SurvivalAnalyse

Survivalanalyse is the statistical analysis of time-to-event data, where the event of interest may be death, relapse, device failure, or another endpoint. Because subjects may not experience the event during the study period, data are often right-censored, meaning the exact event time is unknown for some individuals. The aim is to describe the distribution of event times and relate them to covariates.

Core concepts include the survival function S(t) = P(T > t), the probability that the event has not occurred by time t, and the hazard function h(t), the instantaneous risk of the event at time t given survival up to t. Censoring reduces information about T but can be accommodated in estimates and models.

Nonparametric methods include the Kaplan-Meier estimator of S(t) and the log-rank test for comparing survival between groups. Life-table methods provide alternative, discretized estimates. Semiparametric models, notably the Cox proportional hazards model, relate the hazard to covariates through h(t|X) = h0(t) exp(β’X) without requiring specification of the baseline hazard h0(t). Assumptions include non-informative censoring and, for Cox, proportional hazards.

Parametric approaches specify a distribution for T, such as exponential, Weibull, or log-normal, enabling explicit survival and hazard functions and potentially extrapolation beyond observed data. Model checking, goodness-of-fit, and diagnostics are used to assess appropriateness of the chosen model.

Applications span medicine and epidemiology (clinical trials, survival forecasting), reliability engineering (time to component failure), and risk assessment. Software commonly used includes R packages like survival, Python lifelines, and statistical packages in SAS or STATA.