Neural Networks for net survival estimation and prediction

Technical Summary: Neural Networks for Net Survival Estimation and Prediction

Problem Statement
Net survival is a critical metric in oncology, representing the survival probability in a counterfactual world where cancer is the only possible cause of death. This metric allows for the comparison of cancer-related mortality across populations with differing life expectancies. While non-parametric estimators like the Pohar-Perme (PP) estimator are unbiased, they are limited to stratified analyses, requiring large sample sizes to maintain statistical power when adjusting for multiple covariates. Conversely, parametric and semi-parametric excess hazard regressions (e.g., spline-based models) offer flexibility but rely on assumptions such as proportional hazards (PH) and log-linearity. Violations of these assumptions, or the omission of important interactions, can lead to model misspecification and biased estimates. Current practice often involves complex stratification to avoid these violations, which reduces the effective sample size for each analysis.

Methodology
The authors propose a machine learning approach based on the Partial Logistic Artificial Neural Network (PLANN), originally developed for overall survival prediction, adapted here for net survival estimation and prediction.

• PLANN Adaptation: The method discretizes time into intervals $\tau_l$. It models the conditional probability of death in an interval given survival up to the previous interval using a neural network with one hidden layer and a logistic activation function. The network inputs include time intervals and explanatory variables ($Z_i$).
• Excess Hazard Decomposition: The model estimates overall survival ($\hat{S}_{oi}$) and derives the overall hazard. The individual excess hazard ($\hat{\lambda}_{ei}$) is calculated by subtracting the population hazard (derived from life tables based on age, sex, and period) from the estimated overall hazard.
• Net Survival Calculation: Individual net survival is computed by integrating the excess hazard. Marginal net survival is then derived by weighting individual estimates.
• Comparative Framework: The study compares PLANN against:
  1. Non-parametric PP Estimator: The reference standard.
  2. Spline-based Excess Hazard Regressions: Both Proportional Hazards (PH) and Non-Proportional Hazards (NPH) models using restricted cubic splines (RCS).
• Validation Strategy: The performance was evaluated through extensive simulations (varying sample sizes, PH/NPH assumptions, and interactions) and a real-world application using a colorectal cancer cohort from Slovenia. Metrics included bias, Root Mean Square Error (RMSE), Log-Likelihood, and the Area Under the net ROC Curve (AUC). Hyper-parameters for PLANN (hidden nodes, decay, time intervals) were optimized via 10-fold cross-validation.

Key Contributions

1. Algorithmic Adaptation: The successful adaptation of PLANN for net survival, allowing for the estimation of excess hazards without strict parametric assumptions regarding hazard proportionality or linearity.
2. Software Development: The introduction of the survivalPLANN R package, which facilitates the training of the algorithm, cross-validation, and the prediction of both overall and net survival.
3. Comprehensive Evaluation: A rigorous comparison demonstrating that neural networks can capture complex data structures (non-proportional hazards and interactions) that traditional spline-based models might miss if not explicitly specified.

Results

• Bias and Accuracy: In simulation studies, PLANN produced low bias comparable to both the PP estimator and flexible spline-based models, even under non-proportional hazards and complex interaction scenarios where standard PH models failed significantly.
• Variance and RMSE: A consistent finding across simulations and the real-world application was that PLANN exhibited slightly higher RMSE values compared to spline-based methods. This indicates higher variance in the neural network estimates.
• Sample Size Sensitivity: The variance issue was exacerbated in smaller sample sizes (e.g., $n=500$). While spline-based models maintained relatively stable performance with reduced data, PLANN showed a more rapid increase in RMSE and signs of overfitting (higher training AUC than validation AUC) in smaller cohorts.
• Real-World Application: In the analysis of 5,971 Slovene colorectal cancer patients, PLANN predictions closely matched the PP estimator and were often closer to the reference than spline-based models, particularly in subgroups. However, PLANN predictions exhibited wider confidence intervals.
• Discrimination: In the real-world application, PLANN demonstrated lower discriminative capacity (AUC) compared to spline-based models, a contrast to the simulation results where AUCs were generally comparable.

Significance and Claims
The paper claims that PLANN represents a flexible alternative for net survival estimation, particularly valuable when the data structure is complex (involving non-proportional hazards or interactions) and when avoiding the loss of power associated with stratified analyses. The authors emphasize that the method's ability to relax assumptions about hazard proportionality and log-linearity is its primary advantage.

However, the authors maintain a modest stance regarding the method's limitations. They explicitly state that the higher variance of the machine learning-based estimates may limit its utility in studies with reduced sample sizes compared to traditional model-based approaches. The paper concludes that while PLANN is effective for estimation and prediction, its application should be weighed against sample size constraints, and it is particularly well-suited for high-definition cohorts with multi-modal data where complex interactions are expected but difficult to model parametrically. The authors do not claim PLANN supersedes existing methods in all contexts but rather offers a complementary tool for scenarios where model misspecification is a primary concern.