Survival analysis often relies on Cox models, assuming both linearity and proportional hazards (PH). This study evaluates machine and deep learning methods that relax these constraints, comparing their performance with penalized Cox models on a benchmark of three synthetic and three real datasets. In total, eight different models were tested, including six non-linear models of which four were also non-PH. Although Cox regression often yielded satisfactory performance, we showed the conditions under which machine and deep learning models can perform better. Indeed, the performance of these methods has often been underestimated due to the improper use of Harrell's concordance index (C-index) instead of more appropriate scores such as Antolini's concordance index, which generalizes C-index in cases where the PH assumption does not hold. In addition, since occasionally high C-index models happen to be badly calibrated, combining Antolini's C-index with Brier's score is useful to assess the overall performance of a survival method. Results on our benchmark data showed that survival prediction should be approached by testing different methods to select the most appropriate one according to sample size, non-linearity and non-PH conditions. To allow an easy reproducibility of these tests on our benchmark data, code and documentation are freely available at https://github.com/compbiomed-unito/survhive.

Probabilistic survival analysis models seek to estimate the distribution of the future occurrence (time) of an event given a set of covariates. In recent years, these models have preferred nonparametric specifications that avoid directly estimating survival distributions via discretization. Specifically, they estimate the probability of an individual event at fixed times or the time of an event at fixed probabilities (quantiles), using supervised learning. Borrowing ideas from the quantile regression literature, we propose a parametric survival analysis method based on the Asymmetric Laplace Distribution (ALD). This distribution allows for closed-form calculation of popular event summaries such as mean, median, mode, variation, and quantiles. The model is optimized by maximum likelihood to learn, at the individual level, the parameters (location, scale, and asymmetry) of the ALD distribution. Extensive results on synthetic and real-world data demonstrate that the proposed method outperforms parametric and nonparametric approaches in terms of accuracy, discrimination and calibration.

When dealing with right-censored data, where some outcomes are missing due to a limited observation period, survival analysis -- known as time-to-event analysis -- focuses on predicting the time until an event of interest occurs. Multiple classes of outcomes lead to a classification variant: predicting the most likely event, a less explored area known as competing risks. Classic competing risks models couple architecture and loss, limiting scalability.To address these issues, we design a strictly proper censoring-adjusted separable scoring rule, allowing optimization on a subset of the data as each observation is evaluated independently. The loss estimates outcome probabilities and enables stochastic optimization for competing risks, which we use for efficient gradient boosting trees. SurvivalBoost not only outperforms 12 state-of-the-art models across several metrics on 4 real-life datasets, both in competing risks and survival settings, but also provides great calibration, the ability to predict across any time horizon, and computation times faster than existing methods.

The paper tackles a timely area of research, namely new approaches for time-to-event data modeling which is common in health data analysis. There is a large body of statistics literature on this topic and some recent ML approaches, such as DeepSurvival (Blei lab) and DeepHit (van der Shaar lab). The work proposes a multi-task model to consider competing risks, similar to DeepHit. The authors do compare with other methods, but I think these comparisons fail in two aspects: a) no comparison with a version of their method that considers each risk independently -- this comparison would be important to include in order to understand whether it is the flexibility from boosting trees or from the simultaneous modeling of multiple risks that leads to the improved performance. Apart from the limitations in the results part, I thought that the presentation of the methodology as a boosting method is not very clear.

Esophageal cancer is a major cause of cancer-related mortality internationally, with high recurrence rates and poor survival even among patients treated with curative-intent surgery. Investigating relevant prognostic factors and predicting prognosis can enhance post-operative clinical decision-making and potentially improve patients' outcomes. In this work, we assessed prognostic factor identification and discriminative performances of three models for Disease-Free Survival (DFS) and Overall Survival (OS) using a large multicenter international dataset from ENSURE study. We first employed Cox Proportional Hazards (CoxPH) model to assess the impact of each feature on outcomes. Subsequently, we utilised CoxPH and two deep neural network (DNN)-based models, DeepSurv and DeepHit, to predict DFS and OS. The significant prognostic factors identified by our models were consistent with clinical literature, with post-operative pathologic features showing higher significance than clinical stage features. DeepSurv and DeepHit demonstrated comparable discriminative accuracy to CoxPH, with DeepSurv slightly outperforming in both DFS and OS prediction tasks, achieving C-index of 0.735 and 0.74, respectively. While these results suggested the potential of DNNs as prognostic tools for improving predictive accuracy and providing personalised guidance with respect to risk stratification, CoxPH still remains an adequately good prediction model, with the data used in this study.

Survival analysis is a valuable tool for estimating the time until specific events, such as death or cancer recurrence, based on baseline observations. This is particularly useful in healthcare to prognostically predict clinically important events based on patient data. However, existing approaches often have limitations; some focus only on ranking patients by survivability, neglecting to estimate the actual event time, while others treat the problem as a classification task, ignoring the inherent time-ordered structure of the events. Furthermore, the effective utilization of censored samples - training data points where the exact event time is unknown - is essential for improving the predictive accuracy of the model. In this paper, we introduce CenTime, a novel approach to survival analysis that directly estimates the time to event. Our method features an innovative event-conditional censoring mechanism that performs robustly even when uncensored data is scarce. We demonstrate that our approach forms a consistent estimator for the event model parameters, even in the absence of uncensored data. Furthermore, CenTime is easily integrated with deep learning models with no restrictions on batch size or the number of uncensored samples. We compare our approach with standard survival analysis methods, including the Cox proportional-hazard model and DeepHit. Our results indicate that CenTime offers state-of-the-art performance in predicting time-to-death while maintaining comparable ranking performance. Our implementation is publicly available at https://github.com/ahmedhshahin/CenTime.

Survival analysis is the problem of estimating probability distributions for future event times, which can be seen as a problem in uncertainty quantification. Although there are fundamental theories on strictly proper scoring rules for uncertainty quantification, little is known about those for survival analysis. In this paper, we investigate extensions of four major strictly proper scoring rules for survival analysis and we prove that these extensions are proper under certain conditions, which arise from the discretization of the estimation of probability distributions. We also compare the estimation performances of these extended scoring rules by using real datasets, and the extensions of the logarithmic score and the Brier score performed the best.

Survival analysis is an essential tool for the study of health data. An inherent component of such data is the presence of missing values. In recent years, researchers proposed new learning algorithms for survival tasks based on neural networks. Here, we studied the predictive performance of such algorithms coupled with different methods for handling missing values on simulated data that reflect a realistic situation, i.e., when individuals belong to unobserved clusters. We investigated different patterns of missing data. The results show that, without further feature engineering, no single imputation method is better than the others in all cases. The proposed methodology can be used to compare other missing data patterns and/or survival models. The Python code is accessible via the package survivalsim.

In this paper, we propose a flexible model for survival analysis using neural networks along with scalable optimization algorithms. One key technical challenge for directly applying maximum likelihood estimation (MLE) to censored data is that evaluating the objective function and its gradients with respect to model parameters requires the calculation of integrals. To address this challenge, we recognize that the MLE for censored data can be viewed as a differential-equation constrained optimization problem, a novel perspective. Following this connection, we model the distribution of event time through an ordinary differential equation and utilize efficient ODE solvers and adjoint sensitivity analysis to numerically evaluate the likelihood and the gradients. Using this approach, we are able to 1) provide a broad family of continuous-time survival distributions without strong structural assumptions, 2) obtain powerful feature representations using neural networks, and 3) allow efficient estimation of the model in large-scale applications using stochastic gradient descent. Through both simulation studies and real-world data examples, we demonstrate the effectiveness of the proposed method in comparison to existing state-of-the-art deep learning survival analysis models.

New methods for time-to-event prediction are proposed by extending the Cox proportional hazards model with neural networks. Building on methodology from nested case-control studies, we propose a loss function that scales well to large data sets, and enables fitting of both proportional and non-proportional extensions of the Cox model. Through simulation studies, the proposed loss function is verified to be a good approximation for the Cox partial log-likelihood. The proposed methodology is compared to existing methodologies on real-world data sets, and is found to be highly competitive, typically yielding the best performance in terms of Brier score and binomial log-likelihood. A python package for the proposed methods is available at https://github.com/havakv/pycox.