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 general sample size framework based on the delta method. Our approach applies to settings in which the target bounds of PoCs can be expressed as finite minima or maxima of linear combinations of experimental and observational probabilities. Through simulation studies, we demonstrate that the proposed sample size calculations lead to stable estimation of these bounds.

翻译：因果概率（PoCs），如必要性与充分性概率（PNS），是决策制定的重要工具，但通常无法实现点识别。现有研究通过结合实验数据与观测数据推导了这些量的边界。然而，关于样本量分析——即需要多少实验与观测样本才能达到期望的误差范围——的研究极为有限。本文提出了一种基于Delta方法的通用样本量分析框架。该方法适用于因果概率的目标边界可表示为实验概率与观测概率线性组合的有限最小值或最大值的情形。通过模拟研究，我们证明了所提出的样本量计算方法能够实现这些边界的稳定估计。