Source-linked AI summary
Neural Fine-Gray: Monotonic neural networks for competing risks
Vincent Jeanselme, Chang Ho Yoon, Brian Tom, Jessica Barrett
TL;DR
Competing risks can preclude the event of interest, and methods that ignore them or rely on parametric assumptions and numerical approximations leave limitations in risk estimation. Neural Fine-Gray models competing-risk cumulative incidence functions with monotonic networks to compute the likelihood exactly. The paper reports competitive performance while avoiding those assumptions or expensive approximations, and discusses competing-risk modelling in medical risk-score development.
Problem
Competing events can preclude the event of interest, yet competing risks remain less studied and applied, while estimation methods may rely on assumptions or approximations.
Source: §1.1 · pp. 1–2, §2.2 · p. 3Method
Neural Fine-Gray uses constrained monotonic neural networks to model cumulative incidence functions and compute the likelihood exactly through automatic differentiation.
Source: §1.2 · p. 2, §2.2 · p. 3Results
The approach achieves competitive performance while avoiding parametric assumptions or computationally expensive approximations used by state-of-the-art survival neural networks.
Source: §6 · p. 11Takeaways & Limitations
The Framingham analysis invites practitioners to consider competing-risk modelling when developing risk scores for medical practice.
Source: §6 · p. 11Performance is lower on the data-limited PBC dataset, where the authors recommend simpler models to avoid overfitting.
Source: §5.2 · p. 9
Abstract
from arXiv · showhide
Time-to-event modelling, known as survival analysis, differs from standard regression as it addresses censoring in patients who do not experience the event of interest. Despite competitive performances in tackling this problem, machine learning methods often ignore other competing risks that preclude the event of interest. This practice biases the survival estimation. Extensions to address this challenge often rely on parametric assumptions or numerical estimations leading to sub-optimal survival approximations. This paper leverages constrained monotonic neural networks to model each competing survival distribution. This modelling choice ensures the exact likelihood maximisation at a reduced computational cost by using automatic differentiation. The effectiveness of the solution is demonstrated on one synthetic and three medical datasets. Finally, we discuss the implications of considering competing risks when developing risk scores for medical practice.
1. Introduction
Competing events can preclude the event of interest, and treating them as censoring can misestimate its risk. The paper introduces a monotonic neural-network approach for competing risks and examines it on synthetic and medical datasets.
1.1. Motivation
- Competing events preclude observing the event of interest, and treating them as censoring can misestimate its risk.Censored patients otherwise inform survival estimation by remaining event-free during their study participation.Source: §1.1 · pp. 1–2
- Cumulative Incidence Functions represent the marginal probability of observing each risk, but their estimation may rely on parametric assumptions or numerical integration.These approaches can optimize a suboptimal target that misrepresents the underlying survival distribution.Source: §1.1 · p. 2
1.2. Contribution
- Neural Fine-Gray generalizes monotonic neural networks to competing risks by modelling cumulative incidence functions for exact likelihood computation at lower computational cost.The approach is presented as an alternative to parametric assumptions and computationally expensive approximations.Source: §1.2 · p. 2
- The study evaluates the approach on one synthetic and three real-world medical datasets and further investigates Framingham cardiovascular risk estimation.The Framingham analysis examines the importance of considering competing risks in risk estimation.Source: §1.2 · p. 2
2. Related work
Prior survival models use proportional hazards, neural networks, discretization, parametric distributions, or numerical approximations. For competing risks, the paper positions monotonic networks that directly model cumulative incidence functions as a way to compute the likelihood exactly.
2.1. Time-to-event modelling
- The Cox model remains widely used and assumes proportional hazards, while neural extensions relax aspects of its covariate-effect or hazard assumptions.DeepSurv uses a neural network for covariate effects but retains proportional hazards; DeepCox (Nagpal et al., 2021c) models subgroups with independent Cox models.Source: §2.1 · p. 2
- DeepHit treats survival prediction as classification over discretized time intervals, using a likelihood and ranking penalty; its infinite-discretization extension resembles an ODE.The ODE extension is proposed by Danks and Yau (2022).Source: §2.1 · pp. 2–3
- Rindt et al. (2022) use a constrained monotonic network to obtain survival functions and exact likelihoods, and this study generalizes that approach to competing risks by modelling CIFs directly.The cited work argues that likelihood is a proper scoring rule because its optimum corresponds to the true underlying survival distribution.Source: §2.1 · p. 3
2.2. Modelling competing risks
- Competing-risk methods include cause-specific Cox and Fine-Gray models, discretized or parametric neural models, numerical integration, pseudovalue approximations, and tree-based methods.Cause-specific modelling may misrepresent relative covariate effects; Fine-Gray relies on proportionality and can yield an ill-defined survival function. Existing approaches may approximate likelihoods or optimize a score that misrepresents the underlying distribution, whereas the proposed constrained networks obtain CIFs as derivatives for exact likelihood maximization by automatic differentiation.Source: §2.2 · p. 3
3. Proposed approach
Neural Fine-Gray models competing cumulative incidence functions (CIFs) directly, using monotonic neural networks and automatic differentiation to compute the likelihood exactly. The approach addresses competing risks while avoiding numerical integration, under a non-informative-censoring assumption conditional on identified competing risks.
3.1. Notation
- For a population of patients with covariates, follow-up times, and event causes, the model focuses on right-censoring and can be extended to left-censoring.It assumes that experiencing one event precludes observing another.Source: §3.1 · pp. 3–4
3.2. Survival quantities
- Under non-informative censoring conditional on identified competing risks, event records contribute λ_r(t_i|x_i)S(t_i|x_i), while censored records contribute S(t_i|x_i).Differentiating the modeled F_r yields λ_rS in one backward pass, enabling exact computation of the quantities of interest.Source: §3.2 · pp. 4–5
3.3. Architecture
- R positive monotonic networks model risk-specific distributions, while a SoftMax balancing network weights each risk by its probability given the covariates.The model combines the distribution and balancing network outputs to model F_r directly and trains end-to-end by maximizing the exact log-likelihood.Source: §3.3 · pp. 5–6
3.4. Computational complexity
- The approach computes the likelihood exactly, whereas methods using integral approximation or outcome discretisation converge toward F_r as estimates increase or discretisation becomes finer.The paper compares the computational complexity of NeuralFG and DeSurv, its closest method.Source: §3.4 · p. 6
DeSurv
- More than n/2 computational gain in likelihood estimation and an n gain in inferring F_r compared with DeSurv.Neural Fine-Gray has O(2N) likelihood complexity, compared with DeSurv’s O(nN).Source: DeSurv · pp. 6–7
4. Experiments
The experiments compare NeuralFG with established survival models across four competing-risk datasets, using cross-validation and discrimination and calibration metrics.
4.1. Datasets
- PBC includes 312 patients and 25 covariates, while Framingham uses first-observed covariates from 4,434 men to model CVD risk.Transplant competes with waiting-list death in PBC; death from other causes competes with CVD in Framingham.Source: §4.1 · p. 7
- Synthetic data comprise 30,000 patients with 12 covariates and nonlinear covariate-dependent exponential event times; SEER includes 658,354 breast-cancer patients and 23 covariates.In SEER, breast-cancer death is the primary outcome and CVD is a competing risk.Source: §4.1 · p. 7
4.2. Baseline models
- Baselines include linear cause-specific Cox and Fine-Gray models, plus the neural competing-risk models DSM, DeepHit, and DeSurv.Cause-specific Cox fits each cause independently; Fine-Gray assumes proportional sub-hazards.Source: §4.2 · pp. 7–8
4.3. Experimental settings
- Models were evaluated with five-fold cross-validation, using 10% of each training fold for tuning and early stopping.Random search covered learning rates, batch sizes, dropout, layer counts, and node counts; all models used Adam, and NeuralFG used Tanh activations.Source: §4.3 · p. 8
- DSM used Log-Normal and Weibull alternatives, DeSurv used 15-degree Gauss-Legendre quadrature, and DeepHit used 15 time splits.Architectures without embedding networks were allowed twice as many layers for a fair comparison.Source: §4.3 · pp. 8–9
4.4. Evaluation metrics
- The study measures discrimination with the time-dependent C-index and calibration with the time-dependent Brier score at three event-time quantiles.Reported means and standard deviations are computed over the five cross-validation folds.Source: §4.4 · p. 9
5. Results
NeuralFG generally matches DeSurv while attaining lower or equal Brier scores than other machine-learning models across most datasets and time horizons. The results also show that modeling competing risks improves CVD estimates, with risk-score reclassification potentially affecting treatment decisions.
5.1. Model’s strengths
- NeuralFG has lower or equal Brier scores than other state-of-the-art machine-learning models across most datasets and time horizons, with performance comparable to DeSurv and no costly likelihood approximation.DSM and DeepHit show trade-offs between discrimination and calibration: DSM's discrimination edge does not carry over to calibration, while DeepHit's ranking penalty improves C-index but hurts calibration.Source: §5.1 · p. 9
5.2. Model’s limitations
- NeuralFG performs worse on limited-data PBC, where simpler models may help avoid overfitting.Linear Fine-Gray and cause-specific Cox models are competitive on PBC but fare worse with complex covariate effects; DSM also underperforms with more complex survival distributions, while DeSurv performs better on PBC.Source: §5.2 · p. 9
5.3. Modelling vs ignoring competing risks
- On Framingham CVD risk, modeling competing risks significantly improves performance over the same architecture trained with cause-specific likelihoods, with no significant difference for the competing risk.The comparison uses a shared embedding and the sum of cause-specific likelihoods for the non-competing alternative.Source: §5.3 · p. 10
- Older patients benefit most from modeling death as a competing risk in Framingham.The comparison concerns calibration differences between competing-risk and non-competing-risk models.Source: §5.3 · p. 10, Table 4 · p. 10
What is the impact on medical practice?
- For patients aged 50 or older, competing-risk and non-competing-risk scores reclassify risk differently, and some patients classified as high risk by the latter are assigned lower risk by the former.The authors note that more accurate simulations of potential lives saved or harmed by reclassification are beyond the article’s scope.Source: What is the impact on medical practice? · pp. 10–11, Table 5 · p. 11
6. Conclusion
Neural Fine-Gray models cumulative incidence functions to compute the likelihood exactly at lower computational cost. The conclusion also connects competing-risk modelling to Framingham cardiovascular risk estimation and identifies future directions.
6. Conclusion
- Neural Fine-Gray models cumulative incidence functions for exact likelihood computation, with competitive performance and lower computational cost than approaches using parametric assumptions or expensive approximations.The conclusion describes the architecture as a monotonic neural network.Source: §6 · p. 11
- The Framingham analysis invites practitioners to consider competing-risk modelling when developing cardiovascular disease risk scores.The conclusion connects this analysis to risk-score development for improved care.Source: §6 · p. 11
- Future work will extend the architecture to time-series modalities and explore medically interpretable survival clusters.The conclusion cites Nagpal et al. (2021a) for time series and Jeanselme et al. (2022); Nagpal et al. (2022a) for survival clusters.Source: §6 · p. 11
A.1. Datasets characteristics
The appendix describes dataset-specific evaluation horizons and two time-dependent metrics. The C-index addresses discrimination, while the Brier score addresses calibration; both account for censoring.
A.1. Datasets characteristics
- Table 6 presents observed outcomes at different evaluation horizons, with evaluation times corresponding to quantiles of the uncensored population and results differentiated by dataset.Source: §A.1 · p. 16, Table 6 · p. 16
A.2. Evaluation metrics
- Time-dependent C-index quantifies discrimination by comparing the ordering of predicted survival probabilities with observed survival times, weighted for censoring using a Kaplan-Meier estimator.The censoring weight is the inverse probability of censoring, ω(t_i).Source: §A.2 · p. 16
- Time-dependent Brier score measures model calibration for risk r and is corrected for censoring; its expression uses an indicator and predicted survival probability.The predicted survival probability is written as Ŝ(t|x).Source: §A.2 · p. 16
A.3. Time specific results
Across the dataset-specific event-time quantiles, the appendix reports that the results echo the paper's conclusions, with competing or better-than-state-of-the-art performance.
A.3. Time specific results
- The time-specific results echo the paper's conclusions, with competing or better-than-state-of-the-art performance across evaluations at dataset-specific event-time quantiles.Tables 7, 8, and 9 report C-index, ROC-AUC, and Brier score, respectively.Source: §A.3 · p. 16
A.4. Cumulative evaluation
Cumulative metrics summarize model performance over the total distribution, but the appendix notes that this single-number view is less connected to discretized clinical risk horizons. Table 10's results echo those in Table 2.
A.4. Cumulative evaluation
- Table 10's cumulative time-dependent C-index and Brier-score results echo the findings from Table 2.Source: §A.4 · p. 16
- Cumulative metrics provide a single summary over the total distribution, but are less connected to medical applications that discretize risk by horizon to inform treatment.Source: §A.4 · p. 16
Appendix B. Using R outcomes vs. R networks
The appendix compares separate risk-specific monotonic networks with a single network having R outputs, then examines how quadrature order affects DeSurv and NeuralFG computational complexity.
Appendix B. Using R outcomes vs. R networks
- The architecture comparison reports primary- and competing-risk performance at q0.25, q0.50, and q0.75.Source: Appendix B · p. 17
- MonoFG uses one monotonic network with R outputs; performance differences from separate risk-specific networks were limited, though multiple networks are encouraged when risks have large distributional differences.The comparison is reported in Table 11.Source: Appendix B · p. 21, Table 11 · p. 21
C.1. Impact of n
- At n = 1, NeuralFG and DeSurv have the same computational complexity; DeSurv benefits from larger n, with limited gain above the recommended 15-degree quadrature.The experiment fixes a three-hidden-layer, 50-node perceptron and tests n = 1, 15, 100, and 1000 on the Synthetic dataset.Source: §C.1 · p. 21
C.2. Training speed
On Framingham, NeuralFG has faster training iterations, especially with MonoFG, while DeSurv trades slower iterations for fewer convergence iterations as quadrature order increases.
C.2. Training speed
- The Framingham comparison used a three-hidden-layer, 50-node architecture, learning rate 0.001, batch size 100, and 100 random splits.DeSurv's n forward passes were parallelised; experiments ran on an Apple M1 Pro with 32 GB of memory.Source: §C.2 · p. 21, Table 13 · p. 22
- As n increases, DeSurv requires fewer convergence iterations but trains more slowly per iteration; at n = 1, each iteration is faster but convergence takes more iterations.Source: §C.2 · p. 21, Table 12 · p. 22
- NeuralFG has faster iterations, especially with MonoFG, while its greater iteration count than DeSurv reflects the more complex convergence of constrained monotonic networks.Table 13 compares convergence iterations and total training time.Source: §C.2 · pp. 21–22, Table 13 · p. 22