It's not just that the tails stop being correlated, it's that there can be a spurious negative correlation. In any of your scatterplots, you could slice off the top right corner (with a diagonal line running downwards to the right), and what was left above the line would look like a negative correlation. This is sometimes known as Berkson's paradox.

IMO this should be in main

So in other words, it's not that the strongest can't also be the tallest (etc), but that someone getting that lucky twice more or less never happens. And if you need multiple factors to be good at something, getting pretty lucky on several factors is more likely than getting extremely lucky on one and pretty lucky on the rest.

I enjoyed this post - very clear.

This is all correct so far as I can tell. Yay! (Posting because of the don't-only-post-cricitism discipline.)

One angle for thinking about why the tails come apart (which seems worth highlighting even more than it was highlighted in the OP) is that the farther out you go in the tail on some variable, the smaller the set of people you're dealing with.

Which is better, the best basketball team that you can put together from people born in Pennsylvania or the best basketball team that you can put together from people born in Delaware? Probably the Pennsylvania team, since there are about 13x as many people in that state so you get to draw from a larger pool. If there were no other relevant differences between the states then you'd expect 13 of the best 14 players to be Pennsylvanians, and probably the two neighboring states are similar enough so that Delaware can't overcome that population gap.

Now, imagine you're picking the best 10 basketball players from the 1,000 tallest basketball-aged Americans (20-34 year-olds), and you're putting together another group consisting of the best 10 basketball players from the next 100,000 tallest basketball-aged Americans. Which is a better group of basketball players? In this case it's not obvious - getting to pick from a pool of 100x as many people is an obvious advantage, but that height advantage could matter a lot too. That's the tails coming apart - the very tallest don't necessarily give you the very best basketball players, because "the very tallest" is a much smaller set than the "also really tall but not quite as tall".

(I ran some numbers and estimate that the two teams are pretty similar in basketball ability. Which is a remarkable sign of how important height is for basketball - one pool has about a 4 inch height advantage on average, the other pool has 100x as many people, and those factors roughly balance out. If you want the example to more definitively show the tails coming apart, you can expand the larger pool by another factor of 30x and then they'll clearly be better.)

Similarly, who has higher arm strength: the one person in our sample who has the highest grip strength, or the most arm-strong person out of the next ten people who rank 2-11 in grip strength? Grip strength is closely related to arm strength, but you get to pick the best from a 10x larger pool if you give up a little bit of grip strength. In the graph in the OP, the person who was 6th (or maybe 5th) in grip strength had the highest arm strength, so getting to pick from a pool of 10 was more important. (The average arm strength of the people ranked 2-11 in grip strength was lower than the arm strength of the #1 gripper, but we get to pick out the strongest arm of the ten rather than averaging them.)

So: the tails come apart because most of the people aren't way out on the tail. And you usually won't find the very best person at something if you're looking in a tiny pool, even if that's a pretty well selected pool.

Thrasymachus's intuitive explanation covered this - having a smaller pool to pick from hurts because there are other variables that matter, and the smaller the pool the less you get to select for people who do well on those other variables. But his explanation highlighted the "other variables matter" part of this more than the pool size part of it, and both of these points of emphasis seem helpful for getting an intuitive grasp of the statistics in these types of situations, so I figured I'd add this comment.

Just as markets are anti-inductive, it turns out that markets reverse the "tails come apart" phenomenon found elsewhere. When times are "ordinary", performance in different sectors is largely uncorrelated, but when things go to shit, they go to shit all together, a phenomenon termed "tail dependence".

Great article overall. Regression to the mean is a key fact of statistics, and far too few people incorporate it into their intuition.

But there's a key misunderstanding in the second-to-last graph (the one with the drawn-in blue and red "outcome" and "factor"). The black line, indicating a correlation of 1, corresponds to nothing in reality. The true correlation is the line from the vertical tangent point at the right (marked) to the vertical tangent point at the left (unmarked). If causality indeed runs from "factor" (height) to "outcome" (skill), that's how much extra skill an extra helping of height will give you. Thus, the diagonal red line should follow this direction, not be parallel to the 45 degree black line. If you draw this line, you'll notice that each point on it has equal vertical distance to the top and bottom of the elliptical "envelope" (which is, of course, not a true envelope for all the probability mass, just an indication that probability density is higher for any point inside than any point outside).

Things are a little more complex if the correlation is due to a mutual cause, "reverse" causation (from "outcome" to "factor"), or if "factor" is imperfectly measured. In that case, the line connecting the vertical tangents may not correspond to anything in reality, though it's still what you should follow to get the "right" (minimum expected squared error) answer.

This may seem to be a nitpick, but to me, this kind of precision is key to getting your intuition right.

This looks cool. My biggest caution would be that this effect may be tied to the specific class of data generating processes you're looking at.

Your framing seems to be that you look at the world as being filled with entities whose features under any conceivable measurements are distributed as independent multivariate normals. The predictive factor is a feature and so is the outcome. Then using extreme order statistics of the predictive factor to make inferences about the extreme order statistics of the outcome is informative but unreliable, as you illustrated. Playing around in R, reliability seems better for thin-tailed distributions (e.g., uniform) and worse for heavy-tailed distributions (e.g., Cauchy). Fixing the distributions and letting the number of observations vary, I agree with you that the probability of picking exactly the greatest outcome goes to zero. But I'd conjecture that the probability that the observation with the greatest factor is in some fixed percentile of the greatest outcomes will go to one, at least in the thin-tailed case and maybe in the normal case.

But consider another data generating process. If you carry out the following little experiment in R

fac <- rcauchy(1000)
out <-  fac + rnorm(1000)
plot(rank(fac), rank(out))
rank(out)[which.max(fac)]

it looks like extreme factors are great predictors of extreme outcomes, even though the factors are only unreliable predictors of outcomes overall. I wouldn't be surprised if the probability of the greatest factor picking the greatest outcome goes to one as the number of observations grows.

Informally (and too evocatively) stated, what seems to be happening is that as long as new observations are expanding the space of factors seen, extreme factors pick out extreme outcomes. When new observations mostly duplicate already observed factors, all of the duplicates would predict the most extreme outcome and only one of them can be right.

I ran some simulations in Python, and (if I did this correctly), it seems that if r > 0.95, you should expect the most extreme data-point of one variable to be the same in the other variable over 50% of the time (even more if sample size <= 100)

http://nbviewer.jupyter.org/github/ricardoV94/stats/blob/master/correlation_simulations.ipynb

Upvoted. I really like the explanation.

In the spirit of Don't Explain Falsehoods, it would be nice to test the ubiquity of this phenomenon by specifying a measure of this phenomenon (e.g. correlation) on some representative randomly-chosen pairs. But I don't mean to suggest that you should have done that before posting this.

I must quibble with some of the epistemological terminology here. Both "graphical explanation" and "geometric explanation" are not properly speaking explanations. They merely restate the original empirical observation in graphical or statistical terms, but do not explain why the tails-coming-apart phenomenon occurs. The "intuitive explanation" on the other hand does explain why the phenomenon occurs (i.e. the distributions of other factors influencing the outcome come into play). Similarly, "regression to the mean" is merely a restatement of the empirical observations in statistical language, which does nothing to elucidate the causes of the phenomenon. It should be obvious that the causes of regression to the mean in a series of coin flips and in inheritance of continuous characters are different: coins do not reproduce and we can't select them for better tails-to-heads ratio.

Good post.

If the ellipse is very narrow, things are indeed well-modeled by a linear relationship, and the biggest Y coordinate for a point is likely to also have close to biggest X coordinate.

If the ellipse is not narrow, that could be for two reasons. Either the underlying truth is indeed linear, but your data is very noisy. Or the underlying truth is not linear, and you should not use a linear model. (Or both, naturally).

If the underlying truth is linear, but your data is very noisy, then what happens to the X coordinate of points with given Y values is mostly determined by the noise.

If the underlying truth is not linear, why should we expect sensible answers from a linear model?

Interesting read! That makes sense.

One little side note, though.

So, ceritus paribus,

Did you mean ceteris paribus?

(Ha, finally a chance for me as a language geek to contribute something to all the math talk. :P )

I'd say that this is regression to the mean. If two variables are correlated with |r| < 1, then extreme values on one variable will be associated with somewhat less extreme values on the other variable. So people who are +4 SD in height will tend to be less than +4 SD in basketball ability, and people who are +4 SD in basketball ability will tend to be less than +4 SD in height.

Hsu: http://infoproc.blogspot.com/2014/07/success-ability-and-all-that.html

Thank you for pointing that high IQ problem is probably a statistical effect rather than "too much of a good thing" effect. That was very interesting.

Let me attempt the problem from a simple mathematical point of view.

Let basketball playing ability, Z, is just a sum of height, X, and agility, Y. Both X and Y are Gaussian distributed with mean 0 and variance 1. Assume X and Y are independent.

So, if we know that Z>4, what is the most probable combination of X and Y?

The probability of X>2 and Y>2 is: P(X>2)P(Y>2)=5.2e-4

The probability of X>3 and Y>1 is: P(X>3)P(Y>1)=2.1e-4

So it is more than two times more likely for both abilities to be +2Std than one them is +3Std and the other is +1Std.

I think it can be shown rigorously that the most probable combination is Z/N for each component if there are N independent identically distributed components of an ability.

Given a correlation, the envelope of the distribution should form some sort of ellipse

That isn't an explanation, but a stronger claim. Why should it form an ellipse?

A model of an independent factor or noise is an explanation of the ellipse, and thus of the main point. But people may find a stumbling block this middle section, with its assertion that we should expect ellipses. Also, regression to the mean and the tails coming apart are much more general than ellipses, but ellipses are pretty common.

It generally vindicates worries about regression to the mean

It is regression to the mean, as you yourself say elsewhere. I'm not sure what you are trying to say here; maybe that people's vague worries about regression to the mean are using the technical concept correctly?

Also: this is very reminiscent of St. Rev's old post about economic inequality. http://st-rev.livejournal.com/383957.html

Isn't the far simpler and more likely scenario that you never have just one variable accounting for all of an outcome? If other variables are not perfectly correlated with the variable you are graphing you will get noise. Why is it surprising that that noise also exists in the most extreme points?

EDIT: misunderstood last few paragraphs.

This post has been a core part of how I think about Goodhart's Law. However, when I went to search for it just now, I couldn't find it, because I was using Goodhart's Law as a search term, but it doesn't appear anywhere in the text or in the comments.

So, I thought I'd mention the connection, to make this post easier for my future self and others to find. Also, other forms of this include:

• Campbell's law: https://en.wikipedia.org/wiki/Campbell%27s_law
• The Cobra effect: https://en.wikipedia.org/wiki/Cobra_effect
• Teaching to the test: https://en.wikipedia.org/wiki/Teaching_to_the_test
• Perverse incentives: https://en.wikipedia.org/wiki/Perverse_incentive
• Principal-Agent Problem: https://en.wikipedia.org/wiki/Principal%E2%80%93agent_problem
• Hypocrisy (optimizing for appearances of good rather than becoming good)
• The importance of quick feedback loops for course correction (Even good estimates have error, and as you get closer to your goal those errors compound, and things come apart at the tails.)

Maybe it would be useful to map out as many of the forms of Goodhart's Law as possible, Turchin style.

Following on your Toy Model concept, let's say the important factors in being (for example) a successful entrepreneur are Personality, Intelligence, Physical Health, and Luck.

If a given person has excellent (+3SD) in all but one of the categories, but only average or poor in the final category, they're probably not going to succeed. Poor health, or bad luck, or bad people skills, or lack of intelligence can keep an entrepreneur at mediocrity for their productive career.

Really any competitive venue can be subject to this analysis. What are the important skills? Does it make sense to treat them as semi-independent, and semi-multiplicative in arriving at the final score?

Statistical point: the variance of forecast error for correctly specified simple regression problems is equal to:

Sigma^2(1 + 1/N + (x_o - x_mean)^2 / (Sigma ( x_i - x_mean) ^2))

So forecast error increases as x_o moves away from x_mean, especially when the variance of x is low by comparison.

Edit: Sub notation was apparently indenting things. I'm going to take a picture from my stats book tonight. Should be more readable.

Edit: Here's a more readable link. http://i.imgur.com/pu8lg0Wh.jpg

![](http://latex.codecogs.com/png.latex?%5Ctextit%7Bvar%7D%28f%29%20%3D%20%5Csigma%20%5E%202%20%5Cleft%20%28%201+%5Cfrac%7B1%7D%7BN%7D%20+%20%5Cfrac%7B%28x_0%20-%20%5Cbar%7Bx%7D%29%5E2%7D%7B%5Csum%20%28x_i-%5Cbar%7Bx%7D%29%5E2%7D%20%5Cright%20%29)

\textit{var}(f) = \sigma ^ 2 \left (  1+\frac{1}{N} + \frac{(x_0 - \bar{x})^2}{\sum (x_i-\bar{x})^2} \right )

For business in particular I think network size and effects are the reason that the very top end of earners are much more deviant in earnings than in intellect. The fact that you can capture entire billions of dollars markets because modern society allows a single product to be distributed worldwide will multiply the value of the "top" product by a lot more than its quality might justify.

Interesting post. Well thought out, with an original angle.

In the direction of constructive feedback, consider that the concept of sample size -- while it seems to help with the heuristic explanation -- likely just muddies the water. (We'd still have the effect even if there were plenty of points at all values.)

For example, suppose there were so many people with extreme height some of them also had extreme agility (with infinite sample size, we'd even reliably have that the best players we're also the tallest.) So: some of the tallest people are also the best basketball players. However, as you argued, most of the tallest won't be the most agile also, so most of the tallest are not the best (contrary to what would be predicted by their height alone).

In contrast, if average height correlates with average basketball ability, the other necessary condition for a basketball player with average height to have average ability is to have average agility -- but this is easy to satisfy. So most people with average height fit the prediction of average ability.

Likewise, the shortest people aren't likely to have the lowest agility, so the correlation prediction fails at that tail too.

Some of the 'math' is that it is easy to be average in all variables ( say, (.65)^n where n is the number of variables) but the probability of being standard deviations extreme in all variables is hard (say, (.05)^n to be in the top 5 percent.) Other math can be used to find the theoretic shape for these assumptions (e. g., is it an ellipse?).

Great post!

The R-square measure of correlation between two sets of data is the same as the cosine of the angle between them when presented as vectors in N-dimensional space

Not R-square, just R:

I've made a visualization tool for that:

https://codepen.io/qbolec/pen/qBybXQe

It generates an elliptical cloud of white points where X is distributed normally, and Y=normal + X*0.3, so the two are correlated. Then you can define a green range on X and Y axis, and the tool computes the correlation in a sample (red points) restricted to that (green) range.

So, the correlation in the general population (white points) should be positive (~0.29). But if I restrict attention to upper right corner, then it is much lower, and often negative.

Fantastic, I wish I'd had this back when almost everyone in LW/EA circles I met was reading the biography of everyone in the ' fortune 400 and trying to spot the common factors. A surprisingly common strategy that's likely not to work for exactly these reasons.

My guess is that there are several variables that are indeed positively correlated throughout the entire range, but are particularly highly correlated at the very top. Why not? I'm pretty sure we can come up with a list.

What is interesting is the strength of these relationships appear to deteriorate as you advance far along the right tail.

I read that claim as saying that if you sample the 45% to 55% percentile you will get a stronger correlation than if you sample the 90% to 100% percentile. Is that what you are arguing?

I don't think there's anything special about the tails.

Take a sheet of paper, and cover up the left 9/10 of the high-correlation graph. That leaves the right tail of the X variable. The remaining datapoints have a much less linear shape.

But: take two sheets of paper, and cover up (say) the left 4/10, and the right 5/10. You get the same shape left over! It has nothing to do with the tail -- it just has to do with compressing the range of X values.

The correlation, roughly speaking, tells you what percentage of the variation is not caused by random error. When you compress the X, you compress the "real" variation, but leave the "error" variation as is. So the correlation drops.