Probabilities of Causation for Continuous Outcomes: Bounds and Identification

2026-05-03

AI Summary

This study addresses the limitation of existing causal attribution methods, which are largely confined to binary or ordinal outcomes and struggle to handle probabilities of necessity under continuous outcomes. To bridge this gap, the paper proposes a Generalized Probability of Necessity (GPN) framework that establishes, for the first time, a formal causal attribution system tailored to continuous outcomes. Under standard ignorability and monotonicity assumptions, the authors derive sharp bounds for GPN using partial identification theory and innovatively incorporate a Copula model to characterize the dependence structure between potential outcomes, thereby tightening these bounds. Both simulation studies and empirical analyses demonstrate that the proposed approach substantially outperforms conventional methods that ignore dependence information, yielding more precise estimates of causal necessity probability bounds for continuous outcomes.

Abstract

The probability of necessity (PN), which quantifies the probability that an observed event would not have occurred in the absence of the treatment, is a central estimand in attribution analysis. While PN has been extensively studied for binary outcomes and has recently been developed for ordinal outcomes, a formal framework for continuous outcomes remains underdeveloped. To address this gap, we propose the general probability of necessity (GPN) for continuous outcomes, a setting that is substantially more challenging than the binary and ordinal cases. Rather than imposing strong identifiability assumptions, we adopt a partial identification perspective and derive sharp lower and upper bounds under standard assumptions of ignorability and monotonicity. We further introduce a copula-based framework that exploits dependence information between potential outcomes to tighten these bounds. Simulation studies and real-world applications demonstrate the effectiveness of our method.

Problem

• probability of causation
• continuous outcomes
• probability of necessity
• partial identification
• causal inference

Innovation

• probability of necessity
• continuous outcomes
• partial identification
• copula
• causal inference