Survival analysis / statistics · BEGINNER LESSON
Kaplan–Meier estimator
The Kaplan–Meier estimator is a nonparametric method for estimating a survivor function, S(t), which is the probability of being event-free beyond time t. It uses observed event times and right-censored follow-up times to build a stepwise estimate.
What it is and what it measures
The Kaplan–Meier estimator, also called the product-limit estimator, is used for survival analysis: the study of time until an event such as failure or death. The measured quantity is the survivor function S(t), meaning the probability that the event has not happened by time t. In one common setting, data come from independent survival times with some observations right-censored, meaning the exact event time is unknown because follow-up ended first. [0,2]
[1] web.stanford.edu › ~lutian › coursepdfSTAT331: Unit 3 Kaplan-Meier (KM) Estimator Introduction [3] STAT331: Unit 3 Kaplan-Meier (KM) Estimator Introduction
How it works
The estimator updates survival time by time, but only at observed event times. At each event time vj, it uses dj, the number of events at that time, and Y(vj), the number still at risk just before that time. The estimated conditional event probability is dj/Y(vj), so the estimated chance of surviving past that time is 1 - dj/Y(vj). Multiplying these survival factors across event times gives the Kaplan–Meier curve. This makes the estimate a step function that drops only when events occur. [0,2]
[1] web.stanford.edu › ~lutian › coursepdfSTAT331: Unit 3 Kaplan-Meier (KM) Estimator Introduction [3] STAT331: Unit 3 Kaplan-Meier (KM) Estimator Introduction
How censoring is handled
Right-censored observations are counted in the risk set only until the time they leave follow-up. They still contribute information because they show the person or unit was event-free up to that censoring time. This is why the method can use incomplete follow-up without treating censored cases as events. In a non-survival setting, the same idea can estimate the proportion who experience an event during a study while keeping dropouts in the analysis as censored observations. [1]
Limitations and boundaries
The method does not continue in a fully determined way beyond the last time where anyone remains at risk. If the final risk set is exhausted by events, the estimate becomes 0 afterward; if not, the curve stops at the last observed point and remaining probability mass beyond that time is not identified from the data. The method also relies on assumptions such as censoring that is completely at random, and it is nonparametric, meaning it does not impose a specific parametric form for the survival curve. [0,1,2]
[1] web.stanford.edu › ~lutian › coursepdfSTAT331: Unit 3 Kaplan-Meier (KM) Estimator Introduction [2] The Kaplan-Meier Method for Estimating and Comparing ... - PMCpmc.ncbi.nlm.nih.gov › articles › PMC6141203 [3] STAT331: Unit 3 Kaplan-Meier (KM) Estimator Introduction
SEE THE IDEA
How the Kaplan–Meier estimator builds survival over time
Each subject contributes an event time or a censoring time.
Number still under observation just before an event time.
Number of events occurring at that event time.
Multiply by 1 - dj/Y(vj) at each event time.
Estimated probability of remaining event-free beyond t.
Data determine risk sets; risk sets and event counts determine each step; the steps multiply to form the curve.
The estimator uses only observed event times for drops, while censored observations affect the risk set until they leave follow-up.
TRY IT YOURSELF
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Sources and further reading
This general lesson was generated from public source material. It explains a concept; it does not interpret a private paper or provide medical advice.