July 29, 2026
The authors published this second article as a two part series about exploring novel approaches to predicting survival outcomes and evaluating model performance. They also provide R code to help in the implementations of these. They then go through and discuss several survival methods and machine learning methods.
Regularized Cox models are Cox proportional hazard models with a form of regularization which aims to reduce complexity and avoid overfitting. One of these is least absolute shrinkage and selection operator (LASSO), which does variable selection by shrinking some coefficients all the way to 0, which removes the variable from the model. Decision trees are ML algorithms that predict outcomes based on a sequence of binary decision-based rules, which result in subgroups of people with mostly homogeneous outcomes. The top of the tree or root node is the variable that best divides the population into two with respect to the outcome. The choice of variable is based on the split that can produce the greatest improvement in purity (how well the split separates subgroups by outcome). Purity is assessed by different statistical criteria depending on the type of tree: log-rank test for survival trees, information gain, or Gini impurity for classification trees, and finally variance reduction for regression trees.
Random forests are ensemble algorithms which extend decision trees from multiple trees in order to improve the predictive performance. They use two sources of randomness to reduce correlation between trees to reduce correlations between trees which then can reduce overfitting and improve generalization. A decision tree gets fitted to each bootstrap sample and in random feature selection, only a random subset of predictors is considered at each spit. At each terminal model, the CHF is estimated using only the survival data that has reached that node. Boosted trees are another ensemble algorithm but these trees are created sequentially with each new tree improving predictions on the previous one until a stopping criterion if met and final predictions are generated by combining all trees using a weighted approach. Boosting also can be adapted by combining it with a Cox model (R CoxBoost function, mboost (model-based boosting), GBM (gradient boosting)). Neural networks or deep learning process information through multiple layers. They require hyperparameters to be fine-tuned before use. They had been more recently extended to survival analysis. In continuous-time modeling, the Cox proportional hazards model is augmented by using a neural network to transform the baseline hazard or directly model the log-risk score, minimizing a loss function derived from the Cox partial log likelihood. While in discrete-time modeling, follow-up time is divided into intervals, and a neural network estimates the probability of surviving each interval. These methods can be implemented in Python via the pycox package, and can also be run in R using the survivalmodels package (through reticulate, which interfaces R with Python).
They then talked about performance metrics explicitly designed to deal with censored data after running different survival models or ML algorithms on the same dataset. These can be either to check discrimination or calibration. They didn’t quite define discrimination properly; this should be the ability of the model to correctly separate those who have the event or outcome from those who did not have the event or outcome. Also, calibration should be how well the predictions from the model closely match the observed observations. They calculated observed-to-expected ratios and calibration slopes as well as visual calibration curves. They then spoke about the concordance index (c-index or c-statistic) which is used to measure discrimination. The Brier score is another measure that reflects both discrimination and calibration. It compares an individual predicted survival with the observed one as the squared mean difference of this across all patients. Lower Brier scores indicate better performance. They also mentioned an Integrated Brier Score which extends the Brier score by summarizing it across the entire follow-up period rather than just reporting it at a specific event time. It becomes the weighted mean of all time-specific Brier scores, where the weights account for censoring to ensure the meaningful risk sets stay representative over time.
They found when running their analyses that the random forests did well in terms of Brier score but the Cox model performed better than several of the ML algorithms when compared by the c-statistic. They attribute this to that ML algorithms work better in larger datasets to achieve stable and well-calibrated datasets, while the traditional models, like Cox model, can work provide more stable estimation in smaller datasets since they have more structure. They then listed the advantages and disadvantages of the ML methods considered. Of noteworthy interest regarding neural networks is that while they may have strong predictive performance, they are a “black box which is not interpretable, computationally demanding, and require larger datasets. However their conclusion is that while traditional survival methods will remain appropriate for many datasets, when predictive accuracy is the primary goal than alternative survival strategies should be considered.
Written by,
Usha Govindarajulu, MS PhD
Keywords: survival analysis, machine learning (ML), neural networks, random forests, Brier score, c-statistic, calibration
References:
Ryan Gately, Dharshana Sabanayagam, Wai H. Lim, Lin Zhu, Farzaneh Boroumand, Shuvo Bakar, Armando Teixeira-Pinto, Germaine Wong (2026) “Advances in survival analyses: machine learning methods and model comparison”, Kidney International, Volume 110, Issue 1,
Pages 78-88,ISSN 0085-2538, https://doi.org/10.1016/j.kint.2026.02.041.
https://www.sciencedirect.com/science/article/pii/S0085253826003017