timetoevent

Time-to-event, often referred to as time-to-event data or survival data, describes measurements where the primary variable is the time elapsed from a defined origin to the occurrence of a specified event. The origin may be treatment start, enrollment, diagnosis, or another starting point. Common events include death, disease progression, relapse, device failure, or recovery. Because the exact event time is not always observed, data are frequently right-censored: the event has not occurred by the end of follow-up or the subject is lost to follow-up. Left and interval censoring occur in some studies as well.

Key quantities in time-to-event analysis are the survival function S(t) = P(T > t), the hazard function h(t) = f(t)/S(t), and the cumulative hazard H(t) = ∫0^t h(u) du. The survival and hazard functions are linked by S(t) = exp(-H(t)). Nonparametric methods, such as the Kaplan-Meier estimator for S(t) and the Nelson-Aalen estimator for H(t), summarize data without assuming a distribution. The log-rank test compares survival curves between groups. Semi-parametric models, notably the Cox proportional hazards model, relate the hazard to covariates via h(t|X) = h0(t) exp(βX), yielding hazard ratios for factors of interest. Parametric survival models (e.g., exponential, Weibull, Gompertz) assume a distribution for T and can extrapolate beyond observed data.

Applications span clinical trials, epidemiology, public health, and reliability engineering, wherever the timing of an event is of interest. Software tools commonly used include R (survival, survminer), Python (lifelines), and SAS.