Evolving Features vs Evolving Entire Trees with GP for Interpretable Survival Analysis

Survival analysis has long been a vital statistical tool in medicine, engineering, and social sciences, aiming to predict the time until an event occurs. Classical methods like Cox proportional hazards (Cox PH) model the hazard function as a product of a baseline hazard and an exponential of a linear combination of covariates. While computationally efficient and interpretable, Cox PH assumes proportional hazards and linear effects, limiting its capacity to model complex, nonlinear interactions. To address these limitations, deep learning models such as DeepSurv have been developed, leveraging neural networks to capture nonlinearities. However, these models often operate as black boxes, reducing their clinical trustworthiness.

In parallel, tree-based models like survival trees offer intuitive interpretability, recursively partitioning data based on covariates. Classical greedy algorithms for tree induction are fast but prone to suboptimal splits, especially in complex relationships. Non-greedy approaches, including dynamic programming, can find optimal trees but are limited by the complexity of split conditions they can handle. Recent efforts have focused on feature engineering to enable shallow trees to model nonlinear interactions, but these are typically handcrafted and lack systematic optimization.

Genetic programming (GP), particularly multi-objective variants like GP-GOMEA, has shown promise in symbolic regression and feature construction. Its ability to evolve complex expressions makes it suitable for generating nonlinear features that can improve model expressiveness. This paper builds on this foundation, proposing a framework that combines GP-based feature evolution with joint optimization of survival tree structures, aiming to produce models that are both accurate and inherently interpretable.

The core challenge lies in balancing predictive accuracy with interpretability in survival models. Traditional shallow survival trees are easy to understand but struggle with complex relationships, often requiring deep, less interpretable structures. Conversely, deep trees or complex models like neural networks achieve high accuracy but lack transparency. Existing feature engineering methods are either manual or limited in capturing nonlinear interactions. Moreover, greedy tree induction methods are prone to local optima, missing globally optimal splits that could better model the data. The problem becomes more acute in clinical settings, where model interpretability is crucial for trust and adoption. Therefore, the key question is how to systematically generate nonlinear features and optimize tree structures simultaneously, ensuring models are both accurate and transparent.

This work introduces several key innovations:

• �� Multi-objective genetic programming (GP-GOMEA) for symbolic nonlinear feature construction, enabling the capture of complex covariate interactions;
• �� A joint evolutionary algorithm that simultaneously optimizes the entire survival tree structure and split logic, overcoming the limitations of greedy algorithms;
• �� A multi-objective framework balancing predictive performance (via IBS) and model complexity (expression length), providing diverse solutions for different interpretability needs;
• �� Validation on real-world datasets demonstrating significant improvements in predictive accuracy while maintaining interpretability, especially in modeling complex relationships like XOR interactions.

These innovations collectively enable the development of shallow, interpretable survival models with performance comparable to deeper or black-box models.

• �� Generate nonlinear features using multi-objective GP-GOMEA, optimizing for IBS and expression complexity, with input features from clinical datasets;
• �� Use these features to induce survival trees with depths of 2 or 3, employing three strategies: greedy, optimal via dynamic programming, and joint evolutionary optimization;
• �� For greedy trees, locally search for the best split using constructed features; for optimal trees, use binary features with dynamic programming to find globally optimal splits; for evolutionary trees, simultaneously evolve the tree structure and split logic via multi-tree GP-GOMEA;
• �� Incorporate multi-objective fitness evaluation considering prediction accuracy (IBS) and model simplicity (feature expression length or tree size);
• �� Validate models through 25-fold cross-validation repeated 30 times, using hyperparameters such as population size 1024, max generations 50, and tree depth constraints;
• �� Compare models across datasets (GBSG and METABRIC) with baseline methods (Cox PH, DeepSurv, Random Survival Forests), analyzing the impact of nonlinear feature construction and joint evolution.

Experiments utilized two breast cancer datasets: GBSG (1546 patients, 37% censored) and METABRIC (1523 patients, 57% censored). Features included clinical variables and gene expression data. Baselines comprised traditional greedy survival trees, deep learning models (DeepSurv, CoxKAN), and Random Survival Forests. The proposed approach involved evolving nonlinear features via GP-GOMEA, then inducing shallow survival trees (depth 2 or 3) with different strategies. Performance was assessed using integrated Brier score (IBS) and Harrell’s C-index, with 25-fold stratified shuffle splits repeated 30 times for robustness. Hyperparameters included a population size of 1024, maximum generations of 50, and feature expression length limits. Ablation studies examined the effect of nonlinear feature evolution, joint versus separate optimization, and tree depth. Results consistently showed that nonlinear features and joint evolution significantly improved predictive accuracy and interpretability.

The experimental results demonstrated that models utilizing GP-evolved nonlinear features outperformed baseline methods, with IBS scores reduced by approximately 8-12%. At depth 3, the joint evolutionary survival tree achieved an IBS of 0.074, compared to 0.106 for the baseline. The models effectively captured complex relationships such as XOR interactions, which traditional greedy trees failed to model. The multi-objective framework allowed selecting models that balance accuracy and simplicity, providing interpretable structures without sacrificing performance. The results validated that combining nonlinear feature construction with joint optimization of tree structure yields superior survival predictions, especially in complex scenarios, confirming the approach’s robustness across datasets and tree depths.

This methodology is directly applicable to clinical risk prediction, personalized treatment planning, and early diagnosis in oncology and other chronic diseases. By providing transparent models that clinicians can interpret and validate, it enhances trust and facilitates clinical decision-making. The approach requires only standard clinical or genomic data, making it accessible for routine use. Its ability to model complex interactions without deep neural networks makes it suitable for settings with limited data or where interpretability is paramount. Future applications could include multi-modal data integration, disease subtyping, and real-time risk monitoring, supporting the broader goal of precision medicine.

Despite its strengths, the approach faces challenges such as high computational costs due to evolutionary algorithms, limiting scalability to very large datasets. The current validation on breast cancer datasets may restrict generalizability; additional testing on diverse diseases and multi-modal data is needed. The feature expression complexity and tree depth constraints may limit modeling of extremely intricate relationships. Furthermore, the interpretability depends on the simplicity of evolved expressions, which may become complex in some cases. Future work should focus on improving computational efficiency, extending validation to other domains, and developing automated model selection tools to facilitate clinical adoption.