{"id":"1c980f81-2e7a-4346-b125-0ee4f7adfacd","arxiv_id":"2607.13293","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"This paper proposes a new summary statistic for clinical trials with competing risks: the average cause-specific hazard, a rate per event-free person-time that does not depend on how long patients are followed. It comes with estimation, confidence intervals, two-group tests, and an application to a cancer blood-clot trial.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Remark 1's simplified covariance formula is algebraically wrong: ξ_{k,ℓ} omits the 1/F_k(τ) factor on S(u)I(ℓ=k), so Eq (5) is not the covariance of the influence functions in Eq (4); using it would invalidate the Wald-type inference.","rationale":"The paper's estimand, η_k(τ)=F_k(τ)/R(τ), is well-defined, censoring-invariant by construction, and the plug-in estimator is standard. The simulations and appendices provide reasonable support for the primary inference based on the sample covariance of the plug-in influence functions. The weakest point is the presentation of the variance formulas: Eq (4) uses S(u) where S(u-) is conventional, and Remark 1's Eq (5) has a concrete algebraic error in the definition of ξ. This error is load-bearing for the simplified covariance formula, but the main methodology can be fixed by inserting the missing 1/F_k(τ) factor; the central scientific claim is not fundamentally threatened. The reader's CONDITIONAL verdict remains appropriate, since these are fixable but real errors in a central inference component.","tokens_in":23510,"tokens_out":40340,"duration_ms":405459,"concrete_test":"Independently re-derive σ_{kk'}(τ) from Eq (4) by expanding Cov(ψ_k, ψ_k') and using the orthogonality of the cause-specific martingales together with E[d<M_ℓ>(u)] = G(u) dΛ_ℓ(u). Compare the resulting expression with Eq (5). If the re-derived coefficient of S(u)I(ℓ=k) carries the extra factor 1/F_k(τ), the concern lands. Optionally, simulate the exponential setting of Scenario (i) with m=2 and compute Wald statistics using the plug-in of Eq (5) as printed versus the sample covariance of the plug-in IFs; the former should show distorted type I error.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim includes asymptotically normal plug-in inference with an influence-function variance estimator. The primary estimator in Eq (4) is essentially correct, up to the usual S(u) versus S(u-) convention. However, Remark 1 states that the (k,k') element of Σ(τ) simplifies to Eq (5), with ξ_{k,ℓ}(u,τ)=S(u)I(ℓ=k)+F_k(u)/F_k(τ). Expanding Cov(ψ_k,ψ_k') from Eq (4) using mutual orthogonality of the cause-specific martingales and E[d<M_ℓ>(u)]=G(u)dΛ_ℓ(u) gives an integrand proportional to [S(u)I(ℓ=k)/F_k(τ)+F_k(u)/F_k(τ)-R(u)/R(τ)], not S(u)I(ℓ=k)+F_k(u)/F_k(τ)-R(u)/R(τ). The missing 1/F_k(τ) on the first term is not a harmless notation slip: it changes the scale of the self-martingale term, so the plug-in covariance from Eq (5) as printed is inconsistent for Σ(τ). A user who implements the global Wald test of Section 2.3 using Eq (5) literally would not obtain the claimed χ²_m null distribution. This does not invalidate the main estimand or the sample-covariance estimator based on Eq (4), but it means a central, explicitly stated inference formula is wrong as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":23807,"tokens_out":14752,"duration_ms":141479,"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":[{"comment":"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","section":"Remark 1, Eq. (5)"}],"minor_comments":[{"comment":"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":"Section 2.2, Eq. (4) and Appendix A"},{"comment":"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":"Section 2.4, asymptotic representation display"},{"comment":"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":"Section 3.1 / Table 2 caption"},{"comment":"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.","section":"Section 5 / software availability"}],"recommendation":"major_revision","confidential_remarks":"The Eq. (5) error is real and should be corrected before publication; it is the only substantive defect I found. The estimand, the primary plug-in estimator based on Eq. (4), and the simulation evidence are sound. This is a fixable issue, not a fundamental flaw, so I recommend major revision rather than rejection. The authors should also carefully check the S(u)/S(u−) convention and the hat in Section 2.4 during revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nWho 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.","headline":"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.","tokens_in":24318,"tokens_out":12105,"would_cite":true,"duration_ms":104035,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62N01","62N02","62P10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["competing risks","cause-specific hazard","average hazard","censoring invariance","cumulative incidence function","restricted mean event-free time","incidence rate","influence function"],"falsifier":"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.","tokens_in":23353,"feed_emoji":"📊","tokens_out":5512,"duration_ms":53588,"temperature":0.7,"pith_summary":"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.","feed_headline":"New event rate for competing risks ignores censoring bias","feed_subtitle":"ACSH divides cumulative incidence by event-free time, staying unbiased where naive rates drift.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["A rate for competing risks that ignores censoring bias","New measure: censoring-invariant event rate under competing risks","Unbiased event rate for competing risks via average cause-specific hazard","Censoring-proof rate for competing-risk studies"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["A rate for competing risks that ignores censoring bias","New measure: censoring-invariant event rate under competing risks","Unbiased event rate for competing risks via average cause-specific hazard","Censoring-proof rate for competing-risk studies"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000187,"raw_usage":{"total_tokens":1156,"prompt_tokens":726,"completion_tokens":430,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":470,"completion_tokens_details":{"reasoning_tokens":376}},"tokens_in":470,"tokens_out":430,"duration_ms":6305,"temperature":1.0,"reasoning_tokens":376,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T05:36:11.619152+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}