Men who leave their mark on the world are very often those who, being gifted and full of nervous power, are at the same time haunted and driven by a dominant idea, and are therefore within a measurable distance of insanity.

**Francis Galton**

I have the best company idea ever. Want to win in Value Based care population management? You can. It’s easy.

*Maybe.*

Here is my amazing venture-funding pitch, if we were still in a low cost of capital environment. Maybe a la 2020 I’d add "**Web3**” and “**psychedelics**.” *Alas, now this great idea is only good for a math think piece.*

## Problem:

Healthcare is expensive. High utilization drives spending. We promise to save you money.

### Solution:

A custom care management solution for high-cost members!

#### Criteria for enrollment:

Enroll the top 1% of claimants who also had the 1% highest cost increase last year.

##### Service provided:

We wait 1 year. Do nothing at all.

###### Our businesses model:

Sample the top 1% of claimants the next year.

For everyone one of our prior year's “highest cost” plan members who aren’t there anymore, we claim we generated the savings!

We graciously split this amount with you.

Clever readers will already suspect something is up.

Yep. It’s an article about math. A love letter to the central limit theorem…and a super lame result, regression to the mean.

“Regression toward the mean. That is, in any series of random events an extraordinary event is most likely to be followed, due purely to chance, by a more ordinary one.”

Sir Francis Galton, whose enviable google search result categorization is—*and this is real*— “English polymath,” was the first to understand this phenomenon. He was also first to find it a *bummer*. First, however, a bit about the Lord of Average:

Many words in our statistical lexicon were coined by Galton. For example,

correlationanddeviateare due to him, as isregression, and he was the originator of terms and concepts such asquartile, decileandpercentile, and of the use ofmedianas the midpoint of a distribution^{2}. Of course, words have a way of developing a life of their own, so that, unfortunatelydecileis increasingly being applied to meantenth.

He was a brilliant dude. He was also a nightmarishly precocious child, as this acutal letter to his sister at *almost* 5 years old makes pretty clear:

My dear Adèle,

I am four years old and can read any English book. I can say all the Latin Substantives and adjectives and active words besides 52 lines of Latin poetry. I can cast up any sum in addition and multiply by

2,3,4,5,6,7,8,(9),10,(11)

I can also say the pence table, I read French a little and I know the clock.

Francis Galton

^{1}

Even then, he was a little *narcissistic* and used the numbers he would go on to live by for obfusation—he was a few days from his 5th birthday, but only 4 at the time.

He was an aristocratic British man, and in keeping with his noblesse-obligeified wont, he endevoured to prove the following: Gentlepeople of station and power had offspring who were, as befiting their station, also *ballers*. Alas, the data didn’t support the conjecture! It turns out that powerful people had… less prominent offspring! Not one to bail on a methodology, he went on to look at more objective data.

Next, he looked at height. Tall parents had tall kids. Short people had short kids. But on average, both the tall and the short had ofstring who were, on average, LESS so in either direction. They trended toward average!

*When Galton named “regression to the mean,” *

*the first pass at the name was *

**“regression to mediocrity!”**

Talk about a 💩poster of his day. Dante, eat your heart out.

The above result is even weirder:

The consequence is that we may expect that an adult child is closer to average height than its parents – but also, paradoxically, that parents are closer to average height than is their child.

Yes. That is true. And here is where statistics and reality run into trouble—arguing that taller-than-average Children *cause* more average-height parents is, at once, both *statistically supported* but *logically flawed. * The arrow of time? It doesn’t fly that way. Causes and co-relations (later re-branded correlation) are not the same.

There are tricks, as it were, of the light. They trick us to this day, when it comes to numbers assessed absent a conceptual anchor. This anchor—the sun rises each day leads to a rooster crowing, not the other way around—is a vitally important point.

Some things are caused by other things. It is also true that determining causation from statistics alone is not possible. We’d need to add something.

Don’t misunderstand me—it’s a vexing problem!

In our day-to-day lives, we often make causal inferences, cause that is how we get thought the day. You put a slice of bread in a toaster, and when it pops up, it's toasted. It's quite reasonable to infer that the toaster *causes* the bread to become toasted.

We are gong to need another english iconoclast, aren’t we? Reverend Thomas Bayes, a 18th-century British mathematician *and* Presbyterian minister formulated the now-famous Bayes' theorem. Not content to just name a theorem after himself, Bayesian statistics is a way to update our beliefs about the world in light of new evidence. And, frankly, blow up Galton’s spot. Bayes put it all together: a way of incorporating our *prior knowledge* and *new information* to better understand the complex web of causality in the world around us. It’s the YES-AND of math.

Now we can come back to my awesome company idea. A naïve approach—which is what the sellers are counting on before the golf game they have planned for you— might be to simply look at the data, observe that some high-cost patients are no longer high-cost, and claim that this program saved the day (and some cash). Now, we know better. We know about regression to the mean, and we suspect that this apparent success might just be a fluke, a statistical quirk, a cruel joke played by the numbers on our bottom line.

By combining our *prior knowledge* about healthcare spending patterns and the *natural tendency for extreme cases to regress to the mean* with the data we collect on our high-cost patients, we can calculate the true savings of our program. We can adjust our beliefs and expectations based on the evidence! We can not get scammed. Praise be to latter-day polymaths like

In the immortal words of David Foster Wallace, "The truth will set you free. But not until it is finished with you." Armed with Bayesian statistics, we can move past the sales pitch and on to the reality: there is a fiduciary duty to uphold, and the time for scams is over.

—Owen Scott Muir, M.D.