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Statistical Methods for Survival Data Analysis Third Edition phần 4

Tỷ lệ sống tương đối được định nghĩa là tỷ lệ tỷ lệ sống (xác suất sống sót một năm) cho một bệnh nhân được nghiên cứu (tỷ lệ quan sát) cho một người nào đó trong dân số chung của cùng độ tuổi, giới tính và chủng tộc (dự kiến tỷ lệ) | GAMMA AND GENERALIZED GAMMA DISTRIBUTIONS 149 Figure 6.13 Lognormal probability plot of the survival time in months plus 4 of 162 male patients with chronic myelocytic leukemia. From Feinleib and MacMahon 1960. Reproduced by permission of the publisher. Suppose that failure or death takes place in n stages or as soon as n subfailures have happened. At the end of the first stage after time Tj the first subfailure occurs after that the second stage begins and the second subfailure occurs after time T2 and so on. Total failure or death occurs at the end of the nth stage when the nth subfailure happens. The survival time T is then TI I I . I I I h fitnoo I I I cnonf in ooun cforro oro oooiimoid 1 1 -r T 2 T r T1. The times T 12 . Ifi spent in each stage are assumed to Figure 6.14 Lognormal probability plot of the survival time in months plus 4 of female patients with two types of leukemia. From Feinleib and MacMahon 1960. Reproduced by permission of the publisher. 150 WELL-KNOWN PARAMETRIC SURVIVAL DISTRIBUTIONS be independently exponentially distributed with probability density function A exp ẰtỊ i 1 . n. That is the subfailures occur independently at a constant rate A. The distribution of T is then called the Erlangian distribution. There is no need for the stages to have physical significance since we can always assume that death occurs in the n-stage process just described. This idea introduced by A. K. Erlang in his study of congestion in telephone systems has been used widely in queuing theory and life processes. A natural generalization of the Erlangian distribution is to replace the parameter n restricted to the integers 1 2 . by a parameter y taking any real positive value. We then obtain the gamma distribution. The gamma distribution is characterized by two parameters y and A. When 0 y 1 there is negative aging and the hazard rate decreases monotonically from infinity to Ằ as time increases from 0 to infinity. When y 1 there is positive aging and the hazard .