## Monday, October 16, 2006

### Fat Stats

I’m currently reading J. Eric Oliver’s Fat Politics, a fascinating book that I’ll probably blog about more in the future. But for now, I just want to point out an interesting mathematical error.

Oliver spends a good bit of time debunking the Body Mass Index (BMI) as a measure of overweightness/obesity or health in general. And rightly so; BMI is flawed in a variety of ways. But I don’t think this is one of them:
Today, the same BMI formula is applied to men and women, despite the fact that, because men are taller on average, the formula automatically gives them a lower BMI. (BMI is a ratio of weight to height-squared, so the taller one is, the lower one’s relative BMI will be.) [p. 94-95]
It would make sense to have a different BMI formulas for men and women simply because they have different physiologies (for instance, women just naturally carry more body fat). But it’s untrue that BMI is lower for taller people, so long as their bodies are proportionate to their height. In fact, BMI will be higher for such people. This is because weight increases with the cube of the proportional increase in height.

Suppose person A has an “ideal” BMI. For simplicity, measure him in three dimensions: height (h), width from side-to-side (a), and width from front-to-back (b). Person A’s total volume is hab.1 Treat weight as a scalar function of volume; that is, weight = d*volume.2 So A’s weight is w = dhab. His BMI is weight over height-squared; thus:
BMI = dhab/(h^2) = dab/h
Now, suppose person B is taller than A, but with the exact same proportions. Given the same proportions, this person should be considered equally healthy and therefore “ideal” as well. If person B’s height is k times B’s height, then B’s side-to-side width must be k times A’s side-to-side width, and B’s front-to-back width must be k times A’s front-to-back width. So B’s total volume is (kh)(ka)(kb) = (k^3)(hab), and B’s weight is (k^3)(dhab). And thus:
BMI = (k^3)(dhab)/(k^2)(h^2) = kdab/h
Note that person B’s BMI is k times larger than person A’s, despite A and B both having exactly the same bodily proportions. This is certainly a problem for BMI, since it gives two people with identical proportions different results; but the error is not in the direction indicated by Oliver.

Notes:
1. Yes, this is basically treating the person like a rectangular-based column, but nothing important hinges on that. If you like, imagine taking the person and breaking him into thousands of little Lego-like pieces and doing the same calculation on each one.
2. Weight = mass x force due to gravity, and mass = volume x density, and so weight = volume x density x force due to gravity. Let the scalar d = density x force due to gravity.

Blar said...

...so long as their bodies are proportionate to their height.

Glen, if you think that this is what's wrong with the BMI then I've got a spherical cow to sell ya. You can't top its milk production.

The point being that the other two dimensions of a person don't increase proportionally when people grow taller, or at least they aren't supposed to. Some people have decided that height squared is a decent approximation to account for the increase in width and breadth, and maybe they're wrong, but that's something that you have to decide based on data about the actual shape of human bodies, not pure math.

Glen Whitman said...

Blar, as far as I can tell, nobody has done any calculation at all to justify the assumption that height-squared is the appropriate way to scale. We probably shouldn't assume that ideal tall people have exactly the same proportions as ideal short ones (we wouldn't expect them to have proportionally larger heads, for example). But a cubic function still seems like a much better approximation than a square function, given that humans have three dimensions all of which will tend to be larger for taller people. (Also note that the BMI doesn't use a baseline and square the proportional difference, but instead just squares the height directly. The formula is problematic all over.)

One thing I learned from the book is that the original BMI formula -- the one we still use today -- was devised by an actuary at an insurance company who noticed the correlation (not causation) between BMI and life expectancy. Of course, for insurance purposes, correlation is what really matters for the setting of premiums. So don't blithely assume that smart doctors and scientists put lots of thought and research into this. There's a shockingly large amount of sheer arbitrariness involved in the field of "obesity research."

Glen Whitman said...

Brandon -- interesting! It's nice when someone actually has the data to answer a question. Well, not quite... as you say, there's a question of actual weights versus ideal weights. Still, it's nice to have something quantitative.

I think the way to address the problem, without making unjustified assumptions about the relationship between weight and health, would be to take non-weight-based measures such as clavicle-length or rib-cage depth and see how they correlate with height. That would give us a sense of how to account for changing proportions.