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A margin-free bounding factor based on relative risks of an unmeasured confounder produces sharp nonparametric bounds on principal causal effects.

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2026-06-28 13:42 UTC pith:A7HNPBIZ

load-bearing objection The paper gives a usable new sensitivity tool for principal stratification by bounding PCEs with two relative-risk parameters for an unmeasured confounder and proving the bounds nest inside the nonparametric worst case. the 1 major comments →

arxiv 2606.01669 v1 pith:A7HNPBIZ submitted 2026-06-01 stat.ME

Beyond principal ignorability: Nonparametric sensitivity bounds for principal stratification

Xinyuan Chen , Michael O. Harhay , Fan Li This is my paper
classification stat.ME
keywords principal stratificationprincipal causal effectssensitivity analysisnonparametric boundsprincipal ignorabilityunmeasured confoundingrelative risksE-values
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Principal stratification defines causal effects inside subgroups formed by an intermediate variable, yet point identification of those effects rests on the untestable principal ignorability assumption. The paper supplies a nonparametric sensitivity analysis that replaces this assumption with a single bounding factor whose value is set by the selection and outcome relative risks attributable to one unobserved confounder. This factor delivers sharp bounds on each principal causal effect; the bounds tighten the usual worst-case limits and remain valid with or without a monotonicity restriction. The same construction yields Cornfield-type thresholds and principal E-values that state the smallest joint confounding strength capable of nullifying an observed effect. The approach is further extended to pairwise comparison estimands defined over a product space.

Core claim

We introduce a margin-free bounding factor parameterized by the selection and outcome relative risks of an unmeasured confounder. Using this bounding factor, we derive sharp nonparametric bounds for each PCE. We prove that these bounds nest within the worst-case nonparametric bounds with and without the monotonicity assumption. We then discuss Cornfield-type conditions and principal E-values that quantify the minimum joint magnitude of unmeasured confounding required to nullify the target PCE. Furthermore, we generalize this methodology to principal generalized causal effects, extending the sensitivity bounds and falsification thresholds to the recent pairwise comparison estimands evaluated

What carries the argument

The margin-free bounding factor, defined solely by the selection and outcome relative risks of a single unmeasured confounder, that converts principal ignorability violations into explicit nonparametric bounds on principal causal effects.

Load-bearing premise

The bounding factor is assumed to capture every relevant violation of principal ignorability without any additional restrictions on the joint distribution of the confounder, stratum membership, and outcome.

What would settle it

A dataset or Monte Carlo experiment in which the true principal causal effect lies strictly outside the derived sensitivity interval when the relative risks of the unmeasured confounder are set to the values used to construct the bounds.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Sharp nonparametric bounds are obtained for every principal causal effect once the bounding factor is specified.
  • The new bounds are strictly contained inside the usual worst-case nonparametric bounds both with and without monotonicity.
  • Cornfield-type conditions and principal E-values give the minimum joint relative-risk magnitude needed to explain away any given principal causal effect.
  • The same bounding construction extends directly to pairwise comparison estimands over a product space.

Where Pith is reading between the lines

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

  • The bounding factor could be estimated or elicited from auxiliary data on selection and outcome associations, turning the sensitivity analysis into a quantitative robustness check rather than a purely qualitative exercise.
  • When multiple unmeasured confounders are plausible, the single-factor bounds remain valid as long as the product of their relative risks does not exceed the supplied margin.
  • The nesting property suggests that existing software for worst-case bounds can be reused by simply replacing the worst-case range with the tighter interval produced by the relative-risk factor.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 2 minor

Summary. The paper develops a nonparametric sensitivity analysis for principal causal effects (PCEs) under violations of principal ignorability (PI). It introduces a margin-free bounding factor parameterized solely by the selection relative risk RR_{S|U} and outcome relative risk RR_{Y|U} of a single unmeasured confounder U. Using this factor, the authors derive sharp nonparametric bounds on each PCE and prove that these bounds nest inside the worst-case nonparametric bounds both with and without the monotonicity assumption. The framework is extended to principal generalized causal effects (pairwise comparisons over a product space), with additional results on Cornfield-type conditions and principal E-values that quantify the minimum strength of unmeasured confounding needed to nullify a target PCE.

Significance. If the sharpness and nesting claims hold, the work supplies a practical, low-dimensional sensitivity tool for principal stratification that avoids parametric assumptions on the outcome or selection models. The explicit nesting result and the generalization to generalized causal effects would be useful for applied researchers who already employ worst-case bounds; the Cornfield and E-value extensions provide interpretable falsification thresholds. The margin-free parameterization is a clear strength relative to more heavily parameterized sensitivity models.

major comments (1)
  1. [§3.2, Theorem 3.1] §3.2 and Theorem 3.1: The sharpness claim for the derived bounds rests on the assertion that every possible violation of PI can be represented by some choice of the two relative-risk scalars together with an arbitrary joint law on (U, stratum, Y). The proof sketch does not explicitly construct the extremal distributions or rule out the possibility that attaining the bound requires functional dependence between U and the stratum indicator that cannot be encoded by the marginal RRs alone; this needs a self-contained verification that the two parameters are sufficient without implicit restrictions on higher-order dependence.
minor comments (2)
  1. [§3] Notation for the bounding factor is introduced without an explicit equation number in the main text; adding a displayed equation would improve traceability when the factor is later used in the nesting proof.
  2. [§5] The data examples in §5 report numerical bounds but do not include the corresponding worst-case bounds for direct visual comparison; adding a side-by-side column would make the nesting property immediately verifiable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The single major comment concerns the sharpness claim and proof of Theorem 3.1; we address it directly below and will revise the manuscript to strengthen the argument.

read point-by-point responses

  1. Referee: [§3.2, Theorem 3.1] §3.2 and Theorem 3.1: The sharpness claim for the derived bounds rests on the assertion that every possible violation of PI can be represented by some choice of the two relative-risk scalars together with an arbitrary joint law on (U, stratum, Y). The proof sketch does not explicitly construct the extremal distributions or rule out the possibility that attaining the bound requires functional dependence between U and the stratum indicator that cannot be encoded by the marginal RRs alone; this needs a self-contained verification that the two parameters are sufficient without implicit restrictions on higher-order dependence.

    Authors: We agree that the current proof sketch in §3.2 and Theorem 3.1 would be strengthened by an explicit, self-contained construction of the extremal joint distributions on (U, stratum, Y) that attain the proposed bounds for any choice of the two relative-risk parameters. Such a construction would directly confirm that the marginal RRs are sufficient to encode all relevant dependence structures without hidden restrictions. In the revision we will add this verification, including the explicit form of the extremal laws and a demonstration that functional dependence between U and the stratum indicator is achievable within the given parameterization. revision: yes

Circularity Check

0 steps flagged

No circularity; bounding factor introduced as explicit sensitivity parameter

full rationale

The paper defines a margin-free bounding factor directly in terms of the two relative-risk parameters (selection and outcome) of an unmeasured confounder and then derives nonparametric bounds for the PCEs from that definition. These steps are forward constructions rather than reductions of the target quantities back to fitted inputs or self-referential definitions. The nesting claim is a proved inclusion between two sets of bounds, not a tautology. No load-bearing step reduces by construction to its own inputs, and the abstract and described framework treat the relative risks as free sensitivity parameters rather than quantities recovered from the observed data.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The framework rests on the standard principal stratification setup plus the new parameterization of sensitivity via two relative-risk quantities; no new entities are postulated and the relative risks function as free sensitivity parameters rather than fitted constants.

free parameters (2)
  • selection relative risk
    Sensitivity parameter measuring association between unmeasured confounder and principal stratum membership
  • outcome relative risk
    Sensitivity parameter measuring association between unmeasured confounder and outcome
axioms (2)
  • domain assumption Principal stratification framework and definitions of principal causal effects
    Standard background assumptions in causal inference for intermediate variables
  • domain assumption Nonparametric identification setting without parametric outcome or selection models
    Bounds are derived nonparametrically

pith-pipeline@v0.9.1-grok · 5677 in / 1388 out tokens · 28896 ms · 2026-06-28T13:42:51.641415+00:00 · methodology

0 comments
read the original abstract

Principal stratification is an effective framework addressing intermediate variables in causal inference. However, point identification of the principal causal effects (PCEs) often requires the untestable principal ignorability (PI) assumption. This article develops a nonparametric sensitivity analysis framework for evaluating PI violations. We introduce a margin-free bounding factor parameterized by the selection and outcome relative risks of an unmeasured confounder. Using this bounding factor, we derive sharp nonparametric bounds for each PCE. We prove that these bounds nest within the worst-case nonparametric bounds with and without the monotonicity assumption. We then discuss Cornfield-type conditions and principal E-values that quantify the minimum joint magnitude of unmeasured confounding required to nullify the target PCE. Furthermore, we generalize this methodology to principal generalized causal effects, extending the sensitivity bounds and falsification thresholds to the recent pairwise comparison estimands evaluated over a product space.

Figures

Figures reproduced from arXiv: 2606.01669 by Fan Li, Michael O. Harhay, Xinyuan Chen.

Figure 1
Figure 1. Figure 1: A directed acyclic graph depicting the relationship among variables for principal [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the sensitivity lower bound surfaces for PCRR [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration via the WHO-LARES example. (a) Principal E-value curve; (b) [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the sensitivity bound surfaces for [PITH_FULL_IMAGE:figures/full_fig_p030_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustration via the U.S. Job Corps example. (a) Generalized principal E-value [PITH_FULL_IMAGE:figures/full_fig_p031_5.png] view at source ↗

discussion (0)

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Reference graph

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