Probabilities of Causation for Continuous Outcomes: Bounds and Identification

TL;DR

This paper introduces the general probability of necessity (GPN) for continuous outcomes, providing bounds and a copula-based framework to improve causal attribution analysis without strong assumptions.

Contribution

It develops a formal framework for GPN in continuous outcomes, deriving sharp bounds and leveraging dependence information to tighten these bounds.

Findings

Derived sharp bounds for GPN under standard assumptions.

Introduced a copula-based method to tighten bounds using dependence information.

Validated the approach through simulations and real-world data.

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.