Linear Extrapolation Fallacy (Ignoring Upper Bounds)

BACK TO THE OBESITY APOCALYPSE

So what percentage of Americans are going to be overweight in 2048? You can guess by now how Youfa Wang and his Obesity coauthors generated their projection. The National Health and Nutrition Examination Survey, or NHANES, tracks the health data of a large, representative sample of Americans, covering everything from hearing loss to sexually transmitted infections. In particular, it gives very good data for the proportion of Americans who are overweight, which for present purposes is defined as having a body-mass index of 25 or higher.* There’s no question that the prevalence of overweight has increased in recent decades. In the early 1970s, just under half of Americans had a BMI that high. By the early 1990s that figure had risen to almost 60%, and by 2008 almost three-quarters of the U.S. population was overweight.

You can plot the prevalence of overweight against time:

And you can generate a linear regression, which will look something like this:

In 2048, the line crosses 100%. And that’s why Wang writes that all Americans will be overweight in 2048, if current trends continue.

But current trends will not continue. They can’t! If they did, by 2060, a whopping 109% of Americans would be overweight.

In reality, the graph of an increasing proportion bends toward 100%, like this:

That’s not an ironclad law, like the gravity that bends the missile’s path into a parabola, but it’s as close as you’re going to get in medicine. The higher the proportion of overweight people, the fewer skinny malinkies are left to convert, and the more slowly the proportion increases toward 100%. In fact, the curve probably goes horizontal at some point below 100%. The thin we have always with us! And indeed, just four years later, the NHANES survey showed that the upward march of overweight prevalence had already begun to slow.

But the Obesity paper conceals a worse crime against mathematics and common sense. Linear regression is easy to do—and once you’ve done one, it’s cake to do more. So Wang and company broke down their data by ethnic group and sex. Black men, for instance, were less likely to be overweight than the average American; and, more important, their rate of overweight was growing only half as quickly. If we superimpose the proportion of overweight black men on the proportion of overweight Americans overall, together with the linear regressions Wang et al. worked out, we get a picture that looks like this.

Nice work, black men! Not until 2095 will all of you be overweight. In 2048, only 80% of you will be.

See the problem? If all Americans are supposed to be overweight in 2048, where are those one in five future black men without a weight problem supposed to be? Offshore?

That basic contradiction goes unmentioned in the paper. It’s the epidemiological equivalent of saying there are −4 grams of water left in the bucket. Zero credit.