Linear Extrapolation Fallacy

The stand-up comic Eugene Mirman tells this joke about statistics. He says he likes to tell people, “I read that 100% of Americans were Asian.”

“But Eugene,” his confused companion protests, “you’re not Asian.”

And the punch line, delivered with magnificent self-assurance: “I read that I was!”

I thought of Mirman’s joke when I encountered a paper in the journal Obesity whose title posed the discomfiting question: “Will all Americans become overweight or obese?” As if the rhetorical question weren’t enough, the article supplies an answer: “Yes—by 2048.”

The Obesity paper got plenty of press, as you might imagine. ABC News warned of an “obesity apocalypse.” The Long Beach Press-Telegram went with the simple headline “We’re Getting Fatter.” The study’s results resonated with the latest manifestation of the fevered, ever-shifting anxiety with which Americans have always contemplated our national moral status.

But not every curve is a line. Linear regression is a statistical technique that finds the line that comes closest to passing through all the points in a scatter plot. It's a marvelous tool, but it works on any data set, which is a weakness as well as a strength. The danger is using linear regression without thinking about whether the phenomenon is actually close to linear.

Take a missile’s trajectory, for instance. If you plot the vertical position of a missile at five points in time and do a linear regression, you get a line that passes almost exactly through the points. But the missile's flight is a parabola, not a line. The line gives a precise model for the missile's motion—for every minute that passes, the missile increases its altitude by some fixed amount. When does it come down? It never comes down! An upward sloping line just keeps on sloping upward.

The most vivid warning against thoughtless linear extrapolation comes from Mark Twain's Life on the Mississippi. The Mississippi River has shortened itself over time. By extrapolating linearly, one could conclude that a million years ago the river was over a million miles long, and in 742 years it will be only a mile and three-quarters long. There is something fascinating about science: one gets such wholesale returns of conjecture out of such a trifling investment of fact.