Basic Statistics

212 INTRODUCTION TO SURVIVAL ANALYSIS

Table 14.3 Computations for the Kaplan-Meier Method

Days Deaths Censored nobs (nObs - d)/nobs s(t)

2 1 8 (8- 1)/8 = ,875 ,875

3 1 7 (7- 1)/7= ,857 ,750

3 1 6

4 2 5 (5-2)/5 = .600 ,450

5 1 3

7 1 2 (2 - 1)/2 = ,500 ,225

9 1 1 (1-l)/l= .ooo 0

The numerical estimate of the hazard function for the second interval would be
15/(.5[29.5 - 15/21) = 1.364. The remaining intervals are computed in the same
fashion.
If a statistical program is used to compute the life table, it will probably contain
other output. This might include standard errors of the estimates of the death den-
sity, survival function, and hazard function (see van Belle et al. [2004] or Gross et
al. [1975]). Also, some programs will compute the 50% survival time from the col-
umn labeled S(t) by interpolation. For example, in Table 14.2 we know that at the
beginning of the second interval .797 have survived, and at the beginning of the third
interval .392 have survived. Thus, at some time between .5 and l.Oyears, 0.50 or
50% of the patients survived. This can be computed using the linear interpolation
formula given in Section 6.2.2. Note that Stata, SPSS, and SAS provide survival
analysis programs.

14.3.2 Kaplan-Meier Estimate

In Section 14.3.1 the computations for the clinical life table were presented. Note that
in making these tables we grouped the data into .5-year intervals even though we knew
the number of days each patient lived. The Kaplan-Meier or product limit method of
estimating the survival function uses the actual length of time to the outcome event
such as death, or to censoring, due to loss to follow-up or withdrawn alive from the
study.
As might be expected, the Kaplan-Meier method is considerably more work to
compute by hand if the sample size is large, so statistical programs are generally
used to perform the computations and graph the results. Here, we illustrate the
computations with a small example.
A study was made of 8 patients who were admitted to a hospital with a life-
threatening condition. The outcome event is death. Two patients were transferred
from the hospital before they died and they are considered to be censored at hospital
discharge. The first step in analyzing the data is to order the observations from
smallest to largest. The ordered times were 2, 3, 3', 4, 4, 5', 7, and 9days. The c
indicates that the third and sixth patients were censored.