On concerns with cause-specific incidence and subdistribution hazard

An event that hinders or changes the possibility of observing the event of interest is called a competing risk. For instance, clinical studies for those with multimorbidity or critically severe illnesses often require consideration of competing risks, as the occurrence of other events may preclude the primary event of interest. Since the one-to-one correspondence between the cause-specific hazard and the cause-specific incidence is lost in the classical competing risk model (Andersen et al., International Journal of Epidemiology 41:861–870, 2012), the Fine-Gray hazard was developed. The cause-specific incidence and the Fine-Gray hazard have been widely used and have become the default methods in competing risk analysis. The cause-specific incidence, however, often fails to represent treatment effects properly due to failures of the competing causes, particularly when the hazards of the competing causes are considerably different between the control and treatment arms. This study introduces a new incidence termed, cause-distinct incidence, which has a one-to-one correspondence with the cause-specific hazard and is less affected by competing causes than the cause-specific incidence. Whilst, recent studies have found unexpected effects of censors on the Fine-Gray hazard analysis. For instance, the estimation of the censoring distribution can affect the accuracy or censoring can complicate the estimation. The root cause of these unexpected phenomena has been uninvestigated. The basic requirement for the hazard is that it be independent of the distribution of independent censoring. Nevertheless, this study verifies mathematically and also numerically that the Fine-Gray hazard depends on the distribution of independent censoring.