{"id":"b7941c82-f2b9-4cd8-b8cc-da34343ff939","arxiv_id":"2507.23060","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A stratified Cox model with time-varying coefficients is estimated from zero-truncated recurrent event data supplemented by census information.","lead":"This paper introduces a statistical method for analyzing repeated events when only individuals who had at least one event are observed, such as youth mental health emergency visits. It uses population census counts to correct for the missing zero-event individuals and estimates how risk factors vary with age.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 1's asymptotic claim is not established: Appendix A relies on unproved Lemmas 1-2 and a functional Taylor expansion that does not account for plug-in estimation of conditional probabilities and census-based denominators.","rationale":"I read the paper in good faith: the method is clearly motivated, the simulations support consistency under the models studied, and the real-data analysis is relevant. The reader's CONDITIONAL verdict is appropriate. Where I would put the stress is slightly different: the most load-bearing concern is not the birthdate-independence assumption (which the authors themselves flag in Section 6) but the fact that Proposition 1, the central theoretical claim, is not proved in the manuscript. Appendix A's Lemma 1 and Lemma 2 are unproved, and the Taylor expansion skips the estimation of the plug-in probabilities and the census-based denominator. This matters because the theorem is what the claim that zero-truncated data plus census information suffice to recover true effects rests on. If the missing proof can be supplied, the conditional verdict should stand with a request for the full derivation; if it cannot, the theoretical contribution is unverified. A complete proof or an explicit counterexample from the functional expansion would settle this. The birthdate independence issue remains a practical limitation that should also be addressed in the application, but it is secondary for assessing the theorem itself.","tokens_in":15405,"tokens_out":22369,"duration_ms":299338,"concrete_test":"Complete the proof of Lemma 1 and Lemma 2 by deriving the influence function of tilde beta(a) from estimating equations (6)-(7), explicitly accounting for the plug-in estimators of P(Y_i^(s)(u)=1|Q1i) and the census-based tilde Z_s(gamma; u). If the derivation requires conditions beyond (I)-(VII), such as uniform consistency of the plug-in conditional probabilities or equicontinuity of the functional derivative, Proposition 1 must be revised; if the remainder is o_p(1), the proof gap is purely presentational.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 1 (Section 3.2) is the central claim, but Appendix A provides only an outline. Lemma 1 and Lemma 2 are stated without proof, and the key Taylor expansion of EFn(theta~(·); a) about theta0(a) treats P(Y_i^(s)(u)=1|Q1i) and tilde Z_s(gamma; u) as known smooth functions. In the actual estimator these quantities are recomputed at each iteration of Algorithm 1 and depend on the entire baseline and coefficient functions through the conditional-probability formulas in Appendix B.2. Conditions (I)-(VII) contain no stochastic-equicontinuity or smoothness condition for this plug-in map, and the remainder term o(theta~ - theta0) is not justified when theta(·) is a functional parameter. The positive-definiteness of Pi(theta0(a)) in Lemma 1 is asserted rather than proved. Thus the claimed sqrt(nh) asymptotic normality cannot be checked from the manuscript even under the stated assumptions. The Section 6 caveat that birthdates may not be independent of the counting process is a separate practical threat; the missing proof is more immediate because it leaves the theorem unverified as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a stratified Cox-type intensity model with time-varying regression coefficients for zero-truncated recurrent event data, where the population denominator is supplied by aggregate census information. The stratification variable depends on the event history but is only partially known because events before the observation window are unobserved; the authors replace the unknown stratum indicators by conditional probabilities computed under the model, and they estimate coefficients and cumulative baseline intensities by jointly solving local estimating equations. The main theoretical claim is Proposition 1, which asserts pointwise strong consistency and asymptotic normality of the time-varying coefficient estimator. The method is illustrated on pediatric mental health emergency department data and evaluated in a simulation study. The paper also proposes a Poisson-multiplier resampling procedure for variance estimation and an iterative algorithm for solving the estimating equations.","tokens_in":15590,"tokens_out":5081,"duration_ms":69635,"significance":"If the asymptotic result were rigorously established, the paper would provide a useful inferential framework for a practically important data structure: zero-truncated recurrent events with partly observed event histories and external population counts. The real-data application to MHED visits is relevant, and the modeling framework generalizes several existing recurrent-event models. The simulation study attempts to assess behavior under model misspecification as well as under the fitted model. The main weakness is that the theoretical foundation is incomplete as written: the proof of Proposition 1 rests on unproved lemmas and an unjustified functional Taylor expansion, and the conditional probability formula used in the estimating equations appears to require verification. These issues are load-bearing for the paper's central claim, so the manuscript needs a major revision.","major_comments":[{"comment":"The asymptotic result is not established as written. Appendix A states Lemmas 1 and 2 without proof, and the Taylor expansion of E_Fn(θ(·); a) about θ0(a) treats the plug-in quantities P(Y_i^(s)(u)=1|Q1i) and the census-based denominator as known smooth functions. In the actual procedure these quantities are recomputed at each iteration of Algorithm 1 and depend on the full coefficient and baseline functions through the formulas in Appendix B.2. Conditions (I)-(VII) contain no stochastic equicontinuity or smoothness condition for this plug-in map, and the remainder term o(θ̃ - θ0) is not justified when θ(·) is a functional parameter. The positive-definiteness of Π(θ0(·); a) in Lemma 1 is also asserted rather than proved. A complete proof with all lemmas and the required conditions is needed before the consistency and asymptotic normality claims can be accepted.","section":"§3.2, Proposition 1 and Appendix A"},{"comment":"The displayed formula for P(N_i(C_Li)=0, N_i(a−)−N_i(C_Li)=0 | Q1i) appears to be independent of a, since the right-hand side involves ai1 and C_Li but not a, apart from the event that defines the 'otherwise' case. If the expression is correct, the conditional probability P(Y_i^(1)(a)=1|Q1i) would be constant for all a before the first observed event, which is not compatible with the model's intensity structure. This quantity is used directly in the estimating equation (6) and in Algorithm 1, so the derivation in Appendix B.2 needs to be checked carefully and either corrected or expanded.","section":"Appendix B.2"},{"comment":"The assumption that the birthdate Bi is independent of the counting process Ni(·) is load-bearing for the censoring and truncation adjustment, and Section 6 itself notes that this may fail because different generations can have distinct event patterns. If Bi is correlated with the event process, the census-based denominators and the conditional probabilities in the estimating equations are no longer valid, and the consistency claim in Proposition 1 would not apply to the real-data setting. The authors should at least provide a concrete diagnostic or sensitivity analysis—for example, comparing estimates across birth cohorts or including a calendar-time covariate—to assess how sensitive the conclusions are to this assumption.","section":"§2 and §6"},{"comment":"The simulation study is reported almost entirely through plots. The text states that the true functions fall within the 95% pointwise confidence intervals 'for most of the time' and treats this as verification of consistency, but no numerical bias, Monte Carlo standard error, or pointwise coverage proportions are reported. With 1,000 simulation repetitions, coverage probabilities and average squared errors could be tabulated for representative ages. Without such summaries, the finite-sample evidence for the central claim is weaker than the text suggests.","section":"§5"}],"minor_comments":[{"comment":"The captions for Figures 4–6 refer to 'Table (1)' instead of 'Table 1'.","section":"Figure captions, §5.1"},{"comment":"The phrase 'Models (1 or SSV)' in the description of the real-data results is ambiguous; it should be 'Model (SSV)' or 'Models (1)/(SSV)' as appropriate.","section":"§4"},{"comment":"The algorithm uses a tolerance τ* but the manuscript does not state what value of τ* was used in the simulations or data analysis; this is needed for reproducibility.","section":"§3.2, Algorithm 1"},{"comment":"The choice of bandwidth h=9 units and truncation points τL=9, τR=105 is described but no sensitivity analysis is provided; since the local estimation results can depend on these tuning parameters, a brief sensitivity check would be helpful.","section":"§4"},{"comment":"No code or data availability statement is provided. Given that the method is implemented in C++ via Rcpp/RcppArmadillo, making the code available would substantially strengthen the reproducibility of the simulation and data analysis.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper is a plausible extension of the authors' earlier work, but the theoretical core is not yet verifiable: the proof of Proposition 1 is an outline with unproved lemmas, and the conditional probability formula in Appendix B.2 needs checking. I would send the manuscript back for a major revision rather than reject it, because the proposed estimating procedure and simulation appear sound in broad strokes and the identified issues may be fixable within the scope of the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plainly: the paper does something sensible and useful. It takes Chen et al.'s stratified intensity model for zero-truncated recurrent events and lets the regression coefficients vary with age, using census-based denominators to correct for the unobserved zero-event population. That fills a real gap and the PMHC data example is appropriate. The modeling is thoughtful; the distinction between fully and partially known stratification is handled clearly.\n\nCredit where due: the census correction is properly anchored in Hu and Lawless (1996a) and Xiong et al. (2024), the algorithm is described concretely, and the authors are explicit about the birthdate-independence assumption being potentially violated. The simulation design, though limited, is honest: true functions lie within pointwise intervals in the reported scenarios.\n\nThe soft spots are real. Proposition 1 is the load-bearing claim, and it is not actually proved in the manuscript. Appendix A gives Lemma 1 and Lemma 2 without proof, and the Taylor expansion treats P(Y_i^(s)(u)=1|Q1i) and the tilde Z term as known smooth functions when in fact they are recomputed from current parameter estimates. Conditions (I)-(VII) contain no equicontinuity or smoothness condition for that plug-in map, so the o(theta~-theta0) remainder is not justified. This does not mean the theorem is false, but it means the paper has not earned the asymptotic normality claim as written.\n\nThe simulation evidence is also thinner than stated: one main scenario with graphical summaries, no tables of bias/coverage/MSE, and no code or data to reproduce. The birthdate independence assumption is a practical threat; the authors flag it, but it could break consistency in real applications.\n\nIn balance: this is a credible methodological contribution that deserves a serious referee. The referee should ask for full proofs of Lemmas 1-2, careful treatment of the plug-in estimation, and more extensive simulation diagnostics. I would not cite it as a methodological foundation until the theory is worked out, but I'd bring it to a reading group for discussion. Set would_accept_peer_review true.","headline":"A useful extension of stratified Cox models to zero-truncated data, but the asymptotic proof is only a sketch and the plug-in details need work.","tokens_in":16158,"tokens_out":2261,"would_cite":false,"duration_ms":27107,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62N01","62N02","62G05","62G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that zero-truncated recurrent-event records, supplemented by census counts, suffice to estimate age-varying covariate effects on event intensity consistently.","keywords":["zero-truncated recurrent events","time-varying coefficients","stratified Cox model","conditional intensity function","population census information","local constant estimation","partially known stratification","mental health emergency department visits"],"falsifier":"Simulate recurrent events under Model (1) with a birth-cohort effect, for example $\\lambda_{0s}(a)\\exp\\{\\gamma B_i\\}$, fit the proposed estimator while ignoring $B_i$, and check whether $\\tilde{\\beta}(a)$ drifts and whether the nominal 95% intervals cover the true functions as $\\gamma$ grows; a systematic bias with coverage dropping would confirm that the birthdate-independence assumption is load-bearing. A data version is to split the MHED cohort by birth period and test whether the estimated coefficient curves differ across periods.","tokens_in":15184,"feed_emoji":"📊","tokens_out":10221,"duration_ms":102008,"temperature":0.7,"pith_summary":"This paper tackles a common data problem: electronic health records contain only people who had at least one event, so the population at risk—those who never went to the emergency department—is invisible. The authors claim that by splicing in aggregate census counts, the missing risk sets can be approximated well enough to estimate how covariate effects change with age in a stratified Cox-type intensity model. They construct a local-constant estimator for the time-varying regression coefficients, prove that it is pointwise consistent and asymptotically normal under regularity conditions, and verify the finite-sample behavior in simulations. The motivating application is pediatric mental-health emergency department visits in Alberta, where the fitted model also supports dynamic prediction of future visits based on a child's visit history.","feed_headline":"Zero-truncated records plus census info recover age-varying risks","feed_subtitle":"Population tables fill in the unobserved zero-event group, yielding consistent estimates of covariate effects.","key_machinery":"The load-bearing element is a system of kernel-weighted estimating equations for the local-constant approximation of $\\beta_s(a)$. Equations (3) and (4) are built from the observed zero-truncated event counts, with the denominators—risk-set sums over the entire population—replaced by sums of census counts $\\sum_l C(l,z,\\lfloor u\\rfloor)$ times model-based stratum probabilities. When the stratification is only partially known because the full event history is unavailable for subjects born before the study window, the indicator $Y_i^{(s)}(u)$ is replaced by its conditional probability $P(Y_i^{(s)}(u)=1 \\mid Q_{1i})$ computed under the model; coefficients and cumulative baselines are then solved iteratively in Algorithm 1, and variances are estimated by Poisson-multiplier resampling.","core_discovery":"The central claim is that for recurrent-event data observed only for subjects with at least one event, the age-specific effects of covariates on the event intensity can still be recovered if aggregate population census counts are used to approximate the unobserved risk sets. Under the stratified intensity model $\\lambda(a \\mid H_i(a), Z_i) = \\lambda_{0s}(a)\\exp\\{\\beta_s(a)'Z_i\\}$, the proposed local-constant estimator $\\tilde{\\beta}(a)$ is pointwise consistent, $\\tilde{\\beta}(a) \\to \\beta_0(a)$ almost surely, and asymptotically normal with $\\sqrt{nh}(\\tilde{\\beta}(a)-\\beta_0(a)) \\to N(0, \\mathrm{AV}(\\beta_0(a)))$ for each age in a chosen interior interval. The same estimating system also yields the cumulative baseline intensity functions, and the fitted model can predict the risk of a future event at any age given covariates and the observed history.","pith_inferences":["An implicit consequence is that the method's accuracy is tied to how finely the census tables discretize age and covariates; yearly age bins make the risk-set denominators approximate, and monthly or daily population counts should reduce bias in $\\tilde{\\beta}(a)$ while narrowing the pointwise intervals.","The birthdate-independence assumption, flagged in Section 6 as fragile, could be tested inside the same framework by adding birth cohort as a stratum or covariate; if cohort effects are present, the census-based denominators would need a cohort-specific correction.","The same estimating-equation strategy should transfer to other zero-truncated administrative registries, such as hospital readmissions or justice-system contacts, wherever population tables exist; the key requirement is that census cells align with the covariates in the intensity model.","Because the partially known stratification is handled through conditional probabilities computed under the model, a sensitivity analysis for misspecified stratum-transition probabilities would be the most natural stress test of the procedure."],"forward_implications":["Health administrators can estimate age-specific risk factors for first versus repeat emergency visits from administrative records alone, as long as population census tables with the same covariate levels are available.","The asymptotic normality result justifies pointwise confidence bands for the time-varying coefficients through Poisson-multiplier resampling, giving the reported uncertainties a theoretical grounding.","The stratified model nests the Andersen-Gill model, the Prentice-Williams-Peterson model, and earlier marginal analyses as special cases, so the new estimator generalizes those analyses to age-varying effects.","Given fitted coefficients and baselines, the model supports dynamic prediction of the probability of a future event at any age, conditional on a subject's observed history and covariates.","Simulation results indicate that when the true process has both baseline and coefficient stratification, only the proposed SSV model recovers the true stratum-specific coefficient and baseline functions, while simpler models blend the strata or deviate at later ages."],"supporting_citations":[{"why":"Supplies the supplementary-information approach for truncated lifetime data that the census-correction step adapts.","marker":"Hu and Lawless (1996a)"},{"why":"Establishes nonparametric estimation of rate and mean functions for zero-truncated recurrent events when the population size is known, the basis of the census approximation.","marker":"Hu and Lawless (1996b)"},{"why":"Provides the local-linear estimation framework and asymptotic conditions that Proposition 1 adapts to the stratified setting.","marker":"Hu and Rosychuk (2016)"},{"why":"Prior analysis combining zero-truncated recurrent event data with census information under time-varying coefficients, which this paper extends to intensity-based stratified models.","marker":"Xiong et al. (2024)"},{"why":"Previous intensity-based model with time-varying stratification and partially known stratification, the direct predecessor extended here to time-varying coefficients.","marker":"Chen et al. (2025)"},{"why":"Supplies the Poisson-multiplier resampling method used for variance estimation of the coefficient curves.","marker":"Dobler, Pauly, and Scheike (2019)"}],"fun_headline_variants":["Census data fill in zero-event gap for recurrent event risk","Time-varying risk recovered from zero-truncated data with census","Zero-truncated events? Census counts recover age-specific risks","Stratified Cox with time-varying coefficients uses census data","Census-assisted estimation for zero-truncated recurrent events"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole procedure assumes a subject's birth date is independent of their event process, and Section 6 acknowledges that different generations could have different event patterns; if birthdate and event intensity are correlated, the census-based denominators and estimating equations are biased and consistency collapses.","fun_headline_variants_meta":{"raw":{"variants":["Census data fill in zero-event gap for recurrent event risk","Time-varying risk recovered from zero-truncated data with census","Zero-truncated events? Census counts recover age-specific risks","Stratified Cox with time-varying coefficients uses census data","Census-assisted estimation for zero-truncated recurrent events"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001356,"raw_usage":{"total_tokens":5447,"prompt_tokens":832,"completion_tokens":4615,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":448,"completion_tokens_details":{"reasoning_tokens":4531}},"tokens_in":448,"tokens_out":4615,"duration_ms":33985,"temperature":1.0,"reasoning_tokens":4531,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T11:04:31.511237+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate recurrent events under Model (1) with a birth-cohort effect, for example $\\lambda_{0s}(a)\\exp\\{\\gamma B_i\\}$, fit the proposed estimator while ignoring $B_i$, and check whether $\\tilde{\\beta}(a)$ drifts and whether the nominal 95% intervals cover the true functions as $\\gamma$ grows; a systematic bias with coverage dropping would confirm that the birthdate-independence assumption is load-bearing. A data version is to split the MHED cohort by birth period and test whether the estimated coefficient curves differ across periods.","supporting_citations":[{"cited_title":"\\ Rosychuk, R J","cited_arxiv_id":null,"evidence_quote":"Provides the local-linear estimation framework and asymptotic conditions that Proposition 1 adapts to the stratified setting."},{"cited_title":"Exploring Differences between Two Decades of Mental Health Related Emergency Department Visits by Youth via Recurrent Events Analyses","cited_arxiv_id":"2407.09761","evidence_quote":"Prior analysis combining zero-truncated recurrent event data with census information under time-varying coefficients, which this paper extends to intensity-based stratified models."},{"cited_title":"Stratified Regression Analysis of Zero-Truncated Recurrent Event Data","cited_arxiv_id":"2505.02996","evidence_quote":"Previous intensity-based model with time-varying stratification and partially known stratification, the direct predecessor extended here to time-varying coefficients."},{"cited_title":", Pauly, M","cited_arxiv_id":null,"evidence_quote":"Supplies the Poisson-multiplier resampling method used for variance estimation of the coefficient curves."}],"review_version":1}