanding the nested counterfactual Y(1,M(0)).

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Counterfactual definitions
How is mediation formalized in the counterfactual framework?
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2. /43
Counterfactual outcome
Yi(a,m)
What would happen to you if you were forced to receive treatment value a and mediator value m?
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Nested (composite) counterfactual outcome
Yi(a,Mi(a*))
What would happen to you if you were forced to receive treatment value a and mediator value M(a*)?
M(a*) is your natural value of the mediator under treatment value a*.
When a = a* the counterfactual reduces to Yi(a) (causal consistency assumption).
When a ≠ a* the counterfactual is a cross-world counterfactual. 7

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Natural effect decomposition
Yi(1,Mi(1)) − Yi(0,Mi(0))
Yi(1,Mi(1)) − Yi(1,Mi(0)) + Yi(1,Mi(0)) − Yi(0,Mi(0))
Yi(1) − Yi(0)
Total Effect
Total Effect
Natural Indirect Effect Natural Direct Effect
Total Effect (as nested counterfactuals)
=
=
8

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Interpretation
Pearl. Psychol Methods 2014;19:459
Naimi et al. Int J Epidemiol 2014;43:1656
Natural Direct Effect: Yi(1,Mi(0)) − Yi(0,Mi(0)) Natural Indirect Effect: Yi(1,Mi(1)) − Yi(1,Mi(0))
- change in the outcome when the
mediator changes as though the
exposure had (but when, in actuality,
the exposure hadn’t)
- change in the outcome if we were to
‘freeze’ the mediator value for each
person at the level it would have
taken had the person’s exposure
been some referent level (but when,
in actuality, the person’s exposure
status changes)
9
Characterizing Yi(a,Mi(a*)) more concretely may help…

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Treatment
Assigned to drug
21 3 4 5
76 8 9 10
Yoshida & Desai A&R 2019 (modified)
Treatment
Assigned to placebo
21 3 4 5
76 8 9 10 Treated world
Untreated world

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Treatment
Assigned to drug
Mediator
Assigned to biomarker as if drug
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
Yoshida & Desai A&R 2019 (modified)
Treatment
Assigned to placebo
Mediator
Assigned to biomarker as if placebo
21 3 4 5
76 8 9 10
11
Mediator under
no treatment
Mediator under
treatment

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Treatment
Assigned to drug
Mediator
Assigned to biomarker as if drug
Outcome
Gout flares
Scenarios
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
Yoshida & Desai A&R 2019 (modified)
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
Treatment
Assigned to placebo
Mediator
Assigned to biomarker as if placebo
Outcome
Gout flares
21 3 4 5
76 8 9 10
E[Y(0,M(0))] = E[Y(0)]
E[Y(1,M(1))] = E[Y(1)]
12
Outcome under no
treatment
Outcome under
treatment

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Treatment
Assigned to drug
Mediator
Assigned to biomarker as if drug
Outcome
Gout flares
Scenarios
Total
Effect
Residual Risk
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
Yoshida & Desai A&R 2019 (modified)
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
Treatment
Assigned to placebo
Mediator
Assigned to biomarker as if placebo
Outcome
Gout flares
21 3 4 5
76 8 9 10
E[Y(0,M(0))] = E[Y(0)]
E[Y(1,M(1))] = E[Y(1)]
13

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Scenarios
Total
Effect
Residual Risk
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
Yoshida & Desai A&R 2019 (modified)
Identical
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
Treatment
Assigned to drug
Mediator
Assigned to biomarker as if drug
Outcome
Gout flares
Treatment
Assigned to placebo
Mediator
Assigned to biomarker as if placebo
Outcome
Gout flares
Exposure
Assigned to drug
E[Y(0,M(0))]
E[Y(1,M(1))]
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Take treatment
from treated world

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Scenarios
Total
Effect
Residual Risk
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
Yoshida & Desai A&R 2019 (modified)
Identical
Identical
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
Exposure
Assigned to placebo
Exposure
Assigned to drug
Mediator
Assigned to biomarker as if placebo
Mediator
Assigned to biomarker as if placebo
Exposure
Assigned to drug
Mediator
Assigned to biomarker as if drug
Outcome
Gout flares
Outcome
Gout flares
E[Y(0,M(0))]
E[Y(1,M(1))]
15
Take mediators
from untreated world

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Scenarios
Total
Effect
Residual Risk
Identical
Identical
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
Yoshida & Desai A&R 2019 (modified)
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
Exposure
Assigned to placebo
Exposure
Assigned to drug
Mediator
Assigned to biomarker as if placebo
Mediator
Assigned to biomarker as if placebo
Exposure
Assigned to drug
Mediator
Assigned to biomarker as if drug
Outcome
Gout flares
Outcome
Gout flares
Outcome
Gout flares
E[Y(0,M(0))]
E[Y(1,M(0))]
E[Y(1,M(1))]
16
Hopefully, we get
some intermediate

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Scenarios
Natural
Direct
Effect
Natural
Indirect
Effect
Total
Effect
Residual Risk
Identical
Identical
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
Yoshida & Desai A&R 2019 (modified)
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
21 3 4 5
76 8 9 10
Exposure
Assigned to placebo
Exposure
Assigned to drug
Mediator
Assigned to biomarker as if placebo
Mediator
Assigned to biomarker as if placebo
Exposure
Assigned to drug
Mediator
Assigned to biomarker as if drug
Outcome
Gout flares
Outcome
Gout flares
Outcome
Gout flares
E[Y(0,M(0))]
E[Y(1,M(0))]
E[Y(1,M(1))]
17
TE partitioned
nicely!