Survival functions in the presence of several events and competing risks : estimation and interpretation Beyond Kaplan-Meier

Evaluation of a therapeutic strategy is complex when the course of a disease is characterized by the occurrence of different kinds of events. Competing risks arise when the occurrence of specific events prevents the observation of other events. A particular case is semi-competing risks when only fatal events can prevent the observation of the non fatal ones. Kaplan-Meier is the most popular method to estimate overall or event free survival. On the other hand when a subset of events is considered and net survival is of concern, different estimators have been proposed. Kaplan-Meier method can be used only under the independence assumptions otherwise estimators based on multivariate distribution of times are needed. If causes of death are unknown, relative survival can approximate net survival only under specific assumptions on the mortality pattern. Kaplan-Meier method cannot be used to estimate crude cumulative incidence of specific events. The aim of this work is to present the survival functions used in competing risks framework, their non parametric estimators and semi parametric estimators for net survival based on Archimedean Copulas. This would be a help for the reader who is not experienced in competing risks analysis. A simulation study is performed to evaluate performances of net survival estimators. To illustrate survival functions in presence of different causes of death and of different kind of events a numerical example is given, a literature dataset on prostate cancer and a case series of breast cancer patients have been analysed.