ACCESS - MATH July 2003 Notes on Body Mass Index and actual national data

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1 ACCESS - MATH July 2003 Notes o Body Mass Idex ad actual atioal data What is the Body Mass Idex? If you read ewspapers ad magazies it is likely that oce or twice a year you ru across a article about the body mass idex (B.M.I.), ad its use i determiig health risk factors for overweight ad uderweight people. If you search the iteret for "body mass idex" you will fid may sites which let you compute your B.M.I., ad which tell you a little bit about it. A perso s B.M.I. is computed by dividig their weight by the square of their height, ad the multiplyig by a uiversal costat. If you measure weight i kilograms, ad height i meters, this costat is the umber oe. Thus, the propoats of the B.M.I idex are claimig that for adults at equal risk levels (but differet heights), weight should be proportioal to the square of height. As we have discussed i class, if people were to scale equally i all directios ("self-similar") whe they grew, volume ad hece weight would scale as the cube of height. That particular power law seems a little high, sice adults do t look like uiformly expaded versios of babies; we seem to get relatively stretched out whe we grow taller. Oe might expect the best predictive power for weight as a fuctio of height to be somewhere betwee 2 ad 3, if oe expected a power law at all. If there is a predictive power, ad if it is much larger tha 2, the oe could argue that the body mass idex might eed to be modified to reflect this fact. (I fact, whe you fid body mass idex tables, they ofte explai that you should modifiy the acceptable BMI values for childre.) We have all collected several heights ad weights, ad hopefully i aggregate we will have a good umber of represetative measuremets, from baby-sized to adult. Each group will use this data, see if it is cosistet with a power law relatig weights to heights, ad decide whether the B.M.I. power of 2 is a good choice. I ofte do this experimet with my liear algebra classes as well as with the Accessors, ad we have gotte powers betwee 2.3 ad 2.7. Also, several years ago I foud a atioal data base at the U.S. Ceter for Disease Cotrol web site. It cotaied a wide variety of body measuremets collected betwee 1976 ad 1980, icludig atioal media heights ad weights for boys ad girls, age By usig oly the atioal medias a lot of the variace has bee take out of the data, compared to what yours will look like. The atioal data is very cosistet with a power law, with power = 2.6. If ay of you ca figure out a valid mathematical model which explais this power law, you ll have a publishable paper. How do you test for power laws? Whe you studied logarithms i high school you might have wodered what they were good for. Well, as Nacy showed you yesterday, oe applicatio is i lookig for power laws. Remember how the discussio wet: Suppose we have a set of "" data poits, which you ca thik of as your height-weight data, but which could really be ay set of paired data: [[ x 1, y 1 ], [ x 2, y 2 ], [ x 3, y 3 ],...,[ x, y ]] We wat to see if there is a power m ad a proportioality costat b so that the formula y= b x m effectively mirrors the real data. Takig (atural) logarithms of the proposed power law yields

2 l( y ) = l( b) + m l( x) So if we write Y = l( y ) ad X = l( x ), B = l( b ), this becomes the equatio of a lie i the ew variables X ad Y: Y = mx + B Thus, i order for there to be a power law for the origial data, the l-l data should (approximately) satisfy the equatio of a lie. Furthermore, this process is reversible; if the l-l data lies o a lie with slope m ad itercept B, the the origial data satisfies a power law with power m ad proportioality costat b = e B. That s because of the rules of expoets: Y = B+ mx e Y = e ( B+ mx) e Y = e B ( e X ) m y= b x m I real experimets, it is ot too hard to see if data is well approximated by a lie, so this trick with the logarithm is quite useful. Natioal data example: I used colos after most of these commads to suppress the output. If you wat to go back ad see what each commad is doig, replace the colos with semicolos. > restart: > with(plots): > boyhw:=[[35.9,29.8],[38.9,34.1],[41.9,38.8],[44.3,42.8], [47.2,48.6],[49.6,54.8],[51.4,60.8],[53.6,66.5], [55.7,76.8],[57.3,82.3],[59.8,93.8],[62.8,106.8], [66.0,124.3],[67.3,132.6],[68.4,142.4],[68.9,145.1], [69.6,155.3],[69.6,153.2]]: #boy heights (iches) weights (pouds): Ntl medias for ages 2-19 > girlhw:=[[35.4,28.0],[38.4,32.6],[41.1,36.8],[43.9,41.8], [46.6,47.0],[48.9,52.5],[51.4,60.8],[53.1,65.5], [55.7,76.1],[58.2,89.0],[61.0,100.1],[62.6,108.1], [63.3,117.1],[64.2,117.6],[64.3,122.6],[64.2,128.8], [64.1,124.5],[64.5,126]]: #girl heights(iches) weights (pouds): Ntl medias for ages 2-19 > boys:=poitplot(boyhw): girls:=poitplot(girlhw): display({boys,girls}, title= plot of [height,weight], Natioal medias ages 2-19 );

3 plot of [height,weight], Natioal medias ages Ad ow for the l-l data. > with(lialg): #liear algebra package > B:=evalm(boyhw): #evaluate matrix, this will #tur our list of poits ito a matrix, which will #be easiser to maipulate i Maple G:=evalm(girlhw): BG:=stackmatrix(B,G): #stack the matrices o top of eachother. > lbga:=map(l,bg): #take l of the boy ad girl height-weights lbg:=map(evalf,lbga): #Get decimal (floatig poit) values. #This speeds up computatios later i the #least squares fit - otherwise Maple tries workig #symbolically. > llplot:=poitplot(lbg): display(llplot,title= l-l data );

4 5 l-l data How do I fid the best lie fit to a collectio of poits? From Calculus, there is a slope m ad itercept B yieldig a lie which miimizes the sum of the squared vertical distaces betwee your data poits ad the poits o a lie. If the data poits are the m ad B solve the system of equatios 2 X i i = 1 X i i = 1 [[ X 1, Y 1 ], [ X 2, Y 2 ], [ X 3, Y 3 ],...,[ X, Y ]] X i i = 1 m = B X i Y i i = 1 Y i i = 1 You could solve this system with several "do-loops" to compute the matrix etries, followed by a "solve" commad to solve the system, but the method is so commo that every decet mathematical software or graphig calculator already has a commad to do all of that work for you. I statistics this procedure is called liear regressio as well as the method of least squares. If you looked through the help directory i your meu bar you would evetually fid MAPLE s versio of this commad livig i the stats library packag, ad called "fit[leastsquare]". Here s how the commad works. I ve used it

5 to check the example we worked by had i class. You must be careful with brackets ad paratheses. > with(stats): > fit[leastsquare[[x,y]]]([[0,2,4],[1,2,1]]);#the sytax here is #to first ame your variables, the give two lists, oe of the first variable #values, ad the secod with the correspodig secod variable values. Thus #we are tryig to fid the best lie fit for the poits [0,1],[2,2],[4,1]. Y = 4 3 Now that we ve tested the commad, we ca use it o the atioal data: > Xs:=covert(col(lBG,1),list): #covert the first colum of #the l-l data ito a list of the "x s" The least squares #commad wats to have lists iput, ot matrix colums, eve #though it s hard for us to see ay differece Ys:=covert(col(lBG,2),list): > fit[leastsquare[[x,y]]]([xs,ys]); Y = X We ca paste i the equatio of the lie ad see how well we did. > lie:=plot( *x, X= ,Y= , color=black): display({lie,llplot}, title= least squares fit );

6 least squares fit Y X > Fially, we ca go back from the least squares lie fit to a power law > m:= ; #power b:=exp( ); #proportioality costat m := b := > powerplot:=plot(b*h^m,h=0..80,w=0..200,color=black): display({powerplot,boys,girls},title= power law approximatio for atioal height-weight data ); #by callig the variables h ad w, ad givig their rages, I #get Maple to label the axes as I wat

7 power law approximatio for atioal height-weight data w h >

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