5.3 Life tables
When only grouped data are available, or when the number of subjects is very large, survival is estimated by intervals. Divide time into intervals \([a_{i - 1}, a_i)\); for each record the number \(n_i\) entering the interval, the number \(d_i\) dying in it and the number \(c_i\) censored in it. The censored subjects were at risk, on average, for about half the interval, so the effective number at risk is taken as \(n_i' = n_i - c_i/2\). The actuarial estimates are then \[ \hat q_i = \frac {d_i}{n_i'}, \qquad \hat p_i = 1 - \hat q_i, \qquad \hat S(a_i) = \prod _{l \leq i}\hat p_l , \] the probability of dying in the interval for those entering it, of surviving it, and of surviving to its end. The hazard in the interval can be estimated by \(d_i/\bigl ((n_i' - d_i/2)(a_i - a_{i - 1})\bigr )\), deaths per unit of time at risk.
Example 5.3.1. Grouping lung into quarter-years of \(91\) days: of the \(228\) patients, \(27\) die in the first quarter and none is censored, so \(\hat q_1 = 0.118\) and \(\hat S(91) = 0.882\). In the second, \(201\) enter, \(38\) die and \(6\) are censored, so \(n_2' = 198\), \(\hat q_2 = 0.192\) and \(\hat S(182) = 0.712\). Continuing, \(\hat S(364) = 0.420\) and \(\hat S(728) = 0.127\), close to the Kaplan–Meier values \(0.708\), \(0.409\) and \(0.116\) at those times (Software 5.4.3). The small differences come from grouping and the half-interval assumption; with finer intervals the life table tends to the Kaplan–Meier estimate.
The same arithmetic underlies the population life tables of demography and actuarial science, which give, for each year of age, the probability \(q_x\) that a person aged \(x\) dies before \(x + 1\). They are built from deaths and population counts in a recent period (a current or period life table) and give the life expectancies quoted in official statistics; Halley’s table of 1693 for Breslau was the first.
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