KaplanMeiernestimatet

The Kaplan-Meier estimator is a non-parametric statistic used to estimate the survival function from lifetime data. It is particularly useful when dealing with censored data, where the exact time of an event (such as death or failure) is not known for all individuals in a study. The estimator works by calculating the probability of survival at each distinct time point where an event occurs.

The survival function, denoted as S(t), represents the probability that an individual survives beyond time t. The Kaplan-Meier estimator, often denoted as $\hat{S}(t)$, provides a step-wise approximation of this function. It begins with a survival probability of 1 at time 0 and decreases only at times when events are observed.

The calculation involves tracking the number of individuals at risk (those who have not yet experienced the event and have not been censored) and the number of events occurring at each time point. The estimated survival probability at a given time t is the product of the conditional probabilities of surviving each preceding interval. If $n_i$ individuals are at risk just before time $t_i$ and $d_i$ events occur at $t_i$, the survival probability is updated by multiplying the previous survival probability by $(1 - d_i/n_i)$.

The Kaplan-Meier curve is a graphical representation of this estimator, plotting the estimated survival probability against time. This curve is widely used in fields such as medicine, engineering, and economics to visualize survival rates and compare outcomes between different groups. It is a fundamental tool for survival analysis.

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