Random Survival Forests

RSFs grow an ensemble of binary survival trees using bootstrap samples of the original data. Each tree is grown by recursively splitting nodes to maximize the difference in survival between daughter nodes, typically using the log-rank test as the splitting criterion. The terminal nodes (leaves) contain at least a pre-specified minimum number of unique event times, ensuring stable hazard estimates. This ensemble approach reduces variance compared to a single tree while capturing complex, non-linear relationships in the data.

Unlike standard regression trees, RSFs natively accommodate right-censored observations—patients who have not experienced the event by the end of follow-up. The splitting rule uses only the rank order of event times, not their absolute values, making the method invariant to monotone transformations of time. The Nelson-Aalen estimator is computed within each terminal node to produce a cumulative hazard function, which is then averaged across all trees to generate the ensemble prediction. This non-parametric approach avoids imposing a specific distribution on the baseline hazard.

At each node split, RSFs introduce a second layer of randomization by selecting a random subset of mtry candidate predictors from the full feature set. This decorrelates the individual trees and prevents dominant features from monopolizing every split. Typical defaults set mtry to the square root of the total number of predictors. This mechanism allows RSFs to handle high-dimensional data—such as genomic or proteomic features—where the number of predictors far exceeds the number of observations, without overfitting.

RSFs provide two primary metrics for ranking predictor relevance:

• VIMP (Variable Importance): Measures the drop in prediction error when a variable is randomly permuted. A large positive VIMP indicates the variable is strongly predictive.
• Minimal Depth: Quantifies how early a variable appears in tree splits. Variables with low minimal depth are closer to the root and more influential. These measures enable biomarker discovery by identifying which clinical or molecular features most strongly drive survival outcomes, without requiring parametric assumptions.

RSFs leverage out-of-bag (OOB) data—the approximately 37% of observations not included in a tree's bootstrap sample—to obtain an unbiased estimate of prediction error without requiring a separate validation set. The OOB ensemble prediction for each observation is computed using only trees where that observation was held out. The C-index or Brier score calculated on OOB predictions provides a robust internal validation metric, making RSFs particularly valuable when data is scarce and cross-validation would be computationally expensive.

The standard RSF framework has been extended to handle competing risks—scenarios where a subject can experience one of several mutually exclusive events, such as death from cancer versus death from cardiovascular disease. In this setting, the splitting rule is modified to maximize the difference in cause-specific cumulative incidence functions between daughter nodes. The ensemble produces separate cumulative incidence estimates for each event type, enabling clinicians to distinguish risk profiles for different failure modes from the same set of predictors.

The foundational principle behind RSF: aggregating multiple weak learners to reduce variance and improve generalization. Bagging (bootstrap aggregating) creates diverse trees by sampling observations with replacement. Random feature selection at each split further decorrelates trees. For survival data, the ensemble averages cumulative hazard functions rather than simple class votes. This approach naturally handles non-linear effects and high-order interactions without pre-specification, making it robust against model misspecification that plagues parametric survival models.

The standard splitting criterion in RSF that determines optimal cutpoints by maximizing survival difference between child nodes. The log-rank test statistic compares observed versus expected events in each group under the null hypothesis of no difference. At each node, the algorithm evaluates all possible splits across all variables, selecting the one with the maximum log-rank statistic. This non-parametric criterion makes no distributional assumptions about survival times and naturally accommodates right-censored observations by using the at-risk process in its calculation.

RSF outputs patient-specific predictions via the Nelson-Aalen estimator within each terminal node. For a new patient, the tree determines which leaf they belong to, and the cumulative hazard is calculated as the sum of event counts divided by the at-risk counts at each unique event time. The ensemble prediction averages these estimates across all trees. This provides a non-parametric baseline hazard that flexibly captures complex temporal patterns, unlike the Cox model which leaves the baseline unspecified or parametric models that impose rigid shapes.

RSF provides interpretable feature rankings through permutation importance and minimal depth. Permutation importance randomly shuffles a variable's values and measures the degradation in prediction error—higher degradation indicates greater importance. Minimal depth quantifies how close a variable first appears to the root node across the forest. These metrics help identify prognostic biomarkers in high-dimensional genomic studies. Unlike Cox model coefficients, they capture non-linear and interaction effects without requiring proportional hazards assumptions.

Standard RSF handles a single event type, but the competing risks RSF extends the framework to multiple mutually exclusive events. Instead of a single cumulative hazard, it estimates the cause-specific cumulative incidence function (CIF) for each event type. Splitting rules are modified to maximize differences in CIFs rather than overall survival. This is critical in oncology where patients may die from causes other than cancer, requiring separation of disease-specific mortality from competing mortality for accurate prognostic modeling.

DeepSurv uses a deep neural network with a Cox proportional hazards output layer, while RSF uses tree ensembles with non-parametric splitting. Key trade-offs:

• DeepSurv excels at learning smooth, hierarchical representations from raw inputs but requires careful hyperparameter tuning and larger sample sizes
• RSF handles mixed data types naturally, requires minimal preprocessing, and provides built-in variable importance
• DeepSurv assumes proportional hazards; RSF does not
• RSF typically outperforms on tabular data with complex interactions; DeepSurv can leverage unstructured inputs like images