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The Kaplan–Meier estimator estimates event-free probability in the presence of right-censored observations.
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Formula
S(t) dimensionless 0…1 · t and tj use a time unit · d and n are counts
| Symbol | Meaning | Unit |
|---|---|---|
| Estimated survival function through time t | fraction | |
| Observed event time | time | |
| Events at time tj | people | |
| Individuals still at risk immediately before tj | people |
\hat S(t)=\prod_{t_j\le t}\left(1-\frac{d_j}{n_j}\right)At each event time, the conditional survival probability 1−dj/nj is multiplied; censoring ends risk observation but is not counted as an event. The estimator is informative under non-informative censoring and a correctly defined time origin. It is not directly appropriate for competing risks, where another event type prevents the event of interest.
Requires right-censored event times and non-informative censoring relative to event risk.
Cumulative survival probability is the product of conditional survival fractions at event times.
Solve for a quantity
Update at an event time.
n=8 at risk, d=2 events: what factor?
1−2/8=0.75; multiply S by 0.75.
Counting censored observations as events.
Censoring removes people from later risk sets but does not directly lower the curve.
Formula cluster
Competing events require a different cumulative-incidence treatment.
Start with n=10. At t1, two events give factor 8/10=0.8. Two later censorings do not change S. At t2, six remain at risk and one event gives S=0.8·5/6=0.667. Check: each factor is at most one, so the curve never rises.
Describing time to event with right censoring and no parametric distribution assumption
Here is how to work through a typical Kaplan–Meier estimator (Ŝ(t) = Π(1 − dj/nj)) task step by step:
Task
n=8 at risk, d=2 events: what factor?
Solution path
1−2/8=0.75; multiply S by 0.75.
A right-censored person contributes to the risk set through the last known event-free time, then leaves later risk sets. They are not counted as an event and do not directly lower the curve at censoring. The method uses available follow-up but assumes censoring is non-informative in the relevant sense. If dropout is related to future event risk, the estimate can be biased. Define time origin, event and censoring. Non-informative censoring and suitable risk sets are needed; competing risks require other estimators. Also show how many people remain observed later.
The estimated median is the earliest time when the estimated survival function falls to 0.5 or below. If the curve never reaches 0.5 during follow-up, the median is not estimable and should not be extrapolated. The median is a time-to-event summary, not the mean event time. A confidence interval matters, especially when few people remain at risk late in follow-up. Define time origin, event and censoring. Non-informative censoring and suitable risk sets are needed; competing risks require other estimators. Also show how many people remain observed later.
Censoring should be independent of future event time conditional on considered information. Administrative study end may plausibly be non-informative; dropout due to deterioration may be informative. Unequal follow-up alone does not invalidate Kaplan–Meier, but different censoring mechanisms can bias group comparisons. Report reasons and patterns of loss to follow-up and numbers at risk rather than showing curves alone. Define time origin, event and censoring. Non-informative censoring and suitable risk sets are needed; competing risks require other estimators. Also show how many people remain observed later.
Reproduce the example calculation: Start with n=10. Kaplan–Meier estimator uses Ŝ(t) = Π(1 − dj/nj). In plain terms, the Kaplan–Meier estimator estimates event-free probability in the presence of right-censored observations. Unit check: S(t) dimensionless 0…1 · t and tj use a time unit · d and n are counts. Requires right-censored event times and non-informative censoring relative to event risk. Values are comparable only when their definitions, denominators and periods match. Record these details next to the result so the reference group is clear. If the direction is implausible, check data preparation and the calculation basis before discussing biological causes.
A key limit of Kaplan–Meier estimator is this: Requires right-censored event times and non-informative censoring relative to event risk. Do not use the metric without its reference; it is useful for Describing time to event with right censoring and no parametric distribution assumption. The Kaplan–Meier estimator estimates event-free probability in the presence of right-censored observations. When comparing populations or experiments, check whether definitions, denominators and observation periods match. If the assumption is implausible, the result shows only what the simplified model calculates from its inputs; it is not a reliable estimate for the target population. State this limitation in the report, and do not infer a causal effect from the formula alone.
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Which formula describes Kaplan–Meier estimator?
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How do you rearrange Kaplan–Meier estimator for Solve for a quantity?
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Which common mistake happens with Kaplan–Meier estimator?
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