Estimating the incidence of an event as a function of follow-up time provides important information on the absolute risk of an event. In the absence of competing risks, the Kaplan-Meier estimate of the survival function is frequently used for estimating the survival function. One minus the Kaplan-Meier estimate of the survival function provides an estimate of the cumulative incidence of events over time. In the case study that follows, we examine the incidence of cardiovascular death in patients hospitalized with heart failure. When the complement of the Kaplan-Meier function was used, the estimated incidence of cardiovascular death within 5 years of hospital admission was 43.0%. However, using the Kaplan-Meier estimate of the survival function to estimate the incidence function in the presence of competing risks generally results in upward biases in the estimation of the incidence function.9 (link),10 (link),12 (link) In particular, the sum of the Kaplan-Meier estimates of the incidence of each individual outcome will exceed the Kaplan-Meier estimate of the incidence of the composite outcome defined as any of the event types. Even when the competing events are independent, the Kaplan-Meier estimator yields biases in the probability of the event of interest. The problem here is that the Kaplan-Meier estimator estimates the probability of the event of interest in the absence of competing risks, which is generally larger than that in the presence of competing risks. Furthermore, the hypothetical population in which competing risks do not exist may not be the population of greatest interest for clinical and/or policy making,13 (link) as in the cardiovascular setting where noncardiovascular death may be an important consideration.
The Cumulative Incidence Function (CIF), as distinct from 1 – S(t), allows for estimation of the incidence of the occurrence of an event while taking competing risk into account. This allows one to estimate incidence in a population where all competing events must be accounted for in clinical decision making. The cumulative incidence function for the kth cause is defined as: CIFk(t) = Pr(T ≤ t,D = k), where D is a variable denoting the type of event that occurred. A key point is that, in the competing risks setting, only 1 event type can occur, such that the occurrence of 1 event precludes the subsequent occurrence of other event types. The function CIFk(t) denotes the probability of experiencing the kth event before time t and before the occurrence of a different type of event. The CIF has the desirable property that the sum of the CIF estimates of the incidence of each of the individual outcomes will equal the CIF estimates of the incidence of the composite outcome consisting of all of the competing events. Unlike the survival function in the absence of competing risks, CIFk(t) will not necessarily approach unity as time becomes large, because of the occurrence of competing events that preclude the occurrence of events of type k. In the case study that follows, when using the CIF, the estimated incidence of cardiovascular death within 5 years of hospital admission was 36.8%. This estimate was 6.2% lower than the estimate obtained using the complement of the Kaplan-Meier function. This illustrates the upward bias that can be observed when naively using Kaplan-Meier estimate in the presence of competing risks.