Risks and rates, and the mathematical link between them

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.