Non-Crossing Deep Quantile Regression for Distributional Survival Prediction

Motivation and gap

• Quantile regression directly models conditional survival quantiles on the original time scale, allowing covariate effects to vary across the event-time distribution rather than collapsing them into hazard or mean summaries.This motivation builds on censored-quantile methods developed by Powell (1986), Ying, Jung and Wei (1995), and Peng and Huang (2008).Source: §1 · p. 2
• Existing neural quantile methods fit separate levels without ordering constraints, allowing logically inconsistent crossing curves despite greater flexibility than classical linear censored quantile regression.Classical censored quantile regression cannot adequately represent nonlinear, interacting covariate effects.Source: §1 · p. 2

Proposed framework

• CNQ jointly estimates multiple conditional survival quantiles for right-censored data and enforces their non-crossing ordering by construction.Its feature extractors use KANs, Transformers, or a KAN–Transformer hybrid to model nonlinearities and covariate interactions.Source: §1 · p. 2

Theoretical contributions

• The framework establishes a joint non-asymptotic excess-risk bound whose rate is {N/log N}−β/(2β+p) for KAN and Transformer sieves over β-Hölder conditional quantiles.The bound depends on the at-risk probability and network covering number and is paired with architecture-specific approximation constructions.Source: §1 · p. 2

Empirical and clinical scope

• Across 27 simulation settings and six cohorts, the study benchmarks CNQ and conducts two clinical analyses that recover distribution-varying covariate effects and coherent individualized survival-time quantiles with calibrated uncertainty.The clinical analyses also examine apparent event-projection bias under inverse-probability-of-censoring training and architecture dependence of spline-based feature attributions.Source: §1 · p. 2

2.1. Real Data Descriptions.

• Six cohorts span clinical settings, covariate modalities, sample sizes from n = 144 to n = 8,873, and moderate-to-heavy censoring.The datasets are SUPPORT, METABRIC, GBSG, GBSG-500, NKI70, and FLCHAIN.Source: §2.1 · p. 3
• SUPPORT and FLCHAIN test distributional prediction in non-oncology settings, including distinct short- versus long-term risks and approximately 72% censoring in FLCHAIN.FLCHAIN includes seven demographic and laboratory covariates and evaluates calibration when most event times are unobserved.Source: §2.1 · p. 3
• The breast-cancer cohorts vary in sample size, censoring, and covariate complexity, with GBSG-500 isolating sample-size effects and NKI70 representing a small, heavily censored genomic setting.METABRIC combines molecular, clinical, and treatment variables; NKI70 has n = 144 subjects and 67% censoring.Source: §2.1 · p. 3

2.2. Motivation: Why Distributional (Quantile) Survival Modeling?.

• Right censoring, heterogeneous prognostic effects, and nonlinear high-dimensional interactions motivate quantile survival modeling beyond hazard- or mean-based summaries.Several survival quantiles remain identifiable under bounded follow-up, but flexible multi-quantile estimation must prevent logically inconsistent crossing curves.Source: §2.2 · pp. 3–4

3.1. Existing Work.

• Censored quantile regression estimates conditional event-time quantiles by minimizing an inverse-probability-weighted check loss using Kaplan–Meier-based censoring weights.The estimator is consistent under the cited framework, including median regression.Source: §3.1 · pp. 4–5
• DeepQuantreg replaces the linear predictor with a flexible nonlinear neural network and estimates parameters by minimizing the corresponding weighted check loss.Unlike the smoothed check function used by Jia and Jeong (2022), this methodology uses the exact check loss.Source: §3.1 · p. 5
• DeepQuantreg’s two-hidden-layer MLP may be insufficient for highly structured or high-dimensional covariates, while separate networks increase computation and can yield quantile crossing.These limitations motivate more expressive and jointly structured approaches for censored distributional prediction.Source: §3.1 · p. 5

What is new relative to CQRNN and DeepQuantreg.

• Unlike DeepQuantreg and CQRNN, CNQ estimates quantiles jointly, enforces non-crossing by construction, and adapts this design to right-censored survival data.The construction builds on Wu et al. (2023) and Shen et al. (2025), with an inverse-probability-of-censoring-weighted objective.Source: What is new relative to CQRNN and… · p. 5
• CNQ replaces shallow MLP backbones with attention across covariates and learnable spline transforms, while establishing a joint finite-sample excess-risk bound covering the complete Trans-CNQ architecture.These backbones target heterogeneous, non-additive clinical and molecular covariates; neither baseline provides a corresponding guarantee.Source: What is new relative to CQRNN and… · p. 5

3.2. Censored Non-Crossing Quantile (CNQ) Framework.

• CNQ jointly estimates multiple conditional quantiles in one optimization problem and guarantees their proper ordering, making individualized survival milestones coherent.Its objective averages inverse-probability-weighted check losses across subjects and quantile levels, using inverse Kaplan–Meier weights.Source: §3.2 · pp. 5–6

Non-Crossing Output Module.

• CNQ uses a structurally constrained parameterization to prevent independently estimated lower quantiles from exceeding higher ones, building on non-crossing quantile networks by Wu et al. (2023) and Shen et al. (2025).This guarantees monotonicity across quantile levels by construction.Source: Non-Crossing Output Module · p. 6
• The shared output module operates on log time, with a base network estimating the τ1-quantile and a steps network producing K−1 unconstrained gaps between consecutive quantiles.Both outputs are formed from the backbone representation Z = g(X;θg) through separate linear heads.Source: Non-Crossing Output Module · p. 6
• Softplus-transformed increments are cumulatively summed, and exponentiation maps the ordered log-time quantiles back to the original time scale.Because each increment is nonnegative, quantile estimates remain automatically ordered without post-hoc rearrangement.Source: Non-Crossing Output Module · pp. 6–7

Shared CNQ architecture

• Three CNQ backbones differ only in feature extraction while sharing the same non-crossing output module, enabling direct comparison with valid quantile ordering.The models are trained separately end-to-end.Source: Model Architectures · p. 7

KAN-CNQ

• KAN-CNQ uses learnable univariate edge functions, including spline components, to build hierarchical covariate representations before non-crossing quantile prediction [Liu et al. (2024); Kolmogorov, 1957].The final KAN representation is passed to two linear heads that produce quantile estimates.Source: Model Architectures · p. 7

Trans-CNQ

• Trans-CNQ adapts Transformer self-attention to tabular data by attending across features, using shared scalar-token embeddings, positional encoding, and stacked encoder layers [Vaswani et al., 2017; Huang et al., 2020].The flattened encoder representation is mapped to quantile estimates through the non-crossing output module.Source: Model Architectures · pp. 7–8

TransKAN-CNQ

• TransKAN-CNQ replaces Transformer feedforward sub-layers with KAN layers, combining self-attention for feature interactions with learnable spline-based nonlinear transformations.Its representation feeds the shared non-crossing output module, preserving monotonicity in predicted quantiles.Source: Model Architectures · p. 8

Joint estimation guarantee

• Theorem 4.1 gives a high-probability excess estimation-risk bound jointly across all K fitted quantiles under (A1)–(A2) and the empirical L2 entropy condition.Its proof transfers the empirical Frobenius metric to the true-weight IPCW loss using Cauchy–Schwarz, symmetrization, a truncated Dudley integral, concentration, and Kaplan–Meier plug-in control.Source: §4 · pp. 8–9

Approximation–estimation decomposition

• The total excess risk decomposes into estimation and approximation terms, so the architecture-specific theorems balance Theorem 4.1 against approximation error without assuming a fixed architecture contains the true quantiles.The framework applies the non-crossing map and complexity-indexed estimation bounds to growing KAN and Transformer sieves.Source: §4 · p. 9

KAN-CNQ rate

• Theorem 4.2 derives an excess-risk convergence rate for a fixed-grid, two-layer spline-KAN sieve under bounded smoothness, positive adjacent-quantile gaps, coefficient bounds, and a fixed output envelope.The theoretical sieve has O(J^p) free coefficients at spline resolution J.Source: §4 · p. 9

Trans-CNQ rate

• Theorem 4.3 derives an excess-risk convergence rate for a fixed-envelope Trans-CNQ sieve under the same smoothness and gap conditions, with an additional restriction 0 < β < (p+3)/2.The result fixes shared-embedding post-LayerNorm encoder dimensions and controls the ReLU readout through its unit count and output-variation budget.Source: §4 · p. 10

Scope and limitations

• The bounds are upper bounds for fixed p and K under hard norm constraints, ideal ERM, positive quantile gaps, a fixed envelope, and (A1)–(A2), and they do not rank fitted architectures.The assumptions exclude some unbounded-support censoring settings; the KAN result concerns a degree-p theoretical sieve rather than fitted EfficientKAN, and TransKAN-CNQ is not covered.Source: §4 · pp. 8, 10

5.1. Evaluation Metrics.

• IPCW-weighted mean pinball loss is the primary measure of overall quantile prediction accuracy, with quantile-specific losses reported to reveal variation across levels.Weights use the Kaplan–Meier censoring-survival estimate and normalized form under marginal independent-censoring and positivity assumptions.Source: §5.1 · p. 10
• IPCW-weighted interval coverage assesses calibration on the original time scale at nominal 80% and 50% prediction intervals.Well-calibrated models should achieve empirical coverage close to the nominal rates.Source: §5.1 · p. 10

5.2. Implementation Setup.

• Models use AdamW, warmup-cosine learning-rate schedules, validation-loss early stopping, and method-specific hyperparameter sweeps before final evaluation.Transformer models use patience 10, whereas KAN-CNQ uses patience 20 because its sweep includes spline-grid resolution.Source: §5.2 · p. 11
• Simulation data use 1:1:1 train-validation-test splits, real datasets use 65%–15%–20% splits, and every experiment is repeated over 25 random seeds.The repeated seeds account for variability in data partitioning.Source: §5.2 · p. 11
• The benchmark compares TransKAN-CNQ, Trans-CNQ, and KAN-CNQ with ten quantile, non-quantile, and traditional survival methods using matched evaluation and training protocols.Deep quantile baselines use identical IPCW weights, optimizer, schedule, and tuning budget; all methods are evaluated in simulations and real-data benchmarks.Source: §5.2 · p. 11

• The simulation spans 27 settings combining Gaussian, Gamma, and Weibull event times, approximately 25%, 50%, and 75% censoring, and n ∈ {150,750,1500} with p = 10.Full design details and per-setting results are provided in the Supplementary Material.Source: §6 · p. 11
• CNQ attains the lowest IPCW pinball loss in 7/9 Weibull and 6/9 Gamma settings, whereas RSF leads in 8/9 symmetric Gaussian settings, marginally.The proposed models lead where flexible distributional modeling matters most; RSF is strongest on Gaussian designs and a close second under Weibull/Gamma.Source: §6 · pp. 11–12
• CQRNN is less accurate than the best proposed model in all 27 settings, and DeepQuantreg is less accurate in 26/27, except under Gamma with 75% censoring and n = 150.Hazard-based and remaining classical baselines are weaker distributional predictors.Source: §6 · p. 12
• TransKAN-CNQ is the most consistent proposed architecture, while Trans-CNQ can be preferable with small samples or heavy censoring; the advantage narrows at n = 150.Source: §6 · p. 12
• CNQ’s measured crossing rate is exactly zero in every setting, compared with 34.2% of DeepQuantreg test subjects having an inverted adjacent pair across settings and 65.2% in the hardest setting.For DeepQuantreg, bq0.1 > bq0.9 occurs in 2.1% of subjects on average and 29.6% in the hardest setting.Source: §6 · p. 12

7.1. Overall Predictive Performance.

• CNQ models attain the lowest IPCW pinball loss on all six cohorts, with TransKAN-CNQ reaching 0.279 on FLCHAIN versus 0.287 for DeepQuantreg and 0.332 for RSF.Figure 3 shows the proposed models generally occupying the lowest-loss region, with DeepQuantreg the closest quantile-based competitor.Source: §7.1 · pp. 12–13
• CNQ guarantees coherent non-crossing quantiles, with a measured crossing rate of zero on every cohort, whereas DeepQuantreg crosses for 18.2% of test subjects on average.DeepQuantreg crossing ranges from 2.4% on SUPPORT to 45.7% on NKI70, while its outer interval inverts for only 0.18% of subjects.Source: §7.1 · p. 13
• CNQ intervals remain close to nominal coverage outside METABRIC without conservative widening, whereas AFT and RSF achieve SUPPORT coverage of 0.898 and 0.888 through wide intervals.On FLCHAIN, TransKAN-CNQ, Trans-CNQ and KAN-CNQ achieve coverage of 0.795, 0.806 and 0.828, compared with RSF 0.693 and CoxTime 0.593.Source: §7.1 · p. 13

7.2. Model Checking and Calibration.

• METABRIC calibration tracks the nominal levels closely, but all architectures undershoot at τ = 0.9, with central 80% coverage of 0.72–0.78.The same checks replicate on FLCHAIN with nominal 80% coverage of 0.80–0.83, indicating tighter calibration on its larger test sets.Source: §7.2 · p. 14
• Changing the quantile grid, IPCW truncation threshold, or data split alters pinball loss by at most ∼0.01 and preserves the ordering TransKAN-CNQ ≲Trans-CNQ < KAN-CNQ.The five-quantile, untruncated-weight results are therefore not artifacts of these choices, although the 80/10/10 split is noisier.Source: §7.2 · p. 14, Table 2 · p. 14

7.3. Overview of the Two Case Studies.

• The real-data evaluation extends beyond simulation to two survival datasets differing in clinical context, sample size, and censoring severity.The broader analysis covers six right-censored cohorts, while the case-study overview focuses on two contrasting real-world datasets.Source: §7.3 · p. 14

• METABRIC pairs molecular markers with treatment indicators under moderate censoring.Source: Datasets · pp. 15–16
• The real-world evaluation covers six datasets, with performance summarized using IPCW pinball loss and 80% nominal interval coverage across 25 random splits.Source: Table 3 · p. 16, Table 4 · p. 16
• FLCHAIN is a treatment-free population cohort four times larger than METABRIC and far more heavily censored.Source: Datasets · p. 16

• Across two datasets, 25 independent random partitions (seeds 41–65) divide observations into training, validation, and test sets for evaluation.Source: Evaluation protocol · p. 16
• Trans-CNQ, TransKAN-CNQ, and KAN-CNQ each predict five non-crossing quantiles at τ ∈{0.1,0.25,0.5,0.75,0.9}.Source: Evaluation protocol · pp. 16–17
• All evaluation metrics use untruncated inverse-probability-of-censoring weighting, with Kaplan–Meier censoring weights estimated from the training set.Sensitivity to weight truncation is reported in the Supplementary Material.Source: Evaluation protocol · p. 17

Analysis plan.

• The study combines feature importance, group contrasts, patient profiles, uncertainty stratification, and event projection to assess how individualized survival quantiles behave across cohorts.METABRIC is the primary application, while FLCHAIN provides secondary validation under heavier censoring.Source: Analysis plan · p. 17

7.4.1. Feature Importance.

• Permutation importance shows METABRIC covariate effects vary by quantile: age and chemotherapy dominate the bulk and upper tail, hormone therapy peaks at τ = 0.75 with ∆= 1.64, and PGR peaks centrally at τ = 0.5 with ∆= 0.79.Hormone therapy is negligible at τ = 0.1 (∆= 0.10), while ERBB2 has negative upper-quantile importance; patterns are consistent across architectures.Source: §7.4.1 · pp. 17–18

7.4.2. Clinically Defined Group Contrasts.

• METABRIC group contrasts widen toward the upper tail for ER-positive versus ER-negative patients, from approximately 22–26 months at τ = 0.1 to 47–80 months at τ = 0.9, while hormone-therapy contrasts become increasingly negative.These exploratory, descriptive contrasts use pointwise 95% bootstrap bands and should not be interpreted causally; treatment patterns plausibly reflect confounding by indication.Source: Figure 4 · p. 18, §7.4.2 · p. 18

7.4.3. Representative Patient Profiles.

• Representative METABRIC profiles distinguish similar medians with different tail risks and uncertainty, including interval widths of 150.9 versus 267.9 months and an observed event within a predicted 80% interval.Supplementary analyses also find the expected monotone relationship between interval width, pinball loss, and empirical coverage.Source: §7.4.3 · pp. 18–19

7.4.4. Event Projection.

• Censoring-aware Trans-CNQ event projections overestimate cumulative events by +18–+21% across cutoffs, outperforming the horizon-dependent Weibull benchmark’s +63.8% to +6.9% overshoot.The overestimation is attributed to an objective that weights censored observations without enforcing Ti > Ci; a censoring-aware lower-bound penalty is proposed for future work.Source: §7.4.4 · p. 19

7.5.1. Secondary Validation Findings.

• In FLCHAIN, age dominates Transformer feature importance with peak ∆= 29.3 at τ = 0.5, whereas KAN-CNQ emphasizes creatinine with ∆≈45.5 versus ≈2.5 for Trans-CNQ despite comparable accuracy.The divergence indicates architecture-dependent attribution and non-identifiability, so spline visualization does not establish architecture-independent scientific conclusions.Source: §7.5.1 · pp. 19–20

7.5.2. Clinically Defined Group Contrasts.

• FLCHAIN group contrasts show consistent age effects, little sex signal, and an inconclusive MGUS pattern because MGUS-positive subjects comprise only about 1.5% of each test split.Older patients have ∆(τ) of approximately −40 to −257 days across quantiles, while MGUS contrasts range from +93 to +174 days at lower quantiles and −116 to −198 days at upper quantiles for Trans-CNQ.Source: §7.5.2 · p. 20, Figure 5 · p. 20

8. Discussion.

• CNQ achieved the lowest IPCW pinball loss on every cohort, matched DeepQuantreg in interval calibration, and outperformed hazard- and tree-based competitors.Across METABRIC and FLCHAIN, monotone parameterization produced non-crossing five-quantile milestones; Transformer variants remained most accurate on small or heavily censored cohorts, although cross-split variability cautions against individual-level predictions.Source: §8 · pp. 20–21
• Quantile-specific importance and group contrasts revealed survival-distribution heterogeneity that a single hazard ratio would obscure, including widening METABRIC ER and persistent FLCHAIN age contrasts.Interval width increased monotonically with pinball loss and empirical coverage, functioning as an internal indicator of predictive difficulty.Source: §8 · p. 21
• The theoretical rates are model-family upper bounds subject to Remark 4.4’s scope conditions, deteriorate as G⋆ decreases, and do not verify (A2) over an entire unbounded event-time support.Source: §8 · p. 21, Appendix C · p. 25
• Feature attributions depended on architecture even when predictive accuracy did not, while real-data subgroup and importance analyses remain descriptive and require confirmation in purpose-designed studies.The discussion recommends comparing attributions across backbones rather than relying on one.Source: §8 · p. 21

A.1. Data provenance and availability.

• The six public cohorts use previously preprocessed data, fixed GBSG-500 subsampling, excluded missing or zero-duration records, and released scripts for regenerating all 25 seeded partitions.Cohorts span oncology, genomic, and population-based settings with sample sizes from 144 to 8,873 and censoring from about 32% to 72%.Source: §A.1 · p. 22, Figure 6 · p. 22

APPENDIX B: TECHNICAL RESULT

• The IPCW identity underlying the marginal Kaplan–Meier weight assumes independent censoring, G(T−)>0 almost surely, and integrable check loss.The left-limit convention is essential when G(t)=P(C>t), because G(t−)=P(C≥t).Source: Appendix B · p. 24

APPENDIX C: PROOF OF THEOREM 4.1

• Theorem 4.1 combines estimated-censoring-weight control, empirical covering, entropy integration, and Kaplan–Meier concentration to obtain the stated finite-sample excess-risk bound.The proof uses the check loss’s one-Lipschitz property and a joint contraction over K outputs.Source: Appendix C · pp. 24–25

APPENDIX D: COMPLEXITY-INDEXED ARCHITECTURE BOUNDS AND APPROXIMATION RATES

• The appendix derives architecture-specific complexity controls: KAN bounds support matched growing sieves and approximation-estimation rates, while Theorem D.6 gives complete sample-wise entropy for deterministic Trans-CNQ.For fixed architecture and budgets, the Trans-CNQ entropy coefficient is independent of N.Source: Appendix D · p. 26, §D.2 · p. 28, §D.3 · p. 31

D.3. Complete deterministic Trans-CNQ entropy.

• The entropy proof constructs parameter covers conditionally across layers, propagates row errors through residual connections and readouts, and obtains the prescribed weights in (41).The argument controls the final representation through the Frobenius error bound before applying readout Lipschitz bounds.Source: §D.3 · p. 31
• The result is asymptotic, and its worst-case upper-bound constant may grow exponentially with depth because LayerNorm Lipschitz factors compound through the encoder layers.The passage explicitly warns that the bound may be numerically vacuous at the sample sizes considered.Source: §D.3 · p. 31
• The deterministic entropy construction leads to a Trans-CNQ estimation corollary with κ_N(η) = O(N^-1/2) at fixed confidence under the stated assumptions and output envelope.The corollary indexes estimation error by the deterministic complexity s_Tr,N through E_Tr,N = N C_Tr(s_Tr,N).Source: §D.3 · p. 31

D.4. Trans-CNQ: approximation and a matched growing sieve.

• Trans-CNQ’s shared-embedding class contains a shallow-ReLU subclass with at most KM readout units, enabling Hölder approximation while keeping encoder parameters independent of M.Theorem D.10 provides an approximant with total absolute output coefficient at most CΛ in each raw coordinate; the construction uses smooth bi-Lipschitz feature maps and coordinatewise networks.Source: §D.4 · p. 32
• The matched growing-readout sieve yields a finite-sample excess-risk convergence rate for the marginal-IPCW Trans-CNQ estimator under the stated regularity, norm, and entropy conditions.Choosing Λ = M^ν balances approximation terms of order M^−β/p with the estimation contribution involving M log N/N.Source: §D.4 · pp. 33–34

APPENDIX E: QUANTILE-SPECIFIC PINBALL LOSS

• Quantile-specific losses peak near the median and are smallest in the upper tail; Transformer variants are at or below KAN-CNQ at every METABRIC level, whereas FLCHAIN models differ by at most 0.004.The analyses use τ ∈ {0.1,0.25,0.5,0.75,0.9} and means ± standard deviations across 25 random splits.Source: Appendix E · p. 35, Table 6 · p. 35, Table 7 · p. 35

F.2. Performance Comparisons.

• The proposed models attain the lowest pinball loss in 7/9 Weibull and 6/9 Gamma settings, while Random Survival Forest wins eight of nine Gaussian settings only marginally.For Weibull with 50% censoring and n = 1500, Trans-CNQ achieves 0.112 versus 0.176 for the best baseline RSF; under Gaussian designs, losses are roughly 0.02–0.06.Source: §F.2 · pp. 35–36, Table 8 · p. 37, Figure 7 · p. 38
• The best proposed model beats DeepQuantreg in 26 of 27 settings, with gains becoming uniform at n ≥750, while TransKAN-CNQ is strongest within the proposed family in 18 settings.Exceptions occur at n = 150 under medium or heavy censoring in Gamma and Weibull designs; at n = 1500 with 75% Weibull censoring, the loss is 0.117 versus 0.321.Source: §F.2 · p. 36, Table 8 · p. 37

F.3. Quantile Crossing.

• The CNQ parameterization guarantees ordered quantiles and yields exactly zero measured crossing rates across all datasets, settings, and seeds.Separately fitted DeepQuantreg curves cross for roughly one in five real-cohort subjects and one in three simulated subjects, reaching 65.2% in the hardest simulation setting.Source: §F.3 · pp. 36, 38, Table 9 · p. 39

APPENDIX G: SUPPLEMENTARY TABLES

• Across METABRIC and FLCHAIN, wider predicted 80% intervals correspond monotonically to higher pinball loss and empirical coverage, providing a label-free indicator of predictive difficulty.Representative patient profiles show coherent, individualized quantile milestones with varying tail risk and interval width, including METABRIC survival in months and FLCHAIN survival in days.Source: Appendix G · pp. 40–41, Appendix H · p. 42, Figure 8 · p. 42

H.1. Event Projection on METABRIC.

• The real-data event-projection analysis compares latent, censoring-aware Trans-CNQ, and marginal Weibull strategies on METABRIC test sets, with projection error reported at 200 months.The supplied passages describe the setup and identify Table 14 as reporting 25-seed mean percentage error relative to the raw observed count; they do not provide the numerical comparison.Source: §H.1 · p. 42, Setup · p. 42, Table 14 · p. 43

Results.

• +18–+21% errors across cutoffs make Trans-CNQ closer to raw observed events than KM-adjusted counts, while Weibull overshoots +63.8% at 24 months but narrows to +6.9% at 96 months.Figure 10 compares the event-projection curves across 24-, 48-, 72-, and 96-month horizons.Source: Results · p. 43, Root cause · p. 44, Figure 10 · p. 44

Root cause.

• IPCW does not explicitly enforce Ti > Ci, so late-censored subjects are predicted to fail too early and projected cumulative event counts become inflated.Likelihood-based Weibull or Cox models incorporate the censored subject’s survival contribution S(Ci | Xi), improving far-tail calibration but remaining imperfect under moderate METABRIC censoring.Source: Root cause · p. 43
• Age and chemotherapy dominate METABRIC attribution, hormone therapy acts mainly in the upper quantiles, and FLCHAIN Transformer models emphasize age and serum free light chains while KAN-CNQ ranks creatinine first.METABRIC rankings are broadly architecture-stable, whereas the FLCHAIN discrepancy shows attribution dependence in the pure-KAN backbone.Source: Root cause · p. 44, Figure 11 · p. 45, Figure 12 · p. 46
• Near-zero Transformer median residuals and near-diagonal FLCHAIN coverage show calibration without systematic median bias, including under approximately 72% censoring.FLCHAIN reproduces METABRIC’s calibration and interval-coverage behavior in a larger, more heavily censored cohort.Source: Root cause · p. 45, Figure 14 · p. 47, Figure 15 · p. 47
• Essentially unchanged pinball loss, 80% interval coverage, and Transformer-over-KAN ordering across IPCW truncation thresholds and split ratios show robustness to these analysis choices.The 80/10/10 split is mildly noisier because its test set is smallest.Source: Root cause · pp. 45, 48, Table 17 · p. 48