## 14.1 Survival Theory

This section reviews the fundamentals of survival analysis, including the hazard probability density, and survival functions.

You can specify the survival distribution function either as a survival function or as a hazard function. Define $$F(t) = Pr(T \le t), \hspace{3mm} 0 < t < \infty$$ as the cumulative risk function, the probability of dying on or before time $$t$$. Then the survival function is the probability of surviving up to time $$t$$,

$S(t) = 1 - F(t) = pr(T > t), \hspace{3mm} 0 < t < \infty.$

The hazard function is the instantaneous death rate given survival up to time $$t$$,

$h(t) = \lim_{\delta \rightarrow 0}{\frac{pr(t < T < t + \delta|T > t)}{\delta}}.$

The survival function and the hazard function are related. The probability of dying during the interval $$(t, t + \delta)$$, $$f(t) = F'(t)$$, is the probability of dying during the interval given survival up to point $$t$$ times the probability of surviving up to point $$t$$, $$f(t) = h(t) S(t)$$.

$$S(t)$$ is also the exponent of the negative cumulative hazard function,

$S(t) = e^{-H(t)}.$

You can use the survival function to estimate the mean and median survival times. The mean survival time is $$E(T) = \int S(t)dt$$. The median survival time is $$S(t) = 0.5$$.