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arxiv: 2605.01883 · v1 · submitted 2026-05-03 · 📊 stat.ME

Probabilities of Causation for Continuous Outcomes: Bounds and Identification

Jile Chaoge , Kesen Han , Fahui Liu , Peng Wu This is my paper

Pith reviewed 2026-05-09 16:38 UTC · model grok-4.3

classification 📊 stat.ME
keywords probability of necessitycontinuous outcomespartial identificationcausal boundscopulamonotonicityignorabilityattribution analysis
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The pith

For continuous outcomes, the probability that a treatment was necessary can be bounded sharply from above and below using only ignorability and monotonicity, and narrowed further with copula models of potential-outcome dependence.

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

The paper introduces the general probability of necessity for outcomes that take any real value rather than just two or a few ordered levels. It shows that, when treatment assignment is ignorable given observed covariates and the treatment effect is monotonic, the probability that the observed outcome would have been different without treatment is only partially identified. Sharp lower and upper bounds are derived for this quantity. The authors then demonstrate how a copula that captures the dependence between the two potential outcomes can be used to produce tighter intervals without adding strong parametric assumptions. This framework matters for any setting in which researchers want to attribute a measured change in a continuous variable to the presence or absence of an intervention.

Core claim

The paper proposes the general probability of necessity (GPN) for continuous outcomes as the probability that an observed value of Y would not have been realized had the treatment not been received. Under the standard assumptions of ignorability and monotonicity, sharp lower and upper bounds on GPN are derived directly from the observed data and the joint distribution of the potential outcomes. A copula-based identification strategy is further introduced that exploits any available information on the dependence structure between Y(1) and Y(0) to obtain strictly narrower bounds.

What carries the argument

The general probability of necessity (GPN) defined on the difference between observed and counterfactual continuous outcomes, together with the sharp bounds obtained from the marginal distributions under ignorability and monotonicity, and the copula representation used to incorporate dependence between potential outcomes.

If this is right

  • GPN supplies an attribution measure that applies directly to blood pressure, income, test scores, and other continuous variables.
  • The bounds are the narrowest possible under the stated assumptions, so any further tightening requires either stronger assumptions or additional data.
  • The copula approach lets analysts use external information or estimates of rank dependence to shrink the interval without assuming a specific functional form for the outcome distributions.
  • Simulation checks confirm that the bounds contain the true GPN whenever the ignorability and monotonicity conditions are satisfied.
  • Real-data examples illustrate that the method yields informative intervals in medical and economic attribution problems where binary PN is inapplicable.

Where Pith is reading between the lines

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

  • Researchers could combine the copula tightening step with existing methods for estimating rank correlations from auxiliary data or theory to make attribution statements more precise in observational studies.
  • The same bounding strategy might be adapted to define and partially identify a continuous analogue of the probability of sufficiency.
  • Because the method is nonparametric except for the copula choice, it naturally lends itself to sensitivity analyses that vary the strength of assumed dependence.
  • In practice the width of the reported interval will often be driven by how much dependence between potential outcomes can be credibly assumed or estimated.

Load-bearing premise

The claimed sharp bounds hold only when treatment is ignorable given the observed covariates and when the treatment cannot reverse the ordering of potential outcomes for any individual.

What would settle it

A randomized experiment with a continuous outcome in which the empirical frequency of cases where the outcome would have differed without treatment falls outside the interval given by the derived lower and upper bounds.

Figures

Figures reproduced from arXiv: 2605.01883 by Fahui Liu, Jile Chaoge, Kesen Han, Peng Wu.

Figure 1
Figure 1. Figure 1: Sensitivity analysis of the average GPN. The truth (star) falls within the Expert region view at source ↗
Figure 2
Figure 2. Figure 2: Sensitivity analysis of the estimated average GPN for Smoking (left) and Alcohol (right). view at source ↗
read the original 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.

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 / 3 minor

Summary. The manuscript proposes the General Probability of Necessity (GPN) for continuous outcomes as an extension of the probability of necessity. Under standard ignorability and monotonicity assumptions, it derives sharp lower and upper bounds on the GPN. It further develops a copula-based framework that incorporates dependence information between potential outcomes to tighten the bounds. The claims are supported by simulation studies and real-data applications.

Significance. If the bounds are verifiably sharp and the copula tightening is shown to be valid under explicit, non-circular assumptions, the work would meaningfully extend partial-identification methods from binary/ordinal to continuous outcomes, an area that remains underdeveloped. The partial-identification framing is appropriate, and the simulation/application results provide useful evidence of practical performance. The explicit use of copulas to exploit dependence is a potentially valuable technical contribution if the identification of the dependence parameter is clarified.

major comments (2)
  1. [§3.2] §3.2, around Eq. (8)–(10): the claim that the derived bounds are sharp under ignorability and monotonicity requires an explicit attainability argument. The current construction appears to rely on the Fréchet–Hoeffding bounds for the joint distribution of potential outcomes, but it is unclear whether monotonicity alone guarantees that these bounds are attained for arbitrary continuous marginals without additional regularity conditions on the conditional distributions.
  2. [§4.2] §4.2, copula tightening step: the framework requires a copula family and dependence parameter to tighten the GPN bounds. Because the joint distribution of (Y(1), Y(0)) is not observed, it is not evident whether the dependence parameter is identified from the data, chosen by the analyst, or calibrated in a way that preserves the partial-identification interpretation. If the parameter is fitted to the same observed data used for the marginals, the tightened bounds may no longer be sharp in the original sense and could introduce circularity.
minor comments (3)
  1. [Introduction] The introduction cites binary and ordinal PN literature but omits a brief comparison of the technical challenges unique to the continuous case (e.g., lack of natural ordering for events).
  2. [Simulation studies] Figure 2 (simulation results) would benefit from error bars or shaded regions indicating variability across replications to allow visual assessment of the bound tightness.
  3. [§2] Notation for the GPN estimand is introduced in §2 but the link to the classical PN formula is not restated explicitly; a one-line reminder would aid readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive comments on our manuscript. We address the major comments point by point below, indicating where revisions will be made to strengthen the paper.

read point-by-point responses

  1. Referee: [§3.2] §3.2, around Eq. (8)–(10): the claim that the derived bounds are sharp under ignorability and monotonicity requires an explicit attainability argument. The current construction appears to rely on the Fréchet–Hoeffding bounds for the joint distribution of potential outcomes, but it is unclear whether monotonicity alone guarantees that these bounds are attained for arbitrary continuous marginals without additional regularity conditions on the conditional distributions.

    Authors: We acknowledge that the attainability of the bounds should be demonstrated more explicitly. In the revised version, we will include a detailed proof in an appendix showing that the lower and upper bounds are attained by specific joint distributions of the potential outcomes that satisfy the observed marginals, the ignorability assumption, and the monotonicity condition. Specifically, we construct the extremal couplings by adjusting the Fréchet-Hoeffding bounds to respect the direction of the treatment effect implied by monotonicity (i.e., Y(1) ≥ Y(0) almost surely). This construction works for arbitrary continuous marginal distributions without requiring further regularity conditions beyond the continuity of the CDFs, as the quantile transformations allow us to achieve the bounds. We believe this addresses the concern and confirms the sharpness. revision: yes

  2. Referee: [§4.2] §4.2, copula tightening step: the framework requires a copula family and dependence parameter to tighten the GPN bounds. Because the joint distribution of (Y(1), Y(0)) is not observed, it is not evident whether the dependence parameter is identified from the data, chosen by the analyst, or calibrated in a way that preserves the partial-identification interpretation. If the parameter is fitted to the same observed data used for the marginals, the tightened bounds may no longer be sharp in the original sense and could introduce circularity.

    Authors: The copula approach is presented as a way to incorporate additional dependence information to obtain tighter bounds within the partial identification framework. The dependence parameter is not claimed to be identified from the observed data; rather, it is a sensitivity parameter that the analyst can specify based on substantive knowledge or external data sources. We will revise the manuscript to explicitly state that the tightened bounds are sharp conditional on the chosen copula family and parameter value. We agree that fitting the parameter directly to the observed data could compromise the partial identification guarantees and introduce circularity, and we will add a discussion warning against this practice and suggesting instead the use of plausible ranges for the dependence parameter. This maintains the integrity of the partial-identification interpretation. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds from standard ignorability and monotonicity assumptions to sharp bounds on GPN, then applies a copula model that incorporates separate dependence information between potential outcomes to tighten those bounds. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the copula step is an explicit modeling choice that adds information rather than renaming or tautologically reproducing the input bounds. The approach is framed as partial identification, consistent with the continuous-outcome setting, and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claim rests on ignorability and monotonicity as domain assumptions plus the existence of a copula that correctly captures dependence between potential outcomes.

axioms (2)
  • domain assumption Ignorability: no unmeasured confounding between treatment and potential outcomes
    Stated as a standard assumption under which sharp bounds are derived.
  • domain assumption Monotonicity: treatment never decreases the outcome
    Required for the bounds to be sharp.
invented entities (2)
  • General Probability of Necessity (GPN) no independent evidence
    purpose: Quantifies probability that observed continuous outcome would not have occurred without treatment
    New estimand defined for continuous case
  • Copula-based tightening framework no independent evidence
    purpose: Uses dependence between potential outcomes to narrow the partial-identification bounds
    Introduced to improve upon the basic bounds

pith-pipeline@v0.9.0 · 5438 in / 1347 out tokens · 35871 ms · 2026-05-09T16:38:58.739268+00:00 · methodology

discussion (0)

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Reference graph

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