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

CHAPTER 9 ST 520, A. TSIATIS and D. Zhang
To illustrate how this estimator is constructed, consider the following example:
Figure 9.4: An illustrative example of Kaplan-Meier estimator
1.0
0.8
0.6
0.4
0.2
0.0
x
4.5
x
7.5
o x
11.5
o x x o x
13.5
Patient time (years)
15.516.5
1 − ˆm(x) : 1 1 1 1 9
10 1 1 8
9 1 1 1 6
7 1 1 1 4
5
ˆS(t) : 1 1 1 1 9
10 . . 8
10 . . . 48
70 . . . 192
350
19.5
o
3
4 1 1 1
2 1 1
144 144 . . . .
350 700
m = d/nr =
number of deaths in an interval
number at risk at beginning of interval
= (1/nr or 0 depending on whether or not a death occurred in interval)
(1 − m) = (1 − d/nr) = ((1 − 1/nr) or 1).
In the limit, the Kaplan-Meier (product-limit) estimator will be a step function taking jumps at
times where a failure occurs. Therefore at any time t, the product-limit estimator of the survival
distribution is computed as the product
�
all deaths
�
�
1
1 −
number at risk
over all death times occurring up to and including time t.
By convention, the Kaplan-Meier estimator is taken to be right-continuous.
Non-informative Censoring
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