# Relationship of incidence rate and hazard function

I like to figure out how I can retrieve an incidence rate based on a hazard function. So background could be that my hazard function models the prob of the event based on the age of a person (t = time since birth). Of course, the current age distribution in my population is relevant.

So my understanding is as follows:

• Hazard function h(t): gives me the probability that the event occurs within the next interval based on the assumption that until t there was no event

Now I like to calculate an incidence rate - so percent of new events within one interval (e.g. one year) but independently of single cases.

What is the relation between the concept of hazard function or maybe cumulative hazard function and a general incidence rate?

The hazard function $$h(t)$$ is the instantaneous incidence rate, and is probably the least ambiguous incidence rate to cite. If the incidence rate isn't constant over time, I find it confusing to think about an average incidence rate over a finite time interval. Such an "incidence rate" depends on the choice of starting time and the length of the time interval beyond that. If you do insist on such a value, however, it's pretty simply related to the cumulative hazard function, as you intuited.

The cumulative hazard function, $$H(t)=\int_0^t h(\tau) d\tau$$, is related to the survival probability starting from $$t=0$$ by $$S(t)=\exp\left(-H(t)\right)$$. So if you've already reached some time $$t_0$$ and want to know the fraction of those still at risk who will have an event before a later time $$t_1$$ one time unit later, you could calculate:

\begin{align} \frac{S(t_0)-S(t_1)}{S(t_0)} & = 1- \frac{S(t_1)}{S(t_0)}\\& =1-\exp\left(H(t_0)-H(t_1)\right)\\& =1-\exp \left(-\int_{t_0}^{t_1} h(\tau) d \tau\right). \end{align}

That gives what you want, expressed in terms of the difference in cumulative hazard between the two time points.

• Maybe my question was not clear enough here. Could also be that I mix up different concepts. I think of an incidence rate as in medical science. There, the incidence rate describes the number of new cases within a population in a defined period of time, for example one year. The hazard function h(t) now describes the probability depending on the age of the test persons. To derive the incidence rate as above, I must sum over all h(t) multiplied by the number of subjects for each t? Does this make sense? Or is there a more elegant way?
– Dirk
Commented Jun 14, 2023 at 6:59
• @Dirk "The incidence rate numerator is... the number of new cases. The denominator... is the total person-time, or the amount of time that all at-risk persons were observed." It's a rate: cases per number of individuals initially at risk per unit time. So you add up the total number of cases over, e.g., 1 year (numerator at the left of the equation) and divide by the number at risk at the beginning of the time period (denominator).
– EdM
Commented Jun 14, 2023 at 12:00
• @Dirk See the warning from Wikipedia: "Use of this measure implies the assumption that the incidence rate is constant over different periods of time." That's why I was at first reluctant to provide an answer at all. If you want an "incidence rate" of any type other than the instantaneous hazard and the hazard isn't constant, then you have to do something like what I show.
– EdM
Commented Jun 14, 2023 at 12:03
• So when you say the hazard may be constant - what do you exactly mean by that? h(t) is not constant as it varies by t, here the age of persons. But if I get you right you mean that h(t) does not change over time, right? So, of course, h(t) is not a constant, but the function h(t) does not change over time.
– Dirk
Commented Jun 14, 2023 at 13:36
• @Dirk you are correct that $h(t)$ is not constant if you are examining death as a function of age in the general population. But in many applications of survival analysis to medicine and epidemiology, particularly over periods of just a few years, then $h(t)$ = constant, independent of $t$, is a reasonable approximation of reality. That leads to an exponential survival model.
– EdM
Commented Jun 14, 2023 at 13:46