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

CHAPTER 9 ST 520, A. TSIATIS and D. Zhang
9.1 Describing the Distribution of Time to Event
We will describe some different, but equivalent, ways to define the distribution of the random
variable, T, “time to event.”
• The distribution function:
• The survival function:
F(t) = P(T ≤ t);
S(t) = P(T ≥ t);
The right-continuous version of the survival function will be denoted by
S(t − ) = P(T > t) = 1 − F(t).
Remark: For the most part, we will assume that T is a continuous random variable in which
case S(t − ) = S(t) = 1 − F(t). We will also assume that T has a density function
Clearly:
and
Hazard rate
f(t) = dF(t)
dt
= −dS(t) .
dt
� t
F(t) = f(u)du,
0
� ∞
S(t) = f(u)du.
t
The hazard rate is a useful way of defining the distribution of a survival time which can also be
used to describe the aging of a population. We motivate the definition of a hazard rate by first
introducing “mortality rate” or discrete hazard rate.
The mortality rate at time t (where t is usually taken to be an integer of some unit of time; i.e.
day, week, month, year, etc.) is the proportion of the population who fail between times t and
(t + 1) among individuals alive (who have not failed) at time t.
m(t) = P(t ≤ T < t + 1|T ≥ t).
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