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[[File:Four_survival_functions.svg|Four survival functions|thumb]] [[File:Survival_function_1.svg|Survival function 1|thumb|left]] [[File:Survival_function_2.svg|Survival function 2|thumb|left]] [[File:Survival_function_2_median_survival.svg|Survival function 2 median survival|thumb]] [[File:Median_survival_greater_than_10_months.svg|Median survival greater than 10 months|thumb]] [[File:Distribution_of_AC_failure_times.svg|Distribution of AC failure times|thumb]] '''Survival function''' is a key concept in [[statistics]], particularly in the field of [[survival analysis]], which is concerned with predicting the time until an event of interest occurs, such as failure in mechanical systems or the time to event such as death in biological organisms. The survival function, denoted as S(t), represents the probability that the time to event is longer than some specified time t. Understanding the survival function is crucial for various applications, including [[clinical trials]], [[reliability engineering]], and [[actuarial science]].

==Definition==
The survival function S(t) is defined as the probability that the time to event T is greater than some time t, mathematically expressed as:
\[ S(t) = P(T > t) \]
where T is a random variable denoting the time until the occurrence of the event of interest.

==Characteristics==
The survival function has several important characteristics:
* It is a non-increasing function, meaning that as time progresses, the probability of survival decreases.
* S(0) is usually equal to 1, assuming that the event of interest has not occurred at the beginning of the observation period.
* As t approaches infinity, S(t) approaches zero, indicating that the event is certain to happen eventually.
* The survival function is related to other functions used in survival analysis, such as the [[hazard function]] and the [[cumulative distribution function]] (CDF). Specifically, the survival function can be expressed in terms of the CDF by \( S(t) = 1 - F(t) \), where F(t) is the CDF of the time to event.

==Estimation==
In practice, the survival function can be estimated using non-parametric methods, such as the [[Kaplan-Meier estimator]], or parametric methods that assume the time to event follows a specific distribution, such as the exponential, Weibull, or log-normal distributions. The choice of method depends on the nature of the data and the assumptions that can be reasonably justified.

==Applications==
The survival function is widely used in various fields for different purposes:
* In [[clinical trials]], it helps in estimating the proportion of patients who are expected to survive past a certain time point, which is crucial for evaluating the efficacy of new treatments.
* In [[reliability engineering]], it assists in predicting the time until failure of systems or components, which is essential for maintenance planning and risk assessment.
* In [[actuarial science]], it is used to model the lifetime of individuals, which is fundamental for designing insurance products and pension plans.

==See Also==
* [[Hazard function]]
* [[Kaplan-Meier estimator]]
* [[Cumulative distribution function]]
* [[Reliability engineering]]
* [[Clinical trials]]

[[Category:Statistics]]
[[Category:Survival analysis]]

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Survival function - Revision history

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