Survivor Function vs. Hazard Function

Question

I'm attempting to understand what the survivor and hazard functions describe under a non-traditional context. I have data comprising distances between successive points on a line ($1D$ vector):

Traditionally in my field, such data is fitted with a gamma-distribution in an attempt to describe the distribution of the points. For example, such data may yield a best-fit (MLE) gamma of $\alpha = 3.5$, $\beta = 450$. In plotting this distribution as a survivor function, I obtain:

And as a hazard function:

Under this context, what is each plot describing? It was my understanding that the hazard function could be understood as describing the probability of finding another point when a point already exists at $x(0)$ i.e. the further away from a pre-existing point you go, the higher the chance is that you will encounter another point. However, I am not entirely sure if this is correct.

With regards to the survivor function, I'm struggling to translate the typical descriptions of it (regarding time, populations etc.) into a more applicable understanding.

Any help would be greatly appreciated.

Answer 1

The theory about survival analysis usually regards time to event data, but I would say that the concepts can be transferred here without loss of generality.

You usually have (referring to the wiki article: wiki )

$S(t) = P(T>t)$ = survival function, which is basically the probability that the event has not happened yet at time t, where the event could be the death of a patient or the malfunction of some machinery and T is the random variable describing it.

$h(t) = lim_{dt \to 0} \frac{P(t \leq T<t+dt)}{dt*S(t)} = - \frac{S'(t)}{S(t)}$ is the hazard function and from wiki Suppose that an item has survived for a time t and we desire the probability that it will not survive for an additional time dt: this is the hazard function/rate.

You usually assume that $T_i$ time of death of patient i, is independent from $T_j$ time of death of patient j, for any $j \neq i$.

So i would Say that you could define a set of independent R.V. $X_i$ = distance where I occur in point i, measured from i-1 with $i=1,..,n$, where $n$ = number of points (not considering the first one).Being independent means that the distance between point $i$ and $j$ does not depend on the distances between any other 2 points.

Given this assumptions I'd say:

$D(x)=P(X>x)$ is your 'survival function'.

$h(x) = -\frac{D'(x)}{D(x)}$ is your hazard rate/function.