4.3 The median and reading the curve

The estimated median survival is the smallest time at which \(\hat S(t) \leq \frac 12\). A confidence interval for it (Brookmeyer and Crowley, 1982) consists of the times at which the pointwise interval for \(S\) contains \(\frac 12\): read it off where the lower and upper confidence curves cross \(\frac 12\). For lung the median is \(310\) days, with interval \(285\) to \(363\) days. If \(\hat S\) never falls to \(\frac 12\), the median is not estimable, a common situation when the treatment is effective or follow-up short, and survival at a fixed time is reported instead.

When reading a Kaplan–Meier plot:

each drop is an event; tick marks show censored times;
the number at risk should be shown below the axis, because the right-hand end of the curve rests on few subjects and is unreliable, however dramatic it looks;
the last step does not mean that everyone dies: if the largest time is censored, the curve stops above zero, and nothing is known beyond it.

The median follow-up time, which tells the reader how mature the data are, is best estimated by the reverse Kaplan–Meier method, in which censoring is treated as the event and deaths as censored: for lung it is \(588\) days.

dˆS0246810001173162962120mwmwa(0000000000.2.5.7388130661eoeoyt)055nmnmsaen:ent m:ried mskieandia27n04d2a6ysdays — Figure 4: Kaplan–Meier curves for men and women in the lung data, with ticks at censored times and the numbers at risk below. The dashed line at \(\frac 12\) meets the curves at the medians, \(270\) and \(426\) days. Women survive longer at almost every time (the curves meet only in the sparse tail); whether the difference could be due to chance is tested by the log-rank test of a later section.

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