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:

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

For attribution, please cite this work as

Bonvini (2021, July 10). Last Week's Potatoes: The Hazard Function. Retrieved from https://lastweekspotatoes.com/posts/2021-07-20-the-hazard-function/

BibTeX citation

@misc{bonvini2021the, author = {Bonvini, Andrea}, title = {Last Week's Potatoes: The Hazard Function}, url = {https://lastweekspotatoes.com/posts/2021-07-20-the-hazard-function/}, year = {2021} }