Identifying and bounding the probability of necessity for causes of effects with ordinal outcomes

TL;DR

This paper introduces a new approach to measure and bound the probability of necessity in causal inference for ordinal outcomes, expanding beyond binary cases and providing practical assumptions and bounds.

Contribution

It proposes a novel definition and identification strategy for the probability of necessity with ordinal outcomes, including bounds when assumptions fail.

Findings

Introduces a monotonic incremental treatment effect assumption.

Provides explicit formulas for sharp bounds on probability of necessity.

Discusses testable implications of the identification assumptions.

Abstract

Although the existing causal inference literature focuses on the forward-looking perspective by estimating effects of causes, the backward-looking perspective can provide insights into causes of effects. In backward-looking causal inference, the probability of necessity measures the probability that a certain event is caused by the treatment given the observed treatment and outcome. Most existing results focus on binary outcomes. Motivated by applications with ordinal outcomes, we propose a general definition of the probability of necessity. However, identifying the probability of necessity is challenging because it involves the joint distribution of the potential outcomes. We propose a novel assumption of monotonic incremental treatment effect to identify the probability of necessity with ordinal outcomes. We also discuss the testable implications of this key identification assumption. When it fails, we derive explicit formulas of the sharp large-sample bounds on the probability of necessity.