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The Kaplan-Meier curve, explained for clinicians

What each step means, why a censored patient is neither a survivor nor a death, and how to read median survival straight off the curve.

In a hurry

Follow a group of patients from a starting point - surgery, diagnosis, the first dose - and record when the event of interest happens: a relapse, a death, a device failure. The Kaplan-Meier curve is the estimated probability of still being event-free as time passes. It starts at 100%, drops a step at every event, and puts a small tick where a patient was censored: observation ended while they were still event-free, and the curve does not drop for them. The time where the curve crosses 50% is the median survival; if the curve never gets that low, the median is not reached. Below are ten patients you can drag around and watch the curve rebuild.

Ten patients, two kinds of exits

Say ten patients finish a cancer treatment and we watch for relapse over the next two years. Each patient's story is a line through time, and only two things can end it: the event itself, or the end of observation - the study closes, the patient moves to another city, the follow-up visit never happens. That second ending is called censoring, and it is the reason this curve exists. A censored patient told you something real: no relapse for as long as you watched. Throwing them out of the analysis wastes that information; counting them as relapse-free forever invents information you never had. The Kaplan-Meier estimator is the standard way to keep exactly what they gave you and nothing more.

Build it by hand

The trick is to update the curve only when an event happens, and to use only the patients still under observation at that moment - the patients at risk. Ten are at risk when the first relapse comes at month 3, so survival falls to 9/10 = 90%. At month 5 one patient moves away: a tick on the curve, no drop, but only eight remain at risk. That makes the relapse at month 8 cost more: 90% times 7/8, about 79%. Every event multiplies the current survival by one minus events over at-risk, and censored patients quietly shrink the at-risk count between the drops. The full ladder for our ten patients:

MonthAt riskEventsFactorSurvival
31019/1090%
8817/879%
10716/768%
15514/554%
18413/441%
22211/220%

Patients censored at months 5, 12, 20 and 24 never get a row of their own: they only thin the at-risk column. The curve crosses 50% at month 18, so the median survival is 18 months.

$$\hat S(t) = \prod_{t_i \le t} \left(1 - \frac{d_i}{n_i}\right)$$

d₁ events at time t₁, among n₁ patients still at risk. Multiply the factors for every event time up to t.

Try it: drag their timelines

Each line below is one patient. Drag along a line to change when their story ends; tap the letter to switch that ending between E, the event, and C, censored. The steps, the ticks and the median respond as you go. Push the events late or censor enough patients and the median disappears: if the curve never reaches 50%, the median is simply not reached - a phrase you will meet in every oncology paper, and it is good news wearing dry language.

012 months24
A tick is not a step: a censored patient never drops the curve, but they leave fewer at risk, so every later step gets taller. That is also why the right-hand end of any survival curve is its least reliable stretch - by then, only a handful of patients remain to carry it.

What surprises people about Kaplan-Meier

The curve is an estimate for a patient, not a headcount of your sample. After censoring, a survival of 41% does not mean that 41% of the ten are event-free; it is the estimated chance for a patient like them to reach that time without the event.
Median, not mean. With censored patients the mean is unknowable until the very last one has had the event, so survival is quoted as the median - the time the curve crosses 50%.
Censoring must be unrelated to the outcome. If patients leave the study because they are getting worse, the curve quietly flatters the treatment. No formula can detect that; only the design of the follow-up can prevent it.
Two curves are compared over their whole length, not at one flattering time point - that is the job of the log-rank test. Reading the gap at a single month is how marketing reads a survival plot.
Five-year survival can improve while nobody lives a day longer: screen earlier and you start every clock earlier. That is lead-time bias, and it deserves a page of its own.

This is the second curve on this site that you can build by hand from raw patients - the first was the ROC curve, drawn cell by cell from a ranking. And once you start watching the clock every survival number runs on, the next stop is what screening does to that clock:

ROC & AUC lesson Why survival misleads

Common questions

What is censoring in a survival curve?

Censoring marks a patient whose event was never observed: they left the study, or it ended, while still event-free. They count as at-risk up to that moment, then drop out of the denominator without making the curve step down. It is why survival data is not simply who had the event by the end.

How do you read the median survival off a Kaplan-Meier curve?

Find where the curve first drops to or below 0.5 (50%) and read the time on the horizontal axis. That time is the median survival: half the patients have had the event by then. If the curve never reaches 0.5, the median is not reached.

Why does the Kaplan-Meier curve fall in steps?

Each step down happens at a time when an event occurs, and its size depends on how many patients are still at risk then. Between events the curve is flat. It is a product of survival fractions, one factor per event time, which is why it is called the product-limit estimator.

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