  Current estimation methods for the probability of causation (PC) make strong
parametric assumptions or are inefficient. We derive a nonparametric
influence-function-based estimator for a projection of PC, which allows for
simple interpretation and valid inference by making weak structural
assumptions. We apply our estimator to real data from an experiment in Kenya,
which found, by estimating the average treatment effect, that protecting water
springs reduces childhood disease. However, before scaling up this
intervention, it is important to determine whether it was the exposure, and not
something else, that caused the outcome. Indeed, we find that some children,
who were exposed to a high concentration of bacteria in drinking water and had
a diarrheal disease, would likely have contracted the disease absent the
exposure since the estimated PC for an average child in this study is 0.12 with
a 95% confidence interval of (0.11, 0.13). Our nonparametric method offers
researchers a way to estimate PC, which is essential if one wishes to determine
not only the average treatment effect, but also whether an exposure likely
caused the observed outcome.
