Average Cause-Specific Hazard: A Censoring-Invariant Measure of Event Burden Under Competing Risks

Deb Schrag; Hajime Uno; Jean M Connors; Khondoker Nazmoon Nabi; Lu Tian; Xiang Meng

REVIEW 1 major objections 4 minor 23 references

The paper defines the Average Cause-Specific Hazard (ACSH) as the ratio of a cause's cumulative incidence to restricted mean event-free time, and shows it is a rate summary that does not depend on the censoring distribution.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

A censoring-invariant cause-specific rate—the ratio of cumulative incidence to restricted mean event-free time—is introduced with nonparametric inference and multi-endpoint tests.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection A useful, sound extension of the average-hazard idea to competing risks; the main inference holds, but the simplified covariance formula in Remark 1 is wrong as printed and needs a mechanical fix. the 1 major comments →

arxiv 2607.13293 v1 pith:M7HNJU25 submitted 2026-07-14 stat.ME

Average Cause-Specific Hazard: A Censoring-Invariant Measure of Event Burden Under Competing Risks

Khondoker Nazmoon Nabi , Xiang Meng , Lu Tian , Jean M Connors , Deb Schrag , Hajime Uno This is my paper

classification stat.ME MSC 62N0162N0262P10

keywords competing riskscause-specific hazardaverage hazardcensoring invariancecumulative incidence functionrestricted mean event-free timeincidence rateinfluence function

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.

The reading

The paper aims to give competing-risks researchers a rate-scale summary that means the same thing regardless of how the data were censored. It defines the Average Cause-Specific Hazard (ACSH) as the cause-specific cumulative incidence divided by the restricted mean event-free time—equivalently, the event-free-probability-weighted average of the cause-specific hazard over a horizon [0, τ]. Because both numerator and denominator are functions of the event-time distribution alone, ACSH is a population estimand that does not involve the censoring distribution, unlike the naive incidence rate whose limit depends on censoring whenever hazards vary over time. The paper supplies a plug-in estimator built from Aalen–Johansen and Kaplan–Meier components, an influence-function variance, two-group difference and ratio contrasts, and a global Wald test across endpoints or causes. If the claims hold, ACSH gives investigators an interpretable, model-free complement to the cumulative incidence function for reporting event burden under competing risks.

Core claim

On the paper's terms, the central discovery is that the quantity η_k(τ) = F_k(τ)/R(τ) = ∫_0^τ S(u)dΛ_k(u) / ∫_0^τ S(u)du is a censoring-invariant incidence rate: it is the average rate at which the event of interest occurs among subjects still free of any event, expressed per unit event-free person-time. The authors prove that the nonparametric estimator η̂_k = F̂_k/R̂ is consistent under independent right censoring and that √n(log η̂_k − log η_k) is asymptotically normal with variance given by an influence function; they derive a simplified covariance formula, delta-method contrasts for differences and ratios, a chi-square global test, and an endpoint-specific extension for multiple non-ter

What carries the argument

The central object is the ratio η_k(τ)=F_k(τ)/R(τ), where F_k is the cumulative incidence for cause k and R is the restricted mean event-free time; equivalently it is the cause-specific hazard Λ_k integrated against the event-free survival function S and normalized by the integral of S. This survival-weighting is the mechanism that cancels censoring: the naive incidence rate weights person-time by the censoring survival function as well as S, which is why it drifts under time-varying hazards, while ACSH removes the censoring survival from both numerator and denominator. The inference machinery is the influence-function representation for log η_k, which combines the Aalen–Johansen influence f

Load-bearing premise

The whole inference rests on the assumption that censoring and the event process are independent (C ⊥ (T,J)); if patients who would have events are censored for reasons related to their risk, the plug-in estimator is no longer consistent, and a positivity condition on the at-risk process also must hold so that the denominators stay nonzero.

What would settle it

Simulate a competing-risks trial where the censoring time C depends on the latent event time T or the cause J (violating the independence assumption), with time-varying hazards; compute the ACSH plug-in estimator over many large samples. If the estimates converge to a value that is not the true η_k(τ) from the underlying event-time distribution, the claim of censoring invariance fails.

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

If this is right

ACSH difference and ratio give between-group contrasts on the rate scale without assuming proportional subdistribution hazards or constant cause-specific hazards.
The global Wald test over a prespecified set of causes or non-terminal endpoints permits a single joint comparison of event burden across multiple event types.
The Total ACSH summary, the sum of endpoint-specific ACSH values, provides a scalar multi-endpoint burden measure with a valid delta-method variance even when endpoints are not mutually exclusive.
If the claims hold, reported incidence rates computed with the naive person-time denominator may be systematically biased when rates change over time and censoring is present; ACSH is an alternative that targets a well-defined estimand.
Because ACSH reduces to the average hazard in the absence of competing events, it links competing-risks reporting to the existing single-event average-hazard methodology.

Where Pith is reading between the lines

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

A natural next step the authors leave implicit is regression or covariate-adjusted modeling of ACSH, analogous to existing frameworks for restricted mean survival time, which would allow treatment effects to be adjusted without returning to proportional hazards.
The endpoint-specific construction means ACSH answers a first-occurrence question per endpoint; a reader should not read Total ACSH as a combined event rate on a common denominator—the paper itself cautions this, and a joint multi-state model would be needed to capture recurrent events.
A testable extension: if informative censoring is suspected, inverse-probability-of-censoring weighting could restore consistency, and one could compare ACSH estimates under a range of sensitivity parameters for the censoring mechanism.
Because both numerator and denominator are event-time functionals, ACSH could serve as a loss–benefit summary in decision analysis, combining event rates and event-free survival into a single rate that can be contrasted across treatments.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

A useful, sound extension of the average-hazard idea to competing risks; the main inference holds, but the simplified covariance formula in Remark 1 is wrong as printed and needs a mechanical fix.

read the letter

The short version: the paper defines the Average Cause-Specific Hazard, eta_k(tau)=F_k(tau)/R(tau), as a censoring-invariant rate summary under competing risks, with nonparametric estimation, influence-function variance, two-sample contrasts, and a multi-endpoint extension. The main estimand and the primary inference hold up, and the paper is a useful, practical complement to CIF and Fine-Gray. One thing to know before citing it: Remark 1's simplified covariance formula is wrong as printed.

What is actually new: ACSH is a direct adaptation of the authors' own average-hazard measure to cause-specific hazards, so it is not a major conceptual leap. But the competing-risks formalism, the endpoint-specific construction with death as the terminal competing event, the Total ACSH summary, and the global Wald test are real extensions, and the motivation is solid - the naive incidence rate demonstrably depends on the censoring distribution when hazards are time-varying, and the simulations make that concrete. The simulation program is thorough: main scenarios, robustness to hazard shape, censoring intensity, and event rarity, two-sample contrasts calibrated to CANVAS, and a controlled frailty study for the off-diagonal covariance. Coverage tracks nominal in most configurations, and Appendix C honestly notes a handful of replicates where the variance estimator is undefined under heavy censoring at N=300. The authors also state the independent-censoring limitation plainly in Section 5.

The soft spots, in proportion. The stress-test note is correct about Remark 1. Expanding the covariance of the influence functions in Eq (4), the diagonal self-martingale term carries S(u)/F_k(tau); the printed xi_{k,ell}(u,tau)=S(u)I(ell=k)+F_k(u)/F_k(tau) drops that divisor, so Eq (5) as printed is not the covariance of the influence functions in Eq (4). A user implementing the global Wald test from Eq (5) literally would get the wrong variance. With the divisor restored, the factorization does go through, including the off-diagonal blocks, so the repair is mechanical, not structural. The sample-covariance estimator actually used in Sections 2.2 and 2.3 is unaffected, so the central inference stands. Minor: the main text writes S(u) where Appendix A's Lin (1997) expansion has S(u-), a convention inconsistency that is asymptotically harmless but should be reconciled; and the survACSH package is "available upon request" while CANVAS data are not posted, which weakens the reproducibility story a bit.

Who it is for: biostatisticians and clinical trialists who report competing-risks endpoints and want a rate-scale alternative to CIF and subdistribution hazards. I would send it to peer review. The main claim holds; the Remark 1 fix is mechanical but mandatory before publication.

Referee Report

1 major / 4 minor

Summary. The manuscript introduces the Average Cause-Specific Hazard (ACSH), defined as η_k(τ) = F_k(τ)/R(τ), where F_k is the cause-specific cumulative incidence function and R(τ) is the restricted mean event-free time over [0, τ]. The estimand is a survival-weighted average of the cause-specific hazard and is claimed to be a censoring-invariant rate summary, unlike the naive person-time incidence rate. The paper develops nonparametric estimation using the Aalen–Johansen estimator for the CIF and the Kaplan–Meier estimator for survival, derives an influence-function representation for log ACSH, and constructs confidence intervals, two-group differences and ratios, a global Wald test, and an extension to multiple non-terminal endpoints with a common terminal event (including a Total ACSH summary). The methods are evaluated in extensive simulations and illustrated on the CANVAS trial.

Significance. If the proposed methods are valid, ACSH fills a genuine gap: standard cause-specific incidence rates can have population limits that depend on the censoring distribution when hazards are time-varying, whereas ACSH is defined solely from the event-time distribution. The estimand is simple, has a clear person-time rate interpretation, and reduces exactly to the average hazard of Uno and Horiguchi in single-event settings. The main derivation is standard, and the simulation program is unusually thorough, covering one-sample estimation, two-sample contrasts, Total ACSH, and a controlled correlated-endpoint design with known cross-endpoint correlation. No fitted parameters or circular prediction-from-fit structure are involved. However, the paper contains a concrete algebraic error in the simplified covariance formula in Remark 1 (Eq. (5)) that must be corrected before the inference machinery as printed can be relied upon.

major comments (1)

[Remark 1, Eq. (5)] As printed, Eq. (5) is not the covariance of the influence functions in Eq. (4). From Eq. (4), the coefficient multiplying dM_{ℓ,i}(u) in ψ_{k,i}(τ) is S(u)I(ℓ=k)/F_k(τ) + F_k(u)/F_k(τ) − R(u)/R(τ), not ξ_{k,ℓ}(u,τ) − R(u)/R(τ) with ξ_{k,ℓ}(u,τ)=S(u)I(ℓ=k)+F_k(u)/F_k(τ). The printed ξ omits the 1/F_k(τ) factor on the first term. This is not a harmless notational slip: a plug-in estimator based on the printed Eq. (5) will not be consistent for Σ(τ), and a global Wald test of Section 2.3 implemented literally with Eq. (5) will not have the claimed χ²_m null distribution. The sample-covariance estimator Σ = (1/n)∑ ψ_i^{⊗2} based on Eq. (4) is unaffected, so the paper's primary inference can be repaired by correcting Eq. (5) (or by removing the simplification and relying on the sample covariance). I recommend correcting the formula, re-deriving the simplification, and checking any reported r

minor comments (4)

[Section 2.2, Eq. (4) and Appendix A] The first integrand in Eq. (4) is printed as S(u), while Appendix A and standard martingale representations use S(u−). For continuous event times the distinction is immaterial, but the notation should be made consistent to avoid confusion about predictability of the integrand.
[Section 2.4, asymptotic representation display] The display for √n{θ^NT(τ)−θ^NT(τ)} writes ψ^NT_i(τ) on the right-hand side, whereas the asymptotic linear representation should involve the true influence function ψ^NT_i(τ), not its plug-in estimate. The hat should be removed (or the equation rewritten as an empirical representation).
[Section 3.1 / Table 2 caption] The symbol N is used for sample size, but N is also used for the counting process in Section 2. Consider using n for sample size in the tables and captions to avoid ambiguity.
[Section 5 / software availability] The statement that the survACSH R package 'will be made available upon request' is less useful for reproducibility than a permanent public repository. A URL or archival version would strengthen the paper.

Circularity Check

0 steps flagged

No significant circularity: ACSH is a self-contained ratio estimand; self-citations to the authors' AH/RMST work are background rather than load-bearing, and the plug-in inference is validated against independently computed simulation truths.

full rationale

The paper's derivation chain is self-contained and non-circular. The central estimand ACSH η_k(τ)=F_k(τ)/R(τ) (Eq. 3) is defined directly as a functional of the event-time distribution; the plug-in estimator replaces F_k and R by Aalen–Johansen and Kaplan–Meier estimators, and consistency plus asymptotic normality follow from standard semiparametric theory developed in Appendix A with the influence function exhibited in Eq. (4). No parameter is fitted to data and then renamed as a prediction: the demonstration that the naïve incidence rate depends on censoring is an exact probability-limit formula (Section 3.2), and simulation truths are obtained independently by numerical integration of the latent hazard model, so the estimator is benchmarked against content external to the estimation procedure. The self-citations (Uno and Horiguchi 2023 for the survival-weighted average hazard; Zhao et al. 2016, which includes the present last author, for the RMST influence function) supply motivation and standard technical tools rather than load-bearing conclusions; ACSH reduces to AH only in the acknowledged limiting case of no competing events, which is an explicit special-case relationship, not an identity built into the definition. No uniqueness theorem is imported, and the survival weighting, though adopted from the authors' own AH framework, is a transparent definitional choice rather than a hidden ansatz that determines the paper's conclusions. The one flagged point — the algebraic slip in Remark 1's simplified covariance (Eq. 5 appears to omit the 1/F_k(τ) factor on the S(u)I(ℓ=k) term relative to Eq. 4's influence function) — is a correctness/validity concern in a derived formula: a Wald test implemented literally from Eq. (5) might not achieve the claimed null distribution. But this is not circularity: it does not reduce any claimed result to its inputs, and the plug-in covariance estimator based on Eq. (4) is unaffected. Section 5's honest acknowledgment that informative censoring breaks consistency further confirms that no result is assumed into existence. Finding: no significant circularity; minor self-citation only, hence score 1.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The method is nonparametric with no fitted parameters; τ is a prespecified design choice, not estimated from data. The estimand is a ratio of two standard quantities (CIF and RMEFT), so the only inputs from outside the paper are standard survival-analysis assumptions and cited asymptotic representations.

axioms (5)

domain assumption Independent right-censoring C⊥(T,J)
Assumed in Section 2.2 and Appendix A (C1); if violated, the plug-in estimator is inconsistent for η_k(τ), acknowledged in Section 5.
domain assumption Positivity G(t)=Pr(X≥t)>0 on [0,τ] and R(τ)>0
Assumption C2 in Appendix A; required for the martingale integrals and for finite variance.
standard math Usual regularity conditions for multiplicative intensity models (càdlàg, finite variation)
Assumption C4, guaranteeing uniform consistency and asymptotic linearity of KM and Aalen–Johansen estimators.
standard math Influence-function representations for CIF (Lin 1997) and RMST (Zhao et al. 2016)
These cited results supply eqs. (8) and (9) in Appendix A that are the starting point of the delta-method derivation.
domain assumption Endpoint-specific risk-set construction for non-terminal events
Section 2.4: other non-terminal events do not remove subjects from the risk set for endpoint ℓ; this modeling choice defines the scientific estimand (first occurrence of endpoint ℓ while alive).

reviewed 2026-08-02 · how reviews work

0 comments

Cite this review

Pith. "Pith review of Average Cause-Specific Hazard: A Censoring-Invariant Measure of Event Burden Under Competing Risks." pith.science (2026). https://pith.science/paper/M7HNJU25

@misc{pith2026260713293,
  author       = {Pith},
  title        = {Pith review of: Average Cause-Specific Hazard: A Censoring-Invariant Measure of Event Burden Under Competing Risks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7HNJU25}},
  note         = {Machine review of arXiv:2607.13293}
}

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read the original abstract

Competing events are common in clinical and epidemiologic studies, including semi-competing risks in which a terminal event such as death may follow a nonfatal event but also competes with it beforehand. Standard summaries include the cumulative incidence function (CIF) and the incidence rate (IR), defined as the number of observed events divided by observed event-free person-time. With competing events, the naive IR generally depends on the censoring-time distribution unless intensities are constant. We propose the Average Cause-Specific Hazard (ACSH), a survival-weighted rate per event-free person-time that preserves the interpretation of an incidence rate and is defined purely from the event-time distribution, without involving the censoring-time distribution. We develop nonparametric estimation and inference for ACSH and, for two-sample comparisons, introduce ACSH differences and ratios that provide interpretable contrasts without requiring a strong model assumption between two groups. Simulation studies examine the finite-sample performance, and an analysis of the CANVAS trial illustrates the proposed methods.

Figures

Figures reproduced from arXiv: 2607.13293 by Deb Schrag, Hajime Uno, Jean M Connors, Khondoker Nazmoon Nabi, Lu Tian, Xiang Meng.

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 4 canonical work pages

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, pmid =

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Lihui Zhao and Brian Claggett and Lu Tian and Hajime Uno and Marc A. Pfeffer and Scott D. Solomon and Lorenzo Trippa and L. J. Wei , title =. Biometrics , volume =. 2016 , doi =

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Comparative effectiveness research with average hazard for censored time-to-event outcomes: simulation study and application to observational data

Xiong, Hong and Connors, Jean and Schrag, Deb and Uno, Hajime. Comparative effectiveness research with average hazard for censored time-to-event outcomes: simulation study and application to observational data. BMC Med. Res. Methodol
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Horiguchi, M. and Uno, H. , title =. 2026 , doi =

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Assessing delayed treatment benefits of immunotherapy using long-term average hazard: a novel test/estimation approach

Horiguchi, Miki and Tian, Lu and Kehl, Kenneth L and Uno, Hajime. Assessing delayed treatment benefits of immunotherapy using long-term average hazard: a novel test/estimation approach. Lifetime Data Anal
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Horiguchi, Miki and Tian, Lu and Hattori, Satoshi and Uno, Hajime. Design and analysis of randomized clinical trials with average hazard: Practical guidance and tools for implementation. Stat. Med

This paper was first reviewed by deepseek-v4-flash on August 2, 2026.

[1] [1]

and Johansen, S

Aalen, Odd O. and Johansen, S. An Empirical Transition Matrix for Non-Homogeneous. Scandinavian Journal of Statistics , year =

[2] [2]

, title =

Cox, David R. , title =. Journal of the Royal Statistical Society, Series B , year =

[3] [3]

Journal of the American Statistical Association , volume=

A proportional hazards model for the subdistribution of a competing risk , author=. Journal of the American Statistical Association , volume=. 1999 , doi=

1999

[4] [4]

Fine, J. P. and Jiang, H. and Chappell, R. , title =. Biometrika , year =

[5] [5]

Statistics in Medicine , volume =

Uno, Hajime and Horiguchi, Miki , title =. Statistics in Medicine , volume =. 2023 , pmid =

2023

[6] [6]

The hazards of hazard ratios , journal =

Hern. The hazards of hazard ratios , journal =. 2010 , volume =. doi:10.1097/EDE.0b013e3181c1ea43 , pmid =

doi:10.1097/ede.0b013e3181c1ea43 2010

[7] [7]

Frontiers in Oncology , year =

Kim, Haesook Teresa , title =. Frontiers in Oncology , year =. doi:10.3389/fonc.2024.1360266 , pmid =

arXiv 2024

[8] [8]

and Gange, Stephen J

Lau, Bryan and Cole, Stephen R. and Gange, Stephen J. , title =. American Journal of Epidemiology , year =. doi:10.1093/aje/kwp107 , pmid =

work page doi:10.1093/aje/kwp107

[9] [9]

D. Y. Lin , title =. Statistics in Medicine , volume =

[10] [10]

Competing risks: A practical perspective

Pintilie, Melania. Competing risks: A practical perspective

[11] [11]

Tutorial in biostatistics: competing risks and multi-state models

Putter, H and Fiocco, M and Geskus, R B. Tutorial in biostatistics: competing risks and multi-state models. Stat. Med

[12] [12]

, title =

Putter, Hein and Schumacher, Martin and van Houwelingen, Hans C. , title =. Biometrical Journal , year =. doi:10.1002/bimj.201800274 , pmcid =

work page doi:10.1002/bimj.201800274

[13] [13]

A novel stratified analysis method for testing and estimating overall treatment effects on time-to-event outcomes using average hazard with survival weight

Qian, Zihan and Tian, Lu and Horiguchi, Miki and Uno, Hajime. A novel stratified analysis method for testing and estimating overall treatment effects on time-to-event outcomes using average hazard with survival weight. Stat. Med

[14] [14]

, pmid =

Schrag, Deborah and Uno, Hajime and Rosovsky, Rachel and Rutherford, Cynthia and Sanfilippo, Kristen and ... , pmid =. Direct Oral Anticoagulants vs Low-Molecular-Weight Heparin and Recurrent VTE in Patients With Cancer: A Randomized Clinical Trial. JAMA

[15] [15]

Journal of Clinical Oncology , volume=

Moving beyond the hazard ratio in quantifying the between-group difference in survival analysis , author =. Journal of Clinical Oncology , volume=. 2014 , doi=

2014

[16] [16]

2024 , pmid =

Uno, Hajime and Tian, Lu and Horiguchi, Miki and Hattori, Satoshi and Kehl, Kenneth L , Journal =. 2024 , pmid =

2024

[17] [17]

Pfeffer and Scott D

Lihui Zhao and Brian Claggett and Lu Tian and Hajime Uno and Marc A. Pfeffer and Scott D. Solomon and Lorenzo Trippa and L. J. Wei , title =. Biometrics , volume =. 2016 , doi =

2016

[18] [18]

Comparative effectiveness research with average hazard for censored time-to-event outcomes: simulation study and application to observational data

Xiong, Hong and Connors, Jean and Schrag, Deb and Uno, Hajime. Comparative effectiveness research with average hazard for censored time-to-event outcomes: simulation study and application to observational data. BMC Med. Res. Methodol

[19] [19]

doi:10.5281/zenodo.17756398 , url =

Xing, Eric and Tian, Lu and Uno, Hajime , title =. doi:10.5281/zenodo.17756398 , url =

work page doi:10.5281/zenodo.17756398

[20] [20]

doi:10.5281/zenodo.17782192 , url =

Uno, Hajime and Horiguchi, Miki and Qian, Zihan , title =. doi:10.5281/zenodo.17782192 , url =

work page doi:10.5281/zenodo.17782192

[21] [21]

and Uno, H

Horiguchi, M. and Uno, H. , title =. 2026 , doi =

2026

[22] [22]

Assessing delayed treatment benefits of immunotherapy using long-term average hazard: a novel test/estimation approach

Horiguchi, Miki and Tian, Lu and Kehl, Kenneth L and Uno, Hajime. Assessing delayed treatment benefits of immunotherapy using long-term average hazard: a novel test/estimation approach. Lifetime Data Anal

[23] [23]

Design and analysis of randomized clinical trials with average hazard: Practical guidance and tools for implementation

Horiguchi, Miki and Tian, Lu and Hattori, Satoshi and Uno, Hajime. Design and analysis of randomized clinical trials with average hazard: Practical guidance and tools for implementation. Stat. Med

Average Cause-Specific Hazard: A Censoring-Invariant Measure of Event Burden Under Competing Risks

The reading

Core claim §

What carries the argument

Load-bearing premise §

What would settle it §

If this is right

Where Pith is reading between the lines

discussion (0)

Core claim

Load-bearing premise

What would settle it