  Massive sized survival datasets are becoming increasingly prevalent with the
development of the healthcare industry. Such datasets pose computational
challenges unprecedented in traditional survival analysis use-cases. A popular
way for coping with massive datasets is downsampling them to a more manageable
size, such that the computational resources can be afforded by the researcher.
Cox proportional hazards regression has remained one of the most popular
statistical models for the analysis of survival data to-date. This work
addresses the settings of right censored and possibly left truncated data with
rare events, such that the observed failure times constitute only a small
portion of the overall sample. We propose Cox regression subsampling-based
estimators that approximate their full-data partial-likelihood-based
counterparts, by assigning optimal sampling probabilities to censored
observations, and including all observed failures in the analysis. Asymptotic
properties of the proposed estimators are established under suitable regularity
conditions, and simulation studies are carried out to evaluate the finite
sample performance of the estimators. We further apply our procedure on
UK-biobank colorectal cancer genetic and environmental risk factors.
