Applied Statistics Using SPSS, STATISTICA, MATLAB and R

9.2 Non-Parametric Analysis of Survival Data 363
2. Median and percentiles of survival time.
Since the density function of the survival times, f(t), is usually a positively skewed
function, the median survival time, t0.5, is the preferred location measure. The
median can be obtained from the survivor function, namely:
F(t0.5) = 0.5 ⇒ S(t0.5) = 1 – 0.5 = 0.5. 9.14
When using non-parametric estimates of the survivor function, it is usually not
possible to determine the exact value of t0.5, given the stepwise nature of the
estimate S ˆ( t)
. Instead, the following estimate is determined:
ˆ 5
{ t ; Sˆ
( t ) ≤ 0.
5}
t 0.
= min i i . 9.15
Percentiles p of the survival time are computed in the same way:
{ t ; Sˆ
( t ) ≤ − p}
ˆ = 1 . 9.16
t p min i i
3. Confidence intervals for the median and percentiles.
Confidence intervals for the median and percentiles are usually determined
assuming a normal distribution of these statistics for a sufficiently large number of
cases (say, above 30), and using the following formula for the standard error of the
percentile estimate (for details see e.g. Collet D, 1994 or Kleinbaum DG, Klein M,
2005):
1 [ tˆ
] s[
Sˆ
( tˆ
) ]
s p = p , 9.17
fˆ
( tˆ
)
p
where the estimate of the probability density can be obtained by a finite difference
approximation of the derivative of S ˆ( t)
.
Example 9.6
Q: Determine the 95% confidence interval for the survivor function of Example
9.3, as well as for the median and 60% percentile.
A: SPSS produces an output containing the value of the median and the standard
errors of the survivor function. The standard values of the survivor function can be
used to determine the 95% confidence interval, assuming a normal distribution.
The survivor function with the 95% confidence interval is shown in Figure 9.5.
The median survival time of the specimens is 100×10 4 = 1 million cycles. The
60% percentile survival time can be estimated as follows:
ˆ 6
{ t ; Sˆ
( t ) ≤ 1−
0.
6}
t 0.
= min i i .