SurvivalAnalysen

SurvivalAnalysen, commonly called survival analysis in English, refers to a family of statistical methods for analyzing time-to-event data, where the primary interest is the time until an event of interest occurs (for example death, disease relapse, or machine failure). Data are often censored, meaning the event has not yet occurred for some subjects at the end of observation or they are lost to follow-up.

Key quantities include the survival function S(t) = P(T > t), representing the probability of surviving beyond time t, and the hazard function h(t), the instantaneous risk of the event at time t given survival to that time. The data yield survival curves and hazard estimates.

Core methods include the Kaplan-Meier estimator for S(t); the log-rank test for comparing survival between groups; and the Cox proportional hazards model for assessing covariate effects on the hazard. Parametric models (for example exponential and Weibull) and other non-parametric or semi-parametric approaches are used depending on assumptions. Extensions cover competing risks, multi-state models, and recurrent events.

Applications span medicine and epidemiology (clinical trials, prognosis), reliability engineering, and business analytics (customer churn, time-to-event in product life cycles). Software commonly used includes R packages such as survival and survminer, Python's lifelines, and tools in SAS, Stata, and SPSS.

The method originates with Kaplan and Meier in 1958, with subsequent formalization and expansion by researchers such as David Cox in the 1970s, contributing to its widespread use in biostatistics, epidemiology, and related fields.