Editing Survival analysis (section)

===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>