A nonidentifiability aspect of the problem of competing risks

For an experimental animal exposed to k greater than 1 possible risks of death R1, R2, ..., Rk, the term i-th potential survival time designates a random variable Yi supposed to represent the age at death of the animal in hypothetical conditions in which Ri is the only possible risk. The probability that Yi will exceed a preassigned t is called the i-th net survival probability. The results of a survival experiment are represented by k "crude" survival functions, empirical counterparts of the probabilities Qi(t) that an animal will survive at least up to the age t and eventually die from Ri. The analysis of a survival experiment aims at estimating the k net survival probabilities using the empirical data on those termed crude. Therorems 1 and 2 establish the relationship between the net and the crude probabilities of survival. In particular, Theorem 2 shows that, without the not directly verifiable assumption that in their joint distribution the variables Y1, Y2, ..., Yk are mutually independent, a given set of crude survival probabilities Qi(t) does not identify the corresponding net probabilities. An example shows that the results of a customary method of analysis, based on the assumption that Y1, Y2, ..., Yk are independent, may have no resemblance to reality.