https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/oby.20408<\/a><\/p>\nThe latter paper develops a formula for the BRI based on a model of the human body as an ellipse (really an ellipsoid of revolution, but they call it an ellipse) with the semi-major axis half the height and the semi-minor axis computed from the waist measurement, treated as the circumference of a circle. This results in the following bizarre looking formula:<\/p>\nBRI=364.2-365.2sqrt{(1-frac{[w\/2pi]^2}{[0.5h]^2})}<\/span>\nwith w<\/span>\u00a0<\/em>the waist and h<\/span>\u00a0<\/em>the height both measured in cm. The number in the radical is just the eccentricity of the ellipse. But where do 364.2 and 365.5 come from? The authors of the Obesity article comment, \u201cThis formula was derived solely to scale eccentricity values to a more accessible range of values.\u201d That explanation really explains nothing. If you use 300 for both of the constants, that would have the same effect. In fact 100 would work as well.<\/p>\nFirst, we remark that with trivial algebra, their formula can be immediately simplified to<\/p>\nBRI= 364.2 - 365.5sqrt{1-(frac r{pi})^2}<\/span>\nwhere r=w\/h<\/span>\u00a0<\/em>is the ratio of the waist to the height and it doesn\u2019t matter whether the waist and height are measured in cm or inches or, for that matter, light-years or Angstroms. More important, since r\/pi<\/span>\u00a0<\/em>is most likely to\u00a0<\/i>be <1\/5<\/span> and its square <1\/25<\/span> we can use the well-known approximation sqrt{1+h}approx 1+h\/2<\/span> when |h|<\/span>\u00a0<\/em>is small. Applying this we get<\/p>\nBRIapprox 364.2-365.2(1-frac12(frac r{pi})^2)= 182.75(frac r{pi})^2-1.3approx 18.5r^2-1.3<\/span>\nMoreover, since the result puts you in a range (>6.8<\/span> is bad), why bother with that odd looking 1.3<\/span>? For that matter, why bother with that 18.5<\/span>? Just use r^2<\/span> and say that >.44<\/span> is bad. If you want to avoid fractions, use 10<\/span> as the multiplier and skip the 1.3<\/span>. Or simply use r<\/span>\u00a0<\/em>and say that r>.66<\/span> is bad. Or even simpler, say you are obese if your waist is more than 2\/3<\/span>\u00a0your height.<\/p>\nI should also mention that, like the BMI, your BRI can also be too low. The longevity curve is U-shaped. Your life expectation goes down significantly if your waist is less than half your height. These conclusions are much more useful than the complicated formula.<\/p>\n
My point here isn\u2019t that what they are doing is necessarily a bad idea. Indeed I think the basic idea is sound. Some ranges are good; some are not. My point is that they have taken a very simple idea and surrounded it by unnecessarily complicated mathematical obfuscation. This may was probably caused by mathematical naivet\u00e9, but it really hides a basically simple concept\u2013study obesity by the eccentricity of a containing ellipsoid\u2013behind an odd formula.<\/p>\n
I would like to thank Robert Dawson who made many useful comments and also found the articles referred to above. I had written the original based only on an article in the NY Times.<\/p>\n","protected":false},"author":11,"template":"","section":[468],"keyword":[],"class_list":["post-18305","article","type-article","status-publish","hentry","section-submissions"],"toolset-meta":{"author-4-info":{"author-4-surname":{"type":"textfield","raw":""},"author-4-given-names":{"type":"textfield","raw":""},"author-4-honorific":{"type":"textfield","raw":""},"author-4-institution":{"type":"textfield","raw":""},"author-4-email":{"type":"email","raw":""},"author-4-cms-role":{"type":"textfield","raw":""}},"author-3-info":{"author-3-surname":{"type":"textfield","raw":""},"author-3-given-names":{"type":"textfield","raw":""},"author-3-honorific":{"type":"textfield","raw":""},"author-3-institution":{"type":"textfield","raw":""},"author-3-email":{"type":"email","raw":""},"author-3-cms-role":{"type":"textfield","raw":""}},"author-2-info":{"author-2-surname":{"type":"textfield","raw":""},"author-2-given-names":{"type":"textfield","raw":""},"author-2-honorific":{"type":"textfield","raw":""},"author-2-institution":{"type":"textfield","raw":""},"author-2-email":{"type":"email","raw":""},"author-2-cms-role":{"type":"textfield","raw":""}},"author-info":{"author-surname":{"type":"textfield","raw":"Barr"},"author-given-names":{"type":"textfield","raw":"Michael"},"author-honorific":{"type":"textfield","raw":""},"author-email":{"type":"email","raw":""},"author-institution":{"type":"textfield","raw":""},"author-cms-role":{"type":"textfield","raw":""}},"unknown":{"downloadable-pdf":{"type":"file","raw":"https:\/\/notes.math.ca\/wp-content\/uploads\/2024\/10\/Keep-it-simple-\u2014the-BRI-\u2013-CMS-Notes.pdf","attachment_id":18536},"article-toc-weight":{"type":"numeric","raw":"5"},"author-surname":{"type":"textfield","raw":"Barr"},"author-given-names":{"type":"textfield","raw":"Michael"}}},"_links":{"self":[{"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/article\/18305","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/article"}],"about":[{"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/types\/article"}],"author":[{"embeddable":true,"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/users\/11"}],"version-history":[{"count":45,"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/article\/18305\/revisions"}],"predecessor-version":[{"id":18441,"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/article\/18305\/revisions\/18441"}],"wp:attachment":[{"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/media?parent=18305"}],"wp:term":[{"taxonomy":"section","embeddable":true,"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/section?post=18305"},{"taxonomy":"keyword","embeddable":true,"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/keyword?post=18305"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}