4.2 Greenwood’s formula and confidence intervals
Theorem 4.2.1 (Greenwood’s formula). The approximate variance of the Kaplan–Meier estimator is \[ \widehat {\operatorname {Var}}\,\hat S(t) = \hat S(t)^2\sum _{t_{(j)} \leq t}\frac {d_j}{n_j(n_j - d_j)} . \]
Derivation. \(\log \hat S(t) = \sum \log (1 - \hat q_j)\) with \(\hat q_j = d_j/n_j\). Given \(n_j\), \(d_j\) is approximately binomial, so \(\operatorname {Var}\hat q_j \approx q_j(1 - q_j)/n_j\), and the terms are approximately uncorrelated. By the delta method, \(\operatorname {Var}\log (1 - \hat q_j) \approx \frac {q_j(1 - q_j)/n_j}{(1 - q_j)^2} = \frac {q_j}{n_j(1 - q_j)}\), which with \(q_j = d_j/n_j\) is \(\frac {d_j}{n_j(n_j - d_j)}\). Summing gives \(\operatorname {Var}\log \hat S\), and a second use of the delta method, \(\operatorname {Var}\hat S \approx \hat S^2\operatorname {Var}\log \hat S\), gives the formula. □
The plain interval \(\hat S \pm 1.96\,\widehat {\operatorname {se}}\) can go outside \([0, 1]\) and behaves badly near the ends. Intervals are therefore computed on a transformed scale and transformed back: on the log scale, \(\hat S\exp (\pm 1.96\,\operatorname {se}(\log \hat S))\), the default in R; or on the log-log scale, the default in some other packages, which always stays inside \((0, 1)\). All three agree when there are many subjects at risk.
Example 4.2.2. For lung, the estimated survival is \(0.722\) at \(6\) months (\(95\%\) interval \(0.666\) to \(0.783\)), \(0.409\) at one year (\(0.345\) to \(0.486\)), and \(0.116\) at two years (\(0.072\) to \(0.187\)). The interval widens as the number at risk falls, from \(160\) at \(6\) months to \(13\) at two years. Greenwood’s formula gives the standard error \(0.0358\) at one year, the value R reports (Software 4.4.2).
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