Penalized weighted proportional hazards model for robust variable selection and outlier detection.

Identifying exceptional responders or nonresponders is an area of increased research interest in precision medicine as these patients may have different biological or molecular features and therefore may respond differently to therapies. Our motivation stems from a real example from a clinical trial where we are interested in characterizing exceptional prostate cancer responders. We investigate the outlier detection and robust regression problem in the sparse proportional hazards model for censored survival outcomes. The main idea is to model the irregularity of each observation by assigning an individual weight to the hazard function. By applying a LASSO-type penalty on both the model parameters and the log transformation of the weight vector, our proposed method is able to perform variable selection and outlier detection simultaneously. The optimization problem can be transformed to a typical penalized maximum partial likelihood problem and thus it is easy to implement. We further extend the proposed method to deal with the potential outlier masking problem caused by censored outcomes. The performance of the proposed estimator is demonstrated with extensive simulation studies and real data analyses in low-dimensional and high-dimensional settings.