Mean vs Median Survival Time in Kaplan-Meier estimate
I've performed a Kaplan-Meier or stratified Kaplan-Meier analysis and in my output, a Mean Survival Time is reported, but there is no corresponding Median Survival Time; why is this?
Resolving the problem
For this sample or stratum, the estimated survival probability must never have reached 50%, that is, the survival step function does not cross the line y=.5. The Kaplan-Meier estimate, especially since it is a non-parametric method, makes no inference about survival times (i.e., the shape of the survival function) beyond the range of times found in the data. Mean survival time, on the other hand, is a statement about the observed times. It shouldn't be taken to mean the length of time a subject can be expected to survive.
A look at the definitions of the mean and median survival times in the Statistical Algorithms manual may help. With t1 < t2 < ... < tk representing the times of observed deaths, and S_hat(t) representing the Kaplan-Meier estimate of the survival function,
the median survival time is defined as
(1) MIN ( ti such that S_hat(ti) <= .5 ) ;
but if S_hat(ti) never reaches .5, the set we are taking the minimum over is null and so the median is necessarily undefined. The mean survival time, on the other hand, is defined as
SUM ( S_hat(ti)(ti+1 - ti) )
if the longest observed survival time is for a case that is not censored; if that longest time TL is for a censored observation, we add S_hat(tk)(TL - tk) to the above sum. From this expression, it is easy to see that the mean survival time is the area under the survival step function when it is plotted. Unlike the case of the median, there is no problem with this number being mathematically well-defined.
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