Kaplan–Meier estimator

Kaplan–Meier estimator, also known as the Kaplan–Meier survival curve, is a non-parametric statistic used to estimate the survival function from lifetime data. In medical research, it is often used to measure the fraction of patients living for a certain amount of time after treatment. The Kaplan–Meier estimator is a way to estimate and plot the survival probabilities over time, despite having some censored data, which is common in clinical trials.

Overview

The Kaplan–Meier estimator is a product-limit estimator, which calculates the probability of survival at different times. It is used when the survival probabilities are not known at all time points. The estimator was introduced by Edward L. Kaplan and Paul Meier in 1958, who each submitted similar papers to the Journal of the American Statistical Association. The papers were combined and published together due to their similarity.

Calculation

The Kaplan–Meier survival curve is a step function that jumps at each event time. For a given time, the survival probability is calculated as follows:

S(t) = ∏ (1 - d_i / n_i)

where S(t) is the probability of survival until time t, d_i is the number of events (e.g., deaths) at time t_i, and n_i is the number of subjects at risk just before time t_i. The product is taken over all times t_i less than or equal to t.

Assumptions

The Kaplan–Meier estimator makes two key assumptions: 1. The survival probabilities are the same for subjects recruited early and late in the study. 2. Censoring is non-informative, meaning that the reason for censoring is unrelated to the likelihood of the event of interest.

Applications

The Kaplan–Meier estimator is widely used in clinical trials to estimate survival rates and compare the effectiveness of treatments. It is also used in other fields, such as engineering and sociology, to model time-to-event data.

Limitations

While the Kaplan–Meier estimator is a powerful tool, it has limitations. It cannot handle competing risks directly and does not account for covariates. For these situations, other methods like the Cox proportional hazards model may be more appropriate.

See also

• Survival analysis
• Cox proportional hazards model
• Censoring (statistics)
• Clinical trial