Some time ago I looked at JayMan’s blog trying to figure out what causes regression to the mean in IQ. He explained it nicely with the breeder’s equation R=h2S, where R is the response, S is the selection differential, and h2 is the narrow-sense heritability. So, if the mother has an IQ of 100+x and the heritability of IQ from the mother is 0.6, then the average IQ of the children is 100+0.6x.
Fine. I agree. I believe it is about like that in average, though I know some very intelligent people who many times wondered which one of their parents was so intelligent as to have such an intelligent child as this equation would imply.
And it is a very simple formula. It is not in any way connected with IQ genes that should be the fundamental reason of heritable intelligence. It just gives a formula that apparently works. Back in the old times when I studied more difficult topics, they called such formulas phenomenological equations. One was supposed to be very careful when applying them, or it was much better not to use them at all. It was said that they do not give any correct insight. Even worse, it was said that they are wrong despite giving correct results in some cases.
The explanation in the post derived a nice result from this equation. If we select a large group of parents whose IQ is 100+x, then the average IQ if the children is 100+0.6x. If the children reproduce among themselves they form a new population with the average IQ of 100+0.6x and there is no more regression to the mean as for future generations the selection differential is zero.
That is what this equation says. Should I believe it just like that?
I will try to make a model starting from IQ genes and see where it leads. A gene influencing one trait, like IQ, can be dominant, partially dominant or recessive. IQ is polygenic. We can assume there are many IQ affecting alleles, each with a small effect, and some of them are dominant, other partially dominant and the rest recessive. Most of these genes are in autosomal chromosomes and in the sex chromosomes, mainly in X, there are many nasty alleles causing mental retardation.
First we look at dominant IQ affecting autosomal gene alleles. Let us make some simplifying assumptions. Let the total number of unique dominant IQ affecting alleles be R1, let each dominant IQ affecting allele have the same frequency p and let both parents have the same number N of dominant IQ genes. Children inherit in average N/2 of father’s alleles and N/2 of mother’s alleles. Let the parents alleles overlap by K. The expected value of expressed dominant IQ affecting alleles in a child is a sum
N-K that is, non-overlapping dominant genes have additive effect
+K-K/4 overlapping alleles the child inherits from at least one parent.
=N-K/4
In random mating the expected overlap of dominant IQ affecting alleles between parents is
K=R1(N/R1)2=N(N/R1).
The number of dominant IQ affecting alleles in a child is in average
N-N2/(4R1)=(1-N/(4R1))N.
The parent has N dominant IQ affecting alleles, but they do not need to be all unique. The child gets in K/4 cases the same IQ affecting allele from both parents. In a steady state we can assume that a parent has as many homozygote alleles as the child. Thus, a parent has
N-K/4=(1-N/(4R1))N
expressed dominant IQ affecting autosomal alleles. This is the same figure as the child has, which is natural for a population in a steady state.
Next we look at recessive autosomal IQ affecting gene alleles. Let R2 denote the total number of recessive IQ genes. We assume for simplicity that each recessive IQ gene has the same frequency q and that both parents have the same number V of recessive IQ genes.
There are three cases. The first case is that the father’s and the mother’s expressed recessive alleles overlap by L. In that case the child expresses L recessive genes. A parent has two alleles of each autosomal gene, thus the V alleles are (like) randomly distributed over two DNA-strings of length R2. The number of expressed recessive alleles in a parent is the number of homozygotes and has the expected value
T=R2((V/2)/R2) 2=(1/4)(V/R2)V.
The number L of overlaps of expressed alleles between parents is
L= R2(T/R2) 2=(1/16)(V/R2) 3V.
The second case is that both parent’s unexpressed recessive alleles overlap by S, then the child expresses in average S/4 of these alleles. The number of unexpressed alleles in a parent is V-2T as each homozygote has two recessive IQ alleles. We do not need to explicitly divide by 4 if we distribute these alleles randomly over two strings of length R2. The number of alleles the child expresses is
S/4=R2((V-2T)/(2R2))2=(1/4)(V/R2)(1-V/(2R2))2V.
The third case is that expressed alleles of one parent overlap another parent’s unexpressed alleles. The child expresses half of those alleles. We do not need to divide by if we think of the problem as one where the heterozygote alleles are randomly distributed over two strings of length R2. In the heterozygote side there is a string of length R2 where the probability of a IQ affecting allele is ((V-2T)/2)/R2. In the homozygote side there is a string of length R2 where the probability of a IQ affecting allele is T/R2. We get an IQ affecting allele in both strings with the probability
(T/R2)(V-2T)/(2R2)
and as the child has two parents, the child gets in this way
R2(T/R2)(V-2T)/R2=(1/4)(V/R2) 2(1-V/(2R2))V.
recessive expressed IQ affecting alleles. The total number of expressed recessive IQ affecting alleles for a child is
(1/16)(V/ R2) 3V+(1/4)(V/R2)(1-V/(2R2))2V+(1/4)(V/R2) 2(1-V/(2R2))V=(1/4)(V/R2)V.
This equals the number T of expressed recessive IQ affecting alleles in a parent, as it should since the system is in a steady state.
There are still the recessive IQ affecting alleles in sex chromosomes, notably in X. These alleles are always expressed in males, but in females only if they are homozygotes. These alleles seem to decrease IQ. A likely reason is that an X chromosome gene which is favorable for males becomes fast fixed in the population even if it is harmful to females as the damage appears only in female homozygotes. Therefore if we see different alleles the new allele is almost always harmful and has been fairly recently created by new mutations. Mutations are almost always harmful. These alleles slowly disappear from the population but new ones are created. The X chromosome seems quite important for intelligence and mutations in it often cause mental retardation.
Females seldom are homozygotes on these strongly IQ-decreasing alleles since it would mean that their father had the allele expressed and mother typically did not. They rarely would make a pair. We may assume that the mother does not express the allele and the father does not have the allele. In that case daughters do not express the allele and sons may do. The effect on sons is a major reduction of IQ. It is not regression to the mean as son’s IQ is only decreased. These X-chromosome (and Y chromosome) IQ affecting alleles are not the cause of regression to the mean and will be omitted.
There are also partially dominant genes. IQ affecting alleles of such genes can be taken into account by counting them twice, both to dominant and to recessive.
Let us now see if we found regression to the mean.
The parents can have more dominant IQ affecting alleles than average, or less. This changes N, but it does not cause regression to the mean. The same is true of recessive IQ affecting alleles. V can be larger or smaller than the average, but it does not cause regression to the mean.
What causes regression to the mean is that the parents have less homozygotes than the expected value. In dominant alleles the expected value of homozygotes from N alleles is –K/4=N2/(4R1). If this number is smaller and these alleles increase IQ, then the parent has a higher IQ than the child. In recessive alleles the expected number of expressed alleles in the parent is T=(1/4)(V/R2)V out of V alleles. If these alleles are IQ decreasing and the parent has less than T expressed alleles, then the parent has a higher IQ than the child.
Both of these mechanisms are cases of random variation in the number of matches in a run of throws. As the number of both dominant and recessive IQ affecting alleles is expected to be quite large, there should not be much variation.
Let us nevertheless assume that there is such variation and high IQ people have fewer than expected homozygotes in both recessive and dominant IQ affecting alleles. Let us collect a group of these people and let them have children. Because of regression to the mean, which does exits, the children have a lower average IQ, but it is still above the original average IQ. These children do not in average have less homozygotes than expected and yet their average IQ is higher than in the population from which the parents came.
This example shows that we can find a subset of people from a population that has a higher average IQ than the population and yet does not have less that expected number of homozygotes. The reason for their higher IQ is therefore a larger number of IQ increasing alleles and a smaller number of IQ decreasing alleles.
Do children of such a group show regression to the mean? Apparently they do not. The breeder’s equation does not work for any subset of people. This conclusion is supported by studies have shown that children from higher social economical status (SES) show higher heritability of IQ than children from lower SAS. SAS correlates positively with IQ. Spouses correlate quite highly in intelligence because of assortative mating. As a result there are subpopulations in the population which mostly mate among themselves and they experience less regression to the mean.
Let us consider an extreme case where the high IQ of an individual comes from having less than expected homozygotes in dominant IQ affecting alleles, that is, the parents do not have homozygotes and all their N IQ affecting alleles are heterozygotes. In that case their IQ can be taken as N while their children’s expected IQ is (1-N/(4R1))N. Setting
1-N/(4R1)=h2
gives the breeder’s equation. Since N cannot be larger than R1, we have h2≥0.75. This is higher than 0.6 but some twin studies suggest heritability 0.85. The average value of N is pR1. Inserting this value yields h2=1-0.25p and selecting h2=0.85 because of some twin studies gives p=0.6 as the average frequency of dominant autosomal IQ gene alleles. It is a testable prediction.
Here we have regression to the mean but it is essentially caused by parents, who had fewer heterozygotes than expected. Such a situation can be a result of recent admixture of different populations. Inbreeding results to more homozygotes in all genes and decreases intelligence. If the children of such parents form a population and inbreed, intelligence can decrease several generations until the population is in a steady state and the number of homozygotes is in its expected value. The new population created by these children can stay at a higher level of average IQ only if they obtained more IQ increasing alleles and less IQ decreasing alleles form their parents.
Thus, it is possible to breed plants, animals and even humans and to create a new population that is more developed in some trait, but it is also true that wherever Europeans have migrated, their average IQ is always very close to 100, with the exception of areas where inbreeding has lowered it.
There must be interplay of variation in the number of IQ affecting alleles and also in the number of homozygotes. We cannot automatically use the breeder’s equation in other populations with the values for heritability of IQ measured for Europeans. It is indeed a phenomenological equation and must be applied with this in mind.
The Flynn effect, or the fraction of it which is not explained by environmental factors like better nutrition, may be partially due to reduced inbreeding. Blogger HBDChick studied reduced inbreeding as the key to the rise of the Western civilization, and there may be much sense in it. The number of homozygotes is one of the central factors. But there are other factors as well: the number of advantageous and disadvantageous autosomal IQ alleles and the role of the X chromosome, which may be the key both to the scientific mind in advantageous alleles (which have become fixed) and to mental retardation. Lastly, there is assortative mating, which has a role in this all.
There is another mysterious issue, it is about assortative mating.
Advantageous genes should eventually get fixed in a population. Why are there so many dominant or partially dominant advantageous alleles so that people can make polygenic scores for different populations? I can understand why recessive alleles causing mental retardation remain. They do not: they are removed, but new are created by mutations. Most mutations are harmful. I would not expect to see a large number of IQ increasing alleles, which have not got fixed.
But it is the same as with height. Height is also a polygenic trait, like IQ, and the population shows normally distributed variation in height. Alleles for greater height do not get fixed. We all agree that being taller is better. Or do we? Women do, but men do not choose the tallest woman, nor do women like to choose a shorter man, given a choice. Great height is at some point a disadvantage to a woman and that keeps these alleles from being fixed.
I think it is the same with IQ. Women prefer men with higher IQ than they have, while for many men it is scary if a woman is more intelligent. IQ is advantageous for men but at some high level it becomes disadvantageous for women. This keeps the alleles from becoming fixed.
Thus, we can thank assortative mating not only for decreasing the IQ differences between spouses but also for the existence of IQ variation in the population. There is much more to this difference between men and women. Jus a while ago I had this great insight that the evolutionary reason why women were not warriors is that the winner warriors used to kill the loser warriors, which were men. Then they mated with the women and the genes of the two populations mixed. Had the women also been fighting, they would have been also killed. Then the genes of the population had vanished. There may have been such tribes, but nothing was heard of them after they once lost a war.