Sample Size Design for Bounds on Discrete Probabilities of Causation

Probabilities of causation (PoCs), such as the probability of necessity and sufficiency (PNS), are important tools for decision making but are generally not point identifiable. Existing work has derived bounds for these quantities using combinations of experimental and observational data. However, there is very limited research on sample size analysis, namely, how many experimental and observational samples are required to achieve a desired margin of error. In this paper, we propose a sample-size design framework for PoC bounds that can be expressed as finite minima or maxima of smooth functions of experimental and observational probabilities — a representation that covers binary PNS, PN, and PS, as well as representative multi-valued PoC families recently shown to subsume the discrete PoC bounds in the current literature. Our framework gives endpoint-specific sample-size formulas that account for the joint covariance structure of the probability estimates: a distribution-free rule for affine bounds, and a pilot-based plug-in rule with theoretical safety guarantees for ratio-type bounds. Simulations show that the proposed rules achieve the target precision and coverage, recover a much smaller size requirement for binary PNS than the existing calculation, and provide sample-size designs for other discrete PoCs where no prior general sample-size formula is available.