SURVIVAL MODEL AND ESTIMATION FOR LUNG CANCER PATIENTS - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 204

The survival function gives the probability of being alive at duration t .
Naturally, when t =0, S ( t )=1and t

0.
An alternative characterization of the distribution of T is given by the
hazard function . Sometimes it is also called the force of mortality ,
the mortality intensity function, or the failure rate. The hazard function
is the probability that an individual will experience an event (for example,
death) within a small time interval, given that the individual has survived
up to the beginning of the interval. It can therefore be interpreted as the
instantaneous risk of occurrence of dying at time t . The hazard function
h ( t ) can be estimated using the following equation:

→∞ , S ( t ) →

≤ |

Pr [ t<T t +∆ t T>t ]

h ( t ) = lim
∆ t → 0

. (2 . 3)

∆ t

The numerator of this expression is the conditional probability that the
event will occur in the interval ( t, t +∆ t ) given that it has not occurred
before, and the denominator is the width of the interval. We obtain a rate
of event occurrence per unit of time. Taking the limit as the width of the
interval decreases to zero, we obtain an instantaneous rate of occurrence.
The conditional probability in the numerator may be written as the
ratio of the joint probability that T is in the interval ( t, t +∆ t )and T>t
(which is, of course, the same as the probability that t is in the interval),
to the probability of the condition T>t . The former may be written as
f ( t )∆ t for small ∆ t , while the latter is S ( t ) by definition. Dividing by dt
and passing to the limit gives the useful result

F ( t )
S ( t )

S ( t ))
S ( t )

S ( t )
S ( t ) .

h ( t )= f ( t ) = (1

−

S ( t ) =

= − (2 . 4)

This equation suggests the relationship between the survival function and
the hazard function. That is, the rate of occurrence of the event at duration
t equals the density of events at t divided by the probability of surviving to
that duration without experiencing the event. Furthermore, equation (2.4)
suggests that

d
dt log S ( t ) ,

h ( t )= − (2 . 5)

then

t

log S ( t )= − h ( z ) dz + C. (2 . 6)

0