Cross-World Assumption and Refining Prediction Intervals for Individual Treatment Effects

Abstract

While average treatment effects (ATE) and conditional average treatment effects (CATE) provide valuable population- and subgroup-level summaries, they fail to capture uncertainty at the individual level. For high-stakes decision-making, individual treatment effect (ITE) estimates must be accompanied by valid prediction intervals that reflect heterogeneity and unit-specific uncertainty. However, the fundamental unidentifiability of ITEs limits the ability to derive precise and reliable individual-level uncertainty estimates. To address this challenge, we investigate the role of a cross-world correlation parameter, $ ρ(x) = cor(Y(1), Y(0) | X = x) $, which describes the dependence between potential outcomes, given covariates, in the Neyman-Rubin super-population model with i.i.d. units. Although $ ρ$ is fundamentally unidentifiable, we argue that in most real-world applications, it is possible to impose reasonable and interpretable bounds informed by domain-expert knowledge. Given $ρ$, we design prediction intervals for ITE, achieving more stable and accurate coverage with substantially shorter widths; often less than 1/3 of those from competing methods. The resulting intervals satisfy coverage guarantees $P\big(Y(1) - Y(0) \in C_{ITE}(X)\big) \geq 1 - α$ and are asymptotically optimal under Gaussian assumptions. We provide strong theoretical and empirical arguments that cross-world assumptions can make individual uncertainty quantification both practically informative and statistically valid.

Figures (9)

• Figure 1: Difference between perfectly negatively dependent ($\rho = -1$) and perfectly positively dependent ($\rho = 1$) potential outcomes. In both cases, the observed variables are the same. However, much larger prediction intervals for ITEs are needed when $\rho = -1$, while $\rho = 1$ leads to the narrowest possible intervals. The solid lines represent conditional expected values.
• Figure 2: Synthetic $d=1$ dataset described in Section \ref{['Section5.1']}. The variance of ITE drastically decreases with larger $\rho$, and our $CW(\rho)$ intervals seem to have correct coverage. We also include exact and inexact prediction method from Lei_2021_Conformal_Inference as a comparison. Note that the observed values (treated and untreated units) are the same on all four figures.
• Figure 3: Box plots comparing the coverage and average interval width of different estimation methods across cross-world correlation levels $\rho \in \{-1, -0.5, 0, 0.5, 1\}$, based on 50 synthetic datasets with $n = 2000$ and $d=1$ and $d=15$. Methods include Lei’s exact and inexact intervals Lei_2021_Conformal_Inference, the conformal DR-meta-learner Alaa_2023_Conformal_Meta_learners, CMC jonkers2024conformalconvolutionmontecarlo, and three versions of our $CW(\rho)$ intervals: the baseline $CW(\rho)$, a misspecified variant $CW(\text{misspec.}\rho)$ using $\tilde{\rho} = \rho - 0.25$ (capped at $-1$), and one with added conformal confidence intervals ($CW^{+CI}(\rho)$). The vertical line marks the desired $90\%$ coverage.
• Figure 4: Average gap between the true cross-world correlation and its estimator as a function of the predictive power of the auxiliary variable $\gamma$. The left panel reports $\mathrm{GAP}=\mathbb{E}_X\!\left[\rho(X)-\widehat{\tilde{\rho}}(X)\right]$, while the right panel reports $\mathrm{GAP}=\mathbb{E}_X\!\left[\rho(X)-\widehat{\rho}(X)\right]$. Shaded bands are $95\%$ confidence intervals computed from repeated simulations as described in Appendix A; the horizontal line at $0$ indicates no bias.
• Figure 5: Average gap between the true cross-world correlation and its estimator as a function of the predictive power of the auxiliary variable $\gamma$ and dimension of the auxiliary variable $Z$. The upper panels report $\mathrm{GAP}(\gamma)=\mathbb{E}_X\!\left[\rho(X)-\widehat{\tilde{\rho}}(X)\right]$, while the lower panels report $\mathrm{GAP}(\gamma)=\mathbb{E}_X\!\left[\rho(X)-\widehat{\rho}(X)\right]$. Shaded bands are $95\%$ confidence intervals; the horizontal line at $0$ indicates no bias.

Theorems & Definitions (31)

• Definition 1: Cross-world assumption
• Lemma 2.1
• Definition 2: $CW(\rho)$ intervals
• Theorem 1: Motivation and optimality under a perfect (asymptotic) scenario
• Theorem 2: Gaussian case
• Definition 3
• Theorem 3: Conditional validity of $CW^{+CI}(\rho)$ intervals when $\rho = 1$
• Lemma A.1: Idea; Full statement in Lemma \ref{['lem_appendix_rho_lower_consistency']}
• Theorem 4: Largest-width Scenario $\rho = -1$
• Theorem 5: Assuming $\rho = 1$ is sufficient