Censored follow-up

How can a study report five-year survival when most of its patients were watched for two?

A trial opens in 2019 and closes its books in 2024. Patients arrive throughout, so the woman who enrolled in the first month has six years of record and the man who enrolled last spring has eighteen months. Some were lost when they moved away. Then the paper reports a five-year survival of 51%.

It looks like an overreach. Most of these people were never watched for five years. And we cannot simply discard the short records, because that would leave only the earliest enrollees — a group defined by when they arrived, not by anything clinical. So what exactly does a patient watched for two years tell us about year five?

Start by asking what a short record actually contains. A man observed for two years and still alive at the cutoff has not told us nothing. He has told us something quite definite: he did not die in the first two years. That is a fact about the first two years, and it is exactly as reliable as any other. What he has not told us is anything about years three to five. His record is not missing — it is right-censored, cut off at a known time with the event still ahead of him.

So the information is real but partial. Here is the move that unlocks everything: stop asking "what fraction survive five years?" and ask instead a chain of smaller questions. Given that you reached day t, what is the chance you get through day t alive? Each of those conditional questions can be answered by whoever was actually under observation on that day — and a man censored at two years is under observation for every one of the questions up to that point, and for none after.

Chain them and you have the estimator. At each time an event occurs, count n still at risk and d events; the conditional survival through that instant is (1 − d/n); multiply those factors together as time advances. Censored patients simply leave the at-risk count on their way out, contributing to every factor before their exit and none after. That is the Kaplan-Meier product-limit estimator, and the reason it can speak about year five is that year five's factor is computed from the handful of people who were genuinely there in year five, while the earlier factors — which multiply into the same product — rest on everybody.

Ten patients make it concrete. One dies at 6 months with all 10 at risk, so the first factor is 9/10 = 0.9. Two are censored at 9 months, leaving 7. One dies at 12 months: factor 6/7, and the running estimate is 0.9 × 6/7 = 0.771. Three more are censored at 18 months, leaving 3. One dies at 24 months: factor 2/3, giving 0.771 × 2/3 = 0.514.

Now compare that with the two obvious shortcuts. Count deaths over enrolments — 3 out of 10 — and you get 70% survival, which quietly treats every censored patient as a survivor to 24 months. Keep only patients with a full 24 months of record and you have 3 people, 1 death, 67%. The product-limit answer, 51%, is lower than both, and it is lower for a reason: the censored patients are neither survivors nor deaths, and both shortcuts resolve that uncertainty in the flattering direction.

That last sentence is the whole load-bearing assumption, and it deserves its own name. Kaplan-Meier is valid under non-informative (independent) censoring: at any moment, those censored must have the same future risk as those who remain under observation. When the study simply closes its books — administrative censoring — that is close to guaranteed by design, since the calendar has no opinion about anyone's tumour. When patients are lost to follow-up, it is an assumption, and often a shaky one: people who stop attending because they have deteriorated are censored by their prognosis, and the estimator will then read the curve off the healthier remainder and report survival too high.

How would you falsify it? Three practical tests. Compare the baseline characteristics of censored and uncensored patients at the moment of censoring — systematic differences are evidence against independence. Look at why each record ended, and separate administrative cutoff from dropout; a curve resting mostly on the second kind is fragile. Best of all, bound it: recompute assuming every censored patient died the day they left, then assuming every one survived to the horizon. If those two extremes still bracket a clinically similar answer, the conclusion is robust; if they run from 30% to 80%, the number is doing less work than its confidence interval suggests.

It is also worth separating this from a neighbouring idea. Survivorship bias is about cases that never enter the file at all, absent because of the outcome under study. Censoring is the opposite posture: the case is in the file, its partial history is used, and the analysis is explicit about where the knowledge stops. That is why censoring is survivable and silent selection is not — not because censored data is better, but because the missingness is dated and declared rather than invisible.

One further honesty: the tail of a Kaplan-Meier curve rests on very few people, so a single late step can move it, and statisticians disagree about where to stop drawing.

• How the Cox proportional-hazards model extends this to compare groups while adjusting for covariates.
• Why competing risks (dying of something else) are not censoring, and what happens when they are treated as such.

Every term the collection defines is gathered in the glossary.