# Ronald Geskus

Contrary to the cause-specific hazard, the subdistribution hazard uniquely determines the cumulative incidence for that cause. Its estimate forms the basis for a nonparametric product-limit type estimate of the cause-specific cumulative incidence. We derive a version using inverse probability weights to correct for right censored and left truncated data that is algebraically equivalent to the classical Aalen-Johansen estimator. Fine and Gray formulated a regression model that assumes proportionality of effects on the subdistribution hazard. When estimating the subdistribution hazard, individuals that experience a competing event remain in the risk set. Therefore, it has been debated whether it is possible to include a time-varying covariable, especially when it is internal: we don't know its value after an individual has died. In the classical survival setting with a single event type, the changing value of a covariable can be represented by creating pseudo-individuals. Each row represents a period during which the value remains constant. The start of this interval can be seen as a form of late entry; it has been called internal left truncation. We can take two different approaches when estimating the subdistribution hazard with time-varying covariables. If we interpret these rows as coming from different pseudo-individuals, we use weights to correct for the late entry. In the other approach, we consider the rows as continuing follow-up form the same individuals and therefore no such weights are used. Using a simple example of a dichotomous time-varying covariable, we contrast the interpretation of the estimates as obtained via both approaches.