NationMaster - Encyclopedia: Survival function

The survival function, also known as a survivor function or reliability function, is a property of any random variable that maps a set of events, usually associated with mortality or failure of some system, onto time. It captures the probability that the system will survive beyond a specified time. The term reliability function is common in engineering while the term survival function is used in a broader range of applications, including human mortality. A random variable is a term used in mathematics and statistics. ... A pocket watch. ... The word probability derives from the Latin probare (to prove, or to test). ... Survival analysis is a branch of statistics which deals with death in biological organisms and failure in mechanical systems. ... Engineering is the application of scientific and technical knowledge to solve human problems. ...

Definition

Let X be a continuous random variable with cumulative distribution function F(t) on the interval [0,∞). Its survival-, or reliability-function is: In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or...

$R(t) = P{Tgeq t} = int_t^{infty} f(u),du = 1-F(t).$

Properties

Every survival function R(t) is monotone decreasing, i.e. $R (u) < R (t)$ for $u > t$ In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ...

The time, t = 0, represents some origin, typically the beginning of a study or the start of operation of some system. R(0) is commonly unity but can be less to represent the probability that the system fails immediately upon operation. A pocket watch. ... The word probability derives from the Latin probare (to prove, or to test). ...

Again, lim_t→∞R(t) is commonly zero but can be greater to represent a system in which eternal life is possible. Immortality is the concept of existing for a potentially infinite or indeterminate length of time. ...

1.3.6.2. Related Distributions (597 words)
	The percent point function (ppf) is the inverse of the cumulative distribution function.
	The hazard function is the ratio of the probability density function to the survival function, S(x).
	The cumulative hazard function is the integral of the hazard function.

Survival/Failure Time Analysis (2842 words)
	An assumption of the proportional hazard model is that the hazard function for an individual (i.e., observation in the analysis) depends on the values of the covariates and the value of the baseline hazard.
	is a function of (1) the baseline hazard
	The Chi-square goodness-of-fit value is computed as a function of the log-likelihood for the model with all parameter estimates (L1), and the log-likelihood of the model in which all covariates are forced to 0 (zero; L0).

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