The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

164 CHAPTER 10 Survival Analysis

This is the Cutler – Ederer method, and just about any life table is gener-

ated in a very similar fashion.

10.3 KAPLAN – MEIER CURVES

The Kaplan – Meier curve is a nonparametric estimate of the survival

function (see Kaplan and Meier 1958 ). It is computed using the same

conditioning principle that we used for the life table. However, here we

estimate the survival at every time point, but only do the iterative com-

putations at the event or censoring times. The estimate is taken to be

constant between points. It has sometimes been called the product limit

estimator, because at each time point, it is calculated as the product of

conditional probabilities. Next, we describe in detail how the curve is

estimated.

10.3.1 The Kaplan – Meier Curve: A Nonparametric

Estimate of Survival

For all time from 0 to t 1 , where t 1 is the time of the fi rst event, the

Kaplan – Meier survival estimate is S (^) km ( t ) = 1. At time t 1 , S (^) km (t 1 ) = S (^) km (0)
( n 1 − D 1 )/ n 1 , where n 1 is the total number at risk, and D 1 is the number
that die (have an event) at time t 1. Since S (^) km (0) = 1, S (^) km ( t 1 ) = ( n 1 − D 1 )/ n 1.
For the example in Table 10.3 , below we see that S (^) km ( t 1 ) =
Table 10.3
Kaplan – Meier Survival Estimates for Example in Table 10.1
Time No. of
deaths
in D j
No.
withdrawals
W j
No. at
risk n j
Est.
prop. of
deaths q j
Est. prop.
surviving
p j = 1 − q j
Est.
cumulative
survival
S (^) km ( t j )
t 1 = 1.5 1 0 10 0.1 0.9 0.9
t 2 = 4.3 1 1 9 0.125 0.875 0.788
t 3 = 5.4 1 0 6 0.143 0.857 0.675
t 4 = 11.8 1 0 6 0.167 0.833 0.562
18 > t > 11.8 0 5 5 0 1.0 0.562
t ≥ 18 0 0 0 0 — —