Regression to the Mean
(This is also published at The Unz Review.)
It seems few understand regression to the mean and how and why it works.
Most people (and by most people, I mean most scholars – i.e., the people who should know better) have a vague understanding that it has something to do with IQ. They seem to have the impression it means that the children of smart parents will be less smart. Even more so when those parents come from a population with a low mean IQ.
And they seem to think this phenomenon goes on forever, such that grandchildren and great-grandchildren continue this march to mediocrity.
Well, a lot of this is confused to plain old wrong.
Let’s start with what regression to the mean is. Above is an illustration of a bell curve. A set of parents that lie out on the curve away from the mean will indeed tend to have children that are closer to the average. Hence, if a set of parents are +2 standard deviations for a trait, their children will be typically some degree closer to the mean.
The first thing to clear up is that regression to the mean operates in both directions. Just as parents +2σ will have children some degree less far off to the right on the curve, parents –2σ will have children some degree less far off to the left. That is, children of parents who are below average for a trait regress up.
To clear up some additional confusion, let’s look at what causes regression to the mean. The root of the phenomenon goes back to behavioral genetics, or more broadly heritability. Human traits have several components that contribute to trait variance. They are:
A: Additive heredity
D: Non-additive heredity (D from “dominance”)
C: Shared environment (C from “common environment”)
E: “Unshared” or “unique” environment.
As we’ve seen before, we know that A is typically 50-60%, D 10-20%, C is 0%, and E the remaining 20-30%.
The key fact is that for the transmission of a trait from parent to child, only A and C carry over to the next generation. But, as we know, C is 0; so that only leaves A. The rest, including non-additive heredity (which is basically fortuitous combinations of alleles) and whatever remaining “non-genetic” factors that constitute E (and all the things that comprise that, see Environmental Hereditarianism) are essentially luck. And since we can’t expect lightning to strike twice (as improper as that metaphor is), regression to the mean happens because luck goes away.
This is governed by the breeder’s equation.
R = h2 S.
R is the response to selection, S is the selection differential, and h2 is the narrow-sense heritability. This is the workhorse equation for quantitative genetics. The selective differential S, is the difference between the population mean and the mean of the parental population (some subset of the total population).
This equation can be used in different ways depending on whether we’re talking about whole populations or individual pairs of parents (though fundamentally for the same reason).
Let’s start with individuals. If two parents (let’s say White Americans) are +2σ for a trait, let’s say IQ, or 130, and we give the additive heritability of IQ to be about 0.6, we can expect their children to collectively have an average IQ of
0.6 • +2 = +1.2σ
…or 118. Now, this was assuming that their families had a mean IQ of 100. If their families had a different mean IQ, lets say 120 (+1.333σ), the breeder’s equation would give
0.6 • (2 – 1.333)σ = +0.6667σ
…or a mean IQ of 125 for the children. (That’s because it’s +0.6667σ plus the family mean IQ of +1.333σ.)
What’s better, here’s another illustration. Let’s say the parents’ families have a mean IQ of 120, but the parents themselves have IQs of 110. Given the breeder’s equation,
0.6 • (+0.6667 – 1.333)σ = –0.6667σ
…for a mean IQ of 114 for their children (family mean IQ of +1.333σ minus 0.6667σ). In other words, even though the parents (with IQs of 110) are above the mean of the population, because their families are well above average, their children regress up.
It’s important to make clear that the breeder’s equation, and hence regression to mean, works the same way for any quantitative trait, not just IQ. This includes political orientation, height, body weight, personality, etc. All you need to know are the values to fit the variables in the equation.
For populations, the equation works similarly. Hence, if a group of people with a mean IQ of 130 (who come from population with a mean IQ of 100) go off somewhere and have children, the next generation will have a mean IQ of 118. Now here’s the part that gives a lot of people trouble: the children of the children of this group, the third generation, will also have a mean IQ of 118. Why? Because the initial event changed the mean. The new “population mean”, as far as the breeder’s equation is concerned, is 118. As long as they mate endogamously, there will be no change in their average IQ thanks to regression (only through selection).
This is should illustrate the flaw in thinking that regression happens forever. If daughter populations regressed back to the mean of their original source populations indefinitely, how could there be any selection for quantitative traits? Think about it.
Now let’s return to individuals. Some think OK, if populations don’t regress forever, surely the descendants of any one pair of parents do, yes? Well, not necessarily. Let’s return to our example of the offspring of IQ 130 parents from mean IQ 120 families. When it comes time for their children (the second generation), with a mean IQ 125, to have children, we do again run the breeder’s equation. But the key fact here is that the mean value the third generation of children are regressing to is the mean of their respective families. If all of the 2nd generation parents mated with spouses from high mean IQ families, there would be little to no regression for the third. In other words, regression to the mean for individuals can be slowed or halted by assortative mating. This is why wealthy parents have concerned themselves with the family backgrounds of their children’s mates. And this is why Gregory Clark found what he found (see The Son Also Rises | West Hunter) – namely, very slow regression (around 10 generations, in many cases) to the population mean for families (indeed, virtually none in Indian castes, who only mate within caste).
Indeed, as I mentioned, the reason for regression is the same for individuals and populations. You see regression in populations because the exceptional individuals who comprise whatever select group in question are going to be coming from families of all different averages for whatever trait under consideration. The sum of all these individual regressions is going to add up to regression towards the mean of the source population. (But as mentioned before, this only happens once.)
Hopefully, this serves to clear up the confusion on regression to the mean.
Clever people might notice that all of HBD is based on just two concepts: behavioral genetics (or again, more broadly, heritability) and the breeder’s equation. Know those two things and most of the rest follows.
JayMan's Blog 
Not bad. Not bad at all. However, I think you should have studied and linked to the paper published by Sir Francis Galton in 1866 ‘Regression Towards Mediocrity in Hereditary Stature’ (Journal of the Anthropological Institute) http://www.galton.org/
Nice work. The way most HBDers think of this concept is that anything beyond h (or in this case h^2) is random/and is more likely to hit the mean.
Of course that’s not true.
For it to hit the mean it would still mean parents would have to be on exact opposite ends of the spectrum, of course.
However, it is more rare to have IQs both at 40 and 160 as opposed to 85 and 115…..
Well done. Makes perfect sense, a lot of the articles I have read in the past as I recall did not account in any real way for family mean, and indeed if that were to be the case it would logically exclude genetic evolution to a large degree, as it virtually ignores selection for traits as an cause for change as well.
Yup.
Two problems:
1. When dealing with regression to the mean, we’re not talking about the normal range in practice. We’re talking about exceptional individuals. And so I don’t know if applying the general heritability numbers to the top or bottom 2% makes sense. With edge cases, those could be, as a class, far more a product of extreme environmental impacts.
I.e. – they were “flung out” by atypical environmental effects, and so the unshared environmentality of their distance from the mean could be, say, 40%. This would also work with non-additive genetic effects, where edge cases have systematically more non-additive genetic effects as a proportion of their variance.
2. I don’t know why one should assume that either the unshared environment or non-additive genetic effects completely wash out in the next generation. They could partially wash out, but that’s an empirical question. It’s not something inherent in how this works. Maybe two people with 130 IQs, when they reproduce, are more likely to maintain the non-additive portion that pushes up their IQ into their offspring.
Both of these are reasons to not assume that regression to the mean, for IQ but in principle any trait, regress only once or, when referring to the edge cases that we actually care about, only regress half the distance.
For any individual, regression works the same: towards the family mean. Very exceptional individuals would then be obvious to evaluate if their traits are not as due to additive genetics: their traits will be far off from the mean of their family.
What does this information even imply? That it doesn’t matter to have a white ethno-state because all we’d have to do is bring in high IQ non-whites?
Is your only reason for a white enthostate IQ? why not just move to israel or china then?
I don’t claim to fully understand the maths, but I think I understand enough to have spotted that there’s something significantly wrong here.
Consider again your hypothetical involving a subpopulation branching off and having kids. Namely:
Given the above (and that 30 IQ points is 2 standard deviations), you tell us that it’s legit to plug in h² = 0.6 and S=2 into the breeder’s equation and compute a next-generation expected IQ of 1.2 SD above the mean, or 118. Okay.
But earlier you told us we could do the same calculation on the per-family level. If we apply the family-level breeder’s equation to each family in the subpopulation, what result do we get? Well, we can suppose that every individual in the 130-IQ subpopulation has 130 IQ and came from a family with a mean family IQ of 100; then the breeder’s equation would tell us that each individual’s offspring will have an expected IQ of 118, which is consistent with the result from the population-level application of the equation.
But here’s the catch: the assumption of a mean family IQ of 100 can’t be correct here. If we randomly take people with 130 IQ from the starting population (that had mean IQ 100), the mean family IQ of the families those people belong to isn’t going to be 100; it’s going to lie somewhere between 100 and 130. And, as you noted earlier, if you increase the family mean while holding the parental IQ constant, then the expected IQ of the children goes up. So the per-family application of the breeder’s equation should actually predict the children’s IQs to be some value greater than 118.
There’s therefore a contradiction here; the result we get from applying the equation to a population in the way you suggest we can do is not the same as the result we’d get if we applied it to every family in that population individually, as you also suggest we can do. Which means that one of these applications must be invalid.
Indeed. And you know what that number is in this instance? 118.
That’s what regression is: non-additive factors going away/averaging out so that the true genetic potential value comes through.
But if the people in our 130-IQ subpop come from families with a mean IQ of 118, then they’re only 12 IQ (0.8 SD) above their family mean, so the breeder’s equation tells us their kids should have
0.6 • 0.8 σ = +0.6667σ = +7.2 points,
which implies their expected IQ should be 118+7.2 = 125.2
That’s the inconsistency I’m pointing to – this calculation comes out at 125 points, whereas the result you got from plugging population stats into equation was 118; one of them has to be wrong.
If you double select, that’s what would happen. If you apply it one time, you get 118.
If you then take the 130+ kids from the next generation and repeat the process, you get 125.2.
Gah, you’re still not getting my point. I’ll try one more framing.
Consider three scenarios:
A. A randomly-selected subpopulation with IQ 130 breaks away from the main population (which has IQ 100) and has kids. You tell us these kids should have an average IQ of 118.
B. A randomly-selected subpopulation with IQ 130 who came from families with a mean IQ of 100 break off and have kids. You tell us (in the section about predicting IQ for individuals) that each individual couple in this subpopulation should expect their kids to have an IQ of 118.
C. A randomly-selected subpopulation with IQ 130 who came from families with a mean IQ of 118 break off and have kids. Based on the same calculation as we did for scenario B, each individual couple in this subpopulation should expect their kids to have an IQ of 125.
Scenario A is the subpopulation scenario you describe in the post. But for the conclusion that the next generation will have a mean IQ of 118 to be true, scenario A would have to be the same as scenario B; the people in the subpopulation that breaks away would have to come from families with a mean IQ of 100.
Yet, as you note yourself in the comments above, that’s not true! Scenario A is in fact the same scenario as scenario C – as you note above, a random person with IQ 130 from a population with mean IQ 100 will on average have come from a family with a mean IQ of 118! So we’re left with a paradox: our calculation is scenario A tells us that the next generation in the subpopulation should have a mean IQ of 118, but our calculation in identical scenario C tells us that each individual in that next generation should have a mean IQ of 125. These can’t both be true.