The Survival Function

Overview

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)} %}

Hazard Rate


The hazard function, {% h(t) %} is defined as
{% h(t) = \lim_{\Delta t \rightarrow 0} \; \frac{1}{\Delta t} P(t < T \leq t + \Delta t | T > t ) %}
The cumulative hazard rate is then defined as
{% \displaystyle A(t) = \int_0 ^t h(s) ds %}

Relationships Between Hazard Rate and Survival Function


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 %}
{% \displaystyle -log S(t) = \int_0 ^t h(s)ds %}
which implies that
{% \displaystyle S(t) = exp \{ -\int_0 ^t h(s)ds \} %}

Topics