Biostatistics

14.2 TIME-TO-EVENT DATA AND CENSORING 755
The probability distribution function, just as defined in Chapter 4, is represented by
the set of probabilities that specify the possible values of a random variable. In the context
of survival analysis, this density function represents the probability of an event occurring
in a defined interval of time. We might ask, for example, what is the probability of
surviving 2 months? Although fully appreciating the intricacies of this probability
distribution requires knowledge of calculus, we can illustrate its meaning conceptually
by remembering a concept from our discussion of the normal distribution in Chapter 4.
When we calculated probabilities for the normal distribution, we were interested in
calculating the area under a curve that was bounded by two values. Similarly, in survival
analysis we are interested in calculating the probability of an event bounded by an interval
of time, say Dt, and then finding our probability as the interval becomes very small, that is
as Dt ! 0. Hence, the probability distribution function, f ðtÞ, is defined by
f ðtÞ ¼
Pðt T < t þ DtÞ
; as Dt ! 0 (14.2.3)
Dt
That is, the set of probabilities of events that occur in an infinitesimally small interval of
time defines the probability function. It is also possible to find this function by examining
what happens during a change in FðtÞ, say DFðtÞ, or a change in S(t), say DSðtÞ, in a given
interval of time. That is
f ðtÞ ¼ DFðtÞ
Dt
¼
DSðtÞ
Dt
(14.2.4)
Finally, a function that is often encountered in survival analysis is the hazard function,
hðtÞ. This function is used to define the instantaneous probability of an event occurring given
that the subject has survived up to a given time, t. This function is defined as
hðtÞ ¼
Pðt T < t þ DtjT tÞ
; as Dt ! 0 (14.2.5)
Dt
Note that this function is based on a conditional probability, wherein we are interested in
calculating the probability of an event occurring given that the subject has already survived
to a defined time. The condition of having already survived to a given time means that the
probability of surviving into the future is influenced by having already survived previous
time periods. This idea can be very important in some instances, where surviving the early
stages of a disease may dramatically decrease the potential of an event occurring in the near
future. As an example, consider cancer where nonrecurrence, or remission, for a period of
5 years generally increases survivorship. This function can also be expressed in terms of
two functions previously defined. This expression is
hðtÞ ¼ f ðtÞ
SðtÞ
(14.2.6)
Because the hazard function can exceed 1, it is not truly a probability, though it is based on
the conditional probability of an event occurring. The hazard function is often defined in
survival analysis by a known distribution such as the lognormal, exponential, or Weibull