{"title":"Risks and rates, and the mathematical link between them","authors":"James A. Hanley","doi":"10.1007/s10654-024-01191-9","DOIUrl":null,"url":null,"abstract":"<p>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.</p>","PeriodicalId":11907,"journal":{"name":"European Journal of Epidemiology","volume":"15 1","pages":""},"PeriodicalIF":5.9000,"publicationDate":"2025-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Epidemiology","FirstCategoryId":"3","ListUrlMain":"https://doi.org/10.1007/s10654-024-01191-9","RegionNum":1,"RegionCategory":"医学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PUBLIC, ENVIRONMENTAL & OCCUPATIONAL HEALTH","Score":null,"Total":0}
引用次数: 0
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.
期刊介绍：
The European Journal of Epidemiology, established in 1985, is a peer-reviewed publication that provides a platform for discussions on epidemiology in its broadest sense. It covers various aspects of epidemiologic research and statistical methods. The journal facilitates communication between researchers, educators, and practitioners in epidemiology, including those in clinical and community medicine. Contributions from diverse fields such as public health, preventive medicine, clinical medicine, health economics, and computational biology and data science, in relation to health and disease, are encouraged. While accepting submissions from all over the world, the journal particularly emphasizes European topics relevant to epidemiology. The published articles consist of empirical research findings, developments in methodology, and opinion pieces.
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