What is a Kaplan-Meier curve?

A Kaplan-Meier curve is a step plot of the product-limit estimate of survival: the probability that a patient has not yet had the event of interest, plotted against time since a defined origin. The curve steps down at each observed event, stays flat between events, and keeps censored patients in the denominator for exactly as long as they were followed.

That last clause is the whole invention. Before Kaplan and Meier, incomplete follow-up was handled by discarding patients or by grouping time into intervals; the product-limit estimator uses every patient for the period they were actually observed and no longer. This page covers the formula, what censoring must satisfy for the estimate to mean anything, a worked calculation with real arithmetic, how to read the tail, what Kaplan-Meier cannot do, how mishandling is detected in a manuscript, and what reviewers write when it has been.

The estimator and its formula

The Kaplan-Meier estimator, published by Edward Kaplan and Paul Meier in the Journal of the American Statistical Association in 1958 under the title “Nonparametric estimation from incomplete observations”, computes survival as a running product over the distinct event times:

S(t) = Π (1 − dᵢ / nᵢ), over all event times tᵢ ≤ t

where dᵢ is the number of events at time tᵢ and nᵢ is the number of patients still at risk immediately before tᵢ. Each factor is a conditional probability of surviving that instant given survival to it; multiplying them chains the conditional probabilities into an unconditional survival probability. This is why the estimator is called the product-limit estimator, and why the curve is a step function rather than a smooth line: no factor exists at a time with no event, so nothing changes there.

Two consequences follow immediately and are frequently misread. First, the curve does not step down when a patient is censored — censoring reduces nᵢ for every later event, which makes each subsequent step larger, but produces no drop of its own. Second, the estimator is nonparametric: it assumes no distributional form for survival times, which is both its strength and the reason it cannot adjust for anything.

What censoring must satisfy

Censoring in a Kaplan-Meier analysis must be independent of the event process — the standard term is non-informative censoring, and the substantive requirement is that a patient censored at time t has the same subsequent risk as a comparable patient still under observation at time t.

Administrative censoring usually satisfies this. A patient who is event-free when the database is locked was censored by the calendar, not by their prognosis. Loss to follow-up frequently does not satisfy it. Patients who stop attending because they have deteriorated, moved to hospice, or transferred care are censored precisely because their hazard rose, and Kaplan-Meier will then overestimate survival — sometimes substantially, and with no diagnostic in the output to reveal it.

Nothing in the data identifies informative censoring, because the counterfactual it concerns is unobserved by construction. The honest treatments are to report the number and timing of losses, compare baseline characteristics of censored and uncensored patients, and run a sensitivity analysis at the extremes: assume everyone lost to follow-up had the event at censoring, then assume none did, and report both bounds. A study in which 30% of patients are lost has an estimate whose plausible range is wider than any confidence interval printed on the figure.

Left truncation is a separate problem

Left truncation, also called delayed entry, occurs when a patient only becomes eligible for observation some time after the time origin — a registry that enrols at diagnosis but analyses time from symptom onset, or a cohort assembled from patients alive at a screening visit. Left truncation is not censoring and is not handled by censoring machinery.

The correction is to enter each patient into the risk set at their entry time rather than at time zero, so that nᵢ counts only those actually under observation. Software supports this through a two-argument time specification: Surv(entry, exit, event) in R’s survival package, stset ... , enter() in Stata. Analysing left-truncated data as though everyone entered at zero builds in a survivorship advantage, because patients who died before they could enrol are absent from the numerator and the denominator alike. This is the structural relative of immortal time bias, and in registry cohorts the two often occur together.

A worked calculation

This Kaplan-Meier calculation uses ten patients followed for 25 months, with event times at months 4, 8, 11, 15, 20 and 25, and censoring at months 6, 13, 17 and 22.

Read the third row carefully. One patient died at month 8, but the risk set had already fallen to 8 — one patient had the event at month 4 and a second was censored at month 6 — so the step is 12.5% of the remaining probability rather than the 11.1% it would have been had nobody been censored. The censored patient contributed to the denominator at month 4 and to nothing afterwards.

The estimated median survival is 20 months: the first time at which S(t) falls to or below 0.5. Note what supports it — three patients at risk and a single death. The step from 0.540 to 0.360 is the largest drop up to that point in the curve, and it is driven by one event among three people. Every later step is larger still, for the same reason: the risk set is nearly exhausted.

The final row is the other instructive one. S(25) = 0, because the last patient at risk had the event. The estimator states that survival is zero beyond 25 months on the evidence of a single patient. Had that patient instead been censored at 25, the curve would have ended at 0.360 and been undefined beyond the largest observation, which is the more common presentation and the reason many published curves stop above zero.

Greenwood’s formula, and why plain confidence intervals fail

The standard variance for a Kaplan-Meier estimate is Greenwood’s formula, published by Major Greenwood in 1926:

Var[S(t)] = S(t)² × Σ dᵢ / (nᵢ (nᵢ − dᵢ))

Apply it to the worked example at month 20. The summed terms are 1/(10×9) + 1/(8×7) + 1/(7×6) + 1/(5×4) + 1/(3×2) = 0.0111 + 0.0179 + 0.0238 + 0.0500 + 0.1667 = 0.2694. Multiplying by S(20)² = 0.1296 gives a variance of 0.0349 and a standard error of 0.187. The naive Wald interval is then 0.360 ± 1.96 × 0.187, or (−0.006, 0.726) — a survival probability below zero.

That failure is generic, not a quirk of small samples: the Wald interval is symmetric on a scale bounded at 0 and 1, so it crosses a boundary whenever the estimate is near either end or the standard error is large. The standard remedy is the complementary log-log transformation, computing the interval for log(−log S(t)) and back-transforming. On the same data that yields (0.063, 0.686) — asymmetric, contained in the unit interval, and honest about how little the estimate is worth at month 20. Software differs in what it does by default, so check before quoting an interval in a manuscript: R’s survfit offers conf.type = "log-log" but does not use it unless asked, while SAS’s PROC LIFETEST applies the log-log transformation as its default. An interval that straddles zero or one is the tell that no transformation was applied at all.

Greenwood’s formula is also known to understate variance in the far tail, where few patients remain. Treat a confidence band in the last decile of follow-up as optimistic even after transformation.

Reading the tail without over-reading it

The right-hand end of a Kaplan-Meier curve is estimated from the fewest patients and drawn with the same line weight as the left-hand end, which is the single most reliable source of over-interpretation in survival figures.

Three habits control it. Print the numbers at risk beneath the time axis, aligned to the tick marks — the good-practice recommendation set out by Pocock, Clayton and Altman in The Lancet in 2002, and since taken up in the statistical guidance of clinical journals — the European Heart Journal, for example, recommends giving the number at risk for each stratum at each labelled time point. Truncate the plotted follow-up where the risk set becomes uninformative, stating the cut-off, rather than extending a line supported by two patients. And do not quote a survival probability at a time point where the risk set has collapsed: “5-year survival was 41%” means something different when 180 patients are at risk at 5 years than when 4 are.

Median follow-up belongs on the figure or in its legend, and it is not the median of observed follow-up times. The accepted method is the reverse Kaplan-Meier estimate described by Schemper and Smith in 1996, which reverses the event indicator — censoring becomes the event — so that patients who died do not drag the estimate down. A cohort with early deaths can have a median observed follow-up of 11 months and a reverse Kaplan-Meier median follow-up of 48 months, and the second figure is the one that tells a reader how mature the data are.

What a Kaplan-Meier curve cannot do

A Kaplan-Meier curve estimates the marginal survival function of one group and does nothing else. Four limitations follow, and each is a common source of reviewer objection.

It does not adjust for covariates. Any difference between two curves is unadjusted and carries every imbalance the study has. In an observational cohort the curve is subject to the full force of confounding, and comparing arms by Kaplan-Meier alone is only defensible under randomisation. Adjusted survival requires a regression model, and the adjusted curves derived from it are a different object with different assumptions.

It does not handle competing risks correctly. When patients can experience an event that prevents the event of interest — death from other causes in a cancer cohort, transplantation in a dialysis cohort — treating those patients as censored violates the independence assumption, because a dead patient is not exchangeable with a living one under follow-up. One minus the Kaplan-Meier estimate then overstates cumulative incidence, and the overstatement grows with the competing hazard and with follow-up time. The correct estimator is the cumulative incidence function computed by the Aalen-Johansen method; the corresponding regression is the Fine-Gray subdistribution hazard model, published in 1999. The competing risks agent checks whether the estimand described in the text matches the estimator actually used.

It does not summarise a treatment effect. The log-rank test compares curves and the Cox model summarises the comparison as a hazard ratio, but both carry assumptions the curve itself does not. Where survival curves cross, proportional hazards is violated, the single hazard ratio averages effects of opposite sign, and the log-rank test is at its least powerful precisely there.

It does not fix a badly defined time origin. The estimator takes the origin as given. If the origin is a post-baseline event — first response, treatment received, biomarker measured — the groups differ in the time they had to become eligible, and the curve inherits that difference in full.

How a mishandled Kaplan-Meier analysis is detected in a manuscript

A mishandled Kaplan-Meier analysis is detected first by three arithmetic checks that need nothing beyond the survival figure and the results text.

Check the numbers at risk against the reported n. The risk set at time zero should equal the analysed cohort, and it should decline monotonically. A risk table that starts below the stated sample size means patients were dropped without explanation; one that increases at any point means left truncation was present and unhandled, or the table was assembled by hand.

Check the median against the curve. A reported median survival must correspond to the time where the plotted curve crosses 0.5. When the curve never falls below 0.5, no median is estimable, and a manuscript that reports one has either taken the mean of observed times or read an extrapolation. The paired tell is a median with an implausibly tight confidence interval; the Brookmeyer-Crowley interval for a median is usually wide, and a narrow one signals it was not computed by that method.

Check the censoring accounting. Events plus censored observations must equal the analysed n. Then look for the sentence naming how many patients were lost to follow-up as distinct from administratively censored. Manuscripts that report only a combined censoring count are hiding the distinction that determines whether the independence assumption is credible.

Beyond arithmetic, four textual patterns predict a statistical objection. A cause-specific outcome in a population with substantial other-cause mortality, analysed by Kaplan-Meier with no mention of competing risks — routine in registry analyses. A hazard ratio reported beside a figure showing crossing or converging curves. Groups defined by something that happened after baseline, particularly response status. And causal vocabulary — “improved survival”, “conferred a benefit” — attached to an unadjusted two-curve comparison in an observational cohort, which is what causal language discipline and the overclaim check look for.

Screening-related cohorts deserve a specific pass. When survival is measured from diagnosis and one group was diagnosed by screening, the curve is affected by lead time bias and by length time bias regardless of how carefully the estimator was computed, because both act on the time origin and the case mix rather than on the arithmetic.

What a peer reviewer says when Kaplan-Meier is mishandled

Peer reviewers object to Kaplan-Meier analyses in a small and predictable set of phrasings, and most of them concern reporting rather than arithmetic.

“Numbers at risk should be added below the x-axis at each labelled time point.” “The Kaplan-Meier method is not appropriate here, as death from other causes is a competing risk; please present cumulative incidence functions.” “Median follow-up should be reported using the reverse Kaplan-Meier method.” “The survival curves are extended to a period in which fewer than ten patients remain at risk; please truncate the plot or justify the extrapolation.” “The proportional hazards assumption has not been examined.” “Groups appear to have been defined by an event occurring after the time origin, which introduces immortal time bias.”

Two further comments recur and decide papers. “The comparison is unadjusted; baseline imbalance in [variable] is likely to account for the observed separation” — the objection that statistical reviewers raise most often against a two-curve figure in an observational study. And “the reported median survival is not consistent with the figure”, which is fatal in a way the others are not, because it tells the editor that the numbers and the plot were produced from different analyses.

Where the underlying method is wrong rather than under-reported, expect the objection to arrive as the wrong statistical test — usually a log-rank test applied to matched or clustered data, or applied where the estimand called for cumulative incidence.

How to report a Kaplan-Meier analysis

A Kaplan-Meier analysis is reported completely when a reader can reconstruct the risk set. Report the time origin explicitly, in words, with the event that defines it. Report the event definition and every reason for censoring, with counts for each. Report the analysis population and confirm that the risk set at time zero equals it.

On the figure: numbers at risk beneath the axis at regular intervals, censoring marks on the curves, a stated maximum follow-up, and the confidence-interval method named in the legend rather than assumed. In the text: median survival with its confidence interval by the Brookmeyer-Crowley method, or an explicit statement that the median was not reached; survival probabilities at pre-specified landmark times with intervals; and median follow-up by reverse Kaplan-Meier.

Where a competing risk exists, present cumulative incidence functions and say which estimator produced them. Where curves cross, present restricted mean survival time or a time-partitioned analysis alongside any hazard ratio. Trials should follow their reporting guideline for the flow of participants; the clinical trials reviewer checks the survival figure against the registered primary endpoint and its analysis time point, which is where discrepancies between protocol and paper most often surface.

Related

My survival curves cross · Competing risks · Immortal time bias · Lead time bias · Confounding

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Frequently asked questions

What does a Kaplan-Meier curve show?

A Kaplan-Meier curve shows the estimated probability of remaining event-free over time, computed as a running product of conditional survival probabilities at each observed event time. The line steps down at events, runs flat between them, and treats patients with incomplete follow-up as contributing until the moment they were censored.

What is the Kaplan-Meier estimator?

The Kaplan-Meier estimator, published by Kaplan and Meier in 1958, is a nonparametric estimate of the survival function: S(t) is the product of (1 − dᵢ/nᵢ) over all event times up to t, where dᵢ is events at that time and nᵢ is the number still at risk. No distribution for survival times is assumed.

How do you interpret a Kaplan-Meier plot?

Read the height of the curve as the estimated proportion still event-free at that time, the vertical drops as events, and the tick marks as censored patients. Check the numbers at risk beneath the axis before trusting any point: the right-hand end of the plot is estimated from the fewest patients and carries the widest uncertainty.

Why is Kaplan-Meier called the product-limit estimator?

Because the survival estimate is a product of terms, one per event time, each giving the probability of surviving that instant given survival up to it. Multiplying those conditional probabilities produces the unconditional survival curve, and the estimator arises as the limit of the older actuarial life-table method as the time intervals shrink to zero.

What is a survival curve in statistics?

A survival curve plots the probability that the event of interest has not yet occurred against time since a defined origin, starting at 1.0 and never increasing. The Kaplan-Meier method is the standard nonparametric way to estimate one from data in which some patients are censored before the event is observed.

What is Kaplan-Meier analysis used for?

Kaplan-Meier analysis is used to estimate time-to-event probabilities, read off median survival, and display group differences when follow-up is incomplete. It is descriptive and unadjusted: it estimates one group’s survival function, does not control for covariates, and is not the correct estimator when competing events are present.

What does the y-axis of a Kaplan-Meier curve mean?

The y-axis is the estimated survival probability, running from 1.0 at the time origin downward. Some papers invert it to show cumulative event probability, which is 1 − S(t). Read the axis label before comparing figures, and note that 1 − S(t) is not a valid cumulative incidence when a competing event is present.