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arxiv: 2402.10537 · v4 · submitted 2024-02-16 · 📊 stat.ME

Quantifying Individual Risk for Binary Outcomes

Peng Wu , Peng Ding , Zhi Geng , Yue Liu This is my paper

Pith reviewed 2026-05-24 03:18 UTC · model grok-4.3

classification 📊 stat.ME
keywords fraction negatively affectedindividual treatment effectsensitivity analysisbinary outcomescausal inferencePearson correlationtreatment effect heterogeneityFréchet-Hoeffding bounds
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The pith

Mild restrictions on the Pearson correlation between potential outcomes produce tighter bounds on the fraction of individuals harmed by treatment.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks to bound the fraction negatively affected by a binary treatment, the share of people who experience worse outcomes under treatment than under control. Standard Fréchet-Hoeffding bounds on this quantity are wide and attainable only in extreme cases, but assuming the Pearson correlation between the two potential outcomes lies in a restricted plausible interval yields narrower bounds. A central result is that a positive conditional average treatment effect does not rule out a positive lower bound on individual harm. The correlation range is used as a sensitivity parameter, and the paper supplies nonparametric estimators shown to be consistent and asymptotically normal under strong ignorability.

Core claim

By invoking mild conditions on the value range of the Pearson correlation coefficient between potential outcomes, improved bounds compared with the Fréchet-Hoeffding bounds are obtained for the fraction negatively affected. Even with a positive CATE the lower bound on FNA can be positive. A nonparametric sensitivity analysis framework is established with the Pearson correlation as the sensitivity parameter, and consistent asymptotically normal estimators for the refined bounds are proposed.

What carries the argument

Refined bounds on the fraction negatively affected that use a restricted but plausible range for the Pearson correlation coefficient between the two potential outcomes as a sensitivity parameter.

If this is right

  • Even when the conditional average treatment effect is positive, the lower bound on the fraction negatively affected can remain strictly positive.
  • The Pearson correlation coefficient between potential outcomes functions as a tunable sensitivity parameter for bounding individual-level risk.
  • Nonparametric estimators of the refined bounds are consistent and asymptotically normal.
  • The framework applies directly to observational data under the strong ignorability assumption.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Policies that select individuals for treatment solely on the basis of positive CATE may still expose a non-zero fraction of those individuals to harm.
  • Domain experts could supply the plausible correlation interval from prior studies or mechanistic knowledge rather than data.
  • The same correlation-based tightening could be explored for continuous outcomes or for other individual-level causal functionals.
  • Direct comparison against realized individual effects in settings where both outcomes are observed would provide an empirical check on the width of the resulting intervals.

Load-bearing premise

The Pearson correlation between the potential outcomes under treatment and control is restricted to a plausible interval rather than allowed to range over the full [-1,1] interval.

What would settle it

In a randomized experiment where both potential outcomes can be observed for the same units, compute the realized FNA and check whether it lies inside the proposed bounds for every correlation value inside the assumed interval.

Figures

Figures reproduced from arXiv: 2402.10537 by Peng Ding, Peng Wu, Yue Liu, Zhi Geng.

Figure 1
Figure 1. Figure 1: Bounds on FNA for various values of ρ in cases (C1)-(C3), based on a simulation with a sample size of 2,000. The dotdash red line represents the true value of FNA, the solid purple line is the true value of βρ, the dotted blue lines denote the Fr´echet–Hoeffding lower and upper bounds in Lemma 1, and the shaded areas depict the 95% pointwise confidence intervals for βˆ ρ. The dimension of the covariates in… view at source ↗
Figure 2
Figure 2. Figure 2: The proposed estimator βˆ ρ for ρ ∈ [−0.450, 0.706], where the solid red line is the value of βˆ ρ, the shaded areas depict the 95% pointwise confidence intervals for βˆ ρ, and the dotted blue lines denote the estimated Fr´echet–Hoeffding lower and upper bounds in Lemma 1. 7. Discussion In the main text, we focus on analyzing the bounds on FNA(x), which is just one of the four principal scores, defined as … view at source ↗
read the original abstract

Understanding treatment effect heterogeneity is crucial for reliable decision-making in treatment evaluation and selection. The conditional average treatment effect (CATE) is widely used to capture treatment effect heterogeneity induced by observed covariates and to design individualized treatment policies. However, it is an average metric within subpopulations, which prevents it from revealing individual risk, potentially leading to misleading results. This article fills this gap by examining individual risk for binary outcomes, specifically focusing on the fraction negatively affected (FNA), a metric that quantifies the percentage of individuals experiencing worse outcomes under treatment compared with control. Even under the strong ignorability assumption, FNA is still unidentifiable, and the existing Fr\'{e}chet--Hoeffding bounds are often too wide and attainable only under extreme data-generating processes. By invoking mild conditions on the value range of the Pearson correlation coefficient between potential outcomes, we obtain improved bounds compared with the Fr\'{e}chet--Hoeffding bounds. We show that paradoxically, even with a positive CATE, the lower bound on FNA can be positive, i.e., in the best-case scenario, many individuals will be harmed if they receive treatment. Additionally, we establish a nonparametric sensitivity analysis framework for FNA using the Pearson correlation coefficient as the sensitivity parameter. Furthermore, we propose nonparametric estimators for the refined FNA bounds and prove their consistency and asymptotic normality. We use simulation to evaluate the performance of the proposed estimators and apply the method to a canonical observational study.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that even under strong ignorability, the fraction negatively affected (FNA) remains unidentifiable and that the Fréchet-Hoeffding bounds are often too wide. By restricting the Pearson correlation ρ between the potential outcomes Y(1) and Y(0) to a 'mild' range treated as a sensitivity parameter, the authors derive narrower bounds on FNA, establish that the lower bound on FNA can remain positive even when CATE > 0, develop a nonparametric sensitivity analysis framework with ρ as the parameter, and propose consistent, asymptotically normal nonparametric estimators for the refined bounds, which are evaluated in simulations and applied to an observational study.

Significance. If the posited range restrictions on ρ can be defended, the work supplies a concrete sensitivity-analysis tool for quantifying individual-level harm in binary-outcome settings that goes beyond CATE and highlights a practically relevant paradox. The nonparametric estimators together with the consistency and asymptotic-normality results constitute a clear methodological contribution.

major comments (2)
  1. [Abstract / sensitivity framework] Abstract and sensitivity-analysis framework: the central claims of improved bounds and a positive lower bound on FNA despite positive CATE rest on restricting ρ = corr(Y(1),Y(0)) to an unspecified 'mild' interval. No data-driven procedure, empirical calibration, or argument for plausibility of the interval in observational settings is supplied; when ρ lies outside the interval (while still satisfying the marginal-probability constraints), the bounds revert to the Fréchet-Hoeffding width and the paradox disappears.
  2. [Estimator section] Estimator section: the stated consistency and asymptotic normality of the nonparametric estimators for the refined bounds presuppose that the chosen correlation range is fixed and known; the paper supplies neither the explicit influence-function derivation nor verification that the range restriction does not invalidate the regularity conditions used in the proofs.
minor comments (2)
  1. Notation for the correlation range and the resulting bound expressions should be introduced with explicit equations rather than descriptive text only.
  2. The simulation design should report the exact correlation values used to generate data and confirm they lie inside the posited 'mild' interval.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive report. We address each major comment below, indicating where we agree and plan revisions.

read point-by-point responses

  1. Referee: [Abstract / sensitivity framework] Abstract and sensitivity-analysis framework: the central claims of improved bounds and a positive lower bound on FNA despite positive CATE rest on restricting ρ = corr(Y(1),Y(0)) to an unspecified 'mild' interval. No data-driven procedure, empirical calibration, or argument for plausibility of the interval in observational settings is supplied; when ρ lies outside the interval (while still satisfying the marginal-probability constraints), the bounds revert to the Fréchet-Hoeffding width and the paradox disappears.

    Authors: We agree that the interval for ρ is a user-specified sensitivity parameter rather than data-driven. The framework's purpose is to let analysts assess how FNA bounds vary with different plausible ranges for the correlation between potential outcomes; a data-driven estimator for the interval would change the nature of the analysis. We will add a dedicated subsection providing practical guidance on selecting the range, drawing on domain knowledge and citing empirical literature on correlations between potential outcomes in observational studies. The fact that bounds revert outside the interval is expected and underscores why sensitivity analysis is needed. revision: partial

  2. Referee: [Estimator section] Estimator section: the stated consistency and asymptotic normality of the nonparametric estimators for the refined bounds presuppose that the chosen correlation range is fixed and known; the paper supplies neither the explicit influence-function derivation nor verification that the range restriction does not invalidate the regularity conditions used in the proofs.

    Authors: The estimators target the bound functionals for a fixed ρ, after which the interval is optimized over; the range restriction is a fixed, deterministic constraint that preserves the regularity conditions. We acknowledge that the main text does not display the explicit influence functions. We will add these derivations to the appendix together with a verification that the maintained assumptions on the propensity score and conditional outcome regressions suffice for the asymptotic results to hold. revision: yes

Circularity Check

0 steps flagged

No significant circularity; sensitivity analysis uses explicit parameter without reduction to tautology

full rationale

The paper frames its contribution as a nonparametric sensitivity analysis for the fraction negatively affected (FNA), with the Pearson correlation coefficient between potential outcomes Y(1) and Y(0) introduced explicitly as the sensitivity parameter. Improved bounds relative to Fréchet-Hoeffding are obtained by restricting the feasible range of this parameter under 'mild conditions,' which is a direct mathematical consequence of the imposed restrictions rather than any self-definitional or fitted-input reduction. No load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation are described. Estimators are proposed and their consistency/asymptotic normality proven separately. The central claims (refined bounds and the positive lower bound on FNA despite positive CATE) are therefore conditional on the sensitivity parameter and do not collapse to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on strong ignorability (standard) plus an externally chosen range for the Pearson correlation between potential outcomes (treated as a sensitivity parameter). No new entities are postulated. The correlation range functions as the key free parameter whose specific bounds are not derived from data.

free parameters (1)
  • Pearson correlation range between potential outcomes
    Used to tighten Fréchet-Hoeffding bounds; treated as a sensitivity parameter whose specific interval is chosen rather than identified.
axioms (1)
  • domain assumption Strong ignorability (no unmeasured confounding)
    Invoked explicitly as the baseline assumption under which FNA remains unidentifiable.

pith-pipeline@v0.9.0 · 5791 in / 1358 out tokens · 42322 ms · 2026-05-24T03:18:29.387606+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Trust Me, I'm a Doctor?

    stat.AP 2026-05 unverdicted novelty 7.0

    Sharp bounds are derived on the proportion of physicians whose personal strategies perform at least as well as the trial's better average treatment, using nested randomized and observational data from the same population.

  2. Trust Me, I'm a Doctor?

    stat.AP 2026-05 unverdicted novelty 5.0

    Using nested randomized and observational data, the paper derives sharp bounds on the proportion of physicians whose personal strategies perform at least as well as the trial's better-performing treatment.

Reference graph

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