Mastering rare event analysis: subsample-size determination in Cox and logistic regressions

In the realm of contemporary data analysis, the use of massive datasets has taken on heightened significance, albeit often entailing considerable demands on computational time and memory. While a multitude of existing works offer optimal subsampling methods for conducting analyses on subsamples with minimized efficiency loss, they notably lack tools for judiciously selecting the subsample size. To bridge this gap, our work introduces tools designed for choosing the subsample size. We focus on three settings: the Cox regression model for survival data with rare events, and logistic regression for both balanced and imbalanced datasets. Additionally, we present a new optimal subsampling procedure tailored to logistic regression with imbalanced data. The efficacy of these tools and procedures is demonstrated through an extensive simulation study and meticulous analyses of two sizable datasets: survival analysis of UK Biobank colorectal cancer data with about 350 million rows and logistic regression of linked birth and infant death data with about 28 million observations.