1.3 Why ordinary methods fail

Suppose we want the mean survival time. Two tempting shortcuts are both wrong.

• Averaging all the recorded times, censored or not, underestimates the mean, because censored times are shorter than the true times.
• Averaging only the uncensored times underestimates it even more, because the subjects who die during the study are disproportionately those with short times; the long survivors are the ones most likely to be censored.

In a simulation of \(10\,000\) exponential lifetimes with mean \(10\), censored at uniform times between \(0\) and \(20\), the first shortcut gives \(5.6\) and the second \(4.7\) (Software 1.5.2). For exponential data the right estimate of the event rate is the number of events divided by the total time at risk, \(\hat \lambda = d/\sum t_i\), and its reciprocal gives \(9.8\); the section on parametric models shows why this is the maximum likelihood estimate. The general lesson is that a censored time carries real information, that the subject survived at least that long, and methods must use it rather than discard it or treat it as an event.

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