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.

Accurate disease risk prediction based on genomic and clinical data can lead to more effective disease screening, early prevention, and personalized treatment strategies. However, despite the identifications of hundreds of disease-associated genomic and molecular features for many disease traits through genome-wide studies in the past two decades, drug resistance often causes the targeted therapies to fail in cancer patients, which is largely due to tumor heterogeneity (Zhang et al., 2022). For advanced cancers, tumor heterogeneity encompasses both the malignant cells and their microenvironment, which makes it challenging to develop accurate prediction models for personalized treatment strategies that account for intratumor heterogeneity. Single-cell technologies provide an unprecedented opportunity for dissecting the interplay between the cancer cells and the associated tumor microenvironment (TME), and the produced high-dimensional omics data should also augment existing survival modeling approaches for identifying tumor cell type-specific genes predictive of cancer patient survival.

Previous studies have shown that hazard ratios between treatment groups estimated with the Cox model are uninterpretable because the indefinite baseline hazard of the model fails to identify temporal change in the risk set composition due to treatment assignment and unobserved factors among multiple, contradictory scenarios. To alleviate this problem, especially in studies based on observational data with uncontrolled dynamic treatment and real-time measurement of many covariates, we propose abandoning the baseline hazard and using machine learning to explicitly model the change in the risk set with or without latent variables. For this framework, we clarify the context in which hazard ratios can be causally interpreted, and then develop a method based on Neyman orthogonality to compute debiased maximum-likelihood estimators of hazard ratios. Computing the constructed estimators is more efficient than computing those based on weighted regression with marginal structural Cox models. Numerical simulations confirm that the proposed method identifies the ground truth with minimal bias. These results lay the foundation for developing a useful, alternative method for causal inference with uncontrolled, observational data in modern epidemiology.

We introduce an innovative approach that incorporates a Distributionally Robust Learning (DRL) approach into Cox regression to enhance the robustness and accuracy of survival predictions. By formulating a DRL framework with a Wasserstein distance-based ambiguity set, we develop a variant Cox model that is less sensitive to assumptions about the underlying data distribution and more resilient to model misspecification and data perturbations. By leveraging Wasserstein duality, we reformulate the original min-max DRL problem into a tractable regularized empirical risk minimization problem, which can be computed by exponential conic programming. We provide guarantees on the finite sample behavior of our DRL-Cox model. Moreover, through extensive simulations and real world case studies, we demonstrate that our regression model achieves superior performance in terms of prediction accuracy and robustness compared with traditional methods.

Concept-based learning enhances prediction accuracy and interpretability by leveraging high-level, human-understandable concepts. However, existing CBL frameworks do not address survival analysis tasks, which involve predicting event times in the presence of censored data -- a common scenario in fields like medicine and reliability analysis. To bridge this gap, we propose two novel models: SurvCBM (Survival Concept-based Bottleneck Model) and SurvRCM (Survival Regularized Concept-based Model), which integrate concept-based learning with survival analysis to handle censored event time data. The models employ the Cox proportional hazards model and the Beran estimator. SurvCBM is based on the architecture of the well-known concept bottleneck model, offering interpretable predictions through concept-based explanations. SurvRCM uses concepts as regularization to enhance accuracy. Both models are trained end-to-end and provide interpretable predictions in terms of concepts. Two interpretability approaches are proposed: one leveraging the linear relationship in the Cox model and another using an instance-based explanation framework with the Beran estimator. Numerical experiments demonstrate that SurvCBM outperforms SurvRCM and traditional survival models, underscoring the importance and advantages of incorporating concept information. The code for the proposed algorithms is publicly available.

This paper explores foundational and applied aspects of survival analysis, using fall risk assessment as a case study. It revisits key time-related probability distributions and statistical methods, including logistic regression, Poisson regression, Exponential regression, and the Cox Proportional Hazards model, offering a unified perspective on their relationships within the survival analysis framework. A contribution of this work is the step-by-step derivation and clarification of the relationships among these models, particularly demonstrating that Poisson regression in the survival context is a specific case of the Cox model. These insights address gaps in understanding and reinforce the simplicity and interpretability of survival models. The paper also emphasizes the practical utility of survival analysis by connecting theoretical insights with real-world applications. In the context of fall detection, it demonstrates how these models can simultaneously predict fall risk, analyze contributing factors, and estimate time-to-event outcomes within a single streamlined framework. In contrast, advanced deep learning methods often require complex post-hoc interpretation and separate training for different tasks particularly when working with structured numerical data. This highlights the enduring relevance of classical statistical frameworks and makes survival models especially valuable in healthcare settings, where explainability and robustness are critical. By unifying foundational concepts and offering a cohesive perspective on time-to-event analysis, this work serves as an accessible resource for understanding survival models and applying them effectively to diverse analytical challenges.

Time-to-event analysis provides insights into clinical prognosis and treatment recommendations. However, this task is more challenging than standard regression problems due to the presence of censored observations. Additionally, the lack of confidence assessment, model robustness, and prediction calibration raises concerns about the reliability of predictions. To address these challenges, we propose an evidential regression model specifically designed for time-to-event prediction. The proposed model quantifies both epistemic and aleatory uncertainties using Gaussian Random Fuzzy Numbers and belief functions, providing clinicians with uncertainty-aware survival time predictions. The model is trained by minimizing a generalized negative log-likelihood function accounting for data censoring. Experimental evaluations using simulated datasets with different data distributions and censoring conditions, as well as real-world datasets across diverse clinical applications, demonstrate that our model delivers both accurate and reliable performance, outperforming state-of-the-art methods. These results highlight the potential of our approach for enhancing clinical decision-making in survival analysis.

Chronic kidney disease (CKD) is a significant public health challenge, often progressing to end-stage renal disease (ESRD) if not detected and managed early. Early intervention, warranted by silent disease progression, can significantly reduce associated morbidity, mortality, and financial burden. In this study, we propose a novel approach to modeling CKD progression using a combination of machine learning techniques and classical statistical models. Building on the work of Liu et al. (2023), we evaluate linear models, tree-based methods, and deep learning models to extract novel predictors for CKD progression, with feature importance assessed using Shapley values. These newly identified predictors, integrated with established clinical features from the Kidney Failure Risk Equation, are then applied within the framework of Cox proportional hazards models to predict CKD progression.

We propose a general approach for encouraging fairness in survival analysis models based on minimizing a worst-case error across all subpopulations that occur with at least a user-specified probability. This approach can be used to convert many existing survival analysis models into ones that simultaneously encourage fairness, without requiring the user to specify which attributes or features to treat as sensitive in the training loss function. From a technical standpoint, our approach applies recent developments of distributionally robust optimization (DRO) to survival analysis. The complication is that existing DRO theory uses a training loss function that decomposes across contributions of individual data points, i.e., any term that shows up in the loss function depends only on a single training point. This decomposition does not hold for commonly used survival loss functions, including for the Cox proportional hazards model, its deep neural network variants, and many other recently developed models that use loss functions involving ranking or similarity score calculations. We address this technical hurdle using a sample splitting strategy. We demonstrate our sample splitting DRO approach by using it to create fair versions of a diverse set of existing survival analysis models including the Cox model (and its deep variant DeepSurv), the discrete-time model DeepHit, and the neural ODE model SODEN. We also establish a finite-sample theoretical guarantee to show what our sample splitting DRO loss converges to. For the Cox model, we further derive an exact DRO approach that does not use sample splitting. For all the models that we convert into DRO variants, we show that the DRO variants often score better on recently established fairness metrics (without incurring a significant drop in accuracy) compared to existing survival analysis fairness regularization techniques.

Optimizing Cox regression and its neural network variants poses substantial computational challenges in large-scale studies. Stochastic gradient descent (SGD), known for its scalability in model optimization, has recently been adapted to optimize Cox models. Unlike its conventional application, which typically targets a sum of independent individual loss, SGD for Cox models updates parameters based on the partial likelihood of a subset of data. Despite its empirical success, the theoretical foundation for optimizing Cox partial likelihood with SGD is largely underexplored. In this work, we demonstrate that the SGD estimator targets an objective function that is batch-size-dependent. We establish that the SGD estimator for the Cox neural network (Cox-NN) is consistent and achieves the optimal minimax convergence rate up to a polylogarithmic factor. For Cox regression, we further prove the $\sqrt{n}$-consistency and asymptotic normality of the SGD estimator, with variance depending on the batch size. Furthermore, we quantify the impact of batch size on Cox-NN training and its effect on the SGD estimator's asymptotic efficiency in Cox regression. These findings are validated by extensive numerical experiments and provide guidance for selecting batch sizes in SGD applications. Finally, we demonstrate the effectiveness of SGD in a real-world application where GD is unfeasible due to the large scale of data.