The survival function, label {% S(t) %} gives the probability that the event has not ocurred by time {% t %}.
{% S(t) = P(T>t) %}
Here, T is the time of the event.
The survival function has the following properties
- S is nonincreasing
- S(0) = 1, i.e. the probability that the event occurs on or after the zero time is 1.
- S goes to zero as t goes to infinity
Conditional Survival Function
The conditional survival function gives the probability that the event has not ocurred by time
t, given that it has not ocurred by time x. That is
{% S_x(t) = P(T>t|T>x) %}
The following property holds for the conditional survival function.
{% S_{x}(t+s) = S_{x+t}(s)S_x(t) %}
The survival function can be seen to be the conditional survival function with {% x=0 %}.
{% S(t) = S_{0}(t) %}
Also, the conditional survival function can be computed from the the Survival function.
{% S_x(t) = \frac{S_{0}(x+t)}{S_{0}(t)} %}
The hazard function, h(t) is defined as
{% h(t) = \lim_{\Delta t \rightarrow 0} \; \frac{1}{\Delta t} P(t \leq T< t + \Delta t | T \geq t ) %}
The cumulative hazard rate is then defined as
{% A(t) = \int_0 ^t h(s) ds %}
The derivative of the cumulative hazard function is
{% A'(t) = h(t) = \lim_{\Delta t \rightarrow 0} \; \frac{1}{\Delta t} (S(t) - S(t+\Delta t)) / S(t) = -S'(t)/S(t) %}
Then by integration and S(0) = 1
{% -log S(t) = \int_0 ^t h(s)ds %}
which implies that
{% S(t) = exp \{ -\int_0 ^t h(s)ds \} %}
