General sample size analysis for probabilities of causation: a delta method approach
Authors: Tianyuan Cheng, Ruirui Mao, Judea Pearl, Ang Li
Organizations: Department of Statistics, Florida State University · Department of Computer Science, Cornell University · Department of Computer Science, University of California Los Angeles · Department of Computer Science, Florida State University
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.
Figures & tables
Figure 1: The Causal model for PNS bound experiment. X and Y are binary, and Z is a set of 20 independent binary confounders.
Figure 2: Scatter plots under different sample sizes for Model 1. Left to right: n=120,481,1921 .
Figure 3: Scatter plots under different sample sizes for Model 2. Top to bottom: n=120,481,1921 .
Figure 4: Average error of estimations VS sample size. Left: Model 1, Right:Model 2
Figure 5: Average error with sample size for 20 replicates