survivalanalyses

Survival analysis is a branch of statistics focused on the time until an event of interest occurs. Common applications include time to death, relapse, failure of a device, or recovery. Data are frequently censored, meaning that the event has not occurred for some subjects by the end of follow-up or is only known to occur within an interval. Key quantities are the survival function S(t) = P(T>t), the hazard function h(t) describing instantaneous risk, and the cumulative hazard H(t).

Nonparametric methods accommodate censoring without assuming a distribution. The Kaplan–Meier estimator provides an estimate of S(t). The log-rank test compares survival curves between groups. Censoring types include right-censoring, left-censoring, and interval censoring; different methods address each type, with right-censoring being most common in clinical data.

Statistical models link survival times to covariates. The Cox proportional hazards model is semi-parametric and relates covariates to the hazard through a baseline hazard without specifying its form; interpretation centers on hazard ratios. Parametric models specify a distribution for T, such as exponential or Weibull, allowing explicit survival and hazard forms. Estimation uses partial likelihood for Cox models and maximum likelihood for parametric models; model fit can be assessed by residuals and information criteria.

Survival analysis is widely used in clinical trials, epidemiology, and reliability engineering to assess treatment effects, prognosis, and product failure. Complications include competing risks, time-varying covariates, non-proportional hazards, and informative censoring. Software implementations include R packages such as survival, Python's lifelines, and commercial tools like SAS PROC PHREG or Stata stcox. Proper analysis requires careful handling of censoring, model assumptions, and follow-up duration.