Basic concepts in Survival Analysis

Survival Analysis is a statistical methodology for the occurrence of discrete events unfolding in continuous (or discrete) time. Such events are experienced one or more times during the lifetime of individual subjects whose life histories are available for examination. Examples  include: death of an individual, acute illness (e.g. myocardial infarction), time to resolution of an infection after antibiotics, marriage and divorce, acceptance of a scientific paper. Still, events are not limited to medical, sociological contexts, humans or even biological systems. In fact,  statistical methods for lifetime data form the core of the field of reliability analysis, which focuses on the failure of industrial systems ranging from light bulbs to nuclear safety systems and from software to airplanes.
In all survival analytic activities the term it survival time or  time-to-event is used to denote the time elapsed from an initiating event until the event of interest. In many cases, the initiating event will not coincide with the “birth” of the individual i.e. the time the latter came into existence, but will be some other type of discrete event that is of interest to the investigator. In medical contexts, this could be the time of diagnosis of a particular disease, while in industrial applications an initiating event may be defined by the beginning of a reliability test. Furthermore, the same body of survival time data may be analyzed from alternative viewpoints which adopt a different definition of the initiating event and an origin of the time scale. Even though the relative nature of the survival time is not usually explicitly acknowledged, its importance for proper statistical modelling of real world problems cannot be overemphasized. In particular the issue of left truncation will arise when multiple initiating points are contemplated. Under left truncation individuals whose survival time is less than a threshold cannot studied because they do not survive long enough to be observed. For example when studying stroke, only those individuals that survive the acute (pre-hospital) phase and reach the hospital can be included in a study.
Yet another feature that distinguishes survival analysis from other statistical methods, is censoring. The latter arises because of the nature of collecting of survival times. Typically one has to wait for individuals to experience the event of interest and when a study is terminated only a subgroup of individuals will have experienced that event. Hence the survival, times will be known exactly only for them; the only thing that is known for the lifetimes of the remaining individuals is that they exceed certain values (right censoring).



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