A Different BMI Formula

A brick representing a persons weight to height

Should there be a different BMI formula? Many have suggested that it would be better to come up with a formula different than either the body mass index which is based on the scaling of weight to the power of two or the formula known as the Ponderal Index which is based on the weight being a factor based on the power of three. The cube root (power of three) is the most natural of relationships for objects that increase in size. For example if you double the side of a 1 inch cube to make it a 2 inch cube it will have a volume of 2*2*2 or 2³ or 8 cubic inches. If that cube was of uniform weight it would have 8 times as much weight. If you double a person’s height would you not expect you would double their weight?


Here are some pictures to represent the problem. Think of one brick as some one 5 feet tall and 120 pounds. How much would this person weigh if they were 10 feet tall?

Eight bricks representing a person who doubles propotionately



If all proportions were exactly the same this person would weigh eight times as much or be 960 pounds.











two bricks representing a tall person who is stretched to twice the height.



But taller people do not have the same proportions as shorter people. Are they simply stretched? If so they would be 240 pounds.



















Would they be squared as the BMI formula suggests? If so they would weigh about 480 pounds.





The BMI formula was first observed using some data that was observed on men hundreds of years ago. Let us look at some more recent data which was taken from the 1960 census a time when obesity was not such a big problem. I will try to show the comparison in a visual form. I will also try to show whether a different number between a square (2) and a cube (3) would be more appropriate to use an exponent. It has been suggested that a number between 2.3 and 2.7 would be better.¹ Of course I realize that this analysis was looking at all of the people in 1960 and to make a decision requires looking only at the weight of healthy people, and I do not know where that data is.

Why do I use the 1960 data? Because as the chart below indicates, there were fewer overweight people in 1960.


Table 2. Age-adjusted* prevalence of overweight, obesity and extreme obesity among U.S. adults, age 20-74 years**

 

NHES I
1960-62
n=6,126

NHANES I
1971-74
n=12,911

NHANES II
1976-80
n=11,765

NHANES III
1988-94
n=14,468

NHANES
1999-2000
n=3,603

NHANES
2001-02
n=3,916

NHANES
2003-04
n=3,756

NHANES
2005-06
n=3,835

Overweight (BMI greater than or equal to 25.0 and less than 30.0)

31.5

32.3

32.1

32.7

33.6

34.4

33.4

32.2

Obese (BMI greater than or equal to 30.0)

13.4

14.5

15.0

23.2

30.9

31.3

32.9

35.1

Extremely obese (BMI greater than or equal to 40.0)

0.9

1.3

1.4

3.0

5.0

5.4

5.1

6.2

*Age-adjusted by the direct method to the year 2000 U.S. Bureau of the Census estimates using the age groups 20-39, 40-59, and 60-74 years.

**NHES: National Health Examination Survey; NHES included adults 18-79 years, NHANES I & II did not include individuals over 74 years of age, thus trend estimates are based on age 20-74 years. Pregnant females were excluded from analyses.

Source:
http://www.cdc.gov/nchs/data/hestat/overweight/overweight_adult.htm


If you look at the next chart, you can see that the BMI formula (power of 2) gives a better fit than the Ponderal Index (power of 3). For the power of 3 to be a better fit it would have to give more similar numbers irrespective of a person’s height.

You can also see that the number that gives the most consistent results irrespective of a persons height is the BMI formula to the power of 1.8 rather than to the power of two.



    BMI of Ave Men (1960 25-34 yrs) (Smoothed average Table 5 NCHS "Weight by Height and Age of Adults")
height weight BMI (power 2) power 2.5 power 3 (PI) power 2.2 power 1.8 power 2.1 power 1.9 power  1.85 power 1.75
74 194 24.90787043 18.16786 13.251687 21.9545 28.25854 23.38461 26.53036 27.38081767 29.164394
73 190 25.06722089 18.4089 13.519157 22.15516 28.36204 23.56625 26.66379 27.49980882 29.2513108
72 186 25.22587612 18.65362 13.793677 22.35697 28.46292 23.74814 26.79556 27.61666238 29.3351161
71 181 25.24411702 18.79811 13.998069 22.43581 28.40394 23.79858 26.77746 27.57871295 29.2538615
70 177 25.39659502 19.04626 14.283799 22.63545 28.49455 23.97631 26.90102 27.68632146 29.3263783
69 172 25.39969916 19.18612 14.492582 22.70346 28.41614 24.01377 26.86562 27.63000693 29.224647
68 168 25.54405047 19.43652 14.789284 22.89925 28.49432 24.18552 26.97889 27.72625148 29.2836644
67 163 25.52914713 19.5696 15.001262 22.9538 28.39344 24.20725 26.92323 27.64856427 29.158383
66 159 25.66300807 19.82069 15.308404 23.14366 28.45661 24.3708 27.02373 27.73091556 29.2012857
65 154 25.62667817 19.9443 15.521913 23.18157 28.32968 24.37348 26.94431 27.62831353 29.0488593
64 150 25.74717699 20.19402 15.838569 23.36291 28.37477 24.52609 27.02906 27.69374392 29.0725443
63 145 25.6853336 20.30477 16.051327 23.38031 28.2176 24.50574 26.92171 27.56203949 28.8887518
62 141 25.78897261 20.55045 16.37603 23.54989 28.24094 24.64402 26.98712 27.60691334 28.8895244
hi - low 0.881102189 2.382586 3.1243436 1.595396 0.276953 1.259411 0.498703 0.35009789 0.44636437
62 -74 0.881102189 2.382586 3.1243436 1.595396 -0.0176 1.259411 0.456763 0.226095667 -0.27486959
ave BMI 25.44813428

Therefore, this data suggests that if we must change the BMI formula it should be to the power of 1.8 rather than the 2.2 or 2.3 that has been suggested. The real kicker is that no one has a good set of data for healthy people just data for average people. When we get better data, then perhaps the formula can be changed. Perhaps we should start keeping records of those people who live to be at least 90 and then go back in time to see how tall and how much they weighed when they were young.

We all know that the shape of women are different. It looks like a power of about 1.9 would describe how the weight increases with an increase of height. See the table below.




    BMI of Ave Women (1960 25-34 yrs) (Smoothed average Table 5 NCHS "Weight by Height and Age of Adults")
height weight BMI (power 2) power 2.5 power 3 (PI) power 1.50 power 1.8 power 2.1 power 1.9 power  1.70 power 1.60 power 1.4
68 156 23.71947544 18.04819 13.732906 31.17284 26.45901 22.45799 25.05182 27.94524163 29.5149554 32.92385
67 152 23.80632125 18.24896 13.988907 31.05607 26.47732 22.57363 25.10632 27.92317806 29.4479928 32.75197
66 148 23.88757984 18.44945 14.249332 30.92865 26.48791 22.68477 25.15417 27.89237191 29.3713029 32.56857
65 144 23.96260816 18.64921 14.513996 30.78985 26.49009 22.79079 25.19468 27.85211682 29.2841697 32.37295
64 140 24.03069852 18.84775 14.782664 30.63891 26.48312 22.89101 25.22713 27.80164556 29.1858181 32.16434
63 136 24.09107151 19.04447 15.055038 30.47497 26.46616 22.98469 25.25071 27.74012391 29.0754084 31.94189
62 132 24.14286798 19.23872 15.330752 30.29714 26.43833 23.07099 25.26454 27.66664397 28.9520301 31.70473
61 128 24.18514007 19.42973 15.609359 30.10443 26.39863 23.149 25.26766 27.58021659 28.814695 31.45189
60 124 24.21684108 19.61666 15.890316 29.89579 26.34599 23.21769 25.25899 27.47976294 28.6623291 31.18232
59 120 24.23681401 19.79852 16.172971 29.67006 26.27923 23.27593 25.23737 27.36410502 28.4937639 30.89491
58 116 24.24377861 19.9742 16.456543 29.426 26.19707 23.32245 25.20151 27.23195511 28.3077267 30.58844
57 112 24.23631669 20.14245 16.7401 29.16224 26.09807 23.35585 25.14997 27.08190375 28.1028294 30.26159
hi - low 0.524303171 2.094257 3.0071938 2.0106 0.392028 0.897868 0.215833 0.863337872 1.41212601 2.662262
68-57 0.51684125 2.094257 3.0071938 -2.0106 -0.36094 0.897868 0.098147 -0.863337872 -1.41212601 -2.66226




More Appropriate Exponent for BMI

Nick Korevaar has suggested that an exponent of between 2.3 and 2.7 would be more accurate than the present exponent of 2.0¹

. The proplem is that his analysis only looked at the BMI of growing children and not adults. I am looking for a better exponent for adults. So far I have only given a visual explaination and a look at the data. I will now use the same math as explained by Korevaar, but using exel to find the best exponent for adults. I will start with the last example of data, women 25 to 34.


    BMI of Ave Women (1960 25-34 yrs) (Smoothed average Table 5 NCHS "Weight by Height and Age of Adults")
height weight ht m wt kg ln ht(m) Ln wt(kg)
68 156 1.72720 70.76041 0.54650 4.25930
67 152 1.70180 68.94604 0.53169 4.23332
66 148 1.67640 67.13167 0.51665 4.20666
65 144 1.65100 65.31730 0.50138 4.17926
64 140 1.62560 63.50293 0.48588 4.15109
63 136 1.60020 61.68856 0.47013 4.12210
62 132 1.57480 59.87419 0.45413 4.09225
61 128 1.54940 58.05982 0.43787 4.06147
60 124 1.52400 56.24545 0.42134 4.02973
59 120 1.49860 54.43108 0.40453 3.99694
58 116 1.47320 52.61671 0.38744 3.96303
57 112 1.44780 50.80235 0.37005 3.92794


This chart shows what we saw by exploring the data. For this category, which is women 25 to 34, an exponent of lower than 2.0 would be better. In this case 1.62 would be better.



I have run this test on other age groups and for both men and women. I get results for and exponent of from 1.1 to 1.8. None of these seem to be of tremendous advantage over 2.0 which never seems to be off by more than about a half a BMI value.



Summary:
It has been shown that for adults an exponent of less than 2.0 seems to be slightly better. Others have show that for children and young people, a BMI exponent of greater than 2.0 would be best. The problem with both studies is that the data includes a variety of body types and not only healthy people but also people who are over weight and underweight. Also there would have to be a seperate exponent for each age group. In conclusion, until there is much better data, It would be best to just keep the exponent at 2.0. Even though it is not a perfectly reliable indicator of obesity, it gives the medical professional a starting point for discussion, and as it does combine height and weight into one number.






Related Resources

¹Korevaar, Nick (July 2003). "Notes on Body Mass Index and actual national data"








Related Resources at Ideal-Weight-Charts.com

Go back to the BMI Chart for Men and Women.

Need more information about how BMI charts for men and BMI charts for women change with age?

Chart for Women only? BMI Chart for Women

More General Information about the Body Mass Index Chart?

See how the BMI chart compares to other ideal weight charts.

See what John Barban has to say about BMI Charts.










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