[Paper Review] New bounds for the Probability of Causation in Mediation Analysis

Rossella Murtas, Alexander Philip Dawid+1 more|arXiv (Cornell University)|Jun 15, 2017

Advanced Causal Inference Techniques3 references3 citations

TL;DR

This paper proposes new bounds for the probability of causation (PC) in mediation analysis by incorporating information about a partial mediator, improving upon existing bounds that ignore mediation. By modeling counterfactual distributions of the outcome and mediator under different exposure levels, the authors derive tighter upper bounds that can be either larger or smaller than those ignoring mediation, depending on the data, offering a more nuanced assessment of individual-level causality in causal inference.

ABSTRACT

An individual has been subjected to some exposure and has developed some outcome. Using data on similar individuals, we wish to evaluate, for this case, the probability that the outcome was in fact caused by the exposure. Even with the best possible experimental data on exposure and outcome, we typically can not identify this "probability of causation" exactly, but we can provide information in the form of bounds for it. Under appropriate assumptions, these bounds can be tightened if we can make other observations (e.g., on non-experimental cases), measure additional variables (e.g., covariates) or measure complete mediators. In this work we propose new bounds for the case that a third variable mediates partially the effect of the exposure on the outcome.

Motivation & Objective

• To address the challenge of estimating individual-level causation when only observational or experimental data on exposure and outcome are available.
• To improve existing bounds on the probability of causation by incorporating information about a partial mediator in the causal pathway.
• To provide a framework that yields tighter, more informative bounds for PC when mediator data are available, without assuming complete mediation.
• To clarify when knowledge of partial mediation improves or worsens bounds compared to ignoring it.
• To extend counterfactual-based causal inference to mediation settings with explicit bounds on individual causation.

Proposed method

• The authors use potential outcomes and counterfactuals to model the joint distribution of the outcome Y and mediator M under different exposure levels X.
• They derive upper and lower bounds for the probability of causation (PC) by decomposing the joint counterfactual distribution into terms involving P(Y*(x,m)=0) and P(M(x)=m) for x ∈ {0,1}, m ∈ {0,1}.
• The upper bound is computed as the sum of four terms corresponding to all combinations of M(0) and M(1) with Y*(0,m) and Y*(1,m), under the constraint that only one of Y(0) or Y(1) can be observed per individual.
• The method accounts for partial mediation by allowing Y*(x,m) to differ across m, unlike in complete mediation where Y*(x,0) = Y*(x,1).
• The bounds are compared to the standard bound that ignores the mediator, and the tighter of the two is selected.
• The approach is validated through two empirical examples using experimental data with known mediator probabilities.

Experimental results

Research questions

• RQ1 Can incorporating information about a partial mediator lead to tighter bounds on the probability of causation than ignoring it?
• RQ2 Under what conditions does knowledge of partial mediation improve or degrade the upper bound for PC?
• RQ3 How do the new bounds compare to existing bounds that do not account for mediators in individual-level causality assessment?
• RQ4 What are the theoretical limits of improvement in PC bounds when mediator data are available?
• RQ5 Can the framework be extended to include both covariates and mediators in bounding PC?

Key findings

• In one example with experimental data (Table 3), accounting for the mediator reduced the upper bound for PC from 1.00 to 0.81, demonstrating a meaningful improvement.
• In another example (Table 4), the upper bound increased from 0.88 to 0.95 when including mediator information, showing that mediator data can sometimes worsen the bound.
• The upper bound under partial mediation is never greater than twice the upper bound from ignoring the mediator, due to the constraint that the sum of the four terms cannot exceed 2×P(Y=0|X←0) or 2×P(Y=1|X←1).
• The lower bound remains unchanged when incorporating mediator information, meaning only the upper bound is affected.
• In the case of complete mediation, the upper bound is always less than or equal to the bound obtained by ignoring the mediator, confirming that complete mediation information is never worse than no mediator information.
• The authors conclude that researchers should compute both bounds—ignoring the mediator and accounting for it—and take the smaller upper bound to ensure the tightest possible interval for PC.