Survival Forests in Censored Data Analysis

• Survival forests are ensemble methods designed for time-to-event data, applying specialized criteria like the log-rank statistic to handle right-censoring.
• They aggregate tree-based estimators nonparametrically, yielding flexible risk predictions without relying on proportional hazards assumptions.
• Advanced extensions, such as balanced and federated survival forests, enhance prediction accuracy in imbalanced and privacy-sensitive biomedical and engineering settings.

Survival forests are a class of ensemble learning methods designed for analyzing time-to-event data, especially under right-censoring, where the precise failure time for some units is unknown. These models generalize classical random forests by using specialized splitting criteria and ensemble aggregation tailored to survival analysis, enabling flexible, nonparametric modeling of complex covariate effects on survival distributions. Survival forests underpin a wide range of contemporary research and practice in biostatistics, reliability engineering, and predictive maintenance.

1. Foundational Concepts and Methodology

Survival forests extend Breiman’s random forest framework to censored survival data by aggregating predictions from multiple survival trees, each constructed from bootstrap resamples of the observed data. In each survival tree, samples are recursively partitioned based on covariate split criteria designed for censored data. The most prevalent splitting criterion is the maximization of survival difference, typically quantified with the log-rank statistic, although alternatives such as maximally selected rank statistics (Wright et al., 2016), concordance index–based splits (Schmid et al., 2015), and full–likelihood derived scores for specialized settings (e.g., length-biased data (Lee et al., 22 Aug 2025)) have been presented.

At the terminal nodes, the cumulative hazard function (CHF) for the samples within the node is estimated using the Nelson-Aalen estimator, which for a node h is

$$\widehat{H}_h(t) = \sum_{t_i \leq t} \frac{d_{i,h}}{n_{i,h}}$$

where $d_{i,h}$ is the number of events at time $t_i$ in node $h$ and $n_{i,h}$ the number at risk. Each survival tree outputs a node-specific estimator, and the ensemble CHF for an individual is formed by averaging over trees, optionally using out-of-bag samples for model validation (Yang et al., 2010, Afrin et al., 2018).

Ensemble aggregation leverages the nonparametric structure of trees to provide locally adapted, individualized survival or hazard estimates. The resulting ensemble does not require proportional hazards or linearity assumptions, and is robust to complex covariate relationships including nonlinearity and high-order interactions (Saha et al., 2019, Korepanova et al., 2019).

2. Splitting Criteria, Variable Selection Bias, and Extensions

The choice of splitting criterion fundamentally shapes the properties of survival forests. The classical log-rank statistic is optimal under proportional hazards and yields high power for shift alternatives but can exhibit strong "end-cut preference" (ECP), selecting unbalanced splits—an effect advantageous when many non-informative predictors are present but detrimental with a high fraction of continuous, informative predictors (Schmid et al., 2015).

To address variable selection bias inherent in standard random survival forests (preference for variables with many possible splits), maximally selected rank statistics have been introduced. For each covariate, maximally selected statistics dichotomize at each cut-point and adjust for multiple testing, enabling unbiased variable selection regardless of covariate scale. Efficient p-value approximations (e.g., Lausen-Schumacher) have been implemented to permit practical use in large data (Wright et al., 2016). This approach enhances the detection of non-linear covariate effects, outperforming linear-rank-based conditional inference forests under strong nonlinearities.

Kernel methods augment the expressive power of the splits by replacing the original data matrix with a kernel matrix where each element is, e.g., a polynomial or Gaussian kernel of the input pair, enabling even axis-aligned splits in the tree to capture arbitrarily complex nonlinearities (Yang et al., 2010, Chen, 2020). In Kernel Induced Random Survival Forests (KIRSF), this transformation was shown to increase prediction accuracy, reduce error rate variability, and enhance the discrimination of patient subgroups in both simulated and real-world clinical data.

3. Advanced Ensemble Strategies, Balancing, and Federated Learning

Recent research has introduced strategies for optimizing the construction and aggregation of survival tree ensembles:

• Balanced Random Survival Forests (BRSF): To address severe event (minority) class imbalance, BRSF augments the minority class using synthetic sampling (e.g., SMOTE), alleviating hazard underestimation for rare events and reducing prediction error. Theoretical results establish that this approach systematically reduces the Brier score relative to conventional RSF, with empirical support demonstrating average prediction error reductions exceeding 50% on highly imbalanced clinical datasets (Afrin et al., 2018).
• Optimal Survival Trees Ensemble (OSTE): Rather than aggregating all grown trees, OSTE ranks trees by out-of-bag prediction error (C-index), builds the ensemble incrementally, and retains only trees that reduce the integrated Brier score on validation data. This approach attains comparable or improved accuracy with far fewer trees than conventional bagging or RSF, decreasing computational burden (Gul et al., 2020).
• Federated Survival Forests: In privacy-sensitive distributed settings (e.g., multi-center healthcare), federated algorithms such as FedSurF and FedSurF++ construct global survival forests by aggregating locally trained trees from individual sites in a single communication round, using performance-weighted sampling. This enables robust, privacy-preserving survival function estimation without centralizing individual data. Empirical benchmarking shows that such methods achieve discrimination and calibration competitive with or exceeding that of deep neural network federated algorithms, especially under non-IID label distributions and low-resource constraints (Archetti et al., 2023, Archetti et al., 2023).

4. Adaptations for Complex Data Structures

Survival forests have been systematically extended to accommodate increasingly complex data modalities:

• Time-Varying and Longitudinal Covariates: Ensemble survival methods have been generalized using counting process approaches to allow time-varying covariate inputs by subdividing trajectories into pseudo-subject intervals with constant covariates. Adapted splitting criteria (e.g., log-rank score for left-truncated data) and recursive survival function combination enable dynamic updates as new covariate information becomes available (Yao et al., 2020, Moradian et al., 2021). Multivariate longitudinal endogenous covariates are handled in DynForest by translating repeated measures into mixed model-derived features within each node; this facilitates scalable dynamic prediction in high-dimensional settings and accounts for informative dropout (Devaux et al., 2022).
• Interval-Censored and Length-Biased Data: Conditional inference forests have been modified for interval-censored data by using specialized log-rank scores and weighted maximum likelihood estimators (e.g., Turnbull's estimator) for the survival function (Yao et al., 2019). New survival trees and forests for length-biased right-censored data utilize full-likelihood-based splitting and composite conditional-likelihood estimation for unbiased survival function prediction, correcting for the overrepresentation of long survivors inherent to prevalence sampling (Lee et al., 22 Aug 2025).
• Oblique and Functional Predictors: Oblique random survival forests split nodes based on linear combinations of predictors determined by Newton-Raphson updates of the Cox partial likelihood, improving sensitivity to correlated features and decision boundaries not aligned with the input axes; computational acceleration and variable importance via the negation of coefficients are key innovations (Jaeger et al., 2022). For functional predictors, graphical and Shapley-based local/global explanation frameworks increase transparency in decision making, and functional principal components are used for summarizing high-dimensional covariate time series (Loffredo et al., 25 Apr 2025).

5. Performance Metrics, Hyperparameter Tuning, and Model Validation

Survival forest performance is primarily assessed via Harrell’s concordance index (C-index)—a measure of the probability that predicted and observed orderings of survival events are concordant across permissible pairs, with explicit accommodation for censoring—and the Integrated Brier Score (IBS), which measures the average squared prediction error over time, accounting for censoring via inverse probability weighting (Yang et al., 2010, Afrin et al., 2018). Time-dependent ROC/AUC metrics and calibration plots are also common for benchmarking.

Rigorous hyperparameter tuning, particularly for ntree (number of trees), mtry (number of candidate variables per split), nodesize, and splitrule, substantially improves both discrimination and calibration. Empirical studies demonstrate consistent gains in C-index (mean increase of 0.0547) and reduction in Brier score (mean decrease of 0.0199) across predictive maintenance datasets upon hyperparameter optimization. Optimal ranges vary by dataset and metric, with ntree and mtry exerting the largest effects on discrimination, while nodesize affects both calibration and discrimination; splitrule can exert negative effects if tuned improperly (Yardımcı et al., 20 Apr 2025).

Confidence band estimation for RSF has been addressed through unbiased variance-covariance estimation via U-statistics and Gaussian process approximation to support finite-sample statistical inference for cumulative hazard or survival function predictions, validating coverage properties through simulations and real-data experiments (Formentini et al., 2022).

6. Applications and Impact in Biomedicine and Engineering

Survival forests have achieved widespread adoption in biomedicine for prognostic modeling in oncology, cardiology, and intensive care, where they offer robust solutions for complex, censored, and high-dimensional omics data. They are critical in personalized risk prediction, dynamic prognosis in dementia (using repeated cognitive and imaging markers), and as a federated analytics tool in privacy-constrained hospital networks (Devaux et al., 2022, Archetti et al., 2023). In engineering, survival forests are central to predictive maintenance—accurately forecasting machine or component failure times in high-dimensional sensor environments (e.g., aircraft engine degradation)—where they offer significant operational gains through more reliable maintenance scheduling (Yardımcı et al., 20 Apr 2025).

Advances in interpretability (e.g., negation importance, SurvSHAP, permutation feature importance) further promote adoption by elucidating feature contributions at both global and individual levels, facilitating actionable insight for clinicians and engineers (Jaeger et al., 2022, Loffredo et al., 25 Apr 2025).

7. Limitations, Controversies, and Future Directions

Despite their flexibility, survival forests can be limited by poor performance under extreme model misspecification (e.g., complex length-biased or interval-censored data if not properly addressed), risk of bias when using variable selection schemes that favor features with greater granularity (Wright et al., 2016), and challenging interpretability, especially in functional or obliquely split models. The choice of splitting rule remains a source of active debate, with different regimes (high dimensionality, prevalence of non-informative predictors, censoring rates) favoring log-rank, C-index, maximally selected rank, or full-likelihood-based splits (Schmid et al., 2015, Lee et al., 22 Aug 2025).

Scaling survival forests to extremely high dimensions and distributed federated environments continues to motivate algorithmic developments, including efficient kernel learning, optimal tree sampling for privacy-aware federated ensembles, and dynamic adaptation to changing covariates and data streams. Future research is likely to integrate advanced optimization and meta-learning for automated tunability, further expand statistically rigorous inference (confidence intervals, variable selection p-values), and deepen the interplay between interpretability, privacy, and computational tractability.