Survival analysis
Academic note · applied
What happens to the patients still alive when a trial ends?
A censored patient still contributes survival information until the moment they leave observation.
You only need What a survival probability means, and basic conditional probability.
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Eight patients are followed for relapse. Five relapse at known times. Three are still relapse-free when their observation stops. Their records read like data with a hole in them, and the two obvious repairs both distort the answer: dropping them discards real evidence, and recording their last known-good month as a relapse invents an event that never happened. Both moves push the estimated survival downwards.
What censoring is, before any formula
An observation is right censored when the event has not happened by the time observation stops. Clark and colleagues list three routes to it: the study closes before the patient has the event, the patient is lost to follow-up, or a different event makes further follow-up impossible 2.
The distinction that matters for the arithmetic is that a censored time is not a missing value. It is a lower bound. A patient censored at month 6 is known to have survived at least six months, and their true event time, if it ever comes, is somewhere beyond that. Recording the censored time as though it were the event time therefore understates the true time to event 2.
Write the eight observations as months to relapse, with E for an observed event and C for a censored observation:
Following the risk set month by month
The Kaplan–Meier estimator, also called the product-limit estimator, uses each patient for exactly as long as they are known to be event-free 12. At each time $t_j$ where an event occurs, it forms the conditional probability of getting through that moment given survival up to it, and multiplies those conditional probabilities together:
where $n_j$ is the number of patients still under observation and event-free just before $t_j$ — the risk set — and $d_j$ is the number of events at $t_j$. The estimate holds still between event times and steps down only when an event occurs 2. A censored observation never appears as an event; its only effect is to reduce the risk set for the intervals that follow 3.
The five event times give five steps; the final row records the last censored observation, where nothing changes. Times are months; $S(t)$ is a probability, so it is unitless.
| $t$ (months) | at risk $n_j$ | events $d_j$ | censored | factor | $S(t)$ |
|---|---|---|---|---|---|
| 2 | 8 | 1 | 0 | 7/8 | 0.8750 |
| 3 | 7 | 1 | 0 | 6/7 | 0.7500 |
| 4 | 6 | 1 | 1 | 5/6 | 0.6250 |
| 6 | 4 | 1 | 1 | 3/4 | 0.4688 |
| 8 | 2 | 1 | 0 | 1/2 | 0.2344 |
| 9 | 1 | 0 | 1 | — | 0.2344 |
Two rows repay a second look. At month 4 the risk set is 6 and includes both the patient who relapses and the patient who is censored, because the censored patient was confirmed event-free right up to that instant; they leave afterwards without ever being counted as an event. At month 9 there is no event at all, so the estimate does not move.
The same arithmetic with the censoring thrown away
Now run the identical product-limit machinery, changing one thing: treat all eight observations as relapses. This is what “just use the recorded times” amounts to.
| $t$ (months) | at risk $n_j$ | events $d_j$ | factor | $S(t)$ |
|---|---|---|---|---|
| 2 | 8 | 1 | 7/8 | 0.8750 |
| 3 | 7 | 1 | 6/7 | 0.7500 |
| 4 | 6 | 2 | 4/6 | 0.5000 |
| 6 | 4 | 2 | 2/4 | 0.2500 |
| 8 | 2 | 1 | 1/2 | 0.1250 |
| 9 | 1 | 1 | 0/1 | 0.0000 |
The two tables agree at months 2 and 3, where there was nothing to disagree about, and then separate. By month 6 the naive estimate is 0.25 against the correct 0.4688, and at month 9 it reports that every patient has relapsed. Nothing in the data supports that. It is produced entirely by converting three “we stopped looking” records into three relapses.
The assumption doing the work
Kaplan–Meier is not a way of recovering unknown outcomes. It works by never asserting one, and it rests on an assumption that has a name: censoring must be uninformative, meaning patients censored at any time have the same survival prospects as those who remain under observation 2. Bland and Altman call that assumption not easily testable, and give a case where it fails: in the conception study they work through, censoring linked to failure to conceive would have biased the estimated probabilities downwards 3. The direction is not fixed in general. What is general is that if leaving observation carries information about the outcome, the estimate moves, and the estimator gives you no warning that it has.
That is worth taking seriously in any setting where people leave a study for reasons connected to how they are doing. Patients who withdraw because they feel worse, customers who churn before a contract ends, or units pulled from a reliability test after showing early signs of wear are all cases where the assumption is doubtful and the estimator’s guarantees weaken with it.
A shorter follow-up makes the naive answer worse, not just noisier Cutting the observation window earlier censors more patients, and the naive method converts each of them into a fabricated event. The distortion therefore grows with how early the study stops, rather than averaging out across a larger sample. The same failure appears far from medicine: an “average customer lifetime” computed by dropping still-active customers, or a reliability figure that counts units still working at the end of a test as failures.
Where the curve stops carrying information
The last event here is at month 8 and the last observation of any kind is the censored patient at month 9. After month 8 the estimate has nothing to update it, so it holds at 0.2344; after month 9 there is no one left under observation at all. Software will happily draw the curve flat past that point, but that is a plotting convention, not a claim that 23.4% of patients never relapse — which is why the figure above stops at month 9. Reading a long flat tail as a cure rate is a common misreading, and it is the reason survival curves are usually annotated with the number still at risk beneath the time axis.
Why does a censored patient still count in the "at risk" number at their own censoring time?Recall
Because they were confirmed event-free right up to that instant, so they legitimately belong in the denominator of that interval’s conditional probability. They leave the calculation afterwards, reducing the risk set for later intervals, without ever being recorded as an event.
What does the uninformative-censoring assumption claim, and when should you doubt it?Recall
That patients censored at a given time have the same survival prospects as those still being followed. Doubt it whenever leaving observation is plausibly connected to the outcome — patients withdrawing because they are deteriorating, or units removed from a test after early signs of failure. When censoring carries information about the outcome, the Kaplan–Meier estimate is biased, and the direction depends on the mechanism: if the patients who leave are at higher risk than those who stay, the curve is too optimistic, while Bland and Altman’s conception study is a case in the other direction. The estimator gives no warning either way.
The correct curve is flat from month 8 to the end of the plot. What does that flat stretch mean?Recall
That no further event was observed, not that survival has settled at that level. The estimate only changes at event times, so it holds at 0.234 from month 8 to month 9. The plot stops there because month 9 is the last observation of any kind; past it nobody remains under observation, and a flat line drawn beyond that point would carry no evidence either way.
Sources and further reading
Sources and further reading
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Nonparametric Estimation from Incomplete Observations
(external source)
Edward L. Kaplan & Paul Meier
Journal of the American Statistical Association 53(282), 457–481; DOI 10.1080/01621459.1958.10501452
1958
Used for: The original product-limit estimator for survival from incomplete observations.
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Survival Analysis Part I: Basic concepts and first analyses
(external source)
Taane G. Clark, Michael J. Bradburn, Sharon B. Love & Douglas G. Altman
British Journal of Cancer 89(2), 232–238
2003
Used for: The definition of right censoring and the three ways it arises, the statement that censored times underestimate the true time to event, the product-limit formula in terms of the number at risk, and the uninformative-censoring assumption.
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Statistics Notes: Survival probabilities (the Kaplan-Meier method)
(external source)
J. Martin Bland & Douglas G. Altman
BMJ 317(7172), 1572
1998
Used for: The worked treatment of conditional probabilities, the effect of a censored observation on the number at risk in the next interval, and the direction of the distortion in the study they work through, where censoring was related to the outcome.