ST 520 Statistical Principles of Clinical Trials - NCSU Statistics ...

CHAPTER 9 ST 520, A. TSIATIS and D. Zhang
Figure 9.1: A typical mortality pattern for human
m(t)
0 20 40 60 80 100
age (years)
In a human population, the mortality rate has a pattern like
The hazard rate λ(t) is the limit of the mortality rate or the instantaneous rate of failure at time
t given the individual is alive at time t. That is,
This can be expressed as
� �
P(t ≤ T < t + h|T ≥ t)
λ(t) = lim
.
h→0 h
λ(t) = lim
h→0
= −dS(t)
dt
S(t)
� �
P(t ≤ T < t + h)/h
P(T ≥ t)
−d log{S(t)}
= .
dt
Integrating both sides of the equation above, we get
� t
− log{S(t)} = λ(u)du = Λ(t),
0
= f(t)
S(t)
where Λ(t) is defined as the cumulative hazard function. Consequently,
� � t �
S(t) = exp − λ(u)du
0
= exp {−Λ(t)}.
Note: Although the mortality rate is a probability, the hazard rate is NOT a probability; thus
it can take on any positive value unlike a mortality rate which must be bounded by 1.
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