(untitled) (Epidemiology, Biostatistics, and Public Health, 2022)
Survival analysis is a powerful statistical tool to study failure-time data. In introductory courses students learn how to describe right-censored survival time data using the product-limit estimator of the survival function on a given end-point relying on a product of conditional survival probabilities. In the case of a composite end-point, the next step is to account for the presence of competing risks. The complement to one of the survival function is decomposed into the sum of cause-specific incidences, which are obtained as sum of unconditional probabilities due to the single competing risk. However, this algebraic decomposition is not straightforward, given the difference between the structure of the involved estimators. In addition, one is tempted to use the Kaplan-Meier estimator, leading to an erroneous decomposition of the overall incidence. Here we discuss a simple reinterpretation of the Kaplan-Meier formula in terms of sum of non-conditional probabilities of developing the end-point in time, adjusted for the presence of censoring. This approach could be used for describing survival data through simple frequency tables which are directly generalized to the case of competing risks. In addition, it makes clear how the estimation of the single cause-specific incidence through the Kaplan-Meier estimator, simply considering the occurrence of competing events as censored data, leads to an overestimation of the cause-specific incidence. Two examples are provided to support the explanation: the first one, could help to clarify the procedure described by the formulas; the second one, simulates real data in order to present graphically the results.