Less Wrong is a community blog devoted to refining the art of human rationality. Please visit our About page for more information.
[I'm unsure how much this rehashes things 'everyone knows already' - if old hat, feel free to downvote into oblivion. My other motivation for the cross-post is the hope it might catch the interest of someone with a stronger mathematical background who could make this line of argument more robust]
Many outcomes of interest have pretty good predictors. It seems that height correlates to performance in basketball (the average height in the NBA is around 6'7"). Faster serves in tennis improve one's likelihood of winning. IQ scores are known to predict a slew of factors, from income, to chance of being imprisoned, to lifespan.
What is interesting is the strength of these relationships appear to deteriorate as you advance far along the right tail. Although 6'7" is very tall, is lies within a couple of standard deviations of the median US adult male height - there are many thousands of US men taller than the average NBA player, yet are not in the NBA. Although elite tennis players have very fast serves, if you look at the players serving the fastest serves ever recorded, they aren't the very best players of their time. It is harder to look at the IQ case due to test ceilings, but again there seems to be some divergence near the top: the very highest earners tend to be very smart, but their intelligence is not in step with their income (their cognitive ability is around +3 to +4 SD above the mean, yet their wealth is much higher than this) (1).
The trend seems to be that although we know the predictors are correlated with the outcome, freakishly extreme outcomes do not go together with similarly freakishly extreme predictors. Why?
Too much of a good thing?
One candidate explanation would be that more isn't always better, and the correlations one gets looking at the whole population doesn't capture a reversal at the right tail. Maybe being taller at basketball is good up to a point, but being really tall leads to greater costs in terms of things like agility. Maybe although having a faster serve is better all things being equal, but focusing too heavily on one's serve counterproductively neglects other areas of one's game. Maybe a high IQ is good for earning money, but a stratospherically high IQ has an increased risk of productivity-reducing mental illness. Or something along those lines.
I would guess that these sorts of 'hidden trade-offs' are common. But, the 'divergence of tails' seems pretty ubiquitous (the tallest aren't the heaviest, the smartest parents don't have the smartest children, the fastest runners aren't the best footballers, etc. etc.), and it would be weird if there was always a 'too much of a good thing' story to be told for all of these associations. I think there is a more general explanation.
The simple graphical explanation
[Inspired by this essay from Grady Towers]
Suppose you make a scatter plot of two correlated variables. Here's one I grabbed off google, comparing the speed of a ball out of a baseball pitchers hand compared to its speed crossing crossing the plate:
It is unsurprising to see these are correlated (I'd guess the R-square is > 0.8). But if one looks at the extreme end of the graph, the very fastest balls out of the hand aren't the very fastest balls crossing the plate, and vice versa. This feature is general. Look at this data (again convenience sampled from googling 'scatter plot') of quiz time versus test score:
Given a correlation, the envelope of the distribution should form some sort of ellipse, narrower as the correlation goes stronger, and more circular as it gets weaker:
The thing is, as one approaches the far corners of this ellipse, we see 'divergence of the tails': as the ellipse doesn't sharpen to a point, there are bulges where the maximum x and y values lie with sub-maximal y and x values respectively:
So this offers an explanation why divergence at the tails is ubiquitous. Providing the sample size is largeish, and the correlation not to tight (the tighter the correlation, the larger the sample size required), one will observe the ellipses with the bulging sides of the distribution (2).
Hence the very best basketball players aren't the tallest (and vice versa), the very wealthiest not the smartest, and so on and so forth for any correlated X and Y. If X and Y are "Estimated effect size" and "Actual effect size", or "Performance at T", and "Performance at T+n", then you have a graphical display of winner's curse and regression to the mean.
An intuitive explanation of the graphical explanation
It would be nice to have an intuitive handle on why this happens, even if we can be convinced that it happens. Here's my offer towards an explanation:
The fact that a correlation is less than 1 implies that other things matter to an outcome of interest. Although being tall matters for being good at basketball, strength, agility, hand-eye-coordination matter as well (to name but a few). The same applies to other outcomes where multiple factors play a role: being smart helps in getting rich, but so does being hard working, being lucky, and so on.
For a toy model, pretend these height, strength, agility and hand-eye-coordination are independent of one another, gaussian, and additive towards the outcome of basketball ability with equal weight.(3) So, ceritus paribus, being taller will make one better at basketball, and the toy model stipulates there aren't 'hidden trade-offs': there's no negative correlation between height and the other attributes, even at the extremes. Yet the graphical explanation suggests we should still see divergence of the tails: the very tallest shouldn't be the very best.
The intuitive explanation would go like this: Start at the extreme tail - +4SD above the mean for height. Although their 'basketball-score' gets a massive boost from their height, we'd expect them to be average with respect to the other basketball relevant abilities (we've stipulated they're independent). Further, as this ultra-tall population is small, this population won't have a very high variance: with 10 people at +4SD, you wouldn't expect any of them to be +2SD in another factor like agility.
Move down the tail to slightly less extreme values - +3SD say. These people don't get such a boost to their basketball score for their height, but there should be a lot more of them (if 10 at +4SD, around 500 at +3SD), this means there is a lot more expected variance in the other basketball relevant activities - it is much less surprising to find someone +3SD in height and also +2SD in agility, and in the world where these things were equally important, they would 'beat' someone +4SD in height but average in the other attributes. Although a +4SD height person will likely be better than a given +3SD height person, the best of the +4SDs will not be as good as the best of the much larger number of +3SDs
The trade-off will vary depending on the exact weighting of the factors, which explain more of the variance, but the point seems to hold in the general case: when looking at a factor known to be predictive of an outcome, the largest outcome values will occur with sub-maximal factor values, as the larger population increases the chances of 'getting lucky' with the other factors:
So that's why the tails diverge.
Endnote: EA relevance
I think this is interesting in and of itself, but it has relevance to Effective Altruism, given it generally focuses on the right tail of various things (What are the most effective charities? What is the best career? etc.) It generally vindicates worries about regression to the mean or winner's curse, and suggests that these will be pretty insoluble in all cases where the populations are large: even if you have really good means of assessing the best charities or the best careers so that your assessments correlate really strongly with what ones actually are the best, the very best ones you identify are unlikely to be actually the very best, as the tails will diverge.
This probably has limited practical relevance. Although you might expect that one of the 'not estimated as the very best' charities is in fact better than your estimated-to-be-best charity, you don't know which one, and your best bet remains your estimate (in the same way - at least in the toy model above - you should bet a 6'11" person is better at basketball than someone who is 6'4".)
There may be spread betting or portfolio scenarios where this factor comes into play - perhaps instead of funding AMF to diminishing returns when its marginal effectiveness dips below charity #2, we should be willing to spread funds sooner.(4) Mainly, though, it should lead us to be less self-confident.
1. One might look at the generally modest achievements of people in high-IQ societies as further evidence, but there are worries about adverse selection.
2. One needs a large enough sample to 'fill in' the elliptical population density envelope, and the tighter the correlation, the larger the sample needed to fill in the sub-maximal bulges. The old faithful case is an example where actually you do get a 'point', although it is likely an outlier.
3. If you want to apply it to cases where the factors are positively correlated - which they often are - just use the components of the other factors that are independent of the factor of interest. I think, but I can't demonstrate, the other stipulations could also be relaxed.
4. I'd intuit, but again I can't demonstrate, the case for this becomes stronger with highly skewed interventions where almost all the impact is focused in relatively low probability channels, like averting a very specified existential risk.