2.4 Summaries: mean, median and residual life
Proof. \(T = \int _0^\infty \mathbf 1\{T > t\}\,dt\); take expectations and exchange the expectation and the integral (allowed since the integrand is non-negative). □
So the mean is the area under the survival curve. In practice the mean is often a poor summary: survival distributions are skewed, and with censoring the tail of the curve is not observed, so its area cannot be computed. The median survival time, the \(m\) with \(S(m) = \frac 12\), is preferred: it is defined as soon as half the subjects have had the event, and it is insensitive to the tail. Other percentiles are defined in the same way. For the exponential with rate \(\lambda \) the median is \(\log 2/\lambda = 0.693/\lambda \), below the mean \(1/\lambda \), reflecting the long right tail.
The mean residual life at time \(t\) is the expected further survival of a subject who has lasted to \(t\), \[ m(t) = \mathbb E[T - t \mid T > t] = \frac {\int _t^\infty S(u)\,du}{S(t)} . \] For the exponential it is constant, \(1/\lambda \); for an increasing hazard it decreases with \(t\), and for a decreasing hazard it increases: a component that has survived its burn-in period is expected to last longer than a new one.
Example 2.4.2. Adult human mortality is well described by the Gompertz hazard \(h(x) = ae^{bx}\) at age \(x\), which increases exponentially. With \(a = 5 \times 10^{-5}\) and \(b = 0.09\), the hazard doubles every \(\log 2/b = 7.7\) years. For a person alive at \(30\), the cumulative hazard to age \(x\) is \(\frac ab(e^{bx} - e^{30b})\); the probability of reaching \(60\) is \(0.892\), of reaching \(80\) is \(0.479\), and the median age at death is \(79.3\). A person alive at \(60\) reaches \(80\) with probability \(S(80)/S(60) = 0.537\), more than the \(0.479\) for a \(30\)-year-old, because the risks between \(30\) and \(60\) have already been survived (Software 2.5.3).
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