Survival analysis
= Survival analysis
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<Survival analysis> studies the <probability distribution> of a <survival time>: how long from a specified starting point until a specified occurrence. The occurrence need not be death. It might be a machine's first breakdown, a customer's first renewal, or a person's recovery. The meaning of "survival" is simply that the chosen occurrence has not happened yet. Both the starting point and the outcome must be defined before interpreting a <survival function>.
The distinctive difficulty is incomplete observation. A machine still working when observation stops supplies a lower bound on its <survival time>, rather than an observed breakdown time. <Right censoring> lets us retain that information. The <Kaplan–Meier estimator> estimates a <survival function> using the changing <risk sets>, without choosing an <exponential distribution> or another particular <survival distribution>. A <hazard function> instead describes the current event rate among those still event-free; its <cumulative hazard> connects those local rates to the whole <survival function>.
This framework turns different follow-up lengths into usable evidence about waiting times. It supports questions such as the chance of remaining event-free for another year, or the <restricted mean survival time> within an observed horizon. Lu Tian's Stanford teaching materials introduce these interpretations in https://web.stanford.edu/~lutian/coursepdf/unit1.pdf[Survival Distributions, Hazard Functions, Cumulative Hazards, Section 1.1].