## Friday, March 30, 2007

### Anchor Ages

Here’s one of my theories that I really can’t justify. In fact, I’m not sure it’s even mathematically possible. But somehow it just seems true, based on personal observations. (Remember how I promised more half-baked posts? Yeah, this is one of those.)

I believe each individual has an “anchor age” at which his apparent age is the same as his chronological age. Prior to this age, he looks old for his age; afterward, he looks young for his age. I’m not saying that a person always looks like his anchor age, only that his physical age advances more slowly than calendar time, and the anchor age is the crossing point. Take someone whose anchor age is 25. At 22, he looks old for his age – maybe 23. At 28, he looks young for his age – maybe 27. So in a six-year time span (from 22 to 28), his apparent age has increased by only four years (from 23 to 27). Graphically, it looks something like this:

So how does the average 28-year-old look older than this particular 28-year-old? While his anchor age is less than 28, others’ anchor ages are greater. Our idea of what the typical 28-year-old looks like results from averaging both groups.

Put differently, my half-baked theory is that the aging process appears faster when viewed in cross-section (many individuals of different ages at a single point in time) than when viewed in time-series (a single individual viewed over a period of years).

Like I said, I’m not sure this is mathematically possible. The main difficulty is there’s an endpoint problem. On the young end of the age spectrum, most people should not yet have reached their anchor age, so the average-of-apparent-ages should be greater than the chronological age. On the old end of the age spectrum, most people should have surpassed their anchor age, so that the average-of-apparent-ages should less than the chronological age. Both of these are impossible, if our notion of what chronological age X looks like is the average-of-apparent-ages of people at age X.

But maybe my theory can be saved via some kind of misperception (like this one) or non-linearity in the physical aging function. Suggestions welcome.

August said...

Well,

I am 32, and people still ask me for I.D.
when I buy wine. Last year I had a teenaged girl burst out in surprise when she read my birthdate on my driver's license.
My experience seems to count in your theory's favor, but I've always thought it was genetics. My father has always looked young for his age.

Jeffrey said...

Those apparent age functions probably go through the origin, so they've got to be nonlinear in order to cross the y=x axis again.

Glen Whitman said...

Jeffrey -- yes, that's true, if we assume everyone looks zero when they are zero. However, I don't think that non-linearity is sufficient to save my theory. I'll still have a "next-to-the-endpoint" problem, epsilon distance to the right of zero.

But hey, maybe some people don't look zero at zero. Premies, for instance.

Blar said...

Half-baked at most, I'd say. Note that premies are actually contrary to the theory, since they begin below the y=x line, although you could get around that by counting from conception rather than birth. But I find it highly implausible to claim that no one has ever looked younger than they were from the beginning. I have known several people who looked young for their age ever since early childhood, although I admit that my memories do not go all the way back to day 0 (especially if we place the origin where I suggested). August, did you ever look old for your age when you were younger? If not, then you are actually one of these counterexamples to Glen's theory.

My anchor age is 20 said...

This seems somehow related.

Ari said...

Ahh, finally a conversation where being a mathematician is helpful. What you need is something called a fixed point theorem, and actually a pretty simple one (the most abstract ones are pretty mind bending).

Let x = actual age and y = apparent age. Instead of measuring in years, measure each of these variables on the scale [0,1], where 0 is newborn and 1 is dead. Then each person has an "apparent age function" f:[0,1]->[0,1], such that y = f(x).

If f is a continuous function, there always exists a "fixed point" where x = f(x) -- that is, where a person looks exactly their age. This point is not necessarily unique, though. For example, you could have someone who always looks exactly their age -- or someone who looks young up through early adulthood, is old-looking in middle age, but then ends up being a very well-preserved 90-year-old. (This could certainly happen for someone who goes bald at a young age, but never gets especially wrinkled or sickly.)

Now, one can argue over whether representing ages on a scale from [0,1] is reasonable, and whether f is continuous or not, but this is the relevant math.

Blar said...

Ari, your model depends on the implausible assumptions that 1) no one can look younger than they are when they're born and 2) no one can look older than their age when they die. The former assumption is contradicted by Glen's premie example, and the latter could even be thought of as implying that no one can ever look older than their age, if we consider that anyone could die at any moment. Alter the model by allowing y (apparent age) to be less than 0 (actual age at birth) or greater than 1 (actual age at death) and the fixed point theorem no longer applies.

Something like the anchor age phenomenon might arise on occasion out of wishful thinking, since many people want to look (or to be) older than they are when they are young and younger than they are when they are old. But as a general rule it seems to be both implausible in the abstract and contradicted by experience.

Evan said...

For an Economics student, I've taken surprisingly little statistics, so maybe this is overly simplistic:

Why are we assuming that apparent age advances less quickly than actual age? Couldn't someone, a smoker for example, start off looking younger than they are, but then age overly quickly?

This would solve the question of where perceived ages come from: they are the mean of appearances of people of that age. Presumably these form a normal distribution (the apparent age of a group of people of the same age form a bell curve centered on their actual age).

There is still the childhood problem, but maybe the model is more usefully focused only on grown adults?