Biostatistics

756 CHAPTER 14 SURVIVAL ANALYSIS
distribution. Excellent descriptions of the various models used to represent hazard functions
are provided by Allison (4) and Kleinbaum and Klein (1).
14.3 THE KAPLAN–MEIER PROCEDURE
Now let us show how we may use the data usually collected in follow-up studies of the type
we have been discussing to estimate the probability of surviving for a specified length of
time. The method we use was introduced by Kaplan and Meier (5) and for that reason is
called the Kaplan–Meier procedure. Since the procedure involves the successive multiplication
of individual estimated probabilities, it is sometimes referred to as the product-limit
method of estimating survival probabilities.
As we shall see, the calculations include the computations of proportions of subjects in a
samplewho survive for various lengths of time. We use these sample proportions as estimates of
the probabilities of survival that we would expect to observe in the population represented by
our sample. In mathematical terms we refer to the process as the estimation of a survivorship
function. Frequency distributions and probability distributions may be constructed from
observed survival times, and these observed distributions may show evidence of following
some theoretical distribution of known functional form. When the form of the sampled
distribution is unknown, it is recommended that the estimation of a survivorship function be
accomplished by means of a nonparametric technique, of which the Kaplan–Meier procedure
is one. Nonparametric techniques are defined and discussed in detail in Chapter 13.
Calculations for the Kaplan–Meier Procedure
We let
n ¼ the number of subjects whose survival times are available
p 1 ¼ the proportion of subjects surviving at least the first time period
(day, month, year, etc.)
p 2 ¼ the proportion of subjects surviving the second time period
after having survived the first time period
p 3 ¼ the proportion of subjects surviving the third time period
after having survived the second time period
.
.
p k ¼ the proportion of subjects surviving the kth time period
after having survived the ðk 1Þth time period
We use these proportions, which we may relabel ^p 1 ; ^p 2 ; ^p 3 ; ...; ^p k as estimates of the
probability that a subject from the population represented by the sample will survive time
periods 1, 2, 3, . . . , k, respectively.
For any time period, t, where 1 t k, we estimate the probability of surviving the
tth time period, p t , as follows:
number of subjects surviving at least t 1
^p t ¼ ð Þtime periods who also survive the tth period
number of subjects alive at end of time period ðt 1Þ
(14.3.1)