Thankfully, James Lee, a real expert in the field, sent me a current best estimate for the probability distribution of offspring IQ as a function of parental midpoint (average between the parents' IQs). James is finishing his Ph.D. at Harvard under Steve Pinker -- you might have seen his review of R. Nesbitt's book Intelligence and how to get it: Why schools and cultures count.

The results are stated further below. Once you plug in the numbers, you get (roughly) the following:

Assuming parental midpoint of n SD above the population average, the kids' IQ will be normally distributed about a mean which is around +.6n with residual SD of about 12 points. (The .6 could actually be anywhere in the range (.5, .7), but the SD doesn't vary much from choice of empirical inputs.)

So, e.g., for n = 4 (parental midpoint of 160 -- very smart parents!), the mean for the kids would be 136 with only a few percent chance of any kid to surpass 160 (requires +2 SD fluctuation). For n = 3 (parental midpoint of 145) the mean for the kids would be 127 and the probability of exceeding 145 less than 10 percent.

No wonder so many physicist's kids end up as doctors and lawyers. Regression indeed! ;-)

Below are some more details; see here for calculations. In my earlier post I arrived at the same formulae as below, but I had rho = 0.

Assuming bivariate normality (and it appears that IQ has been successfully scaled to produce this), the offspring density function is normal with mean n*h^2 and variance 1-(1/2)(1+rho)h^2, where rho is the correlation between mates attributable to assortative mating and h^2 is the narrow-sense heritability.

I put h^2 between .5 and .7. Bouchard and McGue found a median correlation between husband and wife of .33 in their review many years back, but not all of that may be attributable to assortative mating. So anything in (.20, .25) may be a reasonable guesstimate for rho.

In discussing this topic with smart and accomplished parents (e.g., at foo camp, in academic science, or on Wall Street), I've noticed very strong interest in the results ...

See related posts mystery of non-shared environment , regression to the mean

**Note**: Some people are confused that the value of h^2 = narrow sense (additive) heritability is not higher than (.5 - .7). You may have seen *broad sense* heritability H^2 estimated at values as large as .8 or .9 (e.g., from twin studies). But H^2 includes genetic sources of variation such as dominance and epistasis (interactions between genes, which violate additivity). Because children are not clones of their parents (they only get half of their genes from each parent, and in a random fashion), the correlation between midparent IQ and offspring IQ is not as large as the correlation between the IQs of identical twins. See here and here for more.

## 30 comments:

It must be nice being called a "regression to the mean" by your father... :)

h and rho here are estimated from a population of which 4 SD parents are a small subpoplation.

Therefore why should this calculation have any interest?

And furthermore how constant are IQ scores at 4 SD?

rho doesn't have a big numerical impact on the results, at least in the plausible range. The main question is what is the narrow sense heritability. Since normality of the population distribution in IQ is a reasonable approximation out to at least n=3 (and possibly n=4 if we carefully factor in different group means***) it is plausible that a measurement of h^2 in the center of the distribution still gives a decent approximation at n=4. In any case, unless h^2 approaches 1 for people in the tail, regression will be a fact of life.

*** Sometimes it is claimed that there is an excess in the tail (a "fat tail" deviation from normality), but a careful look suggests it could be a consequence of a small subpopulation with higher mean that starts to have a significant impact in the tail.

My own guess is that h and the stability of IQ are less at the extremes, but that there are subpopulations where this is not the case. I don't doubt, however, that the Cyprus of Brave New World could be produced within two generations. I have never heard a good argument against eugenics.

For what it's worth, counterintuitively, graduate philosophy students have the highest GRE scores *after* physicists. Too bad they choose to waste so much time & mental energy in that field.

The thing with far right part of the tail that its likely caused by rare beneficial mutations. So thats why probability of inheriting such high IQ even if both your parents had is so low.

It also explains why there are many more males at extreme ends of the spectrum. - Males DNA have more variance from the mean, they are less stable. On the other hand there is large hereditary part. Even in your "example" children of physicists become doctors and lawyers not janitors! It would be better if you investigated correlation of parents/children IQ at other ranges ( 1,2 and 3 STD )

> far right part of the tail that its likely caused by rare beneficial mutations

Not sure why you are confident of this. It's also possible that thousands of individual genes control almost all of the variation in IQ with more or less additive effect (see Plomin's GWAS results and the height GWAS results). With the right experiments we could know the answer within 5-10 years.

I'm told that Shockley (Nobelist and Caltecher who invented the transistor) referred to his son, an honors physics graduate at Stanford, as an example of regression :-)

You might think that we know nothing about the n>3 population, but I think we do. The average at Caltech is probably around n=3 and there is definitely an elite layer of people there (i.e., 90th or 99th percentile for brainpower within that population) whose careers can be tracked. In my own experience I know the SAT and GRE scores (retests 3-4 years apart) for a bunch of such people, and I know what happened to them in their careers. So stability I can attest to, although h^2 I dunno. I think the minimum threshold to be accepted to a top PhD program in theoretical physics or math is n>3. There is also the SMPY population that has been tracked for 40+ years.

This model assumes that the parents are drawn from the general population; in other words that each of them is a four-sigma lucky event.

But if assortive mating has been going on for a while (as I believe it has), the parents may be each be only 2-sigma from their familial background. The resulting expected IQ of their offspring would be higher.

Possible, but do you really think there are isolated breeding sub-populations with average IQ = 130? Only a few generations ago there were not good filtering mechanisms for women as they were not allowed at elite universities, almost never pursued graduate education, etc. If you look at biographies of famous scientists it's clear that many of them weren't IQ-focused in choosing their mates. I should also add that there is a lot more to life than IQ, so it's reasonable for people to focus on more than this in mate selection!

Yup! Terry Tao's wife is pretty unexceptional given his own accomplishments, and his kid isn't (so far) a prodigy of any kind.

Another big story, which will herald the downfall of Chinese civilization and Mathematical greatness as we know it ;)

http://www.3news.co.nz/Porn-now-available-to-Chinese-web-users/tabid/417/articleID/167070/Default.aspx

I'd be interested in the stability between the GRE scores four years later than SAT scores, if you can tell.

There was rough agreement between how people scored on the SAT and then the GRE years later. People who were above +3 on the (old) SAT tended to hit the GRE ceiling, etc. Keep in mind this is just a small, anecdotal data set.

In our study of U Oregon students the correlation between SAT and GRE scores was very high, like .75. (I think this is about as high as the correlation between any two different (distinct types of) IQ tests taken 4 years apart, but I'm not entirely sure.) There was actually an academic paper published by ETS researchers on this topic many years ago and the old SAT had a slightly higher correlation with GRE scores than the recentered (post 1995) SAT. Interestingly, doing a math-intensive major had a slightly positive score on GRE-Q at fixed SAT-M, but it wasn't a huge effect.

Cool beans. Thanks!

Thanks. Back in the 1990s, I'd been told +.5n or +.4n for regression toward the mean calculations, so this is a little higher. Using +.4n, I calculated that the benefit Jodie Foster would get from using her sperm donor with his 160 IQ rather than a 100 IQ donor would be only, on average, 12 IQ points in her sons, but at +.6n, that comes out to 18 IQ points.

I suspect +0.6n includes a fair amount of assortative mating among grandparents, which I think has been common for at least a couple of centuries, if not forever. I keep finding out more and more examples of intellectual dynasties, like the Darwin-Galton-Wedgewoods and the Huxley-Arnolds. Everybody knows that Aldous Huxley and Julian Huxley were brothers, but their less famous half-brother Arnold Huxley won the Nobel in Chemistry. In the U.S., the Eliots produced at least four famous intellectuals (T.S. Eliot, Charles Eliot, Charles Eliot Norton, and Samuel Eliot Morison).

I expect that these two interbreeding subpopulations have mean verbal IQ 130:

Ultra-orthodox Mitnagdic Jews living in the US and Israel.

The hereditary aristocracy of Europe.

and that this is almost entirely environmental. The heritability for adopted children does not apply to these subpopulations.

Two interbreeding low IQ populations where the low IQ is also mostly environmental: the whites of Eastern Kentucky and the Bajan Redlegs

Is the IQ variance the same for males and females?

I think it's larger for males but don't quote me on that as I don't want to get in trouble ;-)

Someone should study the intellectual dynasties in China during the late imperial period. Even in the 20th century, despite paradigmatic changes in the education system starting in 1905 and the Communists' efforts to destroy intellectuals in the 1960s, many famous writers, scientists and scholars came from families that had an excellent record of producing top civil examination candidates in late imperial times. These families selected brides from similar families, many of which educated their daughters as well as sons.

Shouldn't age be factored into this discussion? If the heritability of IQ rises with age, wouldn't the genetic children of geniuses become progressively higher IQ as they got older, and the adopted children of geniuses become progressively lower IQ (as the environmental effects of being raised by geniuses can no longer impact them)? Also, a lot of the regression to the mean seen in children is actually the parent's own IQ regressing to the mean. Someone who scores an IQ equivalent of 160 on the SAT is likely to score IQ 145 on the GRE (and vice verca) because there is a certain degree of good luck involved in scoring in the stratosphere on a given test, causing one to regress to the mean on a second testing.

The higher values of h^2 are coming from late adulthood measures of offspring IQs. If you tried to measure h^2 using younger offspring you would get a lower correlation.

Thanks for the clarification, but your notation is confusing to me. h^2 is the symbol for herritability, is it not? Herritability is not the same thing as the correlation between the IQ of parents and their offspring or the square of such a correlation. Obviously the more herritable a trait is, the higher the IQ's of parents and offsprings will correlate, but the parent-offspring correlation will always be lower than the herritability or its square root because kids are not genetic clones of their parents. Rather, the square root of herritability is roughly equivalent to the correlation between the IQ's of identical twins reared apart.

h^2 is the narrow sense (additive) heritability, which is estimated from twin/adoption data. In order to get a good estimate of h^2 one should use adult IQs of the offspring. See James Lee's notes (link) if you need more clarification.

I could be wrong again but isn't your formula contingent on the variances for both parents being the same? It's probably a small effect though.

Or, if I'm to believe Wikipedia, nicked the idea of the transistor from a Canadian.

Well, there are historical examples like the Wedgewood/Darwin family but I was contemplating the known history of of my own family. Given that my mother's mother was born in 1894 and went to college and I have adult children, your "few" generations is four.

I believe that while there may not have been formal filtering mechanisms in the past (by which I mean 1860, not 1960), there was still filtering. That's part of what courtship was (and is) for. Think of it courting as an extended job interview which attempts to let people discover features beyond surface beauty. I don't mean to imply that IQ was the only issue but that a "dull" girl might not generate as much long-term interest in a smart young man as a "bright" one would.

And "isolated" wasn't what I was thinking. Any person only has two parents, four grandparents, etc. So the genes you get can only come from your ancestors, not from the general population. If there's any selection for a certain trait in your ancestors, shouldn't that be reflected in the model?

Finally, I used two sigma (IQ = 130) as an example for clarity. I suspect a real number for "some family lines" would be likely to be 1.5 than 3.

No. Variance on practically every trait is higher for males, although the difference is usually pretty small.

It's thought to be a consequence of men having only one X-chromosome, while women are all genetic chimeras with (roughly) half their cells expressing one of their two Xs and half expressing the other.

Shouldn't there be too formulas? One for O sub F and one for O sub M. The standard deviation for O sub F would be smaller and the standard deviation for O sub M would be higher.

Jumping in here, because this is the first item that comes up in Google for 'parental midpoint'. I am specifically interested in this because my wife and I have the same grade-school IQ result [142], though I suspect our part scores are slightly different. Stella is now 7 and still surprising us with her smarts [though, for a while she has been all about using her very exact memory to be a social angle shooter, "Do you remember three weeks ago when you said we would go out for ice cream? I want to go now."] Cal is very different than Stella, and we'll see where he ends up.

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