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Non-Crossing Deep Quantile Regression for Distributional Survival Prediction

Shuai Huang, Zhe Qu, Zhaowei Hua, Guohao Shen, Rui Tang, Hongtu Zhu

arXiv:2608.16864v1stat.MLcs.LGstat.AP

TL;DR

Survival hazards and means can obscure covariate effects that differ across the event-time distribution, while existing censored quantile methods may be inflexible or produce crossing curves. CNQ jointly estimates ordered survival quantiles for right-censored data using flexible neural backbones, and it achieves strong benchmark performance while recovering distribution-varying effects and coherent individualized milestones.

  • Problem

    Existing survival summaries can collapse distributional heterogeneity, while censored quantile methods may lack nonlinear flexibility or produce logically inconsistent crossing quantiles.

  • Method

    CNQ jointly estimates multiple conditional survival quantiles with a monotone non-crossing output module and KAN, Transformer, or hybrid feature extractors.

  • Results

    CNQ attained the lowest IPCW pinball loss on every cohort, with interval calibration matching DeepQuantreg and bettering hazard- and tree-based competitors.

  • Takeaways & Limitations

    In METABRIC and FLCHAIN, quantile-specific contrasts recover covariate effects that vary across the survival distribution and yield coherent individualized quantile milestones.

  • Takeaways & Limitations

    Theoretical sieve rates require fixed p and K, hard norm bounds, positive quantile gaps, ideal ERM, and assumptions that can fail under bounded-uniform censoring with unbounded event-time support.

Abstract

from arXiv · show

In survival analysis the way covariates act on the risk of an event often differs between early and late failure times, yet hazard- and mean-based summaries collapse this variation into a single number. Quantile-based modeling instead describes the full conditional distribution on the original time scale, but existing censored-data methods are either inflexible or produce logically inconsistent crossing quantile curves. We propose a Censored Non-crossing Quantile (CNQ) framework for right-censored data that jointly estimates several conditional survival quantiles and guarantees valid ordering by construction, with flexibility supplied by Kolmogorov-Arnold and Transformer backbones, and we establish a finite-sample excess-risk bound holding jointly across all fitted quantile levels. Across 27 simulation settings and six cohorts the framework attains lower pinball loss than quantile-, hazard- and tree-based competitors whenever the conditional distribution is asymmetric, with interval coverage closer to nominal on all six. In two clinical case studies (METABRIC, breast cancer; FLCHAIN, population mortality) it recovers covariate effects that vary across the survival distribution and would be hidden by a single hazard ratio, and yields coherent individualized quantile milestones. Code: https://github.com/BIG-S2/deepcnq

1. Introduction.

Existing survival models often summarize risk through hazards or mean survival, while censored quantile methods can be inflexible or produce crossing curves. CNQ addresses this gap with flexible backbones and a monotone joint quantile estimator.

  • Hazard- and mean-based summaries may miss heterogeneous risk and the full conditional distribution of event times.Quantile regression instead models survival quantiles directly on the original time scale, allowing covariate effects to vary across the distribution.
  • Classical censored quantile regression is limited by linear predictors, whereas neural methods may fit separate quantiles without ordering constraints.Separate unconstrained fits can yield a lower predicted quantile exceeding a higher one, producing an inconsistent conditional distribution.
  • CNQ jointly estimates multiple survival quantiles and enforces non-crossing through a monotone output module for right-censored data.KAN, Transformer, and hybrid feature extractors provide flexibility for nonlinear and interacting covariate effects.
  • The framework combines methodological, theoretical, and empirical contributions, including a jointly valid finite-sample excess-risk bound and analyses across 27 simulation settings and six cohorts.The paper also studies two clinical datasets for distribution-varying covariate effects and individualized survival-time quantiles with calibrated uncertainty.

2. Data and Motivation.

The paper motivates distributional survival modeling using diverse right-censored cohorts and clinical settings where prognostic factors may act differently across survival-time quantiles. Quantile estimates are intended to support calibrated individualized milestones, including in small or heavily censored datasets.

  • Real Data Descriptions: Six real-world right-censored datasets span hospitalized adults, population mortality, and breast-cancer cohorts with varied sample sizes and censoring.The datasets include SUPPORT, METABRIC, GBSG, GBSG-500, NKI70, and FLCHAIN.
  • Real Data Descriptions: FLCHAIN tests calibration under approximately 72% censoring, where most event times are unobserved.Its outcome is all-cause mortality in 7,874 adults, using demographic and laboratory covariates without treatment variables.
  • Motivation: The datasets capture nonlinearity, high-dimensional interactions, and prognostic heterogeneity across survival-time distributions.SUPPORT separates short-term mortality from longer-term survival, while METABRIC contains molecular, clinical, and treatment covariates.
  • Motivation: Why Distributional (Quantile) Survival Modeling?: Quantile modeling is useful because bounded follow-up can make mean survival non-identifiable while clinically relevant survival quantiles remain identifiable.Estimating several quantiles still requires an ordering constraint to avoid logically inconsistent outputs for treatment planning.
  • Motivation: The study asks whether individualized quantiles remain coherent, calibrated, and informative about tail-specific covariate effects in settings including small, heavily censored oncology cohorts.NKI70 is cited as an example with n = 144 and 67% censoring.

3. Methodology.

The methodology extends censored quantile regression with inverse-probability weighting and flexible neural predictors. It retains the check-loss formulation while replacing linear or shallow predictors with learned nonlinear functions.

  • Right-censored observations are represented by the minimum observed time and event time together with an event indicator.For subject i, the observed time is Yi = min(Ti,Ci) and δi indicates whether the event occurs by the censoring time.
  • Censored quantile regression estimates conditional event-time quantiles by minimizing an inverse-probability-weighted check loss.The weights use a Kaplan–Meier estimator of the censoring distribution.
  • DeepQuantreg replaces the linear predictor with a nonlinear neural network and estimates parameters using the corresponding weighted check loss.Its check function is Huber-smoothed, whereas this paper uses the exact check loss throughout.
  • The exact check loss is retained because Huber smoothing changes the loss for residuals within a small bandwidth around zero.The paper presents this choice while targeting complex nonlinear relationships under right censoring.

What is new relative to CQRNN and DeepQuantreg.

Relative to CQRNN and DeepQuantreg, CNQ jointly fits multiple censored quantiles with guaranteed ordering and uses alternative feature-extraction backbones. Its shared output module operates on log time and encodes ordered quantile gaps.

  • What is new relative to CQRNN and DeepQuantreg: CNQ jointly estimates all quantiles in one optimization problem instead of fitting one network per quantile level.Its inverse-probability-of-censoring-weighted objective averages check losses across subjects and quantile levels.
  • What is new relative to CQRNN and DeepQuantreg: CNQ guarantees non-crossing predictions through a shared monotone output architecture, unlike methods without ordering constraints.This makes individualized quantile milestones coherent as a conditional distribution.
  • Censored Non-Crossing Quantile (CNQ) Framework: KAN-CNQ, Trans-CNQ, and TransKAN-CNQ differ in their feature-extraction components while sharing the same non-crossing output architecture.The three architectures are compared directly as illustrated in Figure 1.
  • Censored Non-Crossing Quantile (CNQ) Framework: The output module models the τ1 quantile as a base log-time prediction and represents later quantiles through unconstrained increments between consecutive log-time levels.The representation from one of the three backbones feeds this shared module.

Non-Crossing Output Module.

CNQ uses a shared output module that jointly predicts multiple log-time survival quantiles while enforcing their ordering by construction.

  • Non-Crossing Output Module: CNQ replaces independently fitted quantile networks with a shared module that jointly estimates multiple conditional survival quantiles.The module uses a base network for the first quantile and a steps network for gaps between consecutive quantiles.
  • Non-Crossing Output Module: The feature representation feeds two linear heads producing a scalar base output and K−1 unconstrained step outputs.The base head produces h(X;θ), while the steps head produces s(X;θ) ∈ R^(K−1).
  • Non-Crossing Output Module: Softplus transforms the raw steps into nonnegative increments before cumulative summation constructs the ordered log-time quantiles.This parameterization prevents negative gaps between adjacent quantiles.
  • Non-Crossing Output Module: Exponentiating the log-time outputs gives original-time quantiles that remain automatically ordered without post-hoc rearrangement.The ordering holds because every increment is nonnegative.

Model Architectures.

The framework compares KAN, Transformer, and hybrid feature extractors, all paired with the same non-crossing output architecture for direct comparison.

  • Model Architectures: KAN-CNQ, Trans-CNQ, and TransKAN-CNQ use independent feature-extraction backbones but share one non-crossing output module.All three models are trained separately end-to-end, isolating backbone differences while preserving valid quantile ordering.
  • KAN-CNQ: KAN represents functions through learned univariate transformations on network edges, building hierarchical representations of covariates.Its final representation is passed to the shared quantile-output heads.
  • Trans-CNQ: The Transformer adapts self-attention to tabular data by modeling dependencies across features rather than temporal positions.Scalar features are embedded into tokens, positionally encoded, normalized, and processed through stacked attention and feedforward layers.
  • TransKAN-CNQ: TransKAN-CNQ replaces Transformer feedforward MLP sub-layers with KAN layers, combining self-attention with spline-based nonlinear transformations.The hybrid retains attention-based feature interactions while increasing the expressiveness of within-block transformations.

4. Theoretical Guarantees.

The theory provides joint finite-sample excess-risk guarantees under censoring assumptions, with separate sieve rates for KAN-CNQ and Trans-CNQ and explicit scope limitations.

  • Estimation Bound: The empirical Euclidean formulation treats all K log-time quantile outputs jointly and supports the finite-sample estimation bound.The proof transfers empirical metrics to the IPCW loss class using empirical-process arguments and entropy control.
  • Excess-Risk Decomposition: The total excess risk decomposes into estimation and approximation terms, allowing fixed-architecture guarantees when the target lies in the sieve and growing-sieve rates otherwise.Theorem 4.1 controls estimation, while Theorems 4.2–4.3 balance approximation against estimation.
  • KAN-CNQ Guarantee: The KAN-CNQ rate assumes bounded covariates, smooth conditional log-time quantiles, uniformly positive adjacent-quantile gaps, and a specified spline-KAN sieve.The result concerns a degree-p theoretical sieve with bounded effective edge coefficients.
  • Trans-CNQ Guarantee: The Trans-CNQ rate adds the smoothness restriction 0 < β < (p+3)/2 and fixes encoder dimensions while varying the flattened ReLU readout.Its sieve uses a bounded output variation budget.
  • Scope: The bounds require fixed p and K, hard norm bounds, a fixed output envelope, ideal ERM, positive quantile gaps, and censoring assumptions (A1)–(A2).They do not cover TransKAN-CNQ, and (A2) fails for bounded-uniform censoring with unbounded event-time support.

5. Experimental Setup.

Experiments evaluate censored quantile accuracy and interval calibration with IPCW metrics, repeated data splits, and comparisons against quantile, hazard, and traditional survival baselines.

  • Evaluation Metrics: Quantile prediction errors are measured on the log-time scale, while interval coverage is assessed on the original time scale.Censoring corrections use IPCW weights based on the Kaplan–Meier estimate of censoring survival.
  • Evaluation Metrics: The primary accuracy metric is the IPCW-weighted mean pinball loss averaged across quantile levels.Quantile-specific losses are also reported to reveal heterogeneity across levels.
  • Evaluation Metrics: Interval calibration uses IPCW-weighted coverage for nominal 80% and 50% prediction intervals.Well-calibrated models should achieve empirical coverage close to the corresponding nominal rate.
  • Data Splitting: Simulation data use 1:1:1 training, validation, and test splits, whereas real-world data use a 65%–15%–20% split.All experiments are repeated over 25 random seeds.
  • Baselines: The proposed models are compared with ten existing methods spanning deep quantile regression, non-quantile deep survival, and traditional survival modeling.The comparison includes DeepQuantreg, CQRNN, DeepSurv, CoxTime, CoxCC, PCHazard, and tree-based and classical methods.

6. Simulation Studies.

Across 27 simulation settings, CNQ performs best when covariates reshape the full, asymmetric survival distribution, while its non-crossing construction prevents invalid quantile orderings.

  • CNQ attains the lowest IPCW pinball loss in 7/9 Weibull and 6/9 Gamma settings, where covariates reshape the full survival distribution.RSF leads on Gaussian designs and is a close second under Weibull and Gamma.
  • CQRNN is less accurate than the best proposed model in all 27 settings, while DeepQuantreg is less accurate in 26 of 27.The sole DeepQuantreg exception is Gamma with 75% censoring at n = 150.
  • TransKAN-CNQ is the most consistent proposed architecture, with Trans-CNQ a close alternative under small samples or heavy censoring.At n = 150, the proposed models’ advantage narrows.
  • 0% of test subjects exhibit quantile crossing for any CNQ architecture in every simulation setting.By contrast, separately fitted DeepQuantreg levels produce inverted adjacent pairs in 34.2% of subjects on average and 65.2% in the hardest setting.

7. Real Data Analysis.

Across six right-censored cohorts, CNQ models achieve the lowest pinball loss and generally favorable interval coverage, with direct METABRIC checks showing close central calibration but upper-tail undercoverage.

  • The proposed methods attain the lowest mean IPCW pinball loss on all six datasets, typically led by the Transformer-based variants.On SUPPORT, TransKAN-CNQ and Trans-CNQ achieve Lpin = 0.486 and 0.487 versus 0.492 for DeepQuantreg and 0.541 for RSF.
  • Across six cohorts, CNQ boxplots generally occupy the lowest IPCW pinball-loss region, with DeepQuantreg the closest quantile-based competitor.The figure uses self-normalized pinball loss on the log-time scale, where lower values indicate better performance.
  • CNQ models generally remain closer to the nominal 80% interval coverage target than most competitors while avoiding systematic overcoverage.DeepQuantreg is the closest baseline on several cohorts and is effectively tied on coverage alone.
  • CQRNN overcovers on most cohorts, whereas AFT and RSF can reach nominal-looking coverage through very wide intervals.On SUPPORT, AFT and RSF achieve 0.898 and 0.888 coverage, while the proposed models are near 0.78.
  • METABRIC per-quantile calibration tracks the 45° line closely at τ = 0.1 and τ = 0.5, but all architectures undershoot at τ = 0.9.Central-interval coverage is 0.43–0.47 for nominal 50% intervals and 0.72–0.78 for nominal 80% intervals.
  • Increasing the quantile grid from K = 5 to K = 11 or 19 changes pinball loss and MACE by at most ∼0.01 while preserving model ordering.The reported changes remain within one split-to-split standard deviation of ≈0.02.

Datasets.

The real-data analysis covers six established right-censored survival cohorts, while the two in-depth case-study datasets contrast clinical context, scale, and censoring severity.

  • Tables 3 and 4 summarize mean IPCW pinball loss and 80% nominal interval coverage across the six datasets.Pinball loss is averaged over τ ∈{0.1,0.25,0.5,0.75,0.9}; values closest to 0.80 are highlighted for coverage.
  • METABRIC combines molecular markers with treatment indicators under moderate censoring, whereas FLCHAIN is a treatment-free population cohort four times larger and more heavily censored.

Evaluation protocol.

The evaluation uses three CNQ architectures predicting five non-crossing quantiles, with all metrics weighted by training-set Kaplan–Meier estimates of censoring probabilities.

  • 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}.
  • The in-depth real-data analysis uses two cohorts summarized in Table 5.
  • All evaluation metrics use inverse-probability-of-censoring weighting, with Kaplan–Meier censoring weights estimated from the training set.Weights are untruncated in the main evaluation, with truncation sensitivity reported separately.

Analysis plan.

The case studies examine quantile-specific feature effects, group contrasts, individualized uncertainty, calibration, event projection, and architectural stability across METABRIC and FLCHAIN. These analyses show distribution-dependent covariate effects and useful individualized forecasts, while highlighting limitations in projection and interpretation.

  • Quantile-specific feature importance: Age and chemotherapy effects in METABRIC concentrate at τ = 0.5–0.9, while their influence at τ = 0.1 is modest.Permutation importance was evaluated separately at each quantile level across architectures and seeds.
  • Clinically defined group contrasts: ER-positive patients have longer predicted survival than ER-negative patients, with the contrast widening from 22–26 months at τ = 0.1 to 47–80 months at τ = 0.9.The reported contrast is consistent across architectures and increases toward the upper tail.
  • Clinically defined group contrasts: Hormone-therapy contrasts shift from approximately −1 to 2 months at τ = 0.1 to −14 to −34 months at τ = 0.9, and the authors caution against causal interpretation.The paper attributes this descriptive pattern more plausibly to confounding by indication.
  • Representative patient profiles: Patient D has a 267.9-month interval width versus 150.9 months for Patient C despite a somewhat longer median prediction, indicating greater prognostic uncertainty.The profiles illustrate how quantile milestones distinguish uncertainty beyond the median.
  • Representative patient profiles: The observed event at 100.3 months lies within 1% of the predicted median 99.8 and inside the predicted 80% interval [34.0,192.4].Supporting analyses report monotone relationships between interval width, pinball loss, and empirical coverage.
  • Event projection: Censoring-aware Trans-CNQ projection errors are +18–+21% across cutoffs, while the Weibull benchmark ranges from +63.8% at 24 months to +6.9% at 96 months.All three projection methods overestimate observed cumulative event counts; the paper links this to the objective’s treatment of censored observations.
  • Secondary validation findings: In FLCHAIN, age dominates feature importance across quantiles, whereas KAN-CNQ ranks creatinine highest despite comparable predictive accuracy from Transformer-based models.At τ = 0.5, creatinine importance is approximately 45.5 for KAN-CNQ versus approximately 2.5 for Trans-CNQ.
  • Clinically defined group contrasts: FLCHAIN age contrasts agree across architectures, ranging from approximately −40 to −257 days, while MGUS contrasts are inconclusive because the subgroup comprises roughly 24 patients per test split.The MGUS pattern differs sharply across architectures and has correspondingly wide bands.

8. Discussion.

Across six cohorts, CNQ achieved the lowest IPCW pinball loss and maintained interval calibration relative to competing methods. The discussion also qualifies the theoretical bounds, interpretation of feature attributions, data provenance, and applicability of the framework.

  • Empirical findings: The proposed CNQ framework attained the lowest IPCW pinball loss on every cohort, with interval calibration matching DeepQuantreg and bettering hazard- and tree-based competitors.The discussion presents this as the principal cross-cohort empirical result.
  • Empirical findings: Monotone parameterization yields non-crossing individualized five-quantile milestones, while interval width increases monotonically with pinball loss and empirical coverage.These findings address coherent individualized prediction and internal uncertainty stratification.
  • Empirical findings: Transformer-based variants remained most accurate on small or heavily censored cohorts, although larger cross-split variability warrants caution for individual-level predictions.The stated examples include GBSG-500 and NKI70.
  • Interpretation and limitations: Feature attributions can depend on architecture even when predictive accuracy does not, so the authors recommend comparing attributions across backbones.This reflects non-identifiability in internal representations across correlated covariates.
  • Interpretation and limitations: Real-data subgroup and feature-importance analyses are descriptive and hypothesis-generating rather than causal, requiring confirmation in purpose-designed studies.The discussion limits interpretation of observational cohort contrasts.
  • Theoretical scope: Assumption (A2) is not verified over an entire unbounded event-time support or at a bounded terminal point where Sτ = 0.The Kaplan–Meier rate applies when the target range is identifiable and survival probability is bounded away from zero.
  • Theoretical scope: Theoretical rates hold for fixed p and K under hard norm bounds, a fixed output envelope, ideal ERM, uniformly positive quantile gaps, and assumptions (A1)–(A2).The bounds are upper bounds and do not rank fitted architectures.
  • Theoretical scope: The KAN approximation and total-rate result concern a degree-p theoretical sieve rather than the fitted cubic-spline EfficientKAN architecture, and do not cover TransKAN-CNQ.The appendix distinguishes fitted architectures from the theoretical constructions used for approximation and rate analysis.

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

The Trans-CNQ analysis combines approximation and excess-risk theory with empirical evidence that non-crossing quantile estimation improves accuracy, coherence, and individualized survival interpretation. Results are strongest when covariates reshape asymmetric distributional features, while projection remains limited by censoring-related early-event prediction bias.

  • Approximation and theory: The shared-embedding Trans-CNQ class admits Hölder approximation through a Transformer-based construction with bounded readout complexity.Theorem D.10 builds a function in the shared-embedding Trans-CNQ family with at most K_M readout units and bounded total absolute output coefficients.
  • Approximation and theory: The Trans-CNQ estimator has a non-asymptotic excess-risk convergence result under the stated regularity and sieve conditions.Theorem D.12 derives the estimator’s rate using the approximation result and κ_N(η) = O(N^-1/2) at fixed confidence.
  • Simulation performance: The proposed models attain the lowest pinball loss in 7/9 Weibull and 6/9 Gamma settings, including 0.112 versus 0.176 for Trans-CNQ and the best baseline in one Weibull setting.Under symmetric Gaussian designs, Random Survival Forest leads in eight of nine settings, but only marginally, such as 0.024 versus 0.025.
  • Simulation performance: The best proposed model beats DeepQuantreg in 26 of 27 settings, with uniform gaps at n ≥750 and a Weibull 75%-censoring example of 0.117 versus 0.321.CQRNN is outperformed by the best proposed model in all 27 settings; exceptions against DeepQuantreg occur only at n = 150 under medium or heavy censoring in Gamma and Weibull designs.
  • Quantile coherence: The fitted CNQ architectures have exactly zero measured crossing rate across all datasets, settings, and seeds because the output layer uses non-negative quantile increments.This construction enforces q̂_τ1 ≤ ··· ≤ q̂_τK for every subject, unlike unconstrained quantile networks.
  • Individualized predictions: Wider predicted intervals correspond to higher pinball loss and empirical coverage in both METABRIC and FLCHAIN, making interval width a label-free indicator of predictive difficulty.Patient-level profiles distinguish similar medians with different tail risks and uncertainty, while the same ordering appears in cohort-level tercile analyses.
  • Event projection: Trans-CNQ projections remain above raw observed event counts but track the reference more closely than KM-based and Weibull projections across follow-up horizons.The Trans-CNQ errors are +18–+21%, whereas Weibull overestimates by +63.8% at 24 months and +6.9% at 96 months.
  • Event projection: IPCW training can produce early event-time predictions for late-censored subjects because censored observations contribute no direct loss beyond their weighting role.This limitation inflates projected cumulative event counts; likelihood-based approaches incorporate censoring information through the censoring model.
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