Bounding the Probability of Causation in Mediation Analysis

Abstract

Given empirical evidence for the dependence of an outcome variable on an exposure variable, we can typically only provide bounds for the "probability of causation" in the case of an individual who has developed the outcome after being exposed. We show how these bounds can be adapted or improved if further information becomes available. In addition to reviewing existing work on this topic, we provide a new analysis for the case where a mediating variable can be observed. In particular we show how the probability of causation can be bounded when there is no direct effect and no confounding.

A Matching Based Theoretical Framework for Estimating Probability of Causation

“…The bounds we obtained for the value of PC is exactly the same as shown by Dawid [1,4]. Pearl uses a different notation, but the values are exactly the same under specific conditions, as was mentioned in [3].…”

“…While Pearl approached the problem from a structural equation modeling perspective, Dawid [9] ap-proached the problems from concept similar to a joint distribution over the different cause-effect situations. In his later works [1,4], Dawid has worked on defining the limits of PC under various confounding and mediation scenarios. Here we are not concerned with the philosophical aspects of causation in objective reality as these, as clearly demonstrated by Kant [10] can never be ascertained with certainty, but with issues arising when applying the formalisms developed in this domain to practical problems.…”

“…If the value of PC is calculated only from the observational data, however, the rate of change is much slower, due in part to the fact that P(x,y) is much smaller for the Non-experimental data. If we change the value under experimental (x ,y) cell from 14 to (14+k), Figure 2 shows how the value of Probability of causation (the minimum limit) changes with k for PN, As a second example, we take the dataset given by Dawid in [4], shown in Table 2. In this case we do not as drastic a change as in the last case, but still PN changes from 1 to 0 in 9 observations, which is still less than 10% of the total observations.…”

“…A measure of the probability for CoE situations was offered by Tian and Pearl [3], which they call probability of necessity (PN); and also by Philip Dawid [1], who call it Probability of Causation. The definition of Probability of Causation/Necessity (in our paper we call it Probability of Causation -PC) is based on a counterfactual definition that is used by both Dawid [1,4], and Pearl [3]. If X and Y are assumed to be binary valued events, the definition of PC then states that given we have observed a positive outcome for both event X and Y, the value of PC is the probability that had the event A been negative, event B would also be negative.…”

Effects of Causes and Causes of Effects

“…which is a testable implication of our assumptions. We For the case that M is observed in the experimental data, but not for Juanita, Dawid et al (2016) showed that this additional information does not change the lower bound l on PC in (30), but does lower the upper bound, u. Dawid & Musio (2021) extend this analysis to cases with additional covariates, while Dawid et al (2019) deal with the case that we have a complete mediation sequence X = M0 → M1 → • • • → Mn−1 → Mn = Y , and know the probabilistic structure of each link in the chain; with all, some, or none of the M s being observed for Juanita.…”