The PHREG Procedure

Failure Time Distribution

Let T be a nonnegative random variable representing the failure time of an individual from a homogeneous population. The survival distribution function (also known as the survivor function) of T is written as

\[ S(t)=\mr{Pr}(T \geq t) \]

A mathematically equivalent way of specifying the distribution of T is through its hazard function. The hazard function $\lambda (t)$ specifies the instantaneous failure rate at t. If T is a continuous random variable, $\lambda (t)$ is expressed as

\[ \lambda (t)= \lim _{\Delta t \rightarrow 0^{+} } \frac{ \mr{Pr}(t \leq T < t + \Delta t\ |\ T \geq t) }{ \Delta t } = \frac{f(t)}{S(t)} \]

where $f(t)$ is the probability density function of T. If T is discrete with masses at $x_{1} < x_{2} < \ldots $ , then survivor function is given by

\[ S(t)= \sum _{x_ j\leq t} \mr{Pr}(T=x_ j) = \sum _{j} \mr{Pr}(T=j) \delta (t- x_{j}) \]

where $ \delta (u)$=0 if u < 0 and $\delta (u)$=1 otherwise. The discrete hazards are given by

\[ \lambda _{j}=\mr{Pr}(T=x_{j}\ |\ T \geq x_{j})=\frac{ \mr{Pr}(T=x_{j}) }{S(x_{j}) } \mbox{~ ~ ~ } j=1, 2, \ldots \]