What is a Hazard Function?
More specifically, the hazard function models which periods have the highest or lowest chances of an event. The function is defined as the instantaneous risk that the event of interest happens, within a very narrow time frame. (Note: If you’re familiar with calculus, you may recognize that this instantaneous measurement is the derivative at a certain point).
Hazard functions and survival functions are alternatives to traditional probability density functions (PDFs). They are better suited than PDFs for modeling the types of data found in survival analysis.
Conditional and Variations
The hazard function is a conditional failure rate, in that it is conditional a person has actually survived until time t. In other words, the function at year 10 only applies to those who were actually alive in year 10; it doesn’t count those who died in previous periods.
There are other variations on the function, other than as a conditional rate. The Kaplan Meier (KM) method uses rates, has no upper limit, and is preferred for clinical trials (Fink & Brown, 2006). Conversely, with the actuarial method, the hazard function is a proportion, with values between 0 and 1.
Formula
The hazard function formula is:
Where:
- fY(y) = the probability density function of survival time Y,
- SY = the Survivor function (the probability of surviving beyond a certain point in time)
References
Der, G. & Everitt, B. (2007). Basic Statistics Using SAS Enterprise Guide: A Primer. SAS Institute.
Fink, S., Brown, R. (2006). Survival Analysis. Gastroenterol Hepatol (N Y). May; 2(5): 380–383. Retrieved May 28, 2018 from here (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5338193/).