Kaplan-Meier estimator

Kaplan-Meier estimator

The 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 fundamental tool in clinical trials and epidemiology, providing a visual representation of survival data that can be crucial for understanding treatment outcomes and patient prognosis.

Overview

The Kaplan-Meier estimator is named after Edward L. Kaplan and Paul Meier, who introduced the method in 1958. It is designed to estimate the probability of survival beyond a certain time point, taking into account that some subjects may have right-censored survival times, meaning they leave the study before an event (e.g., death) occurs, or the study ends before the event occurs.

Calculation

The Kaplan-Meier survival curve is calculated by dividing the time into intervals between observed events. For each interval, the probability of surviving past the interval is estimated by the proportion of individuals surviving divided by the number at risk at the beginning of the interval. These probabilities are then multiplied together to give the survival probability at each time point.

Formula

The formula for the Kaplan-Meier estimator is:

\[ S(t) = \prod_{t_i < t} \left(1 - \frac{d_i}{n_i}\right) \]

where \(S(t)\) is the probability of survival until time \(t\), \(d_i\) is the number of event occurrences (e.g., deaths) at time \(t_i\), and \(n_i\) is the number of individuals at risk of the event at time \(t_i\).

Applications

The Kaplan-Meier estimator is widely used in medical research to analyze the survival times of patients across different treatment groups. It is particularly useful in studies where the follow-up time varies among participants or when there are censored observations. The estimator can also be used to compare the survival functions of two or more groups using the log-rank test.

Limitations

While the Kaplan-Meier estimator is a powerful tool for survival analysis, it has limitations. It assumes that censored observations are randomly distributed and does not account for factors that may influence survival times. Additionally, the estimator can become less reliable with small sample sizes or a high proportion of censored data.

See also

• Survival analysis
• Cox proportional hazards model
• Log-rank test