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29-01-2025 | ESSAY

Risks and rates, and the mathematical link between them

Author: James A. Hanley

Published in: European Journal of Epidemiology | Issue 1/2025

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Abstract

The risk over a given time span can be calculated as one minus the exponentiated value of the negative of the integral of the incidence density function (or hazard rate function) over that time span. This relationship is widely used but, in the few instances where textbooks have presented it, the derivations of it tend to be purely mathematical. I first review the historical contexts, definitions, distinctions and links. I then offer a more intuitive heuristic approach that draws on the conceptualization of a person-year in Edmonds’ 1832 definition of the force of mortality, and on the number of replacements in a dynamic population. Similarly I show how the Nelson-Aalen risk estimator can be seen in the context of this historical conceptualization of a person-year, scaled to the experience of a dynamic population of (constant) size 1.
Appendix
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Footnotes
1
Feinstein, the ‘clinical’ epidemiologist, did not work with person-time denominators. Instead of including the 1/time dimension in his incidence statistics, he limited himself to fractions and proportions, and when confronted with their limitations, and the confusion they caused [3840], he deflected the criticisms by deriding statisticians and epidemiologists who blindly use existing data to compute ‘the number of new events per unit of person-time of observation.’ [41]
 
2
Although many publications do not, it is important to distinguish between an estimator of (i) the survival function and (ii) the integral encountered above. Both Nelson and Aalen used the integrated hazard exclusively for graphically assessing the suitability of various parametric models, and it was not until Fleming and Harrington that the exponential formula was used to estimate the survival function.
 
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Metadata
Title
Risks and rates, and the mathematical link between them
Author
James A. Hanley
Publication date
29-01-2025
Publisher
Springer Netherlands
Published in
European Journal of Epidemiology / Issue 1/2025
Print ISSN: 0393-2990
Electronic ISSN: 1573-7284
DOI
https://doi.org/10.1007/s10654-024-01191-9