Saturday, May 08, 2010

Assortative mating and the far tail of intelligence

A Fermi estimate of the increase in the tail population due to elitism, mobility and assortative mating. My conclusion is that there has been a significant increase in the number of kids in the far tail relative to, say, just one or two generations ago. I think my assumptions below are fairly realistic. (Note to non-scientists: this is only an order of magnitude calculation!)

My original estimate was off -- see erratum 2 below and this updated post

US birth cohort roughly 4 million.

Roughly few hundred kids at +4 SD ability (taking into account an excess over the normal distribution prediction; past assortative mating is probably one of the main causes of this excess ;-). What are these kids like? See here or here.

Number of +3 SD graduates per year from elite universities (average at these universities is above +2 SD): 10,000
(Note: a significant chunk of the roughly 40k +3 SD population!)

Plausible additional number of +3 SD / +3 SD marriages due to elitism and assortative mating, relative to one or two generations ago: + 1000 per annum

Number of additional +4 SD children produced, assuming, e.g., heritability of .8 (so, +4 SD kids are +1.6 SD from parental midpoint adjusted for regression): + 100 per annum

This ignores any environmental advantage to a child from having two +3 SD parents; but that effect might be small.

This enhancement of the tail population doesn't imply anything about the average intelligence of the overall population -- it is due to a concentration of good genes in a small sub-population that has been filtered by psychometric exams like the SAT, GRE and LSAT. I suspect that immigration (e.g., foreign graduate students at top programs, engineers coming to Silicon Valley, etc.) produces an effect of similar or even larger size.

Erratum 1: I kept screwing up the most important part of the calculation: what the mean IQ of children of +3/+3 SD parents would be. (See comments below from Carson Chow and Henry Harpending.)

Rather than continuing to try to do it in my head I looked up the result in Gillespie's Population Genetics (p.113) and the result is that the correlation between offspring and midparent is h^2 = heritability.

If the parental midpoint is +3 SD, then the children are distributed around an average of +3 h^2.

I had originally assumed h^2 = .8 which leads to +3 (.8) = +2.4, which requires a +1.6 fluctuation to produce a +4 kid. The odds of that are about 1/20.

If heritability is much lower, like .5, then it requires a +2.5 fluctuation to get a +4 SD kid and the effect of assortative mating is small.

Note, I have also been assuming that the residual variance among the +3/+3 kids is the full 15 point SD of the general population distribution. Probably the variance should be lower, but this requires us to go beyond the simplest models of additive genetic variance. ...

Erratum 2: After a bit more research, I would guess that the additive portion of genetic variance is no more than .5 or so. The high values of heritability typically quoted (e.g., .8) are broad sense heritabilities which include both additive and non-additive variance. The non-additive part results from dominance, gene interactions, etc. and does not allow a simple prediction of offspring qualities from the parental midpoint. So the result given above for h^2 = .5 is probably the largest effect we can expect from assortative mating, and it is not as large as I originally thought. A second point is that the residual variance once parental midpoint is fixed (i.e., exhibited by siblings) is probably smaller than 15 points. If narrow sense (additive) heritability is .5 then the remaining variability would have SD of about 11 points. This further reduces the chances of getting a +4 SD kid.

For parental midpoint n (in SD units), the probability of a child who exceeds this midpoint is

1 - erf( n sqrt ( 1 - h^2) )

where h^2 is the narrow sense or additive heritability. To see this, note that the mean among the offspring would be n h^2 , so the required upward fluctuation is n ( 1 - h^2 ) , but relative to the remaining variance, which has SD = sqrt ( 1- h^2 ). The probability of a child surpassing superior parents gets smaller the more exceptional the parents, however the probability is much higher than for a random child selected from the general population: 1 - erf(n). This would also be the result for any choice of parents if narrow sense heritability is zero. On the other hand if h^2 = 1 then all children would equal their parental midpoint: 1 - erf(0) = 1.

Note, throughout this discussion I have assumed no environmental boost from being raised by exceptional parents.


ccc1685 said...

I believe heritability is defined in terms of R^2 (portion of variance explained) rather than correlation coefficient, so I think you've actually done your calculation for heritability of 0.64.


steve hsu said...

Type your reply...Oops -- I always get h and h^2 mixed up!

steve hsu said...

Oops, thanks. I've fixed it now. Strangely, when talking about validity of psychometric tests one always gives the correlation (and critics then complain that the fraction of variance explained is smaller). For some reason, for "heritability" they quote the portion of variance rather than the correlation. I always get the two mixed up.

Carson Chow said...

In population genetics, the interesting question is how much some variable explains another variable, so R^2 is the natural variable. However, for psychometric tests perhaps they want to know correlations because the sign matters.

steve hsu said...

Either convention is OK as long as it is used consistently (although as you note it's important to know whether things are correlated or anti-correlated). But if I quoted the "validity" of a test as .6 a non-expert wouldn't automatically know whether I was talking about the correlation or portion of variance; same thing with "heritability".

What do you think about the estimate? :-)

Eric Foss said...

I'm doing some work looking at molecular details (protein and transcript levels of 354 genes) from a population of yeast derived from crossing two strains that differ at ~ 1% of their base pairs. That's about the degree of difference between humans and chimps. Several things make heritability high here:

1. simple genetics (all alleles come from just two individuals);

2. enormous genetic differences (human versus chimp equivalent);

3. I'm focused on high abundance proteins, which means transcripts and proteins are both high and measurements are therefore better than for average genes (I know that for transcripts that this subset is enormously enriched for high heritability);

4. environment was entirely controlled for; and

5. my measurement is something far more straight forward and likely to have just a small number of genetic factors behind it than something like intelligence in humans (and I also know that there are a relatively small number for most of these since I've identified mapped regulatory loci).

So I've got everything going for super high heritability - way higher than intelligence in an outbred human population. Average heritability for both transcripts and proteins is 0.7. So if your conclusions depend on your heritability being correct and you've assumed heritability to be 0.64, I'd say kiss your conclusions good bye.

steve hsu said...

But heritability of IQ in twins studies is generally reported in this range. In fact, for IQ in late adulthood I believe numbers like .8 or higher have been reported. This specifically refers to the "humane range" of environments required by adoption agencies, but that is what would apply to the elite population described in the post.

From wikipedia:

Estimates in the academic research of the heritability of IQ have varied from below 0.5[3] to a high of 0.9.[6] A general range of 0.4 to 0.8 was given by the "Mainstream Science on Intelligence", a 1994 declaration of 52 scientists in the field.[7] A 1996 statement by the American Psychological Association gave about .45 for children and about .75 during and after adolescence.[8] A 2004 meta-analysis of reports in Current Directions in Psychological Science gave an overall estimate of around .85 for 18-year-olds and older.[9] The New York Times Magazine has listed about three quarters as a figure held by the majority of studies.[10]

Carson Chow said...

Hi Eric,

Are you arguing that epigenetic effects, which certainly affect protein amounts, will dominate any genetic effects? I think there are many examples where mRNA levels and proteins are very uncorrelated. However, it could be that some complex traits, especially in development, depends more on timing than amount of protein expressed.

Steve, all I'll say is that the number you get is consistent with the numbers you entered.

Carson Chow said...

I'll add that you are predicting an almost 50% increase in 4 SD people per cohort. That should be testable, no?

steve hsu said...

Sadly, I don't think anyone is keeping careful track of this population. The SMPY project doesn't have anything like national coverage -- the kids are typically encouraged by parents to get tested. Keep in mind the +100 I estimated comes from effects that have emerged over generational timescales: elite universities going fully coed, increased demand for elite education, emergence of "geek" subculture, etc.

I can track this sort of thing informally, though, and in my opinion the estimates are not aggressive. Compared to the university town where I grew up, Eugene has a lot more +3/+3 pairings. There is also a surplus of very smart kids here (e.g., who take most or all of an undergrad math major while in HS, are finalists in the Intel/Siemens, make the IMO team, score 1600 on the SAT), most of them children of professors. Some of the math stuff is explainable by the presence of opportunity for training (coaches), but there might be a residual effect. At the national level, who knows?

Carson Chow said...

Has the number of kids who max out on the SAT's changed over time? Also, what sigma gets a perfect score?

steve hsu said...

The old SAT could resolve this population, but (post-1995 recentering) I don't think the newer SAT can. The newer version goes out to maybe 3.5 SD. It would be hard to know whether there has been a shift in the far tail taking place over, say, 20 years, but if anyone could do it, ETC (CollegeBoard) could. Of course, you'd have to accept the SAT as a good indicator of really high level ability, and I'm not sure it is.

MarkyMark said...

I remember reading a few years back about an article in the Lancet which suggested that genes for intelligence were located largely on the X-chromosome meaning that the IQ of a female was determined by both parents whereas the IQ of a boy was determined solely by the mother (the father providing a Y chromosome). This was spun in the popular press as 'men if you want to have smart sons, pick a smart wife'.

If this is correct then it would suggest that assortive mating wouldn't increase the number of +4SD males. In fact if an elite education meant that the woman was less likely to have children (compared to earlier generations) then the number of +4SD men could even decline.

Eric Foss said...

Hi Carson,

No, I'm not arguing that. Two things:

1. As noted in my response to Steve, I made an error in thinking about heritability as a measurement of the demonstrably genetic contribution to a phenotype. So within the framework of that incorrect thinking, I was arguing that when I skew everything to a system where the genetic component is as high as I can get it in a genetically diverse population, I'm still getting heritability values around 0.7, so to assume a heritability value of 0.64 for something like intelligence in a genetically diverse population of humans seemed silly. (Again, I realize I got into the wrong mind set of thinking about the definition of heritability.)

2. To now segue from Steve's intelligence discussion to my biology (with regard to your epigenetics question), no, I'm not arguing what you said. In my work, the argument that I'm pushing is directly related to your comment about the correlation between protein and transcript levels. My view is this: In experimental situations when you put a gene under the control of a very strong promoter and overexpress the transcript, say, 1000 fold, then yes, you will see an increase in the level of the corresonding protein. However, transcript array studies in genetically diverse populations are dealing with variations in transcript levels that typically aren't anywhere near 1000 fold - they're stuff like 2 or 3 fold. Because protein levels can be controlled not only by controlling transcription, but also by controlling translation and protein stability, it's only when the former mode of regulation overwhelms the latter two that one should even expect a good correlation between protein and transcript levels. And then I make the argument (in the manuscript I'm currently writing) that examining this in a genetically diverse population leads me to believe that the correlation is crappy, not because genetic regulation of transcript and protein levels is not important (which I show that it clearly is) but because the effect on protein levels of modest genetic variation in transcript levels is easily obscured by genetic variation in the control of translation and protein stability. Again, this has nothing to do with Steve's comments about intelligence and assortative mating.

catperson said...

Are you sure you're not just detecting the Flynn Effect? Also, on professional IQ tests like the Wechsler, it's not possible to assert that there are more high IQ people than would be predicted by the normal curve because IQ's are defined by the normal curve. There's really no other way to assign IQ's than to assume the curve is normal because standardized tests are almost always ordinal scales so the standard deviation of the raw scores are not meaningful.

catperson said...

Also, assortive mating would probably only have the effect of increasing the standard deviation for intelligence causing the 4 sigma level to be more deviant than in the past rather than increasing the actual frequency of people who are 4 SD above the mean.

Carson Chow said...

Hi Eric,

These are things that my group has a tangential interest in. I just published a paper recently in PNAS showing mathematically how graded control of transcription could occur. I punted on the translational control. However, understanding that is crucial.
Are the genes for controlling transcription of a particular gene located near to the genes controlling translation and protein stability? I guess the question I'm going for is whether the control of transcription and translation cooperate.

catperson said...

I've heard that both the old and new SAT do not discriminate well beyond 1400.

Eric Foss said...

Hi Carson,

I'd say no, they don't - at least in a simple sense. Broadly speaking, I see two large groups of functionally related co-regulated groups of genes. In one, transcripts and proteins behave entirely differently - entirely different regulatory loci for the two data sets, correlation coefficients between protein and transcript that are no better than random, etc. For the other group, proteins and transcripts behave more similarly - similar regulatory loci, plenty of correlation coefficients higher than could easily be achieved by chance, etc. Even here, however, I don't think that transcription is the primary cause for the variation in protein levels because normalizing proteins for transcript levels (regress proteins on transcripts) does not destroy linkage between protein and regulatory locus - in fact normalization often does almost nothing. So I think that what's happening is that I have polymorphisms that affect some broad aspect of cellular physiology (e.g. intracellular concentration of amino acids) and then the cellular response to this has both a transcriptional and translational branch (e.g. TOR pathway). If the translational branch is far more important than the transcriptional branch in controlling protein levels, you'll see a decent correlation between proteins and transcripts but transcription still isn't causal. And then finally I see a handful of proteins that are clearly regulated by transcription, but these are a distinct minority.

steve hsu said...


Perhaps I misunderstand your point, but the definition you give for heritability is the one that intelligence researchers have in mind. They are also aware that the heritability is only well-defined if you also specify the range of environments. The case I'm interested in is one where IQ heritability should be very high: affluent homes with highly educated, intelligent parents; kids in such cases have every chance to express their full genetic potential.

BTW, I should have mentioned that the estimates of heritability for human intelligence are obtained through adoption experiments, not just including identical twins but also fraternal twins or siblings, where the shared genetic information is smaller.

My estimate would not change very much if one took the heritability value for intelligence to be .5, which is the lowest value most researchers in the field today would accept.

There are really just two things to consider:

How much more likely are +3/+3 pairings today than, say, in 1960? I think most people would agree they are far more likely (MIT PhD marries HLS grad, etc.).

Given +3/+3 parents, what are the odds of a +4 SD kid? Pretty high, even if you use the low .5 value for heritability: 1 in 30. Compare that to the 1 in 30k probability of producing one from the general population, and you have an idea of why an increase in the number of +3/+3 pairings can give a large effect.

Henry Harpending said...

I don't think one should square the heritability. Consider two people in a Fisher (1918) world with a relationship coefficent of r. The correlation between their breeding values is rVg/(Vg+Ve), that is the relationship multiplied by the heritability. Here Vg is the additive genetic variance and Ve is the environmental variance. Then to estimate the heritability between pair of relationship r, one computes the correlation and divides by the relationship coefficient. There is no squaring anything. Having pointed that out, I claim no understanding of it.

Henry Harpending

Guest said...

It's interesting to ponder if and to what extent assortative mating is really higher now than in the past. It does seem plausible that widespread standardized testing manages to collect and concentrate higher quality genetic material from the larger population, but there are other mechanisms that could work in the other direction. For example there used to be significant geographical concentrations of high IQ populations in Europe which were eliminated in WWII. How much of a +4 SD deficit did that create, and are we still just making up lost ground? Then you also tend to have lower birthrates in your more elite populations, so who knows, it could all be a wash.

steve hsu said...

Hi Henry,

Thanks for pointing that out. Rather than continuing to try to do it in my head I looked up the result in Gillespie's Population Genetics (p.113) and the result is that the correlation between offspring and midparent is h^2 = heritability.

If the parental midpoint is +3 SD, then the children are distributed around an average of +3 h^2.

I had originally assumed h^2 = .8 which leads to +3 (.8) = +2.4, which requires a +1.6 fluctuation to produce a +4 kid. The odds of that are about 1/20. It turns out that is the correct answer.

So I was correct originally not to take the square root. If heritability is much lower, like .5, then it requires a +2.5 fluctuation to get a +4 SD kid and the effect of assortative mating is small.

Henry Harpending said...

Thx Steve:

The larger point of your original post is interesting and correct. Here I am arguing a tiny minor point but I can't help myself. The correlation between parent and offspring (with random mating etc.) is (h^2)/2, so the correlation between offspring and midparent should be (h^2)*(sqrt(2)) or ~.707*heritability. I don't have my Gillespie here so I can't check.

As a check we can go to the boundary where the heritability is one. Then if the Gillespie formula is right the offspring would be exactly equal to mid-parent (since the correlation would be one) and we would have blending inheritance with consequent halving of the variance every generation.


Best, Henry

Shawn said...

I wish there was a formal study relating to females' sexual attractiveness at in the 3 SD to 5 SD range. Are they uglier than "normal" girls? Since facial and somatype attractiveness can be mathematically quantified quite easily (yes, even attractiveness is a mathematical formula), I can't imagine it would be much of a problem. My guess is that super high IQ women are just too interested in learning to pay much attention to their looks (e.g. not wearing makeup) but they may have inherited a nice mathematical formula. But when you are an average 104-108 IQ guy like me, who is really skinny (yet 6'), I don't I have much chance -- especially considering the fact that I am in a no-name MBA program.

Eric Foss said...

Hi Steve,

With the significant caveat that I know nothing about these sorts of intelligence studies, and actual knowledge of facts might rule out various obvious non-genetic ideas I’d have about how you could get more +3/+3 pairings now than in 1960 …

The “heritability” calculation seems unlikely to be a reliable metric of the fraction of a phenotype that is due to genetics in situations where environment is really difficult to control and the phenotype is likely a multi-locus complex trait with who knows what kind of epistatic interactions between loci, as is almost certainly the case with studies of human intelligence. Heritability, at least the way I calculate it, is a measure of the degree to which the total variation in your data is due to the fact that you’re collecting your data from different individuals as opposed to multiple measurements from the same individual – it’s high when you have significant variation in your population even though repeated measurements of single individuals are pretty constant. In my work, I measure phenotypes in a population of 100 progeny of two parental strains. If I were to look at a trait that is entirely determined by genetics, but is determined by ten unlinked loci, all with approximately equal effect size, I’m going to see very low heritability even though genetics is everything. Conversely, if I am mistaken in thinking that my environment is perfectly controlled – e.g. I could have 100 genetically identical individuals and I accidentally raise 50 in rich media and 50 in poor media - heritability will be huge, even though genetics is nothing.

Then leaving heritability aside, I’m skeptical of predictions of enrichments in something as complex as intelligence that depend on the assumption that intelligence has simple additive genetics – for each “smart allele” of an “intelligence gene” that you inherit, you add x to your total intelligence. If you have to make an assumption about how intelligence is inherited, that’s certainly the best one to make, but complex traits don’t usually behave so simply. A colleague is working on genes that extend the lifespan of a microorganism. She found an allele of a gene that extends the lifespan of one strain by 30% - which is huge. However, that same allele *decreases* the lifespan of another strain of the same species by 20%. If intelligence has weird epistatic relationships like that – which it certainly will to some extent – then it’s not a safe bet to think that smart man plus smart woman gives even smarter child.

I certainly believe that there is a big genetic component to intelligence. We have a three year old daughter whom we adopted at age 4 days, and there are clearly tons of things that are intrinsic to her in personality, intelligence, etc. I’m just skeptical that one can figure out a reliable measure of heritability of something as complex as intelligence (in a real sense – not the mathematical formula) and skeptical that a phenotype so complex is likely to have changed its frequency in the general population in two generations because of factors like increased mobility. But again, I’m completely unfamiliar with the actual data in the field.

steve hsu said...


I've modified the post to take into account additive and non-additive genetic variance, as you suggest. The original model I was using was too simplistic. Breaking up the genetic influence into the two components reduces the effect substantially.

Researchers in this area can estimate the additive and non-additive components of heritability using twin and adoption studies. I'd be interested in your opinions on their methodologies and results, if you care to do a bit more reading.

I do think marriage patterns have changed significantly in the last 50 years, but the effect on intelligence in the tail is probably smaller than I originally thought.

Ohwilleke said...

A couple of observations:

1. The number of kids who are +4SD is not a very interesting number. Major studies (one of California school students), and minor ones (performance of elite law school graduates ethnographies of high IQ societies, etc.) show that IQ has greatly disminishing returns for both the individual in life outcomes, and for society, beyond about +2SD. Contrary to researcher expectations, someone at +3SD or +4SD is not significantly more likely to make great contributions to society or have better life outcomes, than someone at +2SD. The higher up the scale one goes, the less it matters.

2. Your prediction of change in assortive marriages is pretty modest (and hence realistic), but I'm not sure that it captures the main impact of the change in assortive marriage trends. Extreme elites have been very assortive for a long time. There has also long been a significant assortive factor to all marriage. But, I think that the big change is probably the declining share of working class to upper middle class marriages.

The executive to secretary; doctor to nurse kind of pair is being replaced by the executive to executive; doctor to doctor pair. Lots of the formal education/job differences before were between people of similar social class with difference in education/job created by sex discrimination; but lots were genuine socio-class leaps because it was easier to get into the pool of prospects.

If this is correct, what we should see from assortive marriage, is warping of the normal distribution into a bipolar distribution with a suppressed middle. The higher order SD effects here may have a bigger effect than assortive marriage at the very high levels, and the assortive marriage effective generally, may be more pronounce at lower SDs.

But, the way the test instruments are designed inherently suppress observation of this kind of change. The question difficulty is fit to a normal distribution since there isn't any obvious absolute scale to hang one's hat upon.

One also has to consider selection effects across the entire range. If, for example, there is selection against IQ of -1SD or less (and there is evidence to suggest that there is) then even though the high SD results depend on much higher SDs from the parents, the effects can be noticeable.

Carson Chow said...

Actually, i was going to comment that my comment was wrong but I see you have already corrected it.


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