Rare event approximation between subdistribution hazard ratio and cause-specific hazard ratio in survival analysis with competing risks

Background Despite the fact that competing risks are inevitable in epidemiological and clinical studies, distinctions between the hazard ratio estimated by handling competing risks as censoring and the subditribution hazard ratio are often overlooked. Methods We derive quantitative relationships between subdistribution hazard ratio and cause -specific hazard ratio, and derive an approximate calculation method to transform the two into each other. Numerical examinations of hypothetical six scenarios and published information of a randomized clinical trial of cholesterol-lowering therapy and a registry of acute myeloid leukemia were provided. Results General and approximate relationships under rare event assumptions between the two types of hazard ratio were given. The approximation formula is based on a survival ratio and has two possible applications. First, one can calculate a subdistribution hazard ratio from published information. Second, this formula allows sample size estimation that takes the presence of competing risks into account. Conclusions The distinction between the two types of hazard ratio can be addressed by focusing on two quantities. One is how the event of interest and competing risk is rare, and the other is the survival ratio.