11.2 The actuarial life table
A cohort life table summarises survival in a group followed from a common starting point, such as diagnosis, with time since entry divided into intervals. For interval \(i\) let \(l_i\) be the number alive and under observation at its start, \(d_i\) the number who have the event in it, and \(w_i\) the number withdrawn alive (censored) in it.
Definition 11.2.1. The actuarial estimate of the conditional probability of the event in interval \(i\), among those who reach it, is \[ \hat q_i = \frac {d_i}{l_i - w_i/2} , \] and the estimated probability of surviving to the end of interval \(i\) is \(\hat S_i = \prod _{h \leq i}(1 - \hat q_h)\).
The denominator \(l_i - w_i/2\), the effective number at risk, counts each withdrawal as at risk for half the interval, which is right on average if withdrawals are spread evenly through it. The survival estimate multiplies conditional probabilities: to survive three intervals one must survive the first, then the second given the first, and so on.
Theorem 11.2.2 (Greenwood’s formula). Treating the \(d_i\) as binomial given the effective numbers at risk \(n_i'\), with the intervals conditionally independent, \[ \widehat {\Var }(\hat S_i) = \hat S_i^2 \sum _{h \leq i} \frac {d_h}{n_h'(n_h' - d_h)} . \]
Proof. \(\log \hat S_i = \sum _{h \leq i}\log (1 - \hat q_h)\). Each \(\hat q_h\) is a binomial proportion with variance \(q_h(1 - q_h)/n_h'\), and by the delta method \(\Var \{\log (1 - \hat q_h)\} \approx q_h/\{n_h'(1 - q_h)\}\). Conditionally independent terms have summed variances. Substituting \(\hat q_h = d_h/n_h'\) gives \(q_h/\{n_h'(1 - q_h)\} = d_h/\{n_h'(n_h' - d_h)\}\). A second application of the delta method, \(\Var (\hat S) \approx S^2\Var (\log \hat S)\), gives the result. □
Example 11.2.3. \(500\) patients are followed for five years after treatment. The life table is:
| Year | \(l_i\) | \(d_i\) | \(w_i\) | \(n_i'\) | \(\hat q_i\) | \(\hat S_i\) | \(\text {SE}(\hat S_i)\) |
| 0–1 | 500 | 40 | 20 | 490.0 | 0.0816 | 0.918 | 0.012 |
| 1–2 | 440 | 35 | 25 | 427.5 | 0.0819 | 0.843 | 0.017 |
| 2–3 | 380 | 30 | 30 | 365.0 | 0.0822 | 0.774 | 0.020 |
| 3–4 | 320 | 20 | 40 | 300.0 | 0.0667 | 0.722 | 0.021 |
| 4–5 | 260 | 15 | 45 | 237.5 | 0.0632 | 0.677 | 0.023 |
The five-year survival is \(0.677\), with interval \(0.677 \pm 1.96 \times 0.023 = (0.632,\ 0.722)\). Simply dividing the \(140\) events by \(500\) would give a five-year risk of \(0.28\), too low, because the \(160\) people withdrawn before five years were not observed long enough to have all their events counted.
The life table assumes that withdrawals are non-informative: those withdrawn had the same future risk as those who stayed. A patient who leaves the study because they feel too ill to attend breaks this assumption, and the life table then overestimates survival.
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