ifcensoring.pdf - Proc Nat Acad Sci USA Vol 72 No 1 pp...

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Proc. Nat. Acad. Sci. USA Vol. 72, No. 1, pp. 20-22, January 1975 A Nonidentifiability Aspect of the Problem of Competing Risks (crude survival probabilities/net survival probabilities) ANASTASIOS TSIATIS* Department of Statistics, University of California, Berkeley, Calif. 94720 Communicated by Jerzy Neyman, October 17, 1974 ABSTRACT For an experimental animal exposed to k > 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 kI "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. Theorems 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, **, kare mutually independent, a given set of crude survival prob- abilities 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 Yi, Y2, ., Ykare independent, may have no resem- blance to reality. As recently summarized by David (1), the customary treat- ment Qf competing risks is based on the model that we shall term the model of potential survival times. Consider an in- dividual living organism born at time t 0, and assume that through its lifetime it is exposed to k > 1 different "risks" or possible causes of death RI, R2, * , Rk. For i = 1, 2, ... , k let Yi denote a random variable described as the "potential survival time" of the individual in hypothetical conditions in which Ri is the only risk of death, and let Hi(t) = P { Yf > t }. The function Hi is described as the i-th net survival proba- bility or the i-th net "decrement" function. The potential survival times Y, are contrasted with the actual survival time, say X, when the individual in question is exposed to all the k > 1 competing risks, so that X = min (Y,, Y2, - * *, Yk). The function Qi(t), described as the i-th crude survival function, is defined as the probability that the individual considered will survive up to age t and then die from cause Ri. Obviously, for 21, 2, * k and t > 0, Qi(t) = P{(Yi > t) n (Yj > Yi)1. [1] joi Ordinarily, the studies of competing risks are based on empirical counterparts of the crude survival functions Qi(t), perhaps derived from observations of a cohort of experimental animals.
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