  A prevalent feature of high-dimensional data is the dependence among
covariates, and model selection is known to be challenging when covariates are
highly correlated. To perform model selection for the high-dimensional Cox
proportional hazards model in presence of correlated covariates with factor
structure, we propose a new model, Factor-Augmented Regularized Model for
Hazard Regression (FarmHazard), which builds upon latent factors that drive
covariate dependence and extends Cox's model. This new model generates
procedures that operate in two steps by learning factors and idiosyncratic
components from high-dimensional covariate vectors and then using them as new
predictors. Cox's model is a widely used semi-parametric model for survival
analysis, where censored data and time-dependent covariates bring additional
technical challenges. We prove model selection consistency and estimation
consistency under mild conditions. We also develop a factor-augmented variable
screening procedure to deal with strong correlations in ultra-high dimensional
problems. Extensive simulations and real data experiments demonstrate that our
procedures enjoy good performance and achieve better results on model
selection, out-of-sample C-index and screening than alternative methods.
