ABSTRACT
Survivor function, S(t): Dened as the probability that the survival time, T, is greater than or equal to t, i.e.,
S t T t( ) = >( )Pr
Survivor function for the exponential distribution and the Weibull distribution: First the exponential with probability density function
for which the survivor function is given by
Now for the Weibull probability density function which is given by
The survivor function of the Weibull is given by
S t u u du t t
( ) exp( ) exp( )= − = −− ∞∫ λγ λ λγ γ γ1
Plots of the survivor function of the Weibull distribution for different values of the scale parameter, λ, and the shape parameter, γ, are shown in Figure 8.2. Estimating the survivor function: When there is no censoring, the survivor function can be estimated as
ˆ( )S t
n n
nt t= =; number of individuals with survival times
the number of individuals
≥
=
t
n
;
in the data set
ˆ( )
S t d
r j
= −
where rj is the number of individuals at risk just before t( j) (including those censored at t( j)) and dj is the number of individuals who experience the terminal event at t( j). For example, the survivor function at the second death time, t(2), is equal to the estimated probability of not dying at t(1) multiplied by the estimated probability, given the individual is still at risk at time t(2), of not dying at time t(2).
