KaplanMeierkurver

Kaplan-Meierkurver, or Kaplan-Meier curves, are graphical representations of the survival function in time-to-event analyses. They are used to estimate the probability of survival beyond a given time, taking into account censored observations.

The estimator is the product-limit estimator. Given observed event times t1 < t2 < ...; at each event time t_j, d_j individuals experience the event and n_j individuals are at risk just before t_j. The survival estimate is S_hat(t) = product over t_j ≤ t of (1 - d_j / n_j).

Censoring: If a subject is censored at time c before experiencing the event, they contribute to the risk set up to c but do not contribute to d_j. The resulting curve is a step function that declines at event times and remains flat between events.

Variance and confidence intervals: Greenwood's formula gives Var[S_hat(t)]. Confidence bands can be constructed using log-minus-log or other transformations. Comparisons between groups are often made with the log-rank test.

Assumptions and use: Kaplan-Meier curves assume non-informative censoring and independence between censoring and survival. They summarize unadjusted survival and are commonly used in clinical trials and medical research. For examining covariate effects, methods such as Cox regression are used.

Limitations: The estimator does not adjust for covariates and may be biased with heavy censoring or small sample sizes. Extrapolation beyond the last observed event times should be avoided; curves are most informative within the observed follow-up.