Beyond Cox models: Assessing the performance of machine-learning methods in non-proportional hazards and non-linear survival analysis

Survival analysis often relies on Cox models, assuming both linearity and proportional hazards (PH). This study evaluates machine and deep learning methods that relax these constraints, comparing their performance with penalized Cox models on a benchmark of three synthetic and three real datasets. In total, eight different models were tested, including six non-linear models of which four were also non-PH. Although Cox regression often yielded satisfactory performance, we showed the conditions under which machine and deep learning models can perform better. Indeed, the performance of these methods has often been underestimated due to the improper use of Harrell’s concordance index (C-index) instead of more appropriate scores such as Antolini’s concordance index, which generalizes C-index in cases where the PH assumption does not hold. In addition, since occasionally high C-index models happen to be badly calibrated, combining Antolini’s C-index with Brier’s score is useful to assess the overall performance of a survival method. Results on our benchmark data showed that survival prediction should be approached by testing different methods to select the most appropriate one according to sample size, non-linearity and non-PH conditions. To allow an easy reproducibility of these tests on our benchmark data, code and documentation are freely available at https://github.com/compbiomed-unito/survhive.