Kaplan-Meier analysis is a popular method used for analysing time-to-event data. In case of competing event analyses such as that of cardiovascular and non-cardiovascular mortality, however, the Kaplan-Meier method profoundly overestimates the cumulative mortality probabilities for each of the separate causes of death. This article provides an introduction to the problem of competing events in Kaplan-Meier analysis. It explains cumulative incidence competing risk analysis and demonstrates on a cohort of elderly dialysis patients that, in contrast to the Kaplan-Meier method, application of this method yields unbiased estimates of the cumulative probabilities for cause-specific mortality.