4.1 The product-limit estimator
Let \(t_{(1)} < t_{(2)} < \cdots < t_{(r)}\) be the distinct times at which events occur. At each \(t_{(j)}\) let \(n_j\) be the number at risk, the subjects still under observation just before \(t_{(j)}\), and \(d_j\) the number of events at \(t_{(j)}\). Surviving beyond \(t\) means surviving each event time up to \(t\) in turn, so, by repeated conditioning, \[ S(t) = \prod _{t_{(j)} \leq t}P\bigl (T > t_{(j)} \mid T \geq t_{(j)}\bigr ) . \] The conditional probability of surviving \(t_{(j)}\), given survival to it, is estimated by the proportion of those at risk who did not have the event, \(1 - d_j/n_j\). This gives the Kaplan–Meier, or product-limit, estimator \[ \hat S(t) = \prod _{t_{(j)} \leq t}\Bigl (1 - \frac {d_j}{n_j}\Bigr ) . \] \(\hat S\) is a step function that drops only at event times. Censored subjects never cause a drop, but they leave the risk set, so later drops are larger: each remaining subject now represents a larger share of the population.
Example 4.1.1. For the ten patients with times \(3, 5^+, 6, 8, 10^+, 12, 15^+, 18, 22^+, 24^+\) months, the calculation is
| \(t_{(j)}\) | \(n_j\) | \(d_j\) | \(1 - d_j/n_j\) | \(\hat S(t_{(j)})\) |
| \(3\) | \(10\) | \(1\) | \(0.900\) | \(0.900\) |
| \(6\) | \(8\) | \(1\) | \(0.875\) | \(0.900 \times 0.875 = 0.7875\) |
| \(8\) | \(7\) | \(1\) | \(0.857\) | \(0.675\) |
| \(12\) | \(5\) | \(1\) | \(0.800\) | \(0.540\) |
| \(18\) | \(3\) | \(1\) | \(0.667\) | \(0.360\) |
At \(6\) months only \(8\) are at risk, because one patient died at \(3\) and one was censored at \(5\). The estimated probability of surviving \(12\) months is \(0.54\); the curve first falls below \(\frac 12\) at \(18\) months, which is the estimated median (Software 4.4.1).
Proposition 4.1.2. With no censoring, \(\hat S(t)\) equals the empirical survival function, the proportion of subjects with times greater than \(t\).
Proof. Without censoring, \(n_{j + 1} = n_j - d_j\), so the product telescopes: \(\prod _{t_{(j)} \leq t}(n_j - d_j)/n_j = n_{\text {after}}/n\), where \(n_{\text {after}}\) is the number still alive after the last event time up to \(t\), that is, the number with times greater than \(t\). □
The Kaplan–Meier estimator is also the non-parametric maximum likelihood estimator of \(S\): among all distributions, it maximises the censored likelihood of the section on parametric models when probability is allowed to sit only at the observed event times. Under non-informative censoring it is consistent, and approximately normal in large samples.
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