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Machine Learning for Survival Analysis: A Survey
Ping Wang, Yan Li, Chandan K. Reddy
TL;DR
Censored time-to-event data require specialized survival-analysis methods rather than standard predictive algorithms. This survey structures statistical and machine-learning approaches, related topics, metrics, implementations, and applications. It concludes with a comprehensive review spanning conventional and machine-learning methods and applications across healthcare, biomedicine, reliability, student retention, and user behavior modeling.
Problem
Censoring makes event outcomes unobservable for some instances, creating a need for methods that predict event times while handling censored survival data.
Method
The survey reviews statistical and machine-learning survival methods, evaluation metrics, related topics, implementations, taxonomies, and applications.
Results
The survey provides a comprehensive review of conventional and machine-learning survival methods, related topics, evaluation metrics, and applications across multiple real-world domains.
Takeaways & Limitations
The paper offers structured coverage and implementation resources intended to help readers understand and apply survival-analysis methods to censored-data problems.
Takeaways & Limitations
A cited support-vector approach has quadratic computational complexity in the number of instances, while separate analysis of competing risks assumes independence and complicates interpretation.
Abstract
from arXiv · showhide
Accurately predicting the time of occurrence of an event of interest is a critical problem in longitudinal data analysis. One of the main challenges in this context is the presence of instances whose event outcomes become unobservable after a certain time point or when some instances do not experience any event during the monitoring period. Such a phenomenon is called censoring which can be effectively handled using survival analysis techniques. Traditionally, statistical approaches have been widely developed in the literature to overcome this censoring issue. In addition, many machine learning algorithms are adapted to effectively handle survival data and tackle other challenging problems that arise in real-world data. In this survey, we provide a comprehensive and structured review of the representative statistical methods along with the machine learning techniques used in survival analysis and provide a detailed taxonomy of the existing methods. We also discuss several topics that are closely related to survival analysis and illustrate several successful applications in various real-world application domains. We hope that this paper will provide a more thorough understanding of the recent advances in survival analysis and offer some guidelines on applying these approaches to solve new problems that arise in applications with censored data.
1. INTRODUCTION
Survival analysis addresses time-to-event prediction when censoring makes some outcomes unobservable. This survey organizes statistical and machine-learning approaches, evaluation measures, related topics, implementations, and applications.
- Censoring occurs when an event is unobserved because of limited study duration or loss of follow-up, making standard predictive methods unsuitable.
- Machine-learning methods have been adapted alongside statistical approaches to handle censored data and other survival-analysis challenges.
- The survey provides a comprehensive, structured overview of statistical and machine-learning survival methods, including evaluation metrics and advanced formulations.
- It presents a taxonomy, implementation and source-code links, and discussions of applications across survival-analysis research.
2. DEFINITION OF SURVIVAL ANALYSIS
Survival analysis models time to an event when observations may be censored, with the goal of estimating future event times and survival-related functions. The section introduces core notation, censoring types, formal inputs, and the method taxonomy.
- Survival Data and Censoring: Censoring means the event is unobserved, and may be right-, left-, or interval-censoring; this paper mainly analyzes right-censored data.
- Survival Data and Censoring: Observed survival data contain features, an observed time, and an event indicator, while the true event time remains latent for censored instances.
- Problem Statement: The survival-analysis goal is to estimate the non-negative, continuous event time for a new instance from its feature predictors.
- Survival and Hazard Function: The survival function gives the probability that the event time is not earlier than t, decreases monotonically, and starts at 1 when t = 0.
- Survival and Hazard Function: The cumulative death distribution satisfies F(t) = 1 − S(t), while f(t) is its derivative for continuous time or a finite difference for discrete time.
- Survival and Hazard Function: The hazard function is the event rate at time t conditional on no earlier event, rather than the event probability.
- Method Taxonomy: Survival-analysis methods broadly divide into statistical and machine-learning branches, with advanced machine-learning categories including ensemble, active, transfer, and multi-task learning.
3. TRADITIONAL STATISTICAL METHODS
Traditional statistical survival methods estimate survival or hazard functions through non-parametric, semi-parametric, or parametric models. Their suitability depends on distributional and proportional-hazards assumptions.
- Statistical survival methods comprise non-parametric, semi-parametric, and parametric categories summarized by their theoretical and experimental advantages and disadvantages.
- Non-parametric Models: Non-parametric methods avoid an underlying event-time distribution and are more efficient when the proportional-hazards assumption does not hold.
- Semi-parametric Models: The Cox model is the most commonly used semi-parametric regression approach and relies on proportional hazards with partial-likelihood estimation.
- Parametric Models: Parametric methods are more efficient and accurate when event times follow a specified distribution, but estimation is tied to that distributional assumption.
3.1. Non-parametric Models
Non-parametric methods estimate survival or cumulative hazard functions without specifying an event-time distribution. Kaplan–Meier, Life Table, and Nelson–Aalen methods address different data structures and targets.
- Kaplan–Meier estimates survival using observed event times, numbers at risk, and censoring between successive event times.
- The at-risk count must subtract both prior events and intervening censored instances rather than only prior events.
- Life Table analysis is more convenient for interval-grouped data, very large subject counts, or large populations.
- Nelson–Aalen estimates the cumulative hazard function for censored data using a counting-process approach.
3.2. Semi-Parametric Models
Semi-parametric survival models, especially Cox regression, avoid specifying the event-time distribution while modeling covariate effects through hazard ratios. Regularized variants address high-dimensionality, feature selection, correlation, and structured relationships.
- Basic Cox model: Cox regression leaves the baseline hazard unspecified while assuming covariates have an exponential influence on the outcome.
- Basic Cox model: The Cox model estimates coefficients through a partial likelihood that excludes the unspecified baseline hazard and incorporates censored observations.
- Basic Cox model: Cox model fitting with Newton-Raphson estimation has time complexity O(NP 2).
- Regularized Cox models: High-dimensional settings motivate sparsity penalties because P may approach or exceed N, increasing overfitting risk.
- Regularized Cox models: Lasso-Cox performs feature selection through an ℓ1 penalty, while Elastic Net combines ℓ1 and squared ℓ2 penalties to address correlated features.
- Regularized Cox models: KEN-Cox combines ℓ1 regularization with a kernel-based quadratic term, whereas OSCAR-Cox can equalize coefficients for similarly related features.
3.3. Parametric Models
Parametric survival models assume a theoretical distribution for survival times or their logarithms, offering simple and efficient prediction. The section covers common distributions and censored regression extensions.
- Parametric models: Parametric censored regression models assume survival times or log survival times follow a particular theoretical distribution.
- Common distributions: Exponential models impose a constant hazard rate, with larger λ indicating higher risk and shorter survival time.
- Common distributions: Weibull models use shape parameter k to represent constant, decreasing, or increasing hazard patterns more flexibly than exponential models.
- Common distributions: Logistic and log-logistic models allow non-monotonic hazard behavior, unlike the Weibull model.
- Censored regression: Tobit regression introduces a Gaussian latent variable linearly dependent on features to extend linear regression to censored observations.
- Censored regression: Buckley-James regression estimates censored responses with Kaplan-Meier methods before fitting an accelerated failure-time model.
- Censored regression: Penalized regression supports simultaneous variable selection and coefficient estimation, especially with multicollinearity or high-dimensionality.
4. MACHINE LEARNING METHODS
Machine learning methods for survival analysis aim to model complex predictive relationships while handling censored information and time-to-event estimation. The survey reviews trees, Bayesian methods, neural networks, and related approaches.
- Overview: Machine learning methods can model non-linear relationships, but survival applications must appropriately handle censored information and time estimation.
- Survival trees: Survival trees recursively partition data using splitting criteria tailored to censored outcomes and event-related similarity.
- Survival trees: Tree ensembles can avoid selecting a final single tree and may perform better than individual survival trees.
- Bayesian methods: Bayesian methods include Naïve Bayes and Bayesian networks, with Bayesian networks representing relationships among features graphically for interpretability.
- Neural networks: Neural-network survival methods predict survival time, extend Cox proportional-hazards models nonlinearly, or output survival or hazard probabilities.
4.4. Support Vector Machines
Support Vector Machines can be adapted to survival analysis, but naïve and constraint-based approaches expose trade-offs involving censoring information, computational cost, ordering, and output values.
- Support Vector Regression: A naïve SVR approach uses only instances with observed events, thereby discarding order information contained in censored instances.
- Support Vector Classification: Constraint-based support vector classification preserves ordering between comparable instances but has quadratic computational complexity in the number of instances.
- Support Vector Classification: The constraint approach focuses on instance ordering and ignores the actual output values.
- Censored-data adaptations: SVRc adapts standard SVR to censored cases with an asymmetric loss function and includes both uncensored and censored instances.
- Relevance Vector Machines: RVM uses Bayesian inference and priors over weights to provide probabilistic estimations in a formulation related to SVM.
4.5. Advanced Machine Learning Approaches
Advanced machine learning methods adapt ensemble, active, and multi-task learning to censored survival data and related practical constraints. The survey describes these approaches and their core procedures.
- Ensemble Learning: Ensemble methods combine multiple learners to improve approximation and reduce the instability of individual models on survival data.Bagging averages survival-tree predictions, while random survival forests add random feature selection during tree construction.
- Ensemble Learning: Bagging survival trees draw bootstrap samples, build survival trees, and aggregate their predicted survival functions by averaging.
- Ensemble Learning: Random survival forests extend survival-tree ensembles by randomly selecting features for node splitting and using out-of-bag samples.The method draws bootstrap samples and builds trees using randomly selected features and survival-based splitting criteria.
- Active Learning: Active learning incorporates expert feedback by selecting censored subjects for survival-status labeling before adding them to the training set.The ARC algorithm combines this mechanism with regularized Cox regression and discriminative gradient sampling.
- Multi-Task Learning: Multi-task learning reformulates survival-time prediction to address incomplete outcome-label matrices caused by censored instances.MTLSA is designed because standard multi-task learning methods are unsuitable for this censored information.
5. PERFORMANCE EVALUATION METRICS
Survival-model evaluation requires metrics that account for censoring rather than standard regression measures. The section explains concordance-based measures, time-dependent AUC, Brier score, and event-time MAE.
- Standard regression metrics such as root mean squared error and R2 are unsuitable for survival data because censoring makes outcome information incomplete.
- C-index: The C-index evaluates whether predicted relative risks agree with the ordering of observed survival outcomes among comparable pairs.Comparable pairs include two uncensored instances or an uncensored event occurring before a censored instance’s censoring time.
- C-index: 6 out of 10 pairwise comparisons remain feasible with censored observations in the five-instance illustration, compared with 10 without censoring.A censored instance can be compared with an earlier uncensored instance but not with later observations whose event times are unknown.
- C-index: The time-dependent C-index over (0, t*) is a weighted average of AUC values across possible observation times.At each time t, AUC(t) is defined using comparable pairs with one observed before t and another surviving beyond t.
- Brier Score: The censored-data Brier score reweights individual contributions using a Kaplan-Meier estimate of the censoring distribution.Instances censored before t receive zero weight, while uncensored instances at t receive weights greater than 1.
- Mean Absolute Error: Survival MAE averages differences between predicted and observed times using only samples for which the event occurs.The metric applies to models that directly predict event time, such as accelerated failure time models.
6. RELATED TOPICS
Related survival-analysis topics address limited early labels, censored-data transformation, calibration, competing risks, and recurrent events. The survey presents methods tailored to these settings and their assumptions.
- Early Stage Prediction: Survival studies require long follow-up to collect sufficient event labels, creating a need for models that forecast future event occurrence from early data.
- Early Stage Prediction: Early Stage Prediction uses Naive Bayes and Bayesian Networks with extrapolation based on Weibull, Log-logistic, and Log-normal distributions.The ESP framework is described as a two-stage algorithm trained using early-stage study data.
- Data Transformation: Censored observations contain incomplete event information but are not equivalent to completely unlabeled classification samples.Naively deleting censored instances or treating censoring as event-free can discard useful information or underestimate model performance.
- Calibration: Calibration imputes time-to-event labels for censored instances by exploiting row-wise and column-wise correlations through regularized inverse-covariance methods.This produces a new representation of the survival data while addressing missing labels.
- Competing Risks: Competing risks involve multiple possible event types when only one event can occur during follow-up.Analyzing each event separately while treating other events as censored assumes independence among competing risks and complicates interpretation of event-specific survival probabilities.
7. APPLICATION DOMAINS
Survival analysis is applied across healthcare, reliability, crowdfunding, bioinformatics, education, marketing, advertising, and economics. These applications use time-to-event outcomes with domain-specific censoring and prediction goals.
- Healthcare: Healthcare applications model events such as death, readmission, discharge, and other incidents after medical interventions or diagnoses.Missing observation traces can occur when patients move to another hospital, while models estimate failure-time distributions and feature prognostic relevance.
- Reliability: Reliability applications predict whether devices fail and, if so, when, using survival models that can provide early warnings about potential failures.These warnings may help prevent or reduce failures and identify their causes.
- Crowdfunding: Crowdfunding survival models estimate project success probability while combining time-to-success information with successful and failed project outcomes.Failed projects do not provide the time at which success would have occurred, creating incomplete duration information.
- Bioinformatics: Bioinformatics studies use gene-expression measurements to investigate associations with patient survival outcomes, including in cancer data with censoring.
- Other Domains: The survey also identifies survival-analysis applications in student retention, customer lifetime value, online advertising, and economic duration data.Examples include returning to university, customer profit over purchase histories, ad clicking, unemployment duration, and time-to-transaction.
8. RESOURCES
The survey links survival-analysis methods to software implementations, with most methods available in R packages or functions. Resources cover statistical models, machine-learning methods, and related survival-analysis tasks.
- Software availability: Most existing survival-analysis methods can be implemented in R.The resources section identifies R as the primary implementation environment across the surveyed methods.
- Statistical methods: R resources cover non-parametric methods through survival-package functions such as coxph and survfit.These functions implement the listed non-parametric methods.
- Statistical methods: Semi-parametric implementations include Cox, regularized Cox, OSCAR-Cox, and likelihood-based boosting models across several R packages.The listed packages include survival, fastcox, RegCox, and CoxBoost.
- Statistical methods: Parametric methods include Tobit regression, Buckley-James regression, and parametric AFT models through R implementations.The survival and bujar packages provide the corresponding functions or implementations.
- Machine-learning methods: Machine-learning resources include Bayesian model averaging, bagging survival trees, random survival forests, boosting, and active learning survival models.These methods are associated with packages including BMA, ipred, randomForestSRC, mboost, and RegCox.
- Related topics: Related resources support early prediction, calibration, competing risks, and recurrent-event survival modeling.The cited packages and functions address preprocessing, calibration, and specialized event structures.
9. CONCLUSION
The survey reviews conventional and machine-learning survival-analysis methods for predicting future events, together with related topics, implementations, and evaluation metrics. It also documents applications beyond healthcare and biomedicine, including reliability, student retention, and user behavior modeling.
- Scope: Survival analysis aims to predict the occurrence of specific events at future time points.The conclusion frames future-event prediction as the field’s primary goal.
- Contributions: The survey reviews conventional and machine-learning survival methods, related topics, and evaluation metrics.Its coverage includes data transformation, early prediction, complex events, implementation details, and performance evaluation.
- Applications: The paper describes applications of survival analysis in healthcare, biomedicine, reliability, student retention, and user behavior modeling.The conclusion explicitly lists these application domains.