Pure Mediation
Why Is Mediation Challenging?
- Most conventional causal questions explore two possible worlds, with a treatment assigned to A=0 or A=1.
Why Is Mediation Challenging?
Most conventional causal questions explore two possible worlds, with a treatment assigned to A=0 or A=1.
Mediation involves each combination of the original treatment (A=0, A=1) with the mediator levels (M=0, M=1).
Why Is Mediation Challenging?
Most conventional causal questions explore two possible worlds, with a treatment assigned to A=0 or A=1.
Mediation involves each combination of the original treatment (A=0, A=1) with the mediator levels (M=0, M=1).
Controlling both the treatment and the mediator may feel unnatural:
\(CDE(m) = Y | [do(A=1),do(m)] - Y|[do(A=0),do(m)]\)
Why Is Mediation Challenging?
Most conventional causal questions explore two possible worlds, with a treatment assigned to A=0 or A=1.
Mediation involves each combination of the original treatment (A=0, A=1) with the mediator levels (M=0, M=1).
Controlling both the treatment and the mediator may feel unnatural:
\(CDE(m) = Y | [do(A=1),do(m)] - Y|[do(A=0),do(m)]\)
- If that is not strange enough, most mediator interventions are not well defined. If smoking (A) leads to TAR (M) which leads to COPD (Y), how can we manipulate TAR independently? One might imagine small creatures cleaning TAR from the lungs (M=0) or adding TAR (M=1).
Why Is Mediation Helpful?
\(CDE(m) = Y | [do(A=1),do(m)] - Y|[do(A=0),do(m)]\)
Mediation moves us closer to “how nature works” by making causal paths explicit, our narrative becomes more aligned with domain expertise. “Smoking causes COPD” is less informative than “Smoking causes COPD by increasing TAR in the lungs”.
Pure mediation sets clear boundaries on which causal paths are allowed.
CDE under Pure Mediation
\(CDE(m) = Y | [do(A=1),do(m)] - Y|[do(A=0),do(m)]\)
Assuming pure mediation:
\(CDE(m) = Y |do(m) - Y|do(m) = 0\)
- Under pure mediation, a controlled direct effect of zero means that all harm from cigarettes is blocked once we fix TAR.
CDE under Pure Mediation
\(CDE(m) = Y | [do(A=1),do(m)] - Y|[do(A=0),do(m)]\)
Assuming Pure Mediation:
\(CDE(M=0) = Y |do(M=0) - Y|do(M=0) = 0\)
Under pure mediation, a controlled direct effect of zero means that all harm from cigarettes is blocked once we fix TAR.
If a small creature cleaned all the TAR from the lungs, smoking would not cause COPD.
CDE under Pure Mediation
\(CDE(m) = Y | [do(A=1),do(m)] - Y|[do(A=0),do(m)]\)
Assuming Pure Mediation:
\(CDE(M=1) = Y |do(M=1) - Y|do(M=1) = 0\)
Under pure mediation, a controlled direct effect of zero means that all harm from cigarettes is blocked once we fix TAR.
If a small creature cleaned all the TAR from the lungs, smoking would not cause COPD.
If a small creature added TAR to the lungs, avoiding smoking would not prevent COPD.
NDE under Pure Mediation
\(NDE = Y|[do(A=1),do(M|do(A=0))] - Y|[do(A=0),do(M|do(A=0))]\)
- Under pure mediation, the NDE blocks all the harm caused by cigarettes by setting the mediator to the value it would take under no smoking, while comparing smoking versus non-smoking with this manipulated M.
NDE under Pure Mediation
\(NDE = Y|[do(M|do(A=0))] - Y|[do(M|do(A=0))] = 0\)
Under pure mediation, the NDE blocks all the harm caused by cigarettes by setting the mediator to the value it would take under no smoking, while comparing smoking versus non-smoking with this manipulated M.
Under pure mediation, once we know the value of the mediator M, the original treatment A no longer matters. The natural direct effect may feel abstract, but once TAR (M) is fixed, it does not matter whether the patient actually smoked.
NIE under Pure Mediation
\(NIE = Y|[do(A=0),do(M|do(A=1))] - Y|[do(A=0),do(M|do(A=0))]\)
- Under pure mediation, the NIE captures all the harm caused by cigarettes by manipulating TAR (the mediator) to the values it would take under smoking versus not smoking.
NIE under Pure Mediation
\(NIE = Y|[do(M|do(A=1))] - Y|[do(M|do(A=0))]\)
Under pure mediation, the NIE captures all the harm caused by cigarettes by manipulating TAR (the mediator) to the values it would take under smoking versus not smoking.
Again, the original treatment assignment A becomes irrelevant.
😇 God-Given Counterfactual Data
We can now explore how the components of the Natural Indirect Effect:
\[Y|[do(M|do(A=0))] = Y|do(A=0)\]
😇 God-Given Counterfactual Data
We can now explore how the components of the Natural Indirect Effect:
\[Y|[do(M|do(A=1))] = Y|do(A=1)\]
NIE under Pure Mediation
\(NIE = Y|do(A=1) - Y|do(A=0) = TE\)
Finally, we see that the Natural Indirect Effect equals the Total Effect, because under pure mediation the entire effect of the treatment on the outcome is transmitted through the mediator.