Actual and actuarial probabilities of competing risks: Apples and lemons

The probability of a type of failure that is not inevitable, but can be precluded by other events such as death, is given by the cumulative incidence function. In cardiac research articles, it has become known as the actual probability, in contrast to the actuarial methods of estimation, usually implemented by the Kaplan-Meier (KM) estimate. Unlike cumulative incidence, KM attempts to predict what the latent failure probability would be if death were eliminated. To do this, the KM method assumes that the risk of dying and the risk of failure are independent. But this assumption is not true for many cardiac applications in which the risks of failure and death are negatively correlated (ie, patients with a higher risk of dying have a lower risk of failure, and patients with a lower risk of death have a higher risk of failure, which is a condition called informative censoring). Recent editorials in two cardiac journals have promoted the use of the KM method (actuarial estimate) for competing risk events (specifically for heart valve performance) and criticized the use of the cumulative incidence (actual) estimates. This report has two aims: to explain the difference between these two estimates and to show why the KM is generally not appropriate. In the process we will rely on alternative representations of the KM estimator (using redistribution to the right and inverse probability weighting) to explain the difference between the two estimates and to show how it may be possible to adjust KM to overcome the informative censoring.