The Hazard Function

What is the Hazard Function in the context of Survival Analysis? How can we apply it in Computational Neuroscience?

In Survival Analysis we are interested in understanding the risk of an event happening at a particular point in time, where time is a continuous variable.

For example, let’s consider the event firing of a neuron: we define the time of firing as \(X\), and time in general as \(t\).

The hazard function, which is a function of time, is defined as:

\[ h(t) = \lim_{\Delta t\to0}\frac{P(t<X<t+\Delta t|X>t)}{\Delta t} \]

We are conditioning on \(X>t\) because we want to condition our probability on the fact that the event hasn’t occurred yet.

Is there a way to rewrite \(h(t)\) in a different way?

\[ h(t) = \lim_{\Delta t\to0}\frac{P(t<X<t+\Delta t|X>t)}{\Delta t}\\ h(t) = \lim_{\Delta t\to0}\frac{P(t<X<t+\Delta t,X>t)}{P(X>t)\Delta t}\\ \]

It is easy to see that \((t<X<t+\Delta t)\) is just a subset of \(X>t\)

    O---------------------- {     X > t    }
    |       o-------------- { t < X < t+Δt }
    |       |       
----.-------.----.---------
    t       X   t+Δt

\[ h(t) = \lim_{\Delta t\to0}\frac{P(t<X<t+\Delta t)}{P(X>t)\Delta t} \]

\(P(X>t)\) is called the survival function and is just \(1\) minus the cumulative distribution function (CDF):

\[ P(X>t) = 1-F(t)=1-\int_{t_0}^tp(t)dt \]

The remaining part is the definition of the derivative of the CDF, which is just the probability density function (PDF) at time \(t\)

\[ \lim_{\Delta t\to0}\frac{P(t<X<t+\Delta t)}{\Delta t}= \lim_{\Delta t\to0}\frac{P(X<t+\Delta t)-P(X <t)}{\Delta t}=\\ \lim_{\Delta t\to0}\frac{F(t+\Delta t)-F(t)}{\Delta t}=p(t) \]

So, finally we can rewrite the hazard function as:

\[ h(t) = \frac{p(t)}{1-\int_{t_0}^tp(t)dt} \]