Measures for Assessing Causal Effect Heterogeneity Unexplained by Covariates

TL;DR

The paper addresses causal effect heterogeneity that remains after conditioning on covariates by introducing P-CACE and N-CACE for binary X with continuous Y, and P-CPICE and N-CPICE for continuous X with continuous Y under stochastic interventions. It develops principled identifications and sharp bounds for these measures within an SCM framework, and shows how CACE decomposes into positive and negative components, with a binary outcome recovery to THR/TBR. Through both theoretical results and a real-world medical dataset, the authors demonstrate that substantial heterogeneity can exist even when CACE is positive, highlighting the value of examining positive/negative subpopulations and stochastic-intervention effects for personalized interventions. The work thus broadens the causal-heterogeneity toolbox, offering identification, bounding, estimation guidance, and practical illustrations for nuanced policy decisions. Overall, the measures enable a more granular understanding of who benefits or suffers from an intervention, guiding targeted and tempered implementation in practice.

Abstract

There has been considerable interest in estimating heterogeneous causal effects across individuals or subpopulations. Researchers often assess causal effect heterogeneity based on the subjects' covariates using the conditional average causal effect (CACE). However, substantial heterogeneity may persist even after accounting for the covariates. Existing work on causal effect heterogeneity unexplained by covariates mainly focused on binary treatment and outcome. In this paper, we introduce novel heterogeneity measures, P-CACE and N-CACE, for binary treatment and continuous outcome that represent CACE over the positively and negatively affected subjects, respectively. We also introduce new heterogeneity measures, P-CPICE and N-CPICE, for continuous treatment and continuous outcome by leveraging stochastic interventions, expanding causal questions that researchers can answer. We establish identification and bounding theorems for these new measures. Finally, we show their application to a real-world dataset.

Figures (3)

• Figure 1: (Decomposition by P-CACE and N-CACE.) The solid line is CACE, the dotted line is N-CACE, and the dashed line is P-CACE varying from $W=0$ to $W=10$. The x-axis represents the value of $W$, and the y-axis represents the values of each measure.
• Figure 2: (Decomposition by $\text{\normalfont THR}_c$ and $\text{\normalfont TBR}_c$.) The dotted line is $\int_{0}^{100}\text{\normalfont TBR}_c(w)dc$, and the dashed line is $\int_{0}^{100}\text{\normalfont THR}_c(w)dc$. The x-axis means the value of $W$, and the y-axis means the values of each measure.
• Figure 3: Illustration of P-CACE and N-CACE. For the conditional CDF $\mathbb{P}(Y<y|X=0,W=w)$ and $\mathbb{P}(Y<y|X=1,W=w)$ shown as the solid and dashed line respectively, P-CACE$(w)$ is given by the light gray region and N-CACE$(w)$ by the dark gray region. The difference of the light and dark gray region gives CACE$(w)$.

Theorems & Definitions (44)

• Definition 1
• Definition 2: Potential outcome types for a continuous outcome
• Lemma 1
• Definition 3: P-CACE and N-CACE
• Proposition 1
• Theorem 1
• Theorem 2
• Theorem 3
• Definition 4: Potential outcome types for a continuous outcome under stochastic interventions $(X^{\pi_0},X^{\pi_1})$
• Lemma 2