This bias becomes particularly noticeable when the study's endpoint can be inferred from the

When a model's predicted number of events within any time interval is similar to the observed number, it is called well-calibrated . A survival model's calibration can be measured using, for instance, distributional calibration (

The NFM framework utilizes the classical idea of multiplicative frailty in survival analysis as a principled way of extending the proportional hazard assumption, at the same time being able to leverage the strong approximation power of neural architectures for handling nonlinear covariate dependence.

Survival analysis has become a standard approach for modelling time to default by time-varying covariates in credit risk. Unlike most existing methods that implicitly assume a stationary data-generating process, in practise, mortgage portfolios are exposed to various forms of data drift caused by changing borrower behaviour, macroeconomic conditions, policy regimes and so on. This study investigates the impact of data drift on survival-based credit risk models and proposes a dynamic joint modelling framework to improve robustness under non-stationary environments. The proposed model integrates a longitudinal behavioural marker derived from balance dynamics with a discrete-time hazard formulation, combined with landmark one-hot encoding and isotonic calibration. Three types of data drift (sudden, incremental and recurring) are simulated and analysed on mortgage loan datasets from Freddie Mac. Experiments and corresponding evidence show that the proposed landmark-based joint model consistently outperforms classical survival models, tree-based drift-adaptive learners and gradient boosting methods in terms of discrimination and calibration across all drift scenarios, which confirms the superiority of our model design.

In epidemiological research, modeling the cumulative effects of time-dependent exposures on survival outcomes presents a challenge due to their intricate temporal dynamics. Conventional spline-based statistical methods, though effective, require repeated data transformation for each spline parameter tuning, with survival analysis computations relying on the entire dataset, posing difficulties for large datasets. Meanwhile, existing neural network-based survival analysis methods focus on accuracy but often overlook the interpretability of cumulative exposure patterns. To bridge this gap, we introduce CENNSurv, a novel deep learning approach that captures dynamic risk relationships from time-dependent data. Evaluated on two diverse real-world datasets, CENNSurv revealed a multi-year lagged association between chronic environmental exposure and a critical survival outcome, as well as a critical short-term behavioral shift prior to subscription lapse. This demonstrates CENNSurv's ability to model complex temporal patterns with improved scalability. CENNSurv provides researchers studying cumulative effects a practical tool with interpretable insights.

We propose Lomax delegate racing (LDR) to explicitly model the mechanism of survival under competing risks and to interpret how the covariates accelerate or decelerate the time to event. LDR explains non-monotonic covariate effects by racing a potentially infinite number of sub-risks, and consequently relaxes the ubiquitous proportional-hazards assumption which may be too restrictive. Moreover, LDR is naturally able to model not only censoring, but also missing event times or event types. For inference, we develop a Gibbs sampler under data augmentation for moderately sized data, along with a stochastic gradient descent maximum a posteriori inference algorithm for big data applications. Illustrative experiments are provided on both synthetic and real datasets, and comparison with various benchmark algorithms for survival analysis with competing risks demonstrates distinguished performance of LDR.

We study the problem of subgroup discovery for survival analysis, where the goal is to find an interpretable subset of the data on which a Cox model is highly accurate. Our work is the first to study this particular subgroup problem, for which we make several contributions. Subgroup discovery methods generally require a "quality function" in order to sift through and select the most advantageous subgroups. We first examine why existing natural choices for quality functions are insufficient to solve the subgroup discovery problem for the Cox model. To address the shortcomings of existing metrics, we introduce two technical innovations: the *expected prediction entropy (EPE)*, a novel metric for evaluating survival models which predict a hazard function; and the *conditional rank statistics (CRS)*, a statistical object which quantifies the deviation of an individual point to the distribution of survival times in an existing subgroup. We study the EPE and CRS theoretically and show that they can solve many of the problems with existing metrics. We introduce a total of eight algorithms for the Cox subgroup discovery problem. The main algorithm is able to take advantage of both the EPE and the CRS, allowing us to give theoretical correctness results for this algorithm in a well-specified setting. We evaluate all of the proposed methods empirically on both synthetic and real data. The experiments confirm our theory, showing that our contributions allow for the recovery of a ground-truth subgroup in well-specified cases, as well as leading to better model fit compared to naively fitting the Cox model to the whole dataset in practical settings. Lastly, we conduct a case study on jet engine simulation data from NASA. The discovered subgroups uncover known nonlinearities/homogeneity in the data, and which suggest design choices which have been mirrored in practice.

Unlike standard prediction tasks, survival analysis requires modeling right censored data, which must be treated with care. While deep neural networks excel in traditional supervised learning, it remains unclear how to best utilize these models in survival analysis. A key question asks which data-generating assumptions of traditional survival models should be retained and which should be made more flexible via the function-approximating capabilities of neural networks. Rather than estimating the survival function targeted by most existing methods, we introduce a Deep Extended Hazard (DeepEH) model to provide a flexible and general framework for deep survival analysis. The extended hazard model includes the conventional Cox proportional hazards and accelerated failure time models as special cases, so DeepEH subsumes the popular Deep Cox proportional hazard (DeepSurv) and Deep Accelerated Failure Time (DeepAFT) models. We additionally provide theoretical support for the proposed DeepEH model by establishing consistency and convergence rate of the survival function estimator, which underscore the attractive feature that deep learning is able to detect low-dimensional structure of data in high-dimensional space. Numerical experiments also provide evidence that the proposed methods outperform existing statistical and deep learning approaches to survival analysis.

Survival analysis relies fundamentally on the semi-parametric Cox Proportional Hazards (CoxPH) model and the parametric Accelerated Failure Time (AFT) model. CoxPH assumes constant hazard ratios, often failing to capture real-world dynamics, while traditional AFT models are limited by rigid distributional assumptions. Although deep learning models like DeepAFT address these constraints by improving predictive accuracy and handling censoring, they inherit the significant challenge of black-box interpretability. The recent introduction of CoxKAN demonstrated the successful integration of Kolmogorov-Arnold Networks (KANs), a novel architecture that yields highly accurate and interpretable symbolic representations, within the CoxPH framework. Motivated by the interpretability gains of CoxKAN, we introduce KAN-AFT (Kolmogorov Arnold Network-based AFT), the first framework to apply KANs to the AFT model. Our primary contributions include: (i) a principled AFT-KAN formulation, (ii) robust optimization strategies for right-censored observations (e.g., Buckley-James and IPCW), and (iii) an interpretability pipeline that converts the learned spline functions into closed-form symbolic equations for survival time. Empirical results on multiple datasets confirm that KAN-AFT achieves performance comparable to or better than DeepAFT, while uniquely providing transparent, symbolic models of the survival process.