Inference for cause-specific cox model absolute risk in cohort subsampling designs

The original case-cohort design obtains detailed covariate information on a random sample of subjects from the cohort (subcohort) and on the subjects who developed the event of interest (cases). Recently, there was some work on case-cohort estimation of pure risk, i.e., the hypothetical probability that the event occurs, assuming it is the only risk. But competing events can preclude the occurrence of the event of interest, and the pure risk thus overestimates the probability of experiencing the event of interest (absolute risk). Under the cause-specific hazard Cox model, methods for case-cohort inference have been published for relative hazards and cumulative baseline hazards; we have not seen treatments of absolute risk, however. In this work we focus on absolute risk inference under the cause-specific hazard Cox model when using a sample of subjects from the cohort. We propose an influence-based variance estimation formula and consider two sampling designs: (1) a case-cohort with exhaustive sampling of subjects who developed the event of interest or a competing event; and (2) an event-stratified sample of the cohort that only includes fractions of these subjects. Our proposed variance estimate properly accounts for the sampling features and allows appropriate analysis of the sampled data. We illustrate our method and designs in simulation and on the Prostate, Lung, Colorectal and Ovarian Cancer Screening Trial. These analyses also suggest that the "robust" variance originally proposed by Barlow (Biometrics, 50:1064-1072, 1994) may be too large for the absolute risk when using a cohort subsampling design.