1. Identifiability of Path-specific Effects

2. AZT Example Indirect Effect Direct Effect AZT Headaches Pneumonia
Painkillers
Antibiotics
Here is the motivating example for the remainder of the slides. It would be appropriate to discuss here, in English,
the medical story this graph represents. Also mentioning natural direct and indirect effects on this graph would be good as a preliminary.
Survival
Direct Effect
Indirect Effect

3. Path-specific Effects
Antibiotics
AZT
Pneumonia
Headaches
Painkillers
Survival
Antibiotics contribution to the total effect of AZT on survival
AZT
Headaches
Pneumonia
Painkillers
Antibiotics
Here we need to discuss the question which necessitates path-specific effects. In this case, the doctor might be interested in the interrations of AZT and painkillers (resp. antibiotics) on survival. Thus he might want to compare the direct effect of AZT to the path specific effect in the situation where painkillers (resp. antibiotics) are administered as if AZT was _not_ given. It would be good to point out that while these two examples look very similar, one of them will turn out to be problematic, while the other will not.
Survival
Painkiller contribution to the total effect of AZT on survival

4. Problem Statement
Experimental Identifiability – can path-specific effects of interest be uniquely determined from results of experiments.
Experimental non-Identifiability – can there be two causal models which are experimentally indistinguishable while producing different path-specific effects?
Here we state what we are after. Perhaps we can discuss more formally what ‘from experiments’ means (i.e. what the set P* means).

5. Methodology
Path-specific effects we are interested in are expressions in counterfactual logic (with associated probabilities).
We convert these expressions into DAGs with sets of ‘cut’ edges.
Our subsequent reasoning and results are stated in terms of these graphs rather than the underlying logic.
Here we went to some lengths to avoid actually introducing counterfactual logic, but perhaps we should.

6. Effect-invariant Graph Transformations
Here we discuss the two graph transformations which we use in our proof. These rules do not change the path-specific effect. Dashed arrows represent directed _paths_.
Rule 1
Rule 2

7. Main Result Applying the two rules results in one of two cases:
Case 1: we obtain a ‘kite pattern.’ Then the path-specific effect is not identifiable. Z
R - Recanting witness Y

8. Main Result (cont.)
Case 2: all blocked edges emanate from the root node. Then the effect is identifiable. X Y Z W Z’ Z”
Here we can say that identifiable path-specific effects are related to natural direct and indirect effects. All of these effects only
‘cut’ root-emanating edges.

9. AZT Example Revisited
AZT
AZT
Headaches
Pneumonia
Headaches
Pneumonia
Painkillers
Antibiotics
Painkillers
Antibiotics
This slide traces the application of the two rules for the two path-specific effects in the original example graph, showing how we obtain a root-emanating edge in the first case, and a kite pattern in the second case.
Survival
Survival
Painkiller contribution to the total effect of AZT on survival
Antibiotics contribution to the total effect of AZT on survival

10. Recanting Witness P(RX,RX*) is not experimental identifiable
AZT
R-Recanting Witness
Headaches
Pneumonia
Painkillers
Antibiotics
R behave as I
Give the intuition where is the problem with the kite structure.
Survival
Antibiotics contribution to the total effect of AZT on survival
R behave as II
P(RX,RX*) is not experimental identifiable