Non-Crossing Deep Quantile Regression for Distributional Survival Prediction

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

stat.MLAug 17, 2026

Non-Crossing Deep Quantile Regression for Distributional Survival Prediction

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

Organizations: Department of Biostatistics, University of North Carolina at Chapel Hill · Servier Pharmaceuticals, Boston, MA · The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China · Astellas Pharma US, Northbrook, IL

Abstract

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

Figures & tables

Fig 1: The architectures of KAN-CNQ, Trans-CNQ, and TransKAN-CNQ.

Data set	Size	Covariates	Censored%
SUPPORT	8,873	14	32
METABRIC	1,904	9	42
GBSG	2,232	7	43
GBSG-500	500	7	42
NKI70	144	8	67
FLCHAIN	7,874	7	72

Table 1: Summary of datasets used in the analysis.

	$\overline{L}_{\mathrm{pin}}$			MACE
Model	$K{=}5$	$K{=}11$	$K{=}19$	$K{=}5$	$K{=}11$	$K{=}19$
TransKAN-CNQ	0.215	0.226	0.223	0.079	0.109	0.090
Trans-CNQ	0.224	0.230	0.223	0.092	0.102	0.091
KAN-CNQ	0.227	0.231	0.231	0.089	0.098	0.099

Table 2: Quantile-grid sensitivity on METABRIC. Models are trained with

K\in\{5,11,19\}

quantiles and evaluated at the five common levels

\{0.1,0.25,0.5,0.75,0.9\}

(mean over 25 splits).

\overline{L}_{\mathrm{pin}}

is the IPCW pinball loss (log scale); MACE is the mean absolute calibration error. The baseline is

K=5

Fig 2: METABRIC model checking (25 seeds, IPCW-weighted). (a) Per-quantile calibration: for each nominal level

\tau

, the IPCW estimate of

\Pr\{T\leq\widehat{Q}_{\tau}(X)\}

is plotted against

\tau

; the dashed line is perfect calibration and error bars are

\pm 1

standard deviation across splits. (b) Central-interval coverage versus nominal level (

50\%

and

80\%

); points below the diagonal indicate mild undercoverage.

Fig 3: Boxplots of IPCW pinball loss (self-normalized, log-time scale) across the six real-world datasets. Lower values indicate better quantile predictive performance. The proposed CNQ models (blue) consistently occupy the most favorable region of the distribution, with DeepQuantreg (purple), the other quantile-based deep learning method, the closest competitor on most cohorts.

Proposed methods
	GBSG	GBSG-500	METABRIC	SUPPORT	NKI70	FLCHAIN
TransKAN-CNQ	.199 $\pm$ .006	.206 $\pm$ .015	.215 $\pm$ .022	.486 $\pm$ .011	.251 $\pm$ .066	.279 $\pm$ .015
Trans-CNQ	.200 $\pm$ .006	.204 $\pm$ .017	.224 $\pm$ .024	.487 $\pm$ .010	.242 $\pm$ .044	.282 $\pm$ .015
KAN-CNQ	.203 $\pm$ .007	.210 $\pm$ .015	.227 $\pm$ .025	.490 $\pm$ .010	.251 $\pm$ .058	.280 $\pm$ .016
Deep learning baselines
DeepQuantreg	.206 $\pm$ .007	.220 $\pm$ .018	.243 $\pm$ .024	.492 $\pm$ .009	.305 $\pm$ .100	.287 $\pm$ .015

Table 3: Mean IPCW pinball loss (

\overline{L}_{\mathrm{pin}}

, averaged over

\tau\in\{0.1,0.25,0.5,0.75,0.9\}

) on six real-world datasets. Each entry shows the mean

\pm

standard deviation across 25 random splits. Lower is better. The best result per dataset is bolded .

Proposed methods
	GBSG	GBSG-500	METABRIC	SUPPORT	NKI70	FLCHAIN
TransKAN-CNQ	.789 $\pm$ .038	.802 $\pm$ .085	.784 $\pm$ .105	.782 $\pm$ .030	.690 $\pm$ .297	.795 $\pm$ .061
Trans-CNQ	.808 $\pm$ .037	.787 $\pm$ .073	.720 $\pm$ .160	.778 $\pm$ .043	.759 $\pm$ .229	.806 $\pm$ .055
KAN-CNQ	.784 $\pm$ .039	.818 $\pm$ .080	.765 $\pm$ .134	.783 $\pm$ .024	.805 $\pm$ .227	.828 $\pm$ .042
Deep learning baselines
DeepQuantreg	.793 $\pm$ .046	.788 $\pm$ .057	.759 $\pm$ .075	.793 $\pm$ .024	.625 $\pm$ .233	.802 $\pm$ .039

Table 4: IPCW interval coverage probability (ICP) at the 80% nominal level on six real-world datasets. Each entry shows the mean

\pm

standard deviation across 25 random splits. Values closest to the nominal 0.80 are bolded .

Dataset	$n$	Censoring	Outcome	Covariates	Domain
METABRIC	1,904	42%	Overall survival (months)	9	Breast cancer
FLCHAIN	7,874	72%	Time to death (days)	7	General population

Table 5: Summary of the two real-data cohorts used for in-depth analysis.

Fig 4: Clinically defined group contrasts

\Delta(\tau)

for METABRIC with 95% bootstrap bands. Non-constant profiles indicate distributional heterogeneity beyond a single proportional-hazards summary.

Fig 5: Clinically defined group contrasts

\Delta(\tau)

for FLCHAIN with 95% bootstrap bands. The three architectures agree closely on age (b) but not on sex (a) or MGUS (c); the MGUS bands are wide for all models because MGUS-positive subjects make up only about

1.5\%

of each test split.

Model	$\tau=0.1$	$\tau=0.25$	$\tau=0.5$	$\tau=0.75$	$\tau=0.9$
TransKAN-CNQ	0.168 $\pm$ 0.014	0.278 $\pm$ 0.024	0.310 $\pm$ 0.031	0.215 $\pm$ 0.033	0.103 $\pm$ 0.018
Trans-CNQ	0.171 $\pm$ 0.013	0.283 $\pm$ 0.022	0.321 $\pm$ 0.034	0.231 $\pm$ 0.037	0.115 $\pm$ 0.026
KAN-CNQ	0.177 $\pm$ 0.013	0.290 $\pm$ 0.020	0.325 $\pm$ 0.033	0.229 $\pm$ 0.043	0.114 $\pm$ 0.032

Table 6: Per-quantile IPCW pinball loss (log-time scale) on METABRIC for the three proposed models, reported separately at each of the five quantile levels

\tau\in\{0.1,0.25,0.5,0.75,0.9\}

rather than averaged; entries are the mean

\pm

standard deviation across 25 random splits (lower is better). This decomposition complements the quantile-averaged results in the main text and shows where in the distribution each model gains: the loss peaks near the median (

\tau=0.5

) and is smallest in the upper tail (

\tau=0.9

), and the two Transformer-based variants (Trans-CNQ, TransKAN-CNQ) are at or below KAN-CNQ at every quantile level. IPCW weights are untruncated, matching the headline tables in the main text; sensitivity to truncation is reported in Table 16 .

Model	$\tau=0.1$	$\tau=0.25$	$\tau=0.5$	$\tau=0.75$	$\tau=0.9$
TransKAN-CNQ	0.278 $\pm$ 0.022	0.396 $\pm$ 0.024	0.378 $\pm$ 0.020	0.237 $\pm$ 0.012	0.107 $\pm$ 0.006
Trans-CNQ	0.280 $\pm$ 0.021	0.399 $\pm$ 0.023	0.382 $\pm$ 0.021	0.239 $\pm$ 0.014	0.108 $\pm$ 0.006
KAN-CNQ	0.277 $\pm$ 0.021	0.396 $\pm$ 0.023	0.380 $\pm$ 0.021	0.238 $\pm$ 0.013	0.107 $\pm$ 0.006

Table 7: Per-quantile IPCW pinball loss (log-time scale) on FLCHAIN for the three proposed models, reported separately at each of the five quantile levels

\tau\in\{0.1,0.25,0.5,0.75,0.9\}

; entries are the mean

\pm

standard deviation across 25 random splits (lower is better). The companion to Table 6 for the larger, more heavily censored (

\approx 72\%

) FLCHAIN cohort: the same qualitative pattern holds—the loss is largest around the median and smallest in the upper tail. On FLCHAIN the three models are within

0.004

of one another at every

\tau

, so the per-quantile ordering is not separated on this cohort. IPCW weights are untruncated, matching the headline tables in the main text; sensitivity to truncation is reported in Table 15 .

Fig 7: Simulation study, Weibull design: inverse-probability-of-censoring-weighted (IPCW) pinball loss (self-normalized, log-time scale, averaged over the five quantile levels

\tau\in\{0.1,0.25,0.5,0.75,0.9\}

) by method. The figure is a

3\times 3

grid of panels indexed by sample size (rows,

n\in\{150,750,1500\}

) and censoring level (columns, light

\approx 25\%

, medium

\approx 50\%

, heavy

\approx 75\%

); within each panel the methods are arranged along the horizontal axis and the vertical axis reports the pinball loss (lower is better), with boxes/markers summarizing the distribution over the 25 replications. Colors distinguish the three proposed CNQ models (blue), DeepQuantreg (purple) and CQRNN (cyan) from the remaining deep-learning and traditional baselines. The proposed models attain the lowest loss in seven of the nine panels; the two exceptions are

n=150

under medium and heavy censoring, where Random Survival Forest (RSF) is slightly lower. The margin over RSF, the strongest competing baseline, widens markedly as the sample size grows (from

-0.096

+0.048

n=150

to between

+0.056

and

+0.073

n=1500

) and is otherwise stable across censoring levels, visualizing the Weibull rows of Table 8 .

	Any adjacent	Adjacent pair	$\widehat{q}_{0.1}>\widehat{q}_{0.9}$	$\widehat{q}_{0.25}>\widehat{q}_{0.75}$
Real data (6 cohorts)	$18.24\%$	$5.07\%$	$0.18\%$	$0.45\%$
Simulation (27 settings)	$34.17\%$	$10.49\%$	$2.11\%$	$6.12\%$

Table 9: Quantile-crossing rates for DeepQuantreg when the five levels are fit separately (mean over 25 seeds; the simulation figure additionally averages the 27 settings). The corresponding rate for all three CNQ models is

0

, both by construction and as measured on the fitted models. “Any adjacent” is the fraction of test subjects with at least one violated adjacent pair; “adjacent pair” is the fraction of (subject, pair) combinations violated.

	$\hat{q}_{0.1}$	$\hat{q}_{0.25}$	$\hat{q}_{0.5}$	$\hat{q}_{0.75}$	$\hat{q}_{0.9}$	$T_{\mathrm{obs}}$	Event	ER	Age
Patient A	28.3	48.7	91.5	138.8	184.8	41.8	Yes	ER+	86
Patient B	34.0	56.6	100.6	153.5	204.4	119.5	Yes	ER+	69
Patient C	36.9	64.7	102.7	149.9	187.8	85.6	Yes	ER+	78
Patient D	29.8	56.3	119.3	211.2	297.7	104.7	Yes	ER $-$	58
Patient E	34.0	56.7	99.8	147.9	192.4	100.3	Yes	ER+	74

Table 10: Predicted quantile milestones for five representative METABRIC patients (Trans-CNQ, seed 42), with survival reported in months. For each patient (A–E) the table lists the five predicted quantiles

\hat{q}_{0.1},\hat{q}_{0.25},\hat{q}_{0.5},\hat{q}_{0.75},\hat{q}_{0.9}

(non-crossing by construction, so they increase from left to right), the observed follow-up time

T_{\mathrm{obs}}

, the event indicator, and two key clinical covariates (estrogen-receptor status ER and age in years). The rows illustrate how the model translates covariates into individualized, interpretable survival-time quantiles—for example, the ER

-

patient D receives a wider predicted interval and a longer upper tail than the ER+ patients—and correspond to the METABRIC profiles plotted in Figure 8 (a).

Model	Tercile	Pinball Loss	80% Cov.	50% Cov.
TransKAN-CNQ	Narrow	20.10	0.759	0.476
	Medium	26.30	0.747	0.451
	Wide	29.30	0.830	0.517
Trans-CNQ	Narrow	20.59	0.713	0.424
	Medium	28.02	0.705	0.396
	Wide	30.78	0.798	0.526

Table 11: Uncertainty stratification on METABRIC (averaged over 25 seeds) for each of the three proposed models. Test subjects are grouped into terciles —narrow, medium, wide—according to the predicted

80\%

interval width

w_{i}=\hat{q}_{0.9,i}-\hat{q}_{0.1,i}

, an internal, label-free difficulty score. For each model and tercile the table reports the pinball loss (survival scale, in months) and the empirical coverage of the nominal

80\%

and

50\%

prediction intervals. Across all three models the pinball loss and both coverages rise monotonically from the narrow to the wide tercile, confirming that predictions the model itself flags as uncertain (wider intervals) are genuinely harder and receive higher coverage; this is the tabular counterpart of Figure 9 (a).

	$\hat{q}_{0.1}$	$\hat{q}_{0.25}$	$\hat{q}_{0.5}$	$\hat{q}_{0.75}$	$\hat{q}_{0.9}$	$T_{\mathrm{obs}}$	Event	Sex	Age
Patient A	439	1,062	1,876	2,815	3,757	2,047	Yes	F	90
Patient B	155	614	1,735	2,965	4,293	2,706	Yes	M	59
Patient C	847	2,086	3,221	4,207	5,363	2,981	Yes	F	50
Patient D	658	1,895	3,533	5,063	6,639	3,347	Yes	F	70
Patient E	614	1,731	3,246	4,586	5,940	3,245	Yes	M	69

Table 12: Predicted quantile milestones for five representative FLCHAIN patients (Trans-CNQ, seed 42), with survival reported in days. As in Table 10 , each row gives the five non-crossing predicted quantiles

\hat{q}_{0.1},\ldots,\hat{q}_{0.9}

, the observed follow-up time

T_{\mathrm{obs}}

, the event indicator, and two covariates (sex and age in years) for one patient (A–E). The examples show individualized milestones on a very different clinical scale (all-cause mortality over thousands of days) and correspond to the FLCHAIN profiles plotted in Figure 8 (b).

Model	Tercile	Pinball Loss	80% Cov.	50% Cov.	Avg. $n_{\text{unc}}$
TransKAN-CNQ	Narrow	354.7	0.773	0.469	234.6
	Medium	389.8	0.856	0.527	105.0
	Wide	400.8	0.884	0.582	94.2
Trans-CNQ	Narrow	359.3	0.778	0.479	224.2
	Medium	393.0	0.864	0.546	106.3
	Wide	403.7	0.891	0.614	103.3

Table 13: Uncertainty stratification on FLCHAIN (averaged over 25 seeds, with IPCW weights truncated at the 95th percentile) for each of the three proposed models. As in Table 11 , subjects are split into narrow, medium, and wide terciles by predicted

80\%

interval width

w_{i}

; for each model and tercile the table reports the pinball loss, the empirical

80\%

and

50\%

interval coverages, and the average number of test subjects per tercile

\overline{n}_{\text{unc}}

. The monotone increase of loss and coverage with interval width—reproduced here in a larger, more heavily censored cohort —confirms that interval width is a reliable internal indicator of predictive difficulty; see also Figure 9 (b).

Fig 8: Representative patient-level quantile profiles for (a) METABRIC (survival in months) and (b) FLCHAIN (survival in days), produced by Trans-CNQ on a single representative split (seed 42). Each row corresponds to one patient; the horizontal axis is predicted survival time. For each patient the horizontal line spans the predicted

80\%

prediction interval

[\hat{q}_{0.1},\hat{q}_{0.9}]

, the colored markers denote the five predicted quantiles

\hat{q}_{0.1},\hat{q}_{0.25},\hat{q}_{0.5},\hat{q}_{0.75},\hat{q}_{0.9}

(increasing left to right, by construction non-crossing), and a cross marks the observed event time for uncensored subjects. The panels show that the model produces individualized, coherent quantile milestones whose spread (interval width) varies substantially across patients, and that observed event times for uncensored subjects typically fall within the predicted interval. These are the same five example patients tabulated in Tables 10 and 12 .

Fig 9: Uncertainty stratification for (a) METABRIC and (b) FLCHAIN (Trans-CNQ, averaged over 25 seeds). Test subjects are partitioned into terciles—narrow, medium, and wide—by the predicted

80\%

interval width

w_{i}=\hat{q}_{0.9,i}-\hat{q}_{0.1,i}

, a purely internal, label-free measure of predictive difficulty. Within each tercile the panels report the IPCW pinball loss and the empirical coverage of the nominal

50\%

and

80\%

intervals; dashed horizontal lines mark the nominal coverage levels. Moving from narrow to wide terciles, both the pinball loss and the empirical coverage increase monotonically, showing that predictions the model flags as more uncertain (wider intervals) are indeed harder and better covered—so interval width can serve as a self-reported reliability signal at the individual level. The underlying numbers appear in Tables 11 and 13 .

Cutoff (months)	Latent	Trans-CNQ $+$ cens.	Weibull $+$ cens.
24	$+35.2$	$+19.9$	$+63.8$
48	$+34.4$	$+20.4$	$+42.1$
72	$+30.4$	$+19.9$	$+16.6$
96	$+25.4$	$+18.5$	$+6.9$

Table 14: METABRIC event projection error (%) at

t=200

months (seeds 41–65, mean). Positive values indicate overestimation of cumulative events.

Fig 10: METABRIC event-projection curves (single representative split, seed 41) evaluated at four calendar cutoffs—24, 48, 72, and 96 months (vertical dotted lines). The horizontal axis is calendar/follow-up time in months and the vertical axis is the cumulative number (or proportion) of events projected to have occurred by each time. Four curves are overlaid: the solid black curve is the raw observed cumulative-event count; the gray dashed curve is the Kaplan–Meier-based latent reference that adjusts for censoring; the green curve is the Trans-CNQ projection obtained with the censoring (IPCW) model; and the purple dash-dot curve is the corresponding parametric Weibull projection with the same censoring model. The figure assesses how closely each method reproduces the true event accrual over time: the Trans-CNQ curve tracks the KM latent reference closely across all four cutoffs, whereas the Weibull projection deviates more where the parametric shape is misspecified, illustrating the benefit of the flexible distributional model for event-count forecasting.

Fig 11: Permutation feature-importance heatmap for METABRIC, averaged over 25 random splits and pooled across the three proposed architectures (Trans-CNQ, TransKAN-CNQ, KAN-CNQ). Rows index the covariates and columns index the five quantile levels $\tau\in\{0.1,0.25,0.5,0.75,0.9\}$ ; the color of each cell encodes the increase in IPCW pinball loss, $\Delta$ , incurred when the corresponding covariate is randomly permuted (breaking its association with the outcome) while all others are held fixed, so that larger (warmer) values indicate greater prognostic importance at that quantile. Reading a row across $\tau$ reveals whether a covariate acts mainly on the lower (early-event), central, or upper (long-survivor) part of the predicted distribution rather than on a single summary. Age and chemotherapy dominate overall, while hormone therapy shows a pronounced effect concentrated in the upper quantiles—a tail-specific signal that a single hazard ratio would average away.

Panel of Figure 24 — Fig 12: Architecture-specific permutation feature-importance heatmaps for METABRIC, shown separately for (a) Trans-CNQ, (b) TransKAN-CNQ, and (c) KAN-CNQ (each averaged over 25 random splits). As in Figure 11 , rows are covariates, columns are the five quantile levels $\tau$ , and cell color encodes the increase in IPCW pinball loss $\Delta$ from permuting that covariate; the three panels share a common color scale to make them directly comparable. Decomposing the pooled heatmap by architecture shows that the three models yield broadly similar qualitative rankings—age and chemotherapy dominate prognosis and hormone therapy again exhibits a pronounced upper-quantile ( $\tau=0.75,0.9$ ) effect —indicating that the recovered prognostic structure is a stable property of the data rather than an artifact of any single architecture.

Panel of Figure 25 — Fig 13: Architecture-specific permutation feature-importance heatmaps for FLCHAIN, shown separately for (a) Trans-CNQ, (b) TransKAN-CNQ, and (c) KAN-CNQ (each averaged over 25 random splits). Rows are covariates, columns are the five quantile levels $\tau$ , and cell color encodes the increase in IPCW pinball loss $\Delta$ from permuting that covariate. The two Transformer-based models produce broadly similar patterns, with age and the serum free light chains (kappa and lambda) dominating prognosis, whereas KAN-CNQ ranks creatinine first. This discrepancy—which is not present for METABRIC in Figure 12 —illustrates the architecture dependence of attribution in the pure-KAN backbone discussed in the main text, and motivates preferring the Transformer-based variants when feature attributions are to be reported.

Fig 14: METABRIC log-time residual diagnostic at the predicted median. Restricting to uncensored subjects (for whom the event time is observed), the histogram shows the log-scale residual

\log Y-\log\widehat{Q}_{0.5}(X)

, i.e., the signed distance between the observed event time and the predicted median survival time on the log scale, pooled over the 25 random splits; the vertical dashed line marks zero. If the median is well calibrated the residuals should be centered at zero with roughly half above and half below. The Transformer-based models (Trans-CNQ, TransKAN-CNQ) produce residual histograms centered near zero with no systematic over- or under-prediction, corroborating the calibration and coverage checks reported in the main text.

Fig 15: FLCHAIN model-checking (Trans-CNQ, averaged over 25 seeds; IPCW weights truncated at the 95th percentile to limit the influence of large inverse-censoring weights). Panel (a), per-quantile calibration, plots the empirical (realized) coverage against each nominal quantile level

\tau\in\{0.1,0.25,0.5,0.75,0.9\}

; points on the

45^{\circ}

diagonal indicate perfect calibration. Panel (b), interval coverage versus nominal, plots the empirical coverage of central prediction intervals against their nominal level, with the

50\%

and

80\%

intervals highlighted and the diagonal marking exact coverage. Calibration is near-nominal across all quantile levels and the

50\%/80\%

interval coverage is close to target, replicating the METABRIC findings in a larger cohort with substantially heavier (

\approx 72\%

) censoring and thereby demonstrating that the calibration is not specific to a single dataset or censoring regime.

	$\overline{L}_{\mathrm{pin}}$				$80\%$ coverage
Model	none	90th	95th	99th	none	90th	95th	99th
TransKAN-CNQ	0.279	0.284	0.282	0.280	0.795	0.817	0.814	0.806
Trans-CNQ	0.282	0.286	0.285	0.283	0.806	0.825	0.822	0.814
KAN-CNQ	0.280	0.284	0.283	0.281	0.828	0.841	0.840	0.835

Table 15: IPCW weight-truncation sensitivity on FLCHAIN (heavy,

\sim

72% censoring). Because inverse-probability-of-censoring weights can become large when the estimated censoring survival

\widehat{G}

is small, we truncate them at a chosen percentile; this table probes how much that choice matters. The two blocks report the mean pinball loss

\overline{L}_{\mathrm{pin}}

(log scale, averaged over the five common quantile levels

\tau\in\{0.1,0.25,0.5,0.75,0.9\}

) and the empirical

80\%

interval coverage for each proposed model, as the weights are truncated at the 90th, 95th, and 99th percentiles or left untruncated (“none”); all values are means over 25 random splits. Both the loss and the coverage—and the Transformer-over-KAN model ordering—are essentially flat across truncation levels, and coverage stays close to the nominal

0.80

, showing that the reported results are not driven by a particular truncation threshold.

	$\overline{L}_{\mathrm{pin}}$				$80\%$ coverage
Model	none	90th	95th	99th	none	90th	95th	99th
TransKAN-CNQ	0.215	0.222	0.221	0.218	0.784	0.822	0.820	0.803
Trans-CNQ	0.224	0.225	0.223	0.225	0.720	0.797	0.795	0.763
KAN-CNQ	0.227	0.231	0.230	0.230	0.765	0.810	0.808	0.792

Table 16: IPCW weight-truncation sensitivity on METABRIC (moderate,

\sim

40% censoring), reported in the same format as Table 15 . Weight concentration does not track the censoring rate: across the six cohorts the effective sample size

(\sum_{i}w_{i})^{2}/\sum_{i}w_{i}^{2}

, expressed as a fraction of the observed events and computed from untruncated weights, is

98\%

on GBSG,

98\%

on GBSG-500,

84\%

on SUPPORT,

84\%

on FLCHAIN,

75\%

on NKI70 and only

35\%

on METABRIC, so METABRIC is where an IPCW-weighted summary rests on the fewest effective observations. Pinball loss is essentially unaffected by truncation, but

80\%

coverage rises from

0.72

–

0.77

0.80

–

0.82

; for reference, the unweighted coverage among observed events is

0.78

. This suggests the mild undercoverage reported for METABRIC in the main text is driven largely by a few extreme weights rather than by the interval widths themselves. All headline tables use untruncated weights.

	$\overline{L}_{\mathrm{pin}}$			$80\%$ coverage
Model	65/15/20	50/25/25	80/10/10	65/15/20	50/25/25	80/10/10
TransKAN-CNQ	0.228	0.226	0.242	0.736	0.738	0.670
Trans-CNQ	0.229	0.224	0.237	0.720	0.739	0.691
KAN-CNQ	0.233	0.230	0.247	0.729	0.731	0.690

Table 17: Split-ratio sensitivity on METABRIC, assessing whether the results depend on how the data are partitioned. Each proposed model is retrained from scratch under three train/validation/test ratios—

65/15/20

50/25/25

, and

80/10/10

(with all other training choices held fixed)—on partitions that are regenerated for each ratio, and evaluated on its own held-out test set at the five common quantile levels

\tau\in\{0.1,0.25,0.5,0.75,0.9\}

; all entries are means over 25 random splits. Because the partitions are drawn afresh, the

65/15/20

column is not the main-text configuration re-scored: it uses different training and test sets, so it is expected to differ from the headline table in the main text rather than reproduce it. The two blocks report the IPCW pinball loss

\overline{L}_{\mathrm{pin}}

(log scale) and the empirical coverage of the nominal

80\%

interval. Both the performance level and the Transformer-over-KAN ordering are stable across the three ratios; only the configuration with the smallest test set (

80/10/10

) is mildly noisier, as expected from the reduced evaluation sample.

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May 15, 2026Anchit Jain, Kevin Zhang, Stephen BatesPost-Hoc Calibration Survival Analysis
EECS, MIT, Cambridge MA, USA.
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School of Statistics and Data Science, Institute of Data Science and Statistics, Shanghai University of Finance and Economics, Shanghai 200433, China · School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China · School of Artificial Intelligence, and Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, 430072, China +1
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Oct 15, 2026Julie Alberge, Linus Bleistein, Judith Abécassis +3Survival Analysis Tabular Foundation Models
Inria de Saclay · EPFL · Inria de Saclay, :probabl.

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Isotonic Survival Regression: Calibrated Survival Distributions from Deep Cox Models

Anchit Jain, Kevin Zhang, Stephen Bates

EECS, MIT, Cambridge MA, USA.

Post-Hoc Calibration Survival Analysis Isotonic Regression

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Time-to-event data is widespread across the life sciences and engineering, but it is typically encountered together with censoring, which complicates the application of standard machine learning methods. Deep Cox models have emerged as a popular method for analyzing time-to-event data because they gracefully handle censoring and can be used with unstructured data such as clinical text reports, genomic sequences, and pathology images. However, their predicted survival probabilities are often poorly calibrated, thus limiting their practical utility. In this paper, we propose a novel post hoc calibration method for Deep Cox models that uses isotonic regression to refine predicted survival probabilities without affecting discriminative power. We establish favorable theoretical guarantees, including a double-robustness property and asymptotic calibration. Experiments on synthetic and real-world clinical data demonstrate the empirical effectiveness of our method.

Abstract

Survival Analysis

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