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

CHAPTER 9 ST 520, A. TSIATIS and D. Zhang
The mortality rate
P(T ≥ t) − P(T ≥ t + 1)
m(t) =
P(T ≥ t)
P(T ≥ t + 1)
= 1 −
P(T ≥ t)
exp{−Λ(t + 1)}
= 1 −
exp{−Λ(t)}
� � t+1 �
= 1 − exp − λ(u)du .
Notice that if the probability of an event occurring in a single time unit is small and the hazard
rate doesn’t change quickly within that time unit, then the hazard rate is approximately the
same as the mortality rate. To see this, note that
� � t+1
m(t) = 1 − exp −
=
t
� t+1
λ(u)du ≈ λ(t).
t
t
� � � t+1 �
λ(u)du ≈ 1 − 1 − λ(u)du
t
Also, by definition, the hazard rate depends on the time scale being used. Therefore, at the same
point in time the hazard rate in days is 1/365 times the hazard rate in years.
Because of the one-to-one relationships that were previously derived, the distribution of a con-
tinuous survival time T can be defined by any of the following:
Exponential distribution
If the hazard rate is constant over time
S(t), F(t), f(t), λ(t).
λ(t) = λ, then
� � t �
S(t) = exp − λ(u)du = exp(−λt).
0
This is an exponential distribution with hazard equal to λ. Sometimes this is referred to as the
negative exponential.
It is sometimes useful to plot the log survival probability over time. This is because − log{S(t)} =
Λ(t).
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