Editing Survival analysis (section)

==General formulation==
{{unreferenced section|date=April 2021}}

===Survival function===
{{main|Survival function}}

The object of primary interest is the '''survival function''', conventionally denoted ''S'', which is defined as

<math display="block">S(t) = \Pr(T > t)</math>
where ''t'' is some time, ''T'' is a [[random variable]] denoting the time of death, and "Pr" stands for [[probability]]. That is, the survival function is the probability that the time of death is later than some specified time ''t''.
The survival function is also called the ''survivor function'' or ''survivorship function'' in problems of biological survival, and the ''reliability function'' in mechanical survival problems. In the latter case, the reliability function is denoted ''R''(''t'').

Usually one assumes ''S''(0) = 1, although it could be less than 1{{nbsp}}if there is the possibility of immediate death or failure.

The survival function must be non-increasing: ''S''(''u'') ≤ ''S''(''t'') if ''u'' ≥ ''t''. This property follows directly because ''T''>''u'' implies ''T''>''t''. This reflects the notion that survival to a later age is possible only if all younger ages are attained. Given this property, the lifetime distribution function and event density (''F'' and ''f'' below) are well-defined.

The survival function is usually assumed to approach zero as age increases without bound (i.e., ''S''(''t'') → 0 as ''t'' → ∞), although the limit could be greater than zero if eternal life is possible. For instance, we could apply survival analysis to a mixture of stable and unstable [[Carbon#Isotopes|carbon isotopes]]; unstable isotopes would decay sooner or later, but the stable isotopes would last indefinitely.

===Lifetime distribution function and event density===

Related quantities are defined in terms of the survival function.

The '''lifetime distribution function''', conventionally denoted ''F'', is defined as the complement of the survival function,

<math display="block">F(t) = \Pr(T \le t) = 1 - S(t).</math>
If ''F'' is [[Differentiable function|differentiable]] then the derivative, which is the density function of the lifetime distribution, is conventionally denoted ''f'',

<math display="block">f(t) = F'(t) = \frac{d}{dt} F(t).</math>
The function ''f'' is sometimes called the '''event density'''; it is the rate of death or failure events per unit time.

The survival function can be expressed in terms of [[probability distribution]] and [[probability density function]]s

<math display="block">S(t) = \Pr(T > t) = \int_t^{\infty} f(u)\,du = 1-F(t).</math>
Similarly, a survival event density function can be defined as

<math display="block">s(t) = S'(t) = \frac{d}{dt} S(t) = \frac{d}{dt} \int_t^{\infty} f(u)\,du = \frac{d}{dt} [1-F(t)] = -f(t).</math>
In other fields, such as statistical physics, the survival event density function is known as the [[first passage time]] density.

===Hazard function and cumulative hazard function===

The '''[[hazard function]]''' <math>h</math> is defined as the event rate at time <math>t,</math> conditional on survival at time <math>t.</math>

Synonyms for ''hazard function'' in different fields include hazard rate, [[force of mortality]] ([[demography]] and [[actuarial science]], denoted by <math>\mu</math>), force of failure, or [[failure rate]] ([[engineering]], denoted <math>\lambda</math>). For example, in actuarial science, <math>\mu(x)</math> denotes rate of death for people aged <math>x</math>, whereas in [[reliability engineering]] <math>\lambda(t)</math> denotes rate of failure of components after operation for time <math>t</math>.

Suppose that an item has survived for a time <math>t</math> and we desire the probability that it will not survive for an additional time <math>dt</math>:

<math display="block">h(t) = \lim_{dt \rightarrow 0} \frac{\Pr(t \leq T < t+dt)}{dt\cdot S(t)} = \frac{f(t)}{S(t)} = - \frac{S'(t)}{S(t)}.</math>

Any function <math>h</math> is a hazard function if and only if it satisfies the following properties:

#<math>\forall x\geq0\left(h(x)\geq0\right)</math> ,
#<math>\int_{0}^{\infty} h(x)dx=\infty</math> .

In fact, the hazard rate is usually more informative about the underlying mechanism of failure than the other representations of a lifetime distribution.

The hazard function must be non-negative, <math>\lambda(t)\geq0</math>, and its integral over <math>[0, \infty]</math> must be infinite, but is not otherwise constrained; it may be increasing or decreasing, non-monotonic, or discontinuous. An example is the [[bathtub curve]] hazard function, which is large for small values of <math>t</math>, decreasing to some minimum, and thereafter increasing again; this can model the property of some mechanical systems to either fail soon after operation, or much later, as the system ages.

The hazard function can alternatively be represented in terms of the '''cumulative hazard function''', conventionally denoted <math>\Lambda</math> or <math>H</math>:

<math display="block">\,\Lambda(t) = -\log S(t)</math>
so transposing signs and exponentiating

<math display="block">\,S(t) = \exp(-\Lambda(t))</math>
or differentiating (with the chain rule)

<math display="block">\frac{d}{dt} \Lambda(t) = -\frac{S'(t)}{S(t)} = \lambda(t).</math>
The name "cumulative hazard function" is derived from the fact that

<math display="block"> \Lambda(t) = \int_0^{t} \lambda(u)\,du</math>
which is the "accumulation" of the hazard over time.

From the definition of <math>\Lambda(t)</math>, we see that it increases without bound as ''t'' tends to infinity (assuming that <math>S(t)</math> tends to zero). This implies that <math>\lambda(t)</math> must not decrease too quickly, since, by definition, the cumulative hazard has to diverge. For example, <math>\exp(-t)</math> is not the hazard function of any survival distribution, because its integral converges to 1.

The survival function <math>S(t)</math>, the cumulative hazard function <math>\Lambda(t)</math>, the density <math>f(t)</math>, the hazard function <math>\lambda(t)</math>, and the lifetime distribution function <math>F(t)</math> are related through
<math display="block">S(t) = \exp [ -\Lambda(t) ] = \frac{f(t)}{\lambda(t)} = 1-F(t), \quad t > 0.</math>

===Quantities derived from the survival distribution===

'''Future lifetime''' at a given time <math>t_0</math> is the time remaining until death, given survival to age <math>t_0</math>. Thus, it is <math>T - t_0</math> in the present notation. The '''expected future lifetime''' is the [[expected value]] of future lifetime. The probability of death at or before age <math>t_0+t</math>, given survival until age <math>t_0</math>, is just

<math display="block">P(T \le t_0 + t \mid T > t_0) = \frac{P(t_0 < T \le t_0 + t)}{P(T > t_0)} = \frac{F(t_0 + t) - F(t_0)}{S(t_0)}.</math>
Therefore, the probability density of future lifetime is

<math display="block">\frac{d}{dt}\frac{F(t_0 + t) - F(t_0)}{S(t_0)} = \frac{f(t_0 + t)}{S(t_0)}</math>
and the expected future lifetime is

<math display="block">\frac{1}{S(t_0)} \int_0^{\infty} t\,f(t_0+t)\,dt = \frac{1}{S(t_0)} \int_{t_0}^{\infty} S(t)\,dt,</math>
where the second expression is obtained using [[integration by parts]].

For <math>t_0 = 0</math>, that is, at birth, this reduces to the expected lifetime.

In reliability problems, the expected lifetime is called the ''[[mean time to failure]]'', and the expected future lifetime is called the ''mean residual lifetime''.

As the probability of an individual surviving until age ''t'' or later is ''S''(''t''), by definition, the expected number of survivors at age ''t'' out of an initial [[population]] of ''n'' newborns is ''n'' × ''S''(''t''), assuming the same survival function for all individuals. Thus the expected proportion of survivors is ''S''(''t'').
If the survival of different individuals is independent, the number of survivors at age ''t'' has a [[binomial distribution]] with parameters ''n'' and ''S''(''t''), and the [[variance]] of the proportion of survivors is ''S''(''t'') × (1-''S''(''t''))/''n''.

The age at which a specified proportion of survivors remain can be found by solving the equation ''S''(''t'') = ''q'' for ''t'', where ''q'' is the [[quantile]] in question. Typically one is interested in the '''[[median]] lifetime''', for which ''q'' = 1/2, or other quantiles such as ''q'' = 0.90 or ''q'' = 0.99.