Survival Analysis

Survival analysis is a collection of methods for the analysis of data that involve the time to occurrence of some event and, more generally, to multiple durations between occurrences of events. Apart from their extensive use in studies of survival times in clinical and health-related studies, these methods have found application in several other fields, including industrial engineering (e.g., reliability studies and analyses of equipment failure times), demography (e.g., analyses of time intervals between successive child births), sociology (e.g., studies of recidivism, duration of marriages), and labor economics (e.g., analysis of spells of unemployment, duration of strikes). The terms duration analysis, event-history analysis, failure-time analysis, reliability analysis, and transition analysis refer essentially to the same group of techniques, although the emphases in certain modeling aspects could differ across disciplines.

Thetimetoevent T is a positive random variable with distribution function

In biostatistics and epidemiology, it is more common to use the survivorship function or survival function S(t) = 1 − F(t). Thus, S(t) is the probability of being free of events at time t. In clinical studies, where T is the time of death of a patient, one refers to T as the survival time and S(t)asthe probability of survival beyond t. Since time and duration must have an origin, the specific context determines an appropriate starting point from which T is measured. For example, consider a clinical trial of competing treatments in which patients entering the study are randomized to treatment conditions. The time origin is the time of randomization or initiation of treatment, and T is the time until the primary endpoint (e.g., death) is reached.