skip to content

When a Kaplan-Meier curve never drops to 0.5, what does 'median survival not reached' mean?

level: middleimportance: should knowfreq 46%

answer

  1. a statement about follow-up, not failure
  2. the 0.5 line is never crossed
  3. report something inside the window
  4. landmark estimate or area under the curve
  5. never extend the tail by eye

basics

~20 s

It means follow-up ended before half the cohort had the event, so no time exists at which estimated survival is 0.5 or lower. The median is not estimable from the data, and extrapolating past the last observed time invents an answer.

solid answer

~50 s

Median survival is defined as the earliest time at which the Kaplan-Meier estimate falls to 0.5 or below. If the curve plateaus above 0.5 — say it ends at 0.78 because only 22 percent of the cohort cancelled during the observation window — no such time exists, and the honest report is `median not reached`. It is a statement about follow-up length, not about the data being broken; a long median is exactly what you would hope for in a subscription business. Report something that *is* estimable instead: survival at a fixed landmark such as `S(90 days)` with a confidence interval, an earlier quantile like the 25th percentile if the curve crosses 0.75, or the restricted mean survival time, which is the area under the curve up to a horizon you choose. What you must not do is fit a line through the tail and read off where it would cross 0.5.

go deeper

for a junior

Recall the definition: median survival is the earliest time the curve falls to 0.5 or below, and if it never does, there is no median to report.

for a middle

Explain why the phrase describes the length of follow-up rather than a failure of the data, and name at least one estimable substitute such as survival at a fixed landmark time.

for a senior

Show judgment about what you would put in front of stakeholders: a pre-specified landmark or a restricted-mean horizon, with the number at risk shown, and a refusal to extrapolate the tail.

for a principal

Own the reporting standard. Decide the landmark times and horizons the organisation reports against so cohorts stay comparable, and resist pressure to publish an extrapolated median because a slide needs one number.

## Definition The median survival time is the quantile of the estimated survival distribution at 0.5: the smallest `t` for which the Kaplan-Meier estimate `S(t) <= 0.5`. Graphically, draw a horizontal line at 0.5 and drop to the time axis where the step curve first crosses it. Because the estimate is a step function, the crossing is a jump rather than a smooth intersection, and a flat segment sitting exactly at 0.5 makes the median an interval rather than a point — software conventionally reports the left endpoint. ## Why the median is the default summary The *mean* survival time is the area under the whole survival curve from zero to infinity. When the largest observation is censored the curve never reaches zero, so that area is undefined and any reported mean is an artefact of how the tail was closed off. The median needs only the middle of the distribution and is unaffected by how long the longest survivors last, which is why time-to-event reports lead with it. ## What 'not reached' means If fewer than half the cohort has had the event by the end of follow-up, the curve never touches 0.5 and the median is outside the range the data can speak to. Reporting `median not reached` is the correct output. Three things follow: - It is **not** an error state. In a healthy subscription product it is the expected result for any recent cohort — most subscribers are still subscribed. - It is **not** the same as an infinite median. The median may well exist in the population; the observation window is simply too short to locate it. - It **does** limit comparisons. Two groups can both report `not reached` and still have visibly different curves, so a comparison must be made on something other than the medians. ## What to report instead **Landmark survival.** Pick a time inside the follow-up window and report the estimate there with an interval: `S(90) = 0.78, 95 percent CI 0.74 to 0.82`. This is estimable, interpretable, and directly comparable across cohorts as long as everyone uses the same landmark. Choose the landmark before looking at the curves, or you have picked the time that flatters the result. **A different quantile.** If the curve reaches 0.75 but not 0.5, the 25th percentile of survival is estimable and the median is not. Reporting the quartile you can actually see is more honest than reporting the one you cannot. **Restricted mean survival time.** RMST is the area under the Kaplan-Meier curve from 0 up to a chosen horizon `tau`, and it has a plain reading: the average event-free time accrued during the first `tau` days. It is always estimable as long as `tau` is inside the follow-up window, it uses the whole curve rather than one point, and it is stated in days, which stakeholders find easier than a probability. The horizon must be fixed in advance and held identical across the groups being compared. ## What not to do - **Extrapolate.** Extending the tail — by eye, by a straight line, or by fitting a parametric shape — produces a number whose value comes from the assumed shape rather than from observed events. If someone insists, the assumption must be stated as the headline caveat, not buried. - **Force the median by shortening the cohort.** Restricting the analysis to old accounts so that half of them have churned changes the population you are describing and usually makes the result worse, not better. - **Read the plateau height as a time.** A curve that flattens at 0.78 has not produced a median of 0.78 of anything. The vertical axis is a probability; the horizontal axis is time. - **Confuse the flat tail with a stable population.** A plateau late in follow-up is often just an absence of events among a handful of remaining subjects, not evidence that the risk has ended. Check the number at risk under the plateau before making that claim. ## In an interview Say the definition first — the smallest time at which the estimate falls to 0.5 or below — then explain that `not reached` describes follow-up length, then offer landmark survival or RMST as the estimable alternative. Candidates who instead extrapolate a median lose the point immediately, because the whole discipline of censoring is about refusing to invent data past the end of observation.

  • Why is mean survival rarely reported from a Kaplan-Meier curve?
    The mean is the area under the survival curve out to infinity, and when the largest observation is censored the curve never reaches zero, so that area is undefined. Any mean you see has been produced by closing the tail with an assumption. Restricted mean survival time avoids this by capping the horizon at a time inside the follow-up window.
  • How do you choose the landmark time for a fixed-horizon survival estimate?
    Pick it before seeing the curves, from the business or clinical question — a billing cycle, an onboarding window, a contract term — and make sure a reasonable number of subjects are still at risk there. Picking the landmark after inspecting the curves is choosing the time that flatters the result, and it inflates the apparent difference between groups.
  • Two cohorts both report median not reached. How do you compare them?
    Compare something estimable inside the window: survival at a shared landmark with intervals, restricted mean survival time to a shared horizon, or the curves as a whole. Comparing two non-existent medians says nothing, and neither does declaring the groups equal because both reports carry the same phrase.

saying these in an interview costs you the question

  • Calls median not reached a data-quality bug
  • Extrapolates the tail to manufacture a median
  • Reads the plateau height as a survival time
  • Reports a mean survival time from a censored tail
  • Declares two groups equal because both medians are not reached

context

resolving links…