{"id":"34744797-5e2b-4c29-b8f0-04b5801a7965","arxiv_id":"2412.14478","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A landmark-based penalized spline method estimates time-varying effects of a functional covariate in a Cox model, scaling to functional predictors with 1440 dimensions.","lead":"This paper develops two ways to estimate how the effect of a functional predictor, such as a person's daily activity pattern, on survival risk changes over follow-up time. A landmark-based version scales to large datasets and is used to study mortality and diurnal movement in NHANES.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central inference claim rests on an unstated Cox-Poisson equivalence for a smooth bivariate coefficient surface; without a derivation or regularity conditions, the nominal coverage results are evidence only for the four simulated scenarios.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: the Cox-Poisson transformation is asserted without derivation, so the inferential validity of the TV-FLCM is not established outside the simulated settings. I agree with that assessment. The paper has real strengths: the simulation study covers four coefficient surfaces, the landmark and non-landmark approaches are compared on AMSE and computation time, the landmark speed advantage is credible, and the NHANES application illustrates the scalability motivation. However, the central claim of nominal coverage depends on an unproven equivalence. This is not an internal inconsistency, but a missing proof; the supplementary material does not supply the derivation, and Section 6 explicitly lists rigorous asymptotic theory as future work. A direct re-derivation or a numerical check against a brute-force partial-likelihood fit would settle the issue. If the equivalence holds, the methodological contribution is stronger; if it fails, the headline inference claim collapses. Because the reader's condition already captures this risk, I recommend no change to the CONDITIONAL verdict.","tokens_in":18183,"tokens_out":9994,"duration_ms":90705,"concrete_test":"Independently re-derive the score equations of the stacked Poisson model used in Section 2.3 and compare them with the Cox partial-likelihood score equations for model (2) under a tensor-product spline basis for gamma(u,t) with no tied event times. If they differ, the proposed TV-FLCM is not the claimed Cox model. As an empirical complement, simulate N=500 with a known bivariate gamma(u,t)=sin(2*pi*u)/(t+0.5), no ties, moderate censoring, and estimate the coefficient using (a) mgcv::gam(family=cox.ph) exactly as coded and (b) a direct numerical maximization of the penalized partial likelihood, for example coxph with an expanded design matrix. Compare point estimates and Wald SEs; agreement to numerical tolerance across several datasets would show the implementation realizes the Cox partial likelihood, while disagreement would invalidate the central inference claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2.3 asserts, without derivation or regularity conditions, that 'by extending the functional term using tensor product splines, the entire Cox model is transformed into a Poisson regression model,' citing Whitehead (1980). The quoted result is a profile-likelihood or conditional-likelihood identity that requires a specific construction: at each event time, the risk-set contributions must enter as a Poisson stratum with an unrestricted offset, and the baseline hazard must be profiled rather than modeled. For the TV-FLCM, eta_i(t) = integral Z_i(u) gamma(u,t) du is a smooth function of t, so it is not obvious that the stacked risk-set rows with event indicators form a valid Poisson likelihood; the paper gives no offset, no proof that the score equations coincide, and no statement of conditions such as no tied event times, censoring mechanism, or compactness of the spline basis. Section 6 then lists 'developing rigorous asymptotic theory' as future work, an explicit admission that the inference guarantee is not established. Tables 2-4 verify coverage only for four gamma surfaces and even show local undercoverage near t=0 for 10cos(4pi(t-u)); the pointwise coverage failures are acknowledged but not resolved. If the Poisson or partial-likelihood equivalence fails, the headline claims of 'valid estimation and inference' and 'nominal coverage' are unsupported beyond the simulated cases. This is the load-bearing assumption for the Cox-Poisson method; the landmark method's accuracy claims are less affected, though its own inference is explicitly not assessed.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two approaches for estimating a time-varying functional linear Cox model, in which the effect of a baseline functional predictor on the hazard is a bivariate smooth coefficient gamma(u,t) of the functional domain u and follow-up time t. The first approach, TV-FLCM, is said to exploit the Cox-Poisson likelihood connection and is intended for small-to-medium datasets; the second, TV-FLCM-L, uses landmark times and short prediction windows to reduce the computational burden for larger datasets. Estimation is carried out with penalized tensor-product splines via the mgcv package. The simulation study with four gamma surfaces and sample sizes 2000-4000 reports lower AMSE for TV-FLCM than for TV-FLCM-L, nominal or near-nominal average coverage for TV-FLCM in several scenarios, and 10-42 times faster computation for the landmark approach with w=0.04. The NHANES application estimates how the association between diurnal motor activity and mortality attenuates over an eight-year follow-up and illustrates the scalability of the landmark method.","tokens_in":18447,"tokens_out":10594,"duration_ms":88563,"significance":"If the Cox-Poisson equivalence and the associated coverage guarantees can be established rigorously, the paper offers a practical and potentially widely used extension of functional Cox regression to time-varying coefficients, building on the stable and popular mgcv infrastructure. The landmark variant addresses a real computational bottleneck for high-dimensional functional predictors such as 1440-minute accelerometry profiles, and the simulation study covers several nontrivial coefficient shapes, including an interaction-like surface. The NHANES application is directly relevant to a large public-health literature. The paper also deserves credit for including the stacked-landmark data construction, explicit model formulas, and a sensitivity-style comparison of landmark window choices. However, the main inference claim depends on an unproven Poisson-likelihood transformation for a continuously varying bivariate coefficient, and the computational pathway for the TV-FLCM fits is not fully documented in the main text.","major_comments":[{"comment":"The central inference step is asserted rather than derived. Section 2.4 states that 'By extending the functional term using tensor product splines, the entire Cox model is transformed into a Poisson regression model,' citing Whitehead (1980). The paper does not give the stacked Poisson likelihood, the offset or stratum construction, or a proof that the score equations coincide with those of the Cox partial likelihood for the bivariate smooth coefficient surface gamma(u,t) entering through the integral term. The standard Whitehead construction is a profile-likelihood/Poisson equivalence that requires a specific treatment of the baseline hazard and risk sets; it is not immediate for the present continuous-time setting. Please add a derivation or state a theorem with sufficient conditions (e.g., no tied event times, independent censoring, compactly supported spline bases), and reconcile the abstract's unconditional 'valid estimation and inference' claim with Section 6's statement that developing rigorous asymptotic theory is future work.","section":"Section 2.4 (with Section 2.3 and Section 6)"},{"comment":"The simulation data-generating mechanism is incompletely specified. The hazard is written as log lambda_i(t|X_i, Z_i(u), s_l) = log lambda_0(t|s_l) + integral Z_i(u) gamma(u,s_l) du, but the baseline hazard lambda_0 is never defined, and the survival curve S_i(t|Z_i) that is inverted to generate event times is not given in equation form. In addition, the text introduces a noisy version Z_{i,real}(u)=Z_i(u)+epsilon_i(u) after saying the predictor is 'measured with bias,' but it is not stated whether the fitted landmark and Poisson models use Z_{i,real} or the true Z_i. Consequently, the AMSE and coverage numbers in Tables 2-5 are not reproducible from the information provided.","section":"Section 5.1 and Tables 2-5"},{"comment":"The R code for the separate landmark model uses `ti(umat, by=Zlmat, bs=c(\"cc\",\"cr\"), k=c(5,5), mc=c(T,F))` with a single variable, while the proposed model of Eq. (3) involves a bivariate smooth in (u,s_l); as written, the code does not implement the tensor-product smooth described in Section 2.5. Furthermore, no R code or dataset construction is shown for the TV-FLCM (Poisson) estimator whose results appear in Tables 2-5. Since the paper advertises stable software implementation, please correct the code snippets and provide the full estimation pathway for TV-FLCM, including the family, offset, stacking rule, or a pointer to an online supplement with complete code.","section":"Section 3.2"},{"comment":"The abstract's claim that 'The Cox-Poisson method provides nominal coverage probabilities' is too strong for the reported results. Table 4 shows average coverages of 95.1%, 94.3%, and 94.0% for N=2000, 3000, and 4000 for gamma(u,t)=10cos(4*pi*(t-u)), with the text acknowledging 'substantial undercoverage for t close to 0.' The stated explanation (many events before t=0.05) describes the event process but does not explain why the Wald intervals are invalid in that region; if the Poisson equivalence is the basis for the intervals, this is a known failure regime for the inference claim. Please qualify the abstract statement to 'empirically nominal in the scenarios considered' and either correct the intervals in regions with heavy early events or provide a theoretical explanation for the undercoverage.","section":"Section 5.3 and Table 4"}],"minor_comments":[{"comment":"Equation (2) writes the scalar covariate term as X_i beta, whereas Section 2.1 and the landmark model in Eq. (3) allow beta to depend on time; please make the notation consistent, for example by writing X_i beta(t).","section":"Equation (2)"},{"comment":"In the two displayed partial-likelihood expressions, the quantity inside the logarithm is written as e^{eta_i} in the sum over risk-set members; it should be e^{eta_j}, with the sum indexed by j.","section":"Section 2.5, displayed partial-likelihood formulas"},{"comment":"The citation list contains 'Bender et al. 2018, ?'; the missing reference or citation key should be completed.","section":"Section 2.1"},{"comment":"The window specification '(0.5, 0.25, 1.2, 0.8, 1.25, 1, 0.5, 0.75, 1, 0.65, 0.1, 0.5) + 0.3' is ambiguous; please clarify whether 0.3 is added to each window length or to each interval endpoint.","section":"Section 4.2"},{"comment":"The reported AMSE and coverage values are averaged over 500 simulations but do not include Monte Carlo standard errors; adding these would help readers judge whether differences across methods and sample sizes are meaningful.","section":"Tables 2-5"},{"comment":"The conclusion that diurnal effects attenuate over the eight-year follow-up is drawn from a visual comparison of landmark curves; a quantitative summary (e.g., the integrated negative-area per landmark time, or a formal trend test) would strengthen the claim.","section":"Section 4.3"},{"comment":"When discussing REML smoothing-parameter selection for the stacked landmark data, the paper notes that observations are treated as independent; because each subject contributes multiple landmark rows, this is a pseudo-likelihood feature, and the possible effect on smoothing-parameter or interval estimates should be stated as a limitation.","section":"Section 2.7"},{"comment":"The 'small loss of accuracy' conclusion applies only to the short window w=0.04; the w=Inf results are dramatically worse (e.g., AMSE 44.11 vs 2.51 in Table 4 at N=4000). The paper should state explicitly that w=Inf is not a generally advisable choice and that the simulation-based advice concerns short windows only.","section":"Section 5.3 and Tables 2-5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's central methodological claim rests on the Cox-Poisson likelihood connection, but the transformation is asserted without derivation or regularity conditions, and the TV-FLCM fitting code is not shown. These gaps are fixable and do not by themselves invalidate the landmark contribution, so I recommend major revision rather than rejection. I would urge the editor to require that the authors either supply a rigorous justification of the Poisson transformation (or a precise reference to an applicable theorem) and document the TV-FLCM estimation workflow, or explicitly reduce the paper's scope to the landmark estimator with coverage claims removed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a solid applied-methods paper with a practical contribution. The TV-FLCM lets the effect of a functional predictor vary smoothly in both the functional domain and event time, which addresses PH violations for functional predictors. The key novelty relative to Leroux (2020) is dropping the u <= t constraint, giving a rectangular coefficient surface. The landmark estimator, implemented through mgcv's cox.ph family, is a genuinely useful computational workaround for large datasets; the 10-42x speedup over the Poisson approach with only modest accuracy loss is credible from the simulations.\n\nWhat the paper does well: the simulation study is careful. Four functional forms, sample sizes 2000-4000, 500 replicates, and both AMSE and pointwise coverage are reported. The Poisson method hits around 95% average coverage for the smooth surfaces, and even for the oscillatory 10cos(4pi(t-u)) surface average coverage stays in the 94-95% range except near t=0, which they acknowledge. The NHANES application shows the landmark method scales to 4445 subjects with 1440-dimensional predictors, which is the kind of real problem that motivates the method.\n\nThe soft spots are real but mostly fixable. The biggest one: Section 2.3 asserts that tensor product splines turn the Cox model into a Poisson regression, citing Whitehead (1980), but gives no derivation, no offset construction, and no regularity conditions. This is a standard equivalence, but for a bivariate smooth coefficient it deserves at least a careful statement of the data-expansion steps, and the paper's own Section 6 says rigorous asymptotic theory is future work. That means the nominal coverage results are evidence only for the simulated scenarios, not a general guarantee. I'd call this a major gap but not a fatal one—the simulations are encouraging, and the mgcv implementation likely does the right thing.\n\nMinor issues: the NHANES results have no confidence intervals, so the headline 'attenuation over time' is unquantified. No code or data are publicly available, which makes the R snippets hard to trust as a full reproducibility package. The writing has some rough edges (typos, the phrase 'these novels').\n\nWho this is for: applied survival analysts working with accelerometry or other functional predictors, and functional data methodologists. It deserves a serious referee; the derivation gap should be a major revision, not a desk reject.","headline":"Useful extension of the functional Cox model with a practical landmark estimator, but inference validity rests on an unproven Poisson equivalence and the paper ships no code.","tokens_in":19051,"tokens_out":3589,"would_cite":true,"duration_ms":29165,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62N01","62G08","62R10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a functional predictor's effect on survival can be modeled as a smooth bivariate surface over the functional domain and follow-up time, estimated either by a Cox-Poisson transformation or a faster landmark…","keywords":["functional regression","Cox model","time-varying effects","landmark approach","penalized splines","functional data analysis","accelerometry","survival analysis"],"falsifier":"Simulate a dataset with a known coefficient surface that changes sharply in time (for instance, a step at $t=0.5$), fit the Cox-Poisson TV-FLCM and an exact partial-likelihood fit with the same spline basis, and compare pointwise coverage: systematic divergence or coverage well below 95% near the change point would disprove the claimed validity of the Poisson approximation for general smooth surfaces.","tokens_in":17884,"feed_emoji":"📈","tokens_out":9839,"duration_ms":73212,"temperature":0.7,"pith_summary":"This paper proposes a Cox survival model in which a functional predictor measured at baseline—such as a full day of physical activity—is allowed to affect the hazard through a coefficient surface that varies smoothly in both the predictor's domain and follow-up time. The paper develops two estimation routes: a Cox-Poisson transformation that yields accurate estimates with roughly nominal pointwise coverage, and a landmark approximation that partitions follow-up into windows and is 10 to 42 times faster with only a small accuracy loss. Both are fit with penalized tensor product splines inside standard penalized regression software. Applied to NHANES accelerometry data, the model finds that the association between diurnal activity patterns and mortality attenuates over an eight-year follow-up.","feed_headline":"Functional Cox model now estimates time-varying risk surfaces","feed_subtitle":"Both a Cox-Poisson and a landmark estimator recover the bivariate coefficient surface; landmark cuts computation 10-42x.","key_machinery":"The object that carries the argument is the bivariate coefficient surface $\\gamma(u,t)$, which multiplies the functional predictor $Z(u)$ inside the log hazard as $\\int_U Z(u)\\gamma(u,t)\\,du$. Estimation rests on two pieces of machinery: the Cox-Poisson likelihood transformation, in which each subject contributes one row per event time while at risk so the partial likelihood becomes a Poisson likelihood, and the landmark decomposition, which replaces continuous time $t$ by landmark times $s_l$ with prediction windows and stratifies the baseline hazard. Both routes expand $\\gamma$ in penalized tensor product splines—cyclic cubic splines over the functional domain and cubic splines over time—with smoothing parameters selected by restricted maximum likelihood.","core_discovery":"The central claim is that a functional linear Cox model with time-varying coefficients—$\\log \\lambda_i(t) = \\log \\lambda_0(t) + \\int_U Z_i(u)\\gamma(u,t)\\,du$—can be estimated and used for inference when $\\gamma(u,t)$ is a smooth bivariate surface. Estimation is carried out by expanding $\\gamma(u,t)$ in penalized tensor product splines and exploiting the Cox-Poisson likelihood connection, or by replacing $t$ with discrete landmark times $s_l$ and fitting a stratified landmark model. Simulations across four surfaces and sample sizes up to 4000 show the Cox-Poisson estimator recovers the surface with approximately nominal pointwise coverage, and the landmark estimator with short windows ($w=0.04$) has AMSE only 5.6% to 91.3% higher while reducing computation time by a factor of 10 to 42. The authors use the landmark model on the NHANES dataset (4445 subjects, 1440 minute-level activity values per subject) and report that the mortality association with diurnal activity attenuates over the follow-up.","pith_inferences":["The landmark window length is an implicit bias-variance dial: shorter windows reduce the bias from assuming a constant effect within the window but inflate variance, so an automated or cross-validated choice of $w$ could replace the manual settings used here.","The undercoverage near $t=0$ for the rapidly oscillating surface suggests that inference may degrade when many events occur very early; boundary corrections or a different parameterization of the time basis could be tested.","Because landmarking conditions on survival to each landmark time, the same machinery could be used for dynamic prediction, updating risk estimates as new covariate or follow-up information accumulates.","The Poisson equivalence might be extended to surfaces with limited smoothness or interactions between time and the functional domain, but the current simulation evidence does not support that generalization."],"forward_implications":["Because the full model can be expressed as a Poisson regression, standard penalized regression software can fit it, making the method immediately usable for small-to-medium datasets.","The landmark version scales to high-dimensional functional predictors (e.g., 1440 minute-level activity values) on a laptop, which was previously infeasible.","Pointwise confidence intervals from the Cox-Poisson route let researchers test where and when a functional predictor's effect is non-zero rather than only whether it is zero.","The NHANES finding that diurnal activity effects attenuate over eight years suggests future studies should model effect decay rather than assume a constant association.","The modeling framework extends naturally to multiple scalar and functional predictors, as the authors note."],"supporting_citations":[{"why":"Supplies the Cox-Poisson likelihood connection that turns the TV-FLCM into a Poisson regression model.","marker":"Whitehead (1980)"},{"why":"Introduces dynamic prediction by landmarking, the basis for the TV-FLCM-L approximation.","marker":"Van Houwelingen 2007"},{"why":"Provides low-rank scale-invariant tensor product smooths used to model the bivariate coefficient γ(u,t).","marker":"Wood (2006)"},{"why":"Supplies the penalized spline smoothing framework and software underlying the mgcv-based implementation.","marker":"Wood 2017"},{"why":"Established Cox regression with functional covariates using penalized B-splines, the setting this paper extends to time-varying effects.","marker":"Gellar et al. (2015)"},{"why":"Earlier work on a functional Cox model whose coefficient surface depends on both survival time and functional domain, including landmark estimation.","marker":"Leroux (2020)"},{"why":"Supplies the landmarking framework for dynamic predictions with time-dependent covariates used in the TV-FLCM-L.","marker":"Rizopoulos et al. 2017"}],"fun_headline_variants":["Landmark estimator speeds time-varying functional Cox 10-42x","Time-varying functional Cox: fast landmark vs accurate Cox-Poisson","Cox-Poisson and landmark: time-varying functional Cox","Functional Cox with time-varying coefficients: two new estimators"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes the Cox-Poisson likelihood transformation remains valid when the functional coefficient is a smoothly varying bivariate function of time and functional domain, but it gives no derivation or regularity conditions for the continuous-time setting, and the nominal coverage claim rests on simulations of four smooth surfaces.","fun_headline_variants_meta":{"raw":{"variants":["Landmark estimator speeds time-varying functional Cox 10-42x","Time-varying functional Cox: fast landmark vs accurate Cox-Poisson","Cox-Poisson and landmark: time-varying functional Cox","Functional Cox with time-varying coefficients: two new estimators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001218,"raw_usage":{"total_tokens":5039,"prompt_tokens":1004,"completion_tokens":4035,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":620,"completion_tokens_details":{"reasoning_tokens":3964}},"tokens_in":620,"tokens_out":4035,"duration_ms":23277,"temperature":1.0,"reasoning_tokens":3964,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:12:25.829431+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a dataset with a known coefficient surface that changes sharply in time (for instance, a step at $t=0.5$), fit the Cox-Poisson TV-FLCM and an exact partial-likelihood fit with the same spline basis, and compare pointwise coverage: systematic divergence or coverage well below 95% near the change point would disprove the claimed validity of the Poisson approximation for general smooth surfaces.","supporting_citations":[{"cited_title":"(2020), Statistical methods for the analysis of functional data under models with complex association structure, PhD thesis, Johns Hopkins University","cited_arxiv_id":null,"evidence_quote":"Earlier work on a functional Cox model whose coefficient surface depends on both survival time and functional domain, including landmark estimation."}],"review_version":1}