arXiv:2605.03944v1 [cs.LG] 5 May 2026
TabSurv: Adapting Modern Tabular Neural Networks to Survival Analysis Stanislav Kirpichenko∗ Andrei Konstantinov∗,† Lev Utkin∗ [email protected], [email protected], [email protected] Higher School of Artificial Intelligence Technologies,
Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia
Abstract
1
Survival analysis on tabular data is a well-studied problem. However, existing deep learning methods are often highly task-specific, which can limit the transfer of new approaches from other domains and introduce constraints that may affect performance. We propose TabSurv, an approach that adapts modern tabular architectures to survival analysis using either the Weibull distribution or non-parametric survival prediction. TabSurv optimizes SurvHL, a novel histogram loss function supporting censored data. In addition to a baseline feed-forward network, we implement deep ensembles of MLPs for survival analysis within TabSurv. In contrast to prior work, the ensemble components are trained in parallel, optimizing survival distribution parameters before averaging, which promotes diversity across ensemble component predictions. We perform a comprehensive empirical evaluation of different proposed architectures on 10 diverse real-world survival datasets. Our results show that TabSurv consistently outperforms on average established classical and deep learning baselines, such as RSF, DeepSurv, DeepHit, SurvTRACE. Notably, deep ensembles with Weibull parametrization instead of non-parametric models achieve the highest average rank by C-index. Overall, our study clarifies how modern tabular neural networks can be adapted and trained to tackle survival analysis problems, offering a strong and reliable approach. The TabSurv implementation is publicly available.
Survival analysis [20], the statistical study of time-toevent data, is a foundational tool for domains where outcomes are not immediate and observations can be incomplete. Its ability to formally incorporate censored data, where the exact event time for some instances may only be known to exceed a certain point, makes it essential for clinical trials, reliability engineering, financial risk assessment, and beyond. Survival analysis delivers quantifiable insights into event timing uncertainty across these fields. While classical methods like the Kaplan-Meier estimator [24] and Cox Proportional Hazards models [8] and Random Survival Forests (RSF) [23] have dominated the field for decades, recent years have seen significant interest in deep learning approaches, including DeepSurv [25], DeepHit [29], and SurvTRACE [46], which aim to capture complex, non-linear relationships in high-dimensional data. In parallel, nonparametric kernel-based methods, such as the Beran estimator [6], provide flexible alternatives by smoothing the empirical survival distribution based on covariate similarity, offering robustness to model misspecification. Despite this progress, a notable gap persists between advancements in generic tabular deep learning, including TabR [14], TabM [13], RealMLP [19], etc., and their application to survival analysis. Many existing deep survival models are highly task-specific, with architectures and loss functions deeply intertwined 1
Introduction
with the survival objective [27]. This specialization can inadvertently limit the field: it restricts the transfer of modern architectures from the broader tabular machine learning domain and may impose modeling constraints that affect flexibility and performance. Some approaches, such as DeepSurv, inherit the proportional hazards assumption from the Cox model, which may not hold in complex real-world scenarios. Others, like DeepHit, relax this assumption but rely on discretizing the time axis, which introduces sensitivity to binning choices and can impair calibration. More recent models, including SurvTRACE, incorporate transformer-based architectures but are often computationally intensive and require large datasets to realize their full potential, conditions rarely met in many survival analysis applications, which typically involve modest sample sizes and high-dimensional covariates. Moreover, these models often lack explicit modular design, making it difficult to swap components (e.g., backbone networks, loss functions, or output parameterizations) without significant re-engineering. To address this, we propose TabSurv, a principled and flexible approach that bridges modern tabular deep learning with survival analysis. Our core contribution is a framework that adapts standard and ensemble-based tabular architectures, such as Multi-Layer Perceptrons (MLPs), to model survival outcomes through two distinct pathways: a Weibull-distribution-based parametric head and a nonparametric survival prediction head. A key peculiarity of TabSurv is the introduction and optimization of a novel histogram-based loss function, called SurvHL, explicitly designed to handle censored data efficiently. Furthermore, we implement a parallel deep ensemble strategy within TabSurv, where each ensemble member independently learns survival distribution parameters before aggregation. Unlike sequential or bagging methods, our ensemble components optimize their respective survival distribution parameters independently before aggregation, ensuring prediction diversity and enhanced robustness. We conduct a comprehensive empirical evaluation of TabSurv variants on several diverse real-world datasets. Our results demonstrate that TabSurv consistently outperforms established classical (e.g., RSF) and deep learning baselines (e.g., DeepSurv, Deep-
Hit, SurvTRACE). The deep ensemble variant with Weibull parametrization achieves the highest average rank by the concordance index (C-index [16]), highlighting the efficacy of combining modern ensemble techniques with a flexible parametric model. By providing a modular, well-engineered, and publicly available implementation, TabSurv not only advances the state of the art in deep survival analysis, but also offers a reliable foundation for future research and practical deployment. Our work clarifies how offthe-shelf tabular architectures can be systematically adapted to handle censored time-to-event data, opening new avenues for cross-domain knowledge transfer in survival modeling. Our contributions can be summarized as follows: 1. We propose TabSurv, a flexible approach that adapts modern tabular deep learning architectures to survival analysis, supporting both parametric (Weibull) and non-parametric survival prediction while naturally handling censored data. 2. We propose SurvHL, a novel histogram-based loss function tailored for censored time-to-event data, and develop a parallel training strategy for deep ensembles of MLPs that optimizes survival distribution parameters before averaging, enhancing diversity and predictive performance. 3. Through extensive experiments on diverse realworld survival datasets, we demonstrate that TabSurv consistently outperforms established classical and deep learning baselines, including RSF, DeepSurv, DeepHit, SurvTRACE, with Weibullbased deep ensembles achieving the highest average ranks of survival performance measures, providing new insights into the effectiveness of parametric modeling in deep survival analysis. To summarize the experimental results, the Cindex [16], Integrated Brier Score (IBS) [15], and time-dependent (cumulative/dynamic) AUC [42] metrics are applied. In summary, this study clarifies how contemporary tabular neural networks can be effectively adapted for survival analysis, providing a strong, reliable, and 2
publicly available approach. The complete implemen- tures [18, 26, 32, 37, 50]. Ensembles of tabular models, tation of TabSurv is open-sourced to facilitate further where multiple neural networks are trained individuresearch and application. ally and their predictions are averaged, were studied in [28, 48]. Despite the effectiveness of these models with tabular data, their application in survival anal2 Related Work ysis requires the development of special “extensions” that would enable the creation of new survival modMachine learning for survival analysis. Sur- els competitive with existing deep learning survival vival analysis concerned with modeling time-to-event models. outcomes under censoring has benefited substantially from advances in machine learning. Early efforts Survival analysis: the probextended classical statistical models (e.g., Cox re- 3 gression) with nonparametric and ensemble methlem statement ods, while recent work increasingly leverages deep learning to capture complex, nonlinear relationships In survival analysis, a dataset is represented by a set of in high-dimensional data. Comprehensive surveys triplets D = {(x1 , δ1 , t1 ), . . . , (xn , δn , tn )}, where xi ∈ by [33, 38, 43] catalog this evolution, highlighting Rd is a d-dimensional feature vector, ti is the observed trends such as scalable deep architectures, handling time, and δi ∈ {0, 1} is a binary censoring indicator. of competing risks, and integration with electronic An observation is uncensored (δi = 1) if the event of health records. Methodological generalizations of sur- interest (e.g., failure or death) was observed at time ti . vival models are discussed in [4], while [11] emphasize It is censored (δi = 0) if the event was not observed benchmarking and interpretability. Most recently, [49] within the study period, meaning the true event time outline emerging challenges and opportunities in the is known only to exceed ti . A survival model is trained field, including robustness, fairness, and uncertainty on D to estimate probabilistic characteristics of the quantification. future event time T for a new subject described by a Transformer-based models capable of modeling long- feature vector x. range interactions in high-dimensional inputs have A fundamental concept in survival analysis is the gained traction in survival analysis: [47] propose trans- survival function S(t | x), defined as the probability formers for competing-risk settings, while others de- that a subject with features x survives beyond time t: velop specialized variants such as ResDeepSurv [44], S(t | x) = P(T > t | x). (1) explainable transformers [40], and general-purpose survival transformers [22]. Applications in clinical Our objective is to estimate the survival function domains include multi-modal survival prediction [51], S(t | x). To this end, we discretize time into a set semi-supervised learning [41], and SurFormer [45]. of intervals and represent the survival function via The discussed transformer survival models, in coninterval probabilities pi (x), defined as trast to the proposed approach TabSurv, are based on learning from images rather than from tabular data. pi (x) = P T ∈ [τi , τi+1 ) | X = x , Modern tabular machine learning models. A the distinctive feature of many machine learning tasks where 0 < τ1 < · · · < τm < τm+1 = +∞ denote m ordered unique event times. Given {p (x)} , i i=1 the is that the data are predominantly tabular, comsurvival function can be approximated as prising lab results, patient characteristics, and simiX lar structured variables. This necessitates the use S(t | x) ≈ 1 − pi (x). of models specifically designed for tabular data. i:τi ≤t Prominent examples include TabR [14], TabM [13], RealMLP [19], TabICL [35], TabPFN [17], Mother- We denote the interval probabilities, predicted by Net [34], along with several other recent architec- a machine learning model as pbi (x), and represent 3
b = This design choice is motivated by the need to balthe estimated survival function as P a vector S i (Sb1 , . . . , Sbm ), whose entries Sbi (x) = j=1 pbj (x) ap- ance ensemble diversity with representation efficiency. proximate S(τi | x). Low-level numerical feature representations are largely task-agnostic and benefit from being learned from the full data signal, whereas higher-level predictions remain independent across ensemble members. Shar4 TabSurv ing the embedding module reduces the number of In general, TabSurv consists of three main compo- trainable parameters, improves sample efficiency, and nents: a numerical feature embedding module, a neu- stabilizes optimization, while still allowing ensemble ral network backbone, and a survival function con- members to maintain diversity through independent struction module. Figure 1 illustrates four proposed downstream networks. architectural variants, starting from the simplest MLP In the LAS architecture, each ensemble member wrapper to deep ensemble, generating Weibull distri- outputs logits. During training, these logits are indebution parameters. Below, we describe these variants pendently transformed into survival functions, and a in detail. separate loss is computed for each member. At inference time, the logits are averaged and subsequently converted into a survival function. Thus, the final 4.1 Architecture details model represents an ensemble, which can potentially Logits → Survival (LS). The simplest approach is improve generalization performance. non-parametric, i.e., it does not assume any predefined family of event-time distributions. A neural network of arbitrary architecture, such as a fully connected Weibull parameters → Survival → Average network or RealMLP, outputs logits corresponding (WSA). Since non-parametric approaches may yield to a discrete set of time intervals. These logits are noisy survival function estimates, we also consider a transformed into a probability distribution over time parametric formulation based on the Weibull distriintervals via the Softmax function, which is then con- bution. We adopt the Weibull distribution as a parametric survival model due to its favorable balance verted into a survival function. between flexibility and inductive bias. The Weibull family can represent increasing, decreasing, and conLogits → Average → Survival (LAS). Unlike stant hazard rates, which makes it expressive enough classical neural-network-based approaches to survival for a wide range of real-world time-to-event processes analysis, we additionally consider deep ensembles, while remaining low-dimensional and stable to optiwhere the neural network consists of m independently mize. Compared to more expressive parametric famtrained components. As shown in [13], such ensem- ilies, the Weibull distribution exhibits well-behaved bles can lead to substantially more accurate models. likelihood surfaces and requires estimating only two Technically, training is performed in parallel by mini- positive parameters, λ and k. This significantly remizing the sum of the loss functions of all ensemble duces estimation variance and improves robustness, members. Importantly, prediction aggregation is not especially in low- and moderate-sample regimes. Fiperformed before loss computation, which is crucial nally, the Weibull distribution admits a closed-form for preserving prediction diversity. Indeed, optimiz- cumulative distribution and survival function, which ing the sum of losses over independent components allows for efficient discretization on arbitrary time is equivalent up to the shared modules to training m grids and seamless integration with standard survival ensemble members independently. losses. Although this parameterization is not univerA key difference from classical ensembles is the pres- sal, in practice one can determine its suitability for a ence of a shared numerical feature embedding module, given task via cross-validation. As demonstrated in which is trained jointly for all ensemble members. our experiments, the Weibull-based approach achieves 4
TabSurvLS (Logits
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Figure 1: Architectural variants of TabSurv. Blue blocks denote learnable neural networks, Green blocks denote intermediate parameters, and Purple blocks denote the specific survival transformation (Softmax or Weibull). Dashed arrows indicate the flow during training (independent losses), while Solid arrows indicate the prediction flow (including ensemble averaging). superior performance on a wide range of real-world datasets. In the WSA architecture, each ensemble member predicts positive Weibull parameters λ(x) and k(x). Based on these parameters, we compute a discretized probability distribution, where each element represents the probability that the event occurs inside the interval [τi , τi+1 ):
to survival function computation results in a final distribution that remains Weibull. While this imposes a strong modeling constraint, it can significantly reduce prediction noise. Empirically we observe that the multimodality produced by WSA is more likely to reflect estimation noise rather than meaningful uncertainty. Since each MLP is trained independently by fitting the whole distribution, aggregated multimodal survival time espbi (x) = FW (τi+1 ; λ(x), k(x)) − FW (τi ; λ(x), k(x)), timates cannot represent a complex multimodal dis(2) tribution. Therefore, WSA should not be regarded as i inherently superior to WAS. X Sbi (x) = 1 − pbj (x), (3) j=1
4.2
where FW (·) denotes the Weibull cumulative distribution function. Training proceeds analogously to LAS: each ensemble member is optimized using its own loss function. At inference time, the estimated survival functions are averaged. Consequently, the final survival estimate is a mixture of the survival functions produced by individual ensemble members.
SurvHL loss function
The idea behind the loss function SurvHL is to extend the discretized likelihood commonly used in survival analysis, which is defined by probabilities on discretized time intervals in accordance with a conditional probability distribution of time to event and represented in survival analysis or a single data point xi as follows: ( − log pbI(t) (x), δ = 1 ℓ(x, t, δ) = (4) Weibull parameters → Average → Survival − log SbI(t) (x), δ = 0, (WAS). The WAS architecture differs from WSA only at the inference stage; however, this difference is where I(t) = max{i | τi ≤ t} is the index of the substantial. Averaging the Weibull parameters prior corresponding time interval. 5
For uncensored events, instead of considering only one discretized time interval (as in classical likelihood), we smooth across several neighboring intervals. This approach ensures a smoother gradient flow through the loss function and makes model training more robust compared to the standard maximum likelihood method. The formula for a single data point x, for which the predicted probabilities and the survival function b vector are denoted for brevity as pb, S: ( P Figure 2: Hyperparameter tuning results with differm − j=1 ωj (I(t)) log(b pj ), δ = 1 ℓ(x, t, δ) = (5) ent loss functions, TabSurvLAS model on METABRIC − log SbI(t) , δ = 0, dataset where ωj (I(t)) are weights derived from the Gaussian distribution centered at I(t) with the standard deviation σ = r/3, and r is a hyperparameter. Gaussian weighting is chosen for smoothness and locality. The final loss function is the negative average over a batch of size n:
where t̂i and t̂j are the expected event times approximations derived from the predicted survival function. A C-index of 1 indicates perfect ranking, while a value of 0.5 corresponds to random ordering. Another important metric is the Integrated Brier Score (IBS), a time-dependent measure that combines discrimination and calibration. For a given time interval [0, tmax ], it is defined as Z tmax 2 1 IBS = E ∆new (t) − Ŝ(t | xnew ) dt. tmax 0 (8) where ∆new (t) = 1(Tnew > t) is the true survival status at t and Ŝ(t | xnew ) is the predicted survival probability. Lower values of the IBS indicate better predictive accuracy. A complementary time-dependent discrimination metric is the cumulative/dynamic area under the ROC curve (AUC). At a given time t, the time-dependent AUC evaluates the ability of a model to distinguish individuals who experience the event by time t from those who remain event-free at t. Formally, it is defined as the probability that, for a randomly selected pair consisting of one individual with T ≤ t and one individual with T > t, the model assigns a higher risk score to the former. In practice, we compute the cumulative/dynamic b | AUC using the predicted survival probabilities S(t x), with appropriate handling of censoring. To obtain a single summary statistic, we report the timeintegrated AUC by averaging the time-dependent
n
L=−
1X ℓ(xi , ti , δi ). n i=1
(6)
Figure 2 presents the distribution of the C-index on the validation set obtained during hyperparameter tuning of the TabSurvLAS model on the METABRIC dataset. Among the considered objectives, the SurvHL consistently achieves higher validation performance than both the standard likelihood-based loss and the sigmoid-smoothed C-index, and is therefore selected as the optimization objective.
5
Experiments
5.1
Evaluation metrics
Evaluating survival models requires metrics that account for censoring and time-to-event outcomes. The most widely used measure is Harrell’s concordance index (C-index), which quantifies a model ability to correctly rank event times. Let J denote the set of all pairs (i, j) with δi = 1 and ti < tj . The C-index is computed as P (i,j)∈J 1 t̂i < t̂j C= , (7) |J | 6
AUC over the evaluation horizon. Higher values of and the five largest datasets have held-out test sets the AUC indicate better discriminative performance. ranging from 572 to 2732 samples (Table 2), making the use of held-out test evaluation appropriate for the benchmark. 5.2 Datasets In the first stage, hyperparameters are tuned using We evaluate the proposed methods on a collection of the Tree-structured Parzen Estimator (TPE) sampler 10 real-world survival datasets. The list of datasets implemented in Optuna [1]. The optimal hyperpaand their main characteristics are summarized in Ta- rameter configuration is selected by maximizing the ble 2 (Appendix A). C-index on the validation set. The number of optiAll datasets are preprocessed in a unified manner mization iterations depends on the number of hyperprior to training. Numerical features are standardized, parameters of a given method and ranges from 50 while categorical features are encoded using a one-hot to 200. At this stage, the validation set effectively scheme. Missing values are imputed with the mean serves as a test set, while models are trained on the for numerical features and with an additional category training set. For methods employing early stopping, for categorical features. Features that are missing for the validation set is sampled as 25% of the training all samples are removed from the dataset. data. The second stage corresponds to the final evaluation. Depending on the method, the validation set is either 5.3 Methodology used for early stopping or merged with the training All real-world datasets are split into training, val- set (as in the case of Random Survival Forests). All idation, and test sets, with stratification based on evaluation metrics are computed on the held-out test censoring indicators. The experimental procedure set. At each experimental run, models are ranked consists of two stages. The design of the experimen- according to their performance, separately for the Ctal protocol follows the methodology commonly used index, IBS, and time-dependent AUC. The reported in modern tabular neural network studies, such as results correspond to the average metric values and TabR [14] and TabM [13]. This protocol is particu- average ranks computed over 20 independent runs. larly appropriate for neural models whose training is computationally expensive and whose performance is 5.4 Results on real-world datasets sensitive to hyperparameter choices. The same protocol is also applied to all competing methods, for The results are summarized using rank-based comwhich hyperparameter tuning is similarly important. parisons across all datasets. For each method, an For each dataset, we use a fixed train–validation–test average rank is computed on each dataset, and the split and repeat training and evaluation with different distribution of these average ranks is visualized using random seeds. Hyperparameters are selected once for boxplots. Figures 3, 4, and 5 report these rank diseach method within the fixed train–validation part tributions for the C-index, IBS, and time-dependent of each dataset, using the validation set, and are AUC, respectively. Detailed result tables for each then kept fixed across repeated runs rather than re- dataset are provided in Appendix D. Methods are optimized for each random seed. Thus, the protocol ordered according to their mean rank aggregated over does not use nested cross-validation. Nevertheless, all all datasets, with lower ranks indicating better overall methods are evaluated under identical conditions, and performance. In all figures, the evaluated methods are the held-out test set is never used for model selection denoted as follows. The proposed models are referred or hyperparameter tuning, preventing adaptation to to as TabSurv, with a subscript indicating the specific the test portion of a particular split. This setup pro- architectural variant (LAS, WSA, or WAS). For the vides reliable comparisons while remaining feasible for LS architecture, the employed backbone is additionexpensive neural baselines. Moreover, most datasets ally specified in parentheses (MLP or RealMLP). The contain at least 500 observations after preprocessing, Random Survival Forest baseline is abbreviated as 7
Figure 3: Comparison of model performance across all datasets based on the C-index. Diamonds indicate mean ranks.
Figure 4: Comparison of model performance across all datasets based on the Integrated Brier Score (IBS). Diamonds indicate mean ranks.
RSF. All remaining methods are denoted by their full names without abbreviations. Across all datasets and evaluation metrics, the best overall performance is achieved by the proposed TabSurvWAS , TabSurvLAS , and TabSurvLS(MLP) models, while all existing baseline methods consistently yield inferior results with respect to C-index, IBS, and time-dependent AUC. The results indicate that employing RealMLP as a backbone without additional architectural adaptations leads to worse performance compared to standard MLPs. Furthermore, the WSA variant underperforms relative to WAS, which is consistent with theoretical considerations regarding noise smoothing properties inherent to the WAS formulation. The LS(MLP) and LAS architectures achieve the lowest Integrated Brier Score, suggesting that these methods provide the most accurate estimates of the survival function shape. In contrast, the WAS variant attains superior C-index and time-dependent AUC values, indicating more accurate ranking of ex-
pected event times. This behavior may be attributed to the comparatively simpler optimization of neural networks predicting Weibull distribution parameters, as opposed to models that predict a large number of logits.
5.5
Additional analysis
The empirical results also provide practical guidance on which TabSurv variant to use. The LS(MLP), LAS, and WAS variants are the most accurate overall, but they are preferable in different regimes. The WAS model is attractive when a compact parametric description of event times is adequate, because the two-parameter Weibull head acts as a strong regularizer. This reduction in variance can improve ranking metrics such as the C-index even when the exact Weibull assumption is imperfect. However, the simulation study in Appendix E shows that non-parametric variants are better suited to genuinely multimodal 8
by changing the distribution used to map predicted parameters to survival probabilities. To assess whether the gains are merely a consequence of using ensembles, we compare TabSurv with independently ensembled DeepSurv and DeepHit baselines in Appendix C.3. The results indicate that external ensembling alone does not explain the performance of TabSurv: LS(MLP) already outperforms DeepSurv and remains competitive with its ensemble, while the independent LS(MLP) ensemble is practically equivalent to LAS but requires substantially more computation. We also compare TabSurv with GBDTbased survival models based on XGBoost survival variants [2]. Table 1 reports Bayesian probabilities that TabSurvLS(MLP) is better, practically equivalent, or worse than two XGBSE variants. TabSurv is either competitive or favorable across the considered metrics, with particularly strong IBS performance against the stacked Weibull variant. Figure 5: Comparison of model performance across 6 Conclusion all datasets based on the time-dependent AUC. Diamonds indicate mean ranks. We proposed a unified neural framework for tabular survival analysis based on the Survival Histogram Loss and introduced a family of models, TabSurv, that difevent-time distributions. Between LS(MLP) and LAS, fer in their survival distribution parameterization and the former is a more economical choice for smaller aggregation strategies. The framework is architecturedatasets, whereas LAS is preferable on larger datasets, agnostic and supports heterogeneous tabular features where its parameter-efficient ensemble can benefit through flexible embedding mechanisms. from more data. We evaluated the proposed methods on a diverse The framework is not tied to a specific backbone set of real-world survival datasets using a rigorous or distributional family. In this paper we instan- experimental protocol with hyperparameter optimizatiate it with MLP, TabM-like, and RealMLP-style tion and repeated runs. Performance was assessed backbones, observing that the MLP and TabM-like using complementary metrics, C-index, Integrated variants are the strongest in the considered survival Brier Score (IBS), and time-dependent AUC, capsetting. RealMLP performs worse without the kind turing both ranking quality and survival function of large-scale pretraining used in its original tabular accuracy. learning context, which suggests that survival-specific Across all datasets, TabSurvWAS , TabSurvLAS , and adaptation remains important. Extending TabSurv to TabSurvLS(MLP) consistently outperformed existing retrieval-based models such as TabR, or to foundation- baselines. LAS and LS(MLP) achieved the lowest style tabular models such as TabPFN and Mother- IBS values, indicating more accurate estimation of Net, is conceptually possible but requires additional survival function shapes, while WAS showed superior work on survival-compatible output heads and train- performance in terms of C-index and time-dependent ing objectives. Similarly, the Weibull head is only one AUC, suggesting more reliable event-time ordering. parametric instance; other families can be substituted The results also indicate that more complex back9
Baseline
Metric
Better
ROPE
Worse
XGBSEKaplanNeighbors XGBSEKaplanNeighbors XGBSEKaplanNeighbors XGBSEStackedWeibull XGBSEStackedWeibull XGBSEStackedWeibull
C-index IBS AUC C-index IBS AUC
0.567 0.293 0.796 0.254 0.968 0.427
0.254 0.605 0.140 0.742 0.032 0.417
0.179 0.102 0.064 0.004 0.000 0.156
Table 1: Bayesian comparison between TabSurvLS(MLP) and XGBSE-based survival baselines. For C-index and AUC, “Better” denotes P (TabSurv > XGBSE); for IBS, it denotes P (TabSurv < XGBSE). The ROPE width is 0.005. bones, such as RealMLP, do not necessarily improve performance without additional adaptation, and that WAS is preferable to WSA, in line with theoretical considerations on noise smoothing. Overall, the proposed framework establishes a strong baseline for neural survival modeling on tabular data and offers a flexible foundation for future extensions, including alternative distributional assumptions and more advanced backbone architectures.
Impact Statement This paper presents work whose goal is to advance the field of Machine Learning. The proposed methods are intended for general-purpose survival modeling and decision support. As with all predictive models applied to sensitive domains, careful validation and responsible use are required.
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A
Datasets
All statistics reported in Table 2 correspond to the datasets after preprocessing. Additional remarks on datasets: • SEER. The processed version of the SEER dataset is obtained from https://dx.doi.org/10.21227/ a9qy-ph35, where records from multiple years are consistently merged. The official documentation for the raw SEER data is available at https://seer.cancer.gov/data-software/documentation/ seerstat/nov2017/. • TCGA-GBM. This dataset is derived from the TCGA Pan-Cancer Atlas [30] and includes only patients diagnosed with Glioblastoma Multiforme (GBM). • For the METABRIC dataset, we use both clinical features and mRNA expression z-scores, whereas for all other datasets only clinical features are employed. Name METABRIC [9] SEER SUPPORT [7] TCGA-GBM [30] Rotterdam [36] FLC [10] GBSG2 [39] WHAS500 [21] Lung [31] PBC [12]
Number of samples Train Validation Test 999 333 572 2112 704 1208 4779 1594 2732 312 105 179 1565 522 895 4133 1378 2363 360 120 206 262 88 150 119 40 69 219 73 126
Number of features Numeric Categorical 494 75 4 21 37 30 5 20 5 6 4 34 5 4 6 8 3 4 11 15
Event rate 42% 15% 68% 82% 43% 28% 44% 43% 72% 39%
Table 2: Summary of the datasets used in the experiments.
B
Experiment setup
During the experiments, SurvTRACE and several TabSurv models (LAS, WAS, WSA) were executed on an NVIDIA Tesla V100 GPU with 16 GB of memory using CUDA 12.1. Other models (TabSurvLS , DeepHit, DeepSurv, and RSF) were trained on a CPU using up to eight threads simultaneously.
B.1
Tuned hyperparameters
The hyperparameters were tuned using the Optuna framework [1] for each evaluated model, with the following hyperparameter domains: • TabSurvLS(MLP) – Number of layers: 1 to 4 – Number of hidden units per layer: 128, 256, or 512 14
– Activation function: ReLU, SiLU, or SELU – Loss parameter r: 1 to 5 – Learning rate: α ∼ LogUniform[10−4 , 5 · 10−3 ] – Batch size: 32, 64, or 96 – Use of layer normalization between layers – Number of bins in Piecewise Linear Embeddings: 32, 48, or 64 – Use of activation function in numerical embeddings – Dimensionality of numerical embeddings: 8, 12, or 16 • TabSurvLAS – Number of blocks: 1 to 3 – Number of hidden units per block: 128, 256, or 512 – Number of neural networks in the ensemble: 8, 16, or 32 – Activation function: ReLU, SiLU, or SELU – Loss parameter r: 1 to 5 – Learning rate: α ∼ LogUniform[10−4 , 5 · 10−3 ] – Batch size: 32, 64, or 96 – Dropout rate: θ ∼ LogUniform[10−2 , 10−1 ] – Number of bins in Piecewise Linear Embeddings: 32, 48, or 64 – Use of activation function in numerical embeddings – Dimensionality of numerical embeddings: 8, 12, or 16 • TabSurvLS(RealMLP) – Number of layers: 1 to 4 – Number of hidden units per layer: 128, 256, or 512 – Activation function: Mish or SELU – Loss parameter r: 1 to 5 – Learning rate: α ∼ LogUniform[10−4 , 5 · 10−3 ] – Batch size: 32, 64, or 96 – Number of frequencies in Periodic Bias Linear DenseNet embeddings: 32, 48, or 64 – Frequency scale in numerical embeddings: ρ ∼ LogUniform[10−2 , 1] – Dimensionality of numerical embeddings: 8, 12, or 16 • TabSurvWAS – Number of blocks: 1 to 3 – Number of hidden units per block: 64, 128, or 256 15
– Number of neural networks in the ensemble: 8, 16, or 32 – Activation function: ReLU or SELU – Loss parameter r: 1, 3, or 5 bins – Learning rate: α ∼ LogUniform[10−4 , 5 · 10−3 ] – Batch size: 32, 64, or 96 – Number of bins in Piecewise Linear Embeddings: 32, 48, or 64 – Use of activation function in numerical embeddings – Dimensionality of numerical embeddings: 8, 12, or 16 • TabSurvWSA Hyperparameters are identical to those of TabSurvWSA . • SurvTRACE (100 optimization trials) – Number of hidden layers: 1 to 4 – Transformer hidden size: 128, 256, or 512 – Transformer intermediate size: 128, 256, or 512 – Number of attention heads: 2, 4, or 8 – Learning rate: α ∼ LogUniform[10−4 , 5 · 10−3 ] – Dropout rate: θ ∼ LogUniform[10−2 , 10−1 ] – Batch size: 32 or 64 – Weight decay: γ ∼ LogUniform[10−4 , 10−1 ] • DeepHit – Number of hidden layers: 1 to 4 – Number of hidden units per layer: 128, 256, or 512 – Activation function: ReLU or Tanh – Batch size: 64, 128, or 256 – Learning rate: αstep ∼ LogUniform[10−4 , 5 · 10−3 ] – Dropout rate: θ ∼ LogUniform[10−2 , 10−1 ] – Use of batch normalization between layers – Loss weighting coefficient α in DeepHit: α ∼ U[0, 1] – Smoothing parameter σ in DeepHit loss: 10−3 , 10−2 , 10−1 , or 1 • DeepSurv – Number of hidden layers: 1 to 4 – Number of hidden units per layer: 128, 256, or 512 – Activation function: ReLU or Tanh – Batch size: 64, 128, or 256 16
P (A > B), C-index
MLP
Emb
Ens
Emb+Ens
SurvHL
SurvHL+Ens
Emb+SurvHL
Emb+SurvHL+Ens+Weib Emb+SurvHL+Ens SurvHL+Ens SurvHL
0.914 0.959 0.785 0.249
0.947 0.968 0.993 0.892
0.734 0.709 0.596 0.140
0.505 0.748 0.742 0.281
0.900 0.883 0.388 –
0.743 0.265 – 0.123
0.862 0.755 0.654 0.180
Table 3: Bayesian ablation of TabSurv components. Rows correspond to model A, columns to model B. – Learning rate: α ∼ LogUniform[10−4 , 5 · 10−3 ] – Dropout rate: θ ∼ LogUniform[10−2 , 10−1 ] – Use of batch normalization between layers • Random Survival Forest – Number of trees in the ensemble: 20, 50, 100, or 200 – Maximum tree depth: 3, 5, or unlimited – Minimum number of samples required to split a node: 2 samples, 1% of the dataset size, or 5% √ – Number of features considered at each split: d or log2 d, where d denotes the number of features
C
Additional ablations and statistical comparisons
We complement the rank-based summary in the main text with Bayesian comparisons following [3]. For each dataset and method, the metric value is averaged over 20 runs with different random seeds. The Bayesian comparison is then performed across datasets with a region of practical equivalence (ROPE) of 0.005. For the C-index and AUC, larger values are better; for IBS, smaller values are better.
C.1
Component ablation
To isolate the contribution of the main components, we start from a baseline MLP and progressively add numerical embeddings (Emb), ensembling (Ens), the SurvHL loss, and the Weibull head. Table 3 reports the posterior probability that the model in the row outperforms the model in the column in terms of the C-index. The results show that SurvHL, embeddings, and ensembling are most effective when combined: the full non-parametric variant and the Weibull variant both outperform the baseline and most partial configurations with high posterior probability. The Weibull head adds a compact parametric bias that further improves C-index over several non-parametric variants.
C.2
Sensitivity of SurvHL
SurvHL introduces a smoothing parameter r and depends on the discretization grid used to represent the event-time distribution. Table 4 and Figure 10 report C-index values on the METABRIC dataset for several grid sizes and values of r. For each grid, there is a preferred smoothing level, and the best value is obtained with the full grid. At the same time, the total variation across this range of settings is small, indicating that SurvHL is not overly sensitive to these hyperparameters. 17
1.0
TabSurvLS(MLP) 0.00 0.22 0.10 0.32 0.56 0.98 0.93 1.00 0.98 0.82 0.57 0.25
TabSurvLAS 0.01 0.00 0.22 0.77 0.01 0.07 1.00 1.00 1.00 0.31 0.19 0.97
0.8
TabSurvWAS 0.10 0.03 0.00 0.65 0.00 0.06 0.98 0.98 0.98 0.27 0.06 0.82
TabSurvWSA 0.04 0.04 0.00 0.00 0.55 0.94 0.80 0.99 0.99 0.52 0.67 0.07 DeepSurv 0.00 0.00 0.03 0.05 0.00 0.00 0.06 0.93 0.98 0.15 0.09 0.04
DeepHit Ensemble 0.00 0.00 0.02 0.06 0.00 0.00 0.00 0.04 0.50 0.00 0.00 0.08 0.4
DeepHit 0.00 0.00 0.02 0.04 0.00 0.00 0.00 0.00 0.46 0.00 0.00 0.06
SurvTRACE 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.55 0.00 0.02 0.01 0.01
RSF 0.00 0.00 0.09 0.54 0.00 0.00 0.99 0.99 0.99 0.00 0.13 0.88
0.2
XGBSEKaplanNeighbors 0.10 0.00 0.21 0.91 0.04 0.08 1.00 1.00 0.99 0.27 0.00 0.90 XGBSEStackedWeibull
RSF
XGBSEKaplanNeighbors
DeepHit Ensemble 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.96 0.02 0.01 0.00 0.00 0.4
DeepHit 0.00 0.00 0.00 0.00 0.00 0.00 0.96 1.00 0.01 0.01 0.00 0.00
0.4
SurvTRACE 0.00 0.00 0.00 0.03 0.00 0.01 0.02 0.01 1.00 0.01 0.01 0.00 RSF 0.54 0.69 0.65 0.37 0.91 0.97 0.01 0.01 0.01 1.00 0.60 0.12
0.2
XGBSEKaplanNeighbors 0.60 0.80 0.73 0.09 0.63 0.81 0.00 0.00 0.01 0.60 1.00 0.10
0.2
(a) Bayesian pairwise comparison of methods by C-index.
XGBSEStackedWeibull
RSF
XGBSEKaplanNeighbors
DeepHit
SurvTRACE
DeepSurv
TabSurvWSA
DeepSurv Ensemble
TabSurvWAS
0.0
TabSurvLAS
XGBSEStackedWeibull 0.03 0.03 0.07 0.04 0.04 0.08 0.00 0.00 0.00 0.12 0.10 1.00 TabSurvLS(MLP)
XGBSEStackedWeibull
RSF
XGBSEKaplanNeighbors
DeepHit
SurvTRACE
DeepSurv
DeepHit Ensemble
TabSurvWSA
DeepSurv Ensemble
TabSurvWAS
TabSurvLAS
0.6
DeepSurv 0.83 0.93 0.86 0.56 1.00 1.00 0.00 0.00 0.01 0.96 0.81 0.08
XGBSEStackedWeibull 0.74 0.91 0.92 0.68 0.42 0.02 0.01 0.02 0.00 0.16 0.21 1.00 TabSurvLS(MLP)
0.8
TabSurvWSA 0.04 0.22 0.34 1.00 0.42 0.56 0.00 0.00 0.02 0.37 0.09 0.04 DeepSurv Ensemble 0.96 0.96 0.96 0.42 1.00 1.00 0.00 0.00 0.00 0.91 0.63 0.04
0.6
SurvTRACE 0.02 0.00 0.00 0.00 0.00 0.02 0.02 0.02 1.00 0.00 0.02 0.01 RSF 0.14 0.16 0.18 0.30 0.32 0.17 0.17 0.01 0.00 1.00 0.05 0.16
DeepHit
TabSurvWAS 0.72 0.77 1.00 0.35 0.96 0.86 0.00 0.00 0.00 0.64 0.74 0.08
DeepHit Ensemble 0.07 0.03 0.05 0.08 0.07 0.77 1.00 0.10 0.02 0.18 0.03 0.01
XGBSEKaplanNeighbors 0.25 0.24 0.08 0.05 0.23 0.11 0.03 0.01 0.02 0.05 1.00 0.21
SurvTRACE
TabSurvLAS 0.99 1.00 0.76 0.23 0.96 0.93 0.00 0.00 0.00 0.69 0.80 0.03
0.8
DeepSurv 0.02 0.09 0.00 0.01 0.15 1.00 0.77 0.05 0.02 0.17 0.11 0.02 DeepHit 0.00 0.00 0.00 0.00 0.00 0.05 0.10 1.00 0.02 0.01 0.01 0.02
DeepSurv
TabSurvLS(MLP) 1.00 0.99 0.72 0.04 0.96 0.83 0.00 0.00 0.00 0.53 0.60 0.03
TabSurvWSA 0.65 0.69 0.81 1.00 0.35 0.01 0.08 0.00 0.00 0.30 0.05 0.68 DeepSurv Ensemble 0.43 0.37 0.16 0.36 1.00 0.15 0.07 0.00 0.00 0.31 0.23 0.43
0.0
1.0
P(rope)
TabSurvLS(MLP) 1.00 0.76 0.86 0.64 0.43 0.02 0.07 0.00 0.02 0.14 0.25 0.74 TabSurvWAS 0.86 0.87 1.00 0.80 0.16 0.00 0.05 0.00 0.00 0.18 0.08 0.92
DeepHit Ensemble
0.0
1.0
TabSurvLAS 0.76 1.00 0.87 0.69 0.37 0.09 0.03 0.00 0.00 0.17 0.23 0.91
0.2
XGBSEStackedWeibull 0.00 0.00 0.10 0.20 0.00 0.00 0.92 0.93 0.98 0.00 0.00 0.00 TabSurvLS(MLP)
XGBSEStackedWeibull
RSF
XGBSEKaplanNeighbors
DeepHit
SurvTRACE
DeepSurv
DeepHit Ensemble
TabSurvWSA
DeepSurv Ensemble
TabSurvLAS
TabSurvWAS
TabSurvLS(MLP)
XGBSEStackedWeibull 0.00 0.03 0.04 0.25 0.44 0.94 0.92 0.98 0.99 0.59 0.23 0.00
P(rope)
0.4
SurvTRACE 0.00 0.00 0.01 0.03 0.00 0.00 0.48 0.53 0.00 0.00 0.00 0.02
DeepHit Ensemble
RSF 0.03 0.11 0.09 0.18 0.35 0.68 0.65 0.95 0.97 0.00 0.28 0.25 XGBSEKaplanNeighbors 0.18 0.28 0.23 0.29 0.42 0.80 0.72 0.98 0.97 0.66 0.00 0.56
0.6
DeepSurv 0.04 0.00 0.07 0.44 0.00 0.00 1.00 1.00 0.99 0.04 0.11 0.92
DeepHit Ensemble 0.00 0.03 0.09 0.12 0.02 0.17 0.00 0.90 0.98 0.17 0.25 0.07 DeepHit 0.00 0.00 0.00 0.00 0.00 0.03 0.00 0.00 0.43 0.05 0.01 0.00
0.8
TabSurvWSA 0.01 0.00 0.00 0.00 0.00 0.00 0.94 0.96 0.94 0.09 0.00 0.75 DeepSurv Ensemble 0.04 0.03 0.04 0.58 0.00 0.00 1.00 1.00 1.00 0.09 0.33 0.96
0.6
TabSurvWSA
DeepSurv Ensemble 0.00 0.12 0.08 0.09 0.00 0.85 0.91 1.00 1.00 0.34 0.35 0.13
DeepSurv Ensemble
TabSurvWAS 0.03 0.05 0.00 0.20 0.75 0.97 0.86 1.00 1.00 0.73 0.69 0.04
TabSurvLS(MLP) 0.00 0.00 0.18 0.95 0.00 0.13 1.00 1.00 1.00 0.47 0.29 0.97
TabSurvLAS
TabSurvLAS 0.01 0.00 0.08 0.26 0.50 0.91 0.94 1.00 1.00 0.72 0.48 0.06
1.0
P(A < B)
TabSurvWAS
P(A > B)
0.0
(b) Bayesian pairwise comparison of methods by IBS.
Figure 6: Bayesian pairwise comparisons of methods: (a) by C-index; (b) by IBS.
18
1.0
P(A > B) Emb+SurvHL+Ens+Weib 0.00 0.05 0.86 0.74 0.90 0.51 0.73 0.95 0.91 Emb+SurvHL+Ens 0.08 0.00 0.75 0.26 0.88 0.75 0.71 0.97 0.96
1.0
P(A < B) Emb+SurvHL+Ens+Weib 0.00 0.02 0.07 0.07 0.16 0.70 0.45 0.78 0.37
0.8
Emb+SurvHL+Ens 0.21 0.00 0.05 0.00 0.10 0.81 0.25 0.83 0.42
Emb+SurvHL 0.07 0.01 0.00 0.03 0.07 0.26 0.18 0.89 0.62
Emb+SurvHL 0.08 0.00 0.00 0.00 0.00 0.54 0.05 0.86 0.08 0.6
SurvHL+Ens 0.24 0.00 0.04 0.00 0.04 0.69 0.18 0.72 0.26
0.4
Emb+Ens 0.09 0.00 0.00 0.00 0.00 0.00 0.00 0.37 0.00
Ens 0.11 0.01 0.18 0.02 0.09 0.16 0.00 0.89 0.29
Ens 0.21 0.00 0.00 0.00 0.00 0.17 0.00 0.59 0.04
Emb 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.2
Emb 0.08 0.00 0.00 0.00 0.00 0.05 0.00 0.00 0.00
SurvHL 0.06 0.07 0.18 0.12 0.00 0.28 0.14 0.89 0.25
Baseline MLP
Ens
Emb
Emb+Ens
Emb+SurvHL+Ens 0.76 1.00 0.95 1.00 0.90 0.19 0.75 0.17 0.58
0.6
SurvHL+Ens 0.69 1.00 0.95 1.00 0.96 0.31 0.82 0.28 0.74
Emb+Ens 0.21 0.18 0.46 0.31 0.42 1.00 0.83 0.59 0.53
Ens 0.16 0.27 0.64 0.38 0.77 0.78 1.00 0.11 0.71
Ens 0.33 0.75 0.95 0.81 0.96 0.83 1.00 0.41 0.94
Emb 0.01 0.03 0.11 0.00 0.11 0.26 0.11 1.00 0.08
0.2
Emb 0.14 0.17 0.14 0.28 0.09 0.59 0.41 1.00 0.37
0.4
0.2
(a) Full Bayesian ablation comparison by C-index.
Baseline MLP
Ens
Emb
Emb+Ens
SurvHL
SurvHL+Ens
Emb+SurvHL
0.0
Emb+SurvHL+Ens
Baseline MLP 0.52 0.57 0.92 0.73 0.97 0.53 0.95 0.37 1.00 Emb+SurvHL+Ens+Weib
Baseline MLP
Ens
Emb
SurvHL
Emb+Ens
SurvHL+Ens
Baseline MLP 0.02 0.02 0.31 0.21 0.76 0.50 0.72 0.08 1.00 Emb+SurvHL
0.6
SurvHL 0.74 0.91 1.00 0.96 1.00 0.42 0.96 0.09 0.97 0.4
Emb+SurvHL+Ens
0.8
Emb+SurvHL 0.85 0.95 1.00 0.95 1.00 0.46 0.95 0.14 0.92
SurvHL 0.03 0.05 0.75 0.49 1.00 0.56 0.77 0.11 0.75
Emb+SurvHL+Ens+Weib
0.0
Emb+SurvHL+Ens+Weib 1.00 0.76 0.85 0.70 0.74 0.21 0.33 0.14 0.52 0.8
Emb+SurvHL 0.06 0.24 1.00 0.32 0.75 0.65 0.64 0.11 0.31
Emb+Ens 0.43 0.25 0.65 0.23 0.57 1.00 0.78 0.27 0.50
0.2
1.0
P(rope)
Emb+SurvHL+Ens+Weib 1.00 0.87 0.06 0.13 0.03 0.42 0.16 0.01 0.02
SurvHL+Ens 0.13 0.64 0.32 1.00 0.49 0.24 0.39 0.00 0.21
SurvHL
SurvHL+Ens
0.0
1.0
Emb+SurvHL+Ens 0.87 1.00 0.24 0.63 0.05 0.25 0.28 0.03 0.02
0.4
Baseline MLP 0.11 0.00 0.00 0.00 0.00 0.47 0.02 0.63 0.00 Emb+SurvHL+Ens+Weib
Baseline MLP
Ens
Emb
SurvHL
Emb+Ens
SurvHL+Ens
Emb+SurvHL
Emb+SurvHL+Ens
Emb+SurvHL+Ens+Weib
Baseline MLP 0.06 0.02 0.07 0.00 0.00 0.11 0.00 0.91 0.00
P(rope)
0.6
SurvHL 0.10 0.00 0.00 0.00 0.00 0.58 0.04 0.91 0.03
Emb+SurvHL
Emb+Ens 0.07 0.00 0.09 0.02 0.16 0.00 0.06 0.73 0.39
Emb+SurvHL+Ens
SurvHL+Ens 0.13 0.10 0.65 0.00 0.39 0.74 0.60 0.99 0.78
0.8
(b) Full Bayesian ablation comparison by IBS.
Figure 7: Full Bayesian ablation comparisons: (a) by C-index; (b) by IBS.
19
0.0
Model
Baseline MLP Emb Ens Emb+Ens SurvHL SurvHL+Ens Emb+SurvHL Emb+SurvHL+Ens Emb+SurvHL+Ens+Weib
2
0 2 Relative C-index, %
4
Model
Figure 8: Relative component ablation results by C-index.
Baseline MLP Emb Ens Emb+Ens SurvHL SurvHL+Ens Emb+SurvHL Emb+SurvHL+Ens Emb+SurvHL+Ens+Weib
20
10
0 10 Relative IBS, %
20
Figure 9: Relative component ablation results by IBS.
Grid
r=1
r=5
r=7
r = 10
r = 15
r = 20
25% 50% 75% 100%
0.81006 0.80938 0.80911 0.80976
0.81313 0.81105 0.81159 0.81368
0.81218 0.81156 0.81288 0.81503
0.81160 0.81287 0.81369 0.81485
0.80970 0.81211 0.81175 0.81447
0.80491 0.80671 0.80889 0.81193
Table 4: C-index on METABRIC for different SurvHL smoothing parameters and time-grid sizes.
20
Figure 10: Sensitivity of the METABRIC C-index to the SurvHL smoothing parameter and discretization grid.
C.3
Comparison with external ensembles
The LAS variant uses parameter-efficient ensembling with a shared embedding module. To verify that its gains are not simply due to averaging many independently trained models, we compare it with external ensembles of DeepSurv, DeepHit, and LS(MLP), using 32 independently trained members where applicable. Table 5 reports posterior probabilities of superiority for the C-index, and Table 6 reports the corresponding ROPE probabilities. LS(MLP) already dominates DeepSurv with high probability and remains competitive with DeepSurv Ensemble. The independent LS(MLP) Ensemble is practically equivalent to LAS with high ROPE probability, while LAS is trained as a single shared model and therefore requires fewer computational resources.
P (A > B)
LS
LAS
LS Ens
DeepSurv
DS Ens
DH Ens
LS LAS LS Ens DeepSurv DS Ens DH Ens
0.000 0.011 0.161 0.000 0.002 0.001
0.225 0.000 0.112 0.000 0.124 0.026
0.162 0.049 0.000 0.000 0.008 0.001
0.978 0.910 0.967 0.000 0.851 0.168
0.564 0.502 0.835 0.000 0.000 0.024
0.926 0.941 0.972 0.059 0.907 0.000
Table 5: Bayesian C-index comparison between TabSurv variants and external ensembles. LS denotes TabSurvLS(MLP) ; DS and DH denote DeepSurv and DeepHit. 21
P (ROPE)
LS
LAS
LS Ens
DeepSurv
DS Ens
DH Ens
LS LAS LS Ens DeepSurv DS Ens DH Ens
1.000 0.760 0.678 0.022 0.433 0.073
0.764 1.000 0.838 0.090 0.373 0.031
0.674 0.840 1.000 0.034 0.157 0.027
0.022 0.090 0.033 1.000 0.149 0.775
0.434 0.370 0.157 0.147 1.000 0.067
0.073 0.033 0.026 0.774 0.068 1.000
Table 6: ROPE probabilities for the ensemble comparison in Table 5.
D
Results of experiments on real data
TabSurvLS(MLP) TabSurvLS(RealMLP) TabSurvLAS TabSurvWAS TabSurvWSA DeepSurv DeepHit SurvTRACE RSF
C-index Mean (↑) Rank (↓) 0.815 ± 0.003 1.4 ± 0.5 0.713 ± 0.004 7.4 ± 1.1 0.812 ± 0.003 1.9 ± 0.7 0.805 ± 0.004 3.3 ± 0.5 0.788 ± 0.006 4.8 ± 0.4 0.709 ± 0.017 7.5 ± 1.1 0.710 ± 0.005 7.8 ± 0.7 0.695 ± 0.072 7.3 ± 1.3 0.800 ± 0.010 3.6 ± 1.1
IBS Mean (↓) 0.093 ± 0.002 0.111 ± 0.002 0.090 ± 0.001 0.104 ± 0.011 0.104 ± 0.004 0.112 ± 0.004 0.171 ± 0.001 0.127 ± 0.015 0.106 ± 0.002
Rank (↓) 1.9 ± 0.3 6.3 ± 0.9 1.1 ± 0.3 4.1 ± 1.8 4.0 ± 0.9 6.2 ± 1.0 8.9 ± 0.2 7.8 ± 0.6 4.7 ± 0.8
AUC Mean (↑) 0.836 ± 0.007 0.755 ± 0.005 0.832 ± 0.003 0.793 ± 0.008 0.779 ± 0.009 0.733 ± 0.016 0.744 ± 0.013 0.722 ± 0.076 0.806 ± 0.023
Rank (↓) 1.4 ± 0.6 6.8 ± 0.8 1.8 ± 0.6 4.0 ± 0.7 5.0 ± 0.9 8.2 ± 0.8 7.5 ± 0.9 6.8 ± 2.1 3.3 ± 1.1
Table 7: Comparison of model performance on the METABRIC dataset across C-index, Integrated Brier Score (IBS), and time-dependent AUC
TabSurvLS(MLP) TabSurvLS(RealMLP) TabSurvLAS TabSurvWAS TabSurvWSA DeepSurv DeepHit SurvTRACE RSF
C-index Mean (↑) Rank (↓) 0.724 ± 0.007 6.7 ± 2.0 0.734 ± 0.002 2.2 ± 1.2 0.733 ± 0.004 3.4 ± 2.2 0.729 ± 0.003 5.3 ± 1.7 0.731 ± 0.003 4.2 ± 2.4 0.729 ± 0.005 4.9 ± 2.1 0.723 ± 0.005 8.0 ± 0.9 0.725 ± 0.008 5.8 ± 2.9 0.731 ± 0.003 4.5 ± 2.0
IBS Mean (↓) 0.083 ± 0.001 0.081 ± 0.001 0.081 ± 0.001 0.082 ± 0.001 0.084 ± 0.004 0.081 ± 0.001 0.209 ± 0.010 0.082 ± 0.002 0.082 ± 0.000
Rank (↓) 6.8 ± 1.5 3.1 ± 1.9 4.0 ± 1.7 4.1 ± 2.1 5.3 ± 2.8 2.4 ± 1.4 9.0 ± 0.0 5.0 ± 2.1 5.3 ± 1.2
AUC Mean (↑) 0.738 ± 0.012 0.752 ± 0.003 0.751 ± 0.004 0.745 ± 0.005 0.748 ± 0.003 0.748 ± 0.005 0.730 ± 0.006 0.739 ± 0.013 0.738 ± 0.004
Rank (↓) 6.1 ± 2.8 2.4 ± 1.2 3.0 ± 1.8 5.0 ± 1.8 4.1 ± 1.6 3.8 ± 1.8 8.1 ± 0.9 5.5 ± 2.9 7.0 ± 0.9
Table 8: Comparison of model performance on the SEER dataset across C-index, Integrated Brier Score (IBS), and time-dependent AUC 22
TabSurvLS(MLP) TabSurvLS(RealMLP) TabSurvLAS TabSurvWAS TabSurvWSA DeepSurv DeepHit SurvTRACE RSF
C-index Mean (↑) Rank (↓) 0.897 ± 0.000 3.9 ± 0.5 0.895 ± 0.001 5.0 ± 0.4 0.898 ± 0.001 2.4 ± 1.0 0.898 ± 0.001 1.6 ± 0.9 0.898 ± 0.000 2.1 ± 0.8 0.886 ± 0.001 7.0 ± 0.0 0.879 ± 0.002 8.0 ± 0.0 0.893 ± 0.001 6.0 ± 0.2 0.858 ± 0.001 9.0 ± 0.0
IBS Mean (↓) 0.096 ± 0.002 0.099 ± 0.002 0.100 ± 0.005 0.097 ± 0.005 0.096 ± 0.002 0.103 ± 0.001 0.107 ± 0.003 0.100 ± 0.004 0.105 ± 0.000
Rank (↓) 2.7 ± 1.2 4.5 ± 1.4 4.6 ± 2.4 3.0 ± 2.5 2.6 ± 1.6 6.5 ± 0.5 8.4 ± 1.0 4.8 ± 2.0 7.8 ± 0.6
AUC Mean (↑) 0.952 ± 0.000 0.951 ± 0.000 0.953 ± 0.001 0.955 ± 0.001 0.954 ± 0.001 0.945 ± 0.001 0.945 ± 0.001 0.951 ± 0.001 0.945 ± 0.001
Rank (↓) 4.0 ± 0.7 5.5 ± 0.6 3.0 ± 0.9 1.4 ± 0.7 1.9 ± 0.6 7.8 ± 0.9 8.3 ± 0.7 5.2 ± 0.9 7.8 ± 0.7
Table 9: Comparison of model performance on the SUPPORT dataset across C-index, Integrated Brier Score (IBS), and time-dependent AUC
TabSurvLS(MLP) TabSurvLS(RealMLP) TabSurvLAS TabSurvWAS TabSurvWSA DeepSurv DeepHit SurvTRACE RSF
C-index Mean (↑) Rank (↓) 0.863 ± 0.003 3.1 ± 1.2 0.853 ± 0.003 6.0 ± 1.0 0.861 ± 0.007 3.8 ± 1.9 0.861 ± 0.003 3.7 ± 1.1 0.862 ± 0.003 3.5 ± 1.0 0.841 ± 0.008 8.0 ± 1.0 0.868 ± 0.003 1.3 ± 0.6 0.837 ± 0.011 8.3 ± 0.9 0.847 ± 0.004 7.2 ± 0.8
IBS Mean (↓) 0.062 ± 0.001 0.063 ± 0.001 0.066 ± 0.006 0.065 ± 0.005 0.074 ± 0.009 0.072 ± 0.002 0.084 ± 0.001 0.077 ± 0.010 0.073 ± 0.001
Rank (↓) 1.9 ± 0.7 2.2 ± 1.0 3.5 ± 2.2 3.2 ± 1.4 6.0 ± 1.8 5.9 ± 1.0 8.7 ± 0.7 7.0 ± 1.3 6.5 ± 1.0
AUC Mean (↑) 0.918 ± 0.003 0.913 ± 0.003 0.915 ± 0.008 0.917 ± 0.003 0.918 ± 0.005 0.892 ± 0.009 0.923 ± 0.003 0.910 ± 0.009 0.924 ± 0.002
Rank (↓) 4.4 ± 1.5 6.5 ± 1.3 5.7 ± 2.4 4.7 ± 1.5 4.5 ± 1.6 8.8 ± 0.5 2.0 ± 1.2 6.7 ± 2.2 1.9 ± 0.8
Table 10: Comparison of model performance on the TCGA-GBM dataset across C-index, Integrated Brier Score (IBS), and time-dependent AUC
TabSurvLS(MLP) TabSurvLS(RealMLP) TabSurvLAS TabSurvWAS TabSurvWSA DeepSurv DeepHit SurvTRACE RSF
C-index Mean (↑) Rank (↓) 0.723 ± 0.002 2.7 ± 1.5 0.721 ± 0.002 3.9 ± 1.4 0.722 ± 0.002 3.4 ± 1.3 0.723 ± 0.002 2.5 ± 1.3 0.723 ± 0.002 2.7 ± 1.4 0.708 ± 0.003 7.3 ± 0.5 0.706 ± 0.003 7.7 ± 0.6 0.675 ± 0.015 9.0 ± 0.0 0.717 ± 0.002 5.8 ± 0.7
IBS Mean (↓) 0.145 ± 0.001 0.148 ± 0.002 0.142 ± 0.000 0.151 ± 0.008 0.155 ± 0.012 0.151 ± 0.001 0.209 ± 0.007 0.191 ± 0.104 0.148 ± 0.001
Rank (↓) 2.5 ± 0.7 4.5 ± 1.6 1.1 ± 0.2 4.8 ± 2.0 6.0 ± 2.2 6.7 ± 0.9 8.8 ± 0.4 5.5 ± 2.4 5.1 ± 1.0
AUC Mean (↑) 0.775 ± 0.003 0.777 ± 0.003 0.780 ± 0.003 0.771 ± 0.005 0.770 ± 0.004 0.754 ± 0.003 0.754 ± 0.004 0.750 ± 0.014 0.773 ± 0.003
Rank (↓) 3.5 ± 1.3 2.6 ± 1.2 1.4 ± 0.9 4.7 ± 1.6 5.3 ± 1.1 7.8 ± 0.8 8.0 ± 0.7 7.8 ± 1.8 4.0 ± 1.0
Table 11: Comparison of model performance on the Rotterdam dataset across C-index, Integrated Brier Score (IBS), and time-dependent AUC 23
TabSurvLS(MLP) TabSurvLS(RealMLP) TabSurvLAS TabSurvWAS TabSurvWSA DeepSurv DeepHit SurvTRACE RSF
C-index Mean (↑) Rank (↓) 0.936 ± 0.001 5.0 ± 1.2 0.935 ± 0.001 6.4 ± 1.0 0.937 ± 0.001 4.3 ± 1.1 0.938 ± 0.000 1.6 ± 0.7 0.938 ± 0.000 2.1 ± 0.7 0.938 ± 0.000 2.5 ± 1.0 0.933 ± 0.001 8.4 ± 0.7 0.930 ± 0.014 7.5 ± 1.5 0.934 ± 0.001 7.0 ± 1.2
IBS Mean (↓) 0.047 ± 0.005 0.047 ± 0.001 0.046 ± 0.000 0.046 ± 0.002 0.047 ± 0.002 0.044 ± 0.000 0.055 ± 0.001 0.055 ± 0.021 0.048 ± 0.000
Rank (↓) 3.5 ± 1.4 5.5 ± 0.9 3.6 ± 1.5 3.8 ± 1.9 4.7 ± 2.1 1.1 ± 0.4 8.8 ± 0.7 6.8 ± 1.6 7.1 ± 0.8
AUC Mean (↑) 0.955 ± 0.001 0.951 ± 0.002 0.955 ± 0.001 0.957 ± 0.000 0.950 ± 0.013 0.955 ± 0.001 0.952 ± 0.001 0.934 ± 0.081 0.953 ± 0.001
Rank (↓) 4.2 ± 1.6 7.8 ± 1.2 4.5 ± 1.4 1.6 ± 0.5 3.0 ± 3.1 3.6 ± 0.9 7.4 ± 1.5 7.1 ± 1.2 5.8 ± 1.5
Table 12: Comparison of model performance on the FLC dataset across C-index, Integrated Brier Score (IBS), and time-dependent AUC
TabSurvLS(MLP) TabSurvLS(RealMLP) TabSurvLAS TabSurvWAS TabSurvWSA DeepSurv DeepHit SurvTRACE RSF
C-index Mean (↑) Rank (↓) 0.719 ± 0.007 2.2 ± 1.4 0.702 ± 0.005 6.6 ± 1.2 0.714 ± 0.007 3.9 ± 1.8 0.716 ± 0.004 3.2 ± 1.4 0.714 ± 0.006 3.8 ± 1.6 0.700 ± 0.014 6.3 ± 2.1 0.669 ± 0.026 8.7 ± 0.6 0.685 ± 0.037 6.9 ± 2.0 0.715 ± 0.005 3.3 ± 1.6
IBS Mean (↓) 0.152 ± 0.010 0.170 ± 0.009 0.159 ± 0.007 0.156 ± 0.005 0.169 ± 0.043 0.152 ± 0.003 0.202 ± 0.002 0.607 ± 0.000 0.152 ± 0.001
Rank (↓) 2.2 ± 1.5 6.5 ± 1.0 4.5 ± 2.0 4.2 ± 1.1 5.4 ± 1.1 2.6 ± 1.5 8.0 ± 0.2 9.0 ± 0.0 2.6 ± 0.9
AUC Mean (↑) 0.756 ± 0.027 0.762 ± 0.015 0.765 ± 0.007 0.788 ± 0.004 0.784 ± 0.006 0.755 ± 0.019 0.689 ± 0.033 0.548 ± 0.007 0.788 ± 0.009
Rank (↓) 5.6 ± 1.1 5.3 ± 1.1 5.0 ± 1.2 1.9 ± 0.7 2.4 ± 0.7 6.0 ± 1.6 7.9 ± 0.3 9.0 ± 0.0 1.9 ± 1.0
Table 13: Comparison of model performance on the GBSG2 dataset across C-index, Integrated Brier Score (IBS), and time-dependent AUC
TabSurvLS(MLP) TabSurvLS(RealMLP) TabSurvLAS TabSurvWAS TabSurvWSA DeepSurv DeepHit SurvTRACE RSF
C-index Mean (↑) Rank (↓) 0.801 ± 0.026 2.0 ± 1.8 0.763 ± 0.017 8.0 ± 1.1 0.782 ± 0.006 6.0 ± 1.1 0.801 ± 0.009 2.7 ± 1.7 0.795 ± 0.013 4.1 ± 2.5 0.783 ± 0.018 5.8 ± 2.1 0.780 ± 0.013 6.2 ± 1.9 0.777 ± 0.024 6.2 ± 2.6 0.793 ± 0.006 4.1 ± 1.2
IBS Mean (↓) 0.163 ± 0.018 0.200 ± 0.016 0.163 ± 0.008 0.153 ± 0.011 0.198 ± 0.074 0.160 ± 0.010 0.194 ± 0.025 0.350 ± 0.153 0.162 ± 0.003
Rank (↓) 3.8 ± 2.0 7.4 ± 0.8 4.5 ± 1.7 2.0 ± 2.1 4.3 ± 2.6 3.5 ± 1.6 6.5 ± 1.7 8.2 ± 1.6 4.8 ± 1.0
AUC Mean (↑) 0.826 ± 0.044 0.803 ± 0.020 0.828 ± 0.008 0.826 ± 0.029 0.809 ± 0.035 0.811 ± 0.020 0.820 ± 0.018 0.801 ± 0.063 0.839 ± 0.008
Rank (↓) 3.4 ± 2.4 7.0 ± 1.8 4.3 ± 1.5 3.8 ± 2.2 5.9 ± 2.6 6.2 ± 2.3 5.7 ± 2.2 5.8 ± 2.7 2.8 ± 1.7
Table 14: Comparison of model performance on the WHAS500 dataset across C-index, Integrated Brier Score (IBS), and time-dependent AUC 24
TabSurvLS(MLP) TabSurvLS(RealMLP) TabSurvLAS TabSurvWAS TabSurvWSA DeepSurv DeepHit SurvTRACE RSF
C-index Mean (↑) Rank (↓) 0.649 ± 0.012 2.3 ± 0.8 0.588 ± 0.045 5.4 ± 1.6 0.621 ± 0.037 3.9 ± 1.8 0.571 ± 0.032 6.2 ± 1.3 0.517 ± 0.042 7.8 ± 1.5 0.624 ± 0.024 4.0 ± 1.4 0.517 ± 0.061 7.5 ± 1.7 0.549 ± 0.082 6.5 ± 2.2 0.668 ± 0.018 1.5 ± 1.2
IBS Mean (↓) 0.166 ± 0.003 0.167 ± 0.002 0.169 ± 0.006 0.295 ± 0.077 0.392 ± 0.081 0.170 ± 0.004 0.165 ± 0.001 0.210 ± 0.082 0.166 ± 0.004
Rank (↓) 2.9 ± 1.5 3.9 ± 1.2 4.3 ± 1.3 7.8 ± 0.7 8.8 ± 0.4 5.0 ± 1.7 2.4 ± 1.1 6.2 ± 2.3 3.7 ± 2.3
AUC Mean (↑) 0.646 ± 0.014 0.604 ± 0.034 0.630 ± 0.022 0.528 ± 0.039 0.486 ± 0.029 0.648 ± 0.032 0.519 ± 0.045 0.556 ± 0.081 0.670 ± 0.023
Rank (↓) 3.0 ± 1.0 4.8 ± 1.2 3.8 ± 1.3 7.1 ± 1.2 8.2 ± 0.9 2.8 ± 1.5 7.4 ± 1.2 6.2 ± 2.6 1.8 ± 1.0
Table 15: Comparison of model performance on the Lung dataset across C-index, Integrated Brier Score (IBS), and time-dependent AUC
TabSurvLS(MLP) TabSurvLS(RealMLP) TabSurvLAS TabSurvWAS TabSurvWSA DeepSurv DeepHit SurvTRACE RSF
C-index Mean (↑) Rank (↓) 0.981 ± 0.013 5.4 ± 1.0 0.969 ± 0.017 7.0 ± 0.5 0.993 ± 0.002 3.4 ± 0.5 0.997 ± 0.001 1.6 ± 0.6 0.997 ± 0.001 1.5 ± 0.5 0.983 ± 0.002 5.3 ± 0.6 0.949 ± 0.029 8.5 ± 0.5 0.977 ± 0.054 4.2 ± 1.7 0.964 ± 0.004 8.1 ± 0.8
IBS Mean (↓) 0.046 ± 0.050 0.057 ± 0.046 0.033 ± 0.004 0.023 ± 0.006 0.029 ± 0.009 0.028 ± 0.002 0.232 ± 0.011 0.219 ± 0.229 0.069 ± 0.003
Rank (↓) 4.5 ± 1.2 6.0 ± 0.7 4.0 ± 0.9 1.7 ± 1.1 2.7 ± 1.5 2.4 ± 0.8 8.6 ± 0.5 8.0 ± 0.8 7.1 ± 0.4
AUC Mean (↑) 0.985 ± 0.013 0.955 ± 0.011 0.992 ± 0.007 0.999 ± 0.001 0.999 ± 0.001 0.991 ± 0.002 0.952 ± 0.042 0.993 ± 0.010 0.967 ± 0.007
Rank (↓) 5.5 ± 1.1 8.6 ± 0.5 3.9 ± 1.1 1.8 ± 0.8 1.6 ± 0.9 4.9 ± 0.9 7.7 ± 0.9 3.5 ± 1.5 7.5 ± 0.7
Table 16: Comparison of model performance on the PBC dataset across C-index, Integrated Brier Score (IBS), and time-dependent AUC
E
Simulation Experiments
As demonstrated in the experiments on real-world datasets, TabSurvWAS achieves the best performance in terms of C-index and time-dependent AUC, while remaining competitive with respect to IBS. However, this behavior may be influenced by the specific characteristics of the considered real datasets. To further investigate the limitations of the proposed methods, we conduct an additional simulation study designed to illustrate scenarios in which TabSurvWAS is clearly less effective than TabSurvLAS with respect to the remaining metrics. Synthetic event times were generated using the covariates of the Rotterdam dataset. To induce a bimodal time-to-event distribution, each feature vector was randomly assigned to one of two latent clusters using a Bernoulli distribution, with approximately equal cluster sizes. Conditioned on the cluster assignment, event times were sampled from a covariate-dependent Weibull distribution following the procedure described in [5], with parameters defined as λ0 = 10−5 , k0 = 4, β0 ∼ U[0, 1],
λ1 = 10−10 , k1 = 6,
(9)
β1 ∼ U[−1, 1].
The random cluster assignment induces a bimodal event-time distribution for each covariate vector x. 25
Figure 11: Results of the simulation study evaluated using the Kolmogorov–Smirnov statistic. Censoring indicators were generated independently using a Bernoulli distribution with a predefined censoring rate. The data were split into training, validation, and test sets using the same protocol as in the original Rotterdam experiments. The goal of this experiment is to assess how well different models approximate the underlying bimodal event-time distribution. To this end, we employ the Kolmogorov–Smirnov (KS) test to statistically quantify the similarity between the empirical distribution of test event times and the distribution obtained by averaging the predicted discrete probability densities across the test set. The KS statistic is computed exclusively on the test subset. Specifically, each model predicts a discrete event-time distribution for each test instance on a time grid defined by the event times observed in the training set. These predicted distributions are then averaged across all test instances to obtain a population-level prior, which is subsequently compared to the empirical distribution of observed test event times. The censoring proportion is varied, and each point in Figure 11 corresponds to the average over 100 independent experimental runs. Standard deviations are also reported to illustrate the variability of the results. Figure 12 visualizes the predicted event-time densities for a single experimental run with 20% censoring, using the SurvHL-TabM, RSF, and SurvHL-WPRM models. Kernel density estimation is used for visualization, as the original discrete distributions consist of approximately 1,000 time intervals, making histogram-based representations difficult to interpret. This figure provides qualitative insight into how different models capture the shape of the simulated event-time distribution. For clarity of presentation, only a subset of models is included in this experiment, namely TabSurvLS(MLP) , TabSurvLAS , RSF, TabSurvWSA , TabSurvWAS , and DeepSurv. These models were selected based on their performance in the real-data experiments. Figure 11 shows that the TabSurvLS and TabSurvLAS models, which are not constrained by a Weibull distributional assumption, consistently achieve superior performance across all censoring proportions. The event-time distributions predicted by these models are the closest to the empirical distribution observed in the test data. Figure 12 highlights substantial differences in the shapes of the distributions predicted by the considered models. Among them, TabSurvLAS most closely matches the ground-truth bimodal distribution, providing the most accurate approximation in this setting.
26
Figure 12: Predicted event-time densities on the simulated dataset for TabSurvLAS , RSF, and TabSurvWAS .
27