What is the implied model behind assortative mating?

In a fascinating new paper by Brant, et.al. on IQ, I read the following bit:

…we found that higher-IQ parents actually showed less assortative mating: the difference between parental IQ scores was positively correlated with mean parental IQ score.

My questions are numerous but I will start with two.

First, is this the correct metric for “less” assortative mating?

Second, do “straightforward” models predict such a result as a matter of course?  For instance, higher-IQ individuals may have greater scope to choose mates on the basis of complementary skills.  That may imply higher IQ gaps.  Furthermore higher-IQ individuals may marry later in life, put more effort into choosing, and encounter a wider variety of potential partners.  That may also imply wider gaps in IQ across partners, even if assortative mating (as defined in an all things considered way) remains strong.

Very often there is more variability at the tops of distributions rather than at the bottoms or in the middles.  Yet “pull away” forces may continue to operate.

To present a simple analogy, income gaps between marrying couples are probably higher at the upper end of the distribution than at the lower end.  Yet assortative mating with respect to income still may reenforce income inequality.


It's paywalled, so I don't want to comment with certainty, but I'd say "to present a simple analogy, income gaps between marrying couples are probably higher at the upper end of the distribution than at the lower end" is on the perfect analogy.

How is "difference" measured? If it's in absolute IQ points, it seems like a rather obvious and completely trivial finding. It should be adjusted for standard deviation within the low and high IQ measured groups.

But from my read it looks like it's "absolute". In absolute terms, the smarter you are the more likely you are to "fall" a greater distance from yourself. Thus is a normal distribution.

My only qualm with the "pull away" comparison is that incomes are log-normally distributed for the bottom 97% but resemble power law at the highest levels. This is not the case with IQ which is normal throughout. All the more that this isn't a very good claim against assortive mating. But that is still to say that you would be much more surprised with a couple earning 20k (full time) and 100k than 250k and 500k. So not correcting for SD here is dangerous (to make nontrivial findings).

I also haven't read the paper, but I think you are generally correct. However I think there are problems with correcting for standard deviation - it's a quantity you have to estimate, and there are several assumption of normality you have to make; as you say income is not normally distributed. Also measurement of IQ at high (and low) levels is less accurate, so I'm a bit sceptical about the validity of the normality assumption of IQ.

A more sensible approach would be to use percentiles. So you consider (p_1,p_2) and (q_1,q_2) where p_i is the percentile of IQ and q_i is the percentile of income for parents i=1, 2. You can then easily plot a scatter graph to see if there is any dependence between D_p = | p_1 - p_2 | and D_q = | q_1 - q_2 |.

Agreed. Percentiles show you how much sorting or absence thereof there is in the sample. IQ score differences show how much absolute difference in IQ there is in couples which may have value but not to show varied degrees of assortive marriage.

I'd agree that if the difference is measured in IQ points, the observation is both trivial and obvious. Frankly, when you're looking at the higher end of the spectrum, the perceived differences (be they functional, social or professional) between people with a given IQ gap are much less than you'd perceive in the middle, or at the bottom, of the distribution.

I was thinking about personal anecdote. My husband has a significantly higher IQ than I do, in excess of 30 points. If he had a normal IQ, I'd wager there is no way on earth he would be married to me. If I had a normal IQ, we might have married but more likely we'd never have met while we were in the market for mates. However, our mean IQ is ... I'll just say it's well over 160. Our situation (one very bright spouse, one frighteningly smart) is not very unusual in our experience. But that gap, shifted lower in the distribution, would involve a normally intelligent person married to someone with significant mental retardation, or an average person married to a fairly brilliant one. Neither seems intuitively as likely as someone at the top of their class at an internationally recognized university marrying someone from the middle of the class.

Regarding income gaps as they track IQ gaps, I'll just note that I earn an order of magnitude more than my husband, and always have. (Maybe that's why he married me - I knew he was smart.)

Does this paper consider the notion that men likely have greater variance in IQ than women?

Thus, at the right-tail, the absolute IQ gap between high IQ couples would increase.

This is a good point. But the same logic would hold at the left tail – to equal effect (if IQs are normal) – implying the positive association emerges from something else.

People on the extreme left of the IQ scala probably aren't getting married in the first place.

Suppose the male standard deviation is bigger, but the probability of getting married is also positively correlated with IQ. Then would we see the results reported in the paper?

And people on the extreme right of the IQ scala probably know better than to get married :)

That was my thought. As to Rao's point that it should also happen at the low end, perhaps women at the low end rationally choose not to marry or even have sex with dolts at the low end.

"the difference between parental IQ scores was positively correlated with mean parental IQ score"

Is this just a statistical oddity? Take a bell curve and some matching probabilities and this is the result?


That sounds right. The higher your IQ, the fewer potential mates you will meet with approximately the same IQ. Of course that applies at the low end also. Does the paper study the full range, or only the upper end.

The tail at the low end would probably be difficult to study because of interference from various forms of brain damage. There is no form of brain damage which puts someone in the tail at the high end.

You obviously haven't dealt with the same very bright people I have...

Whatever it is, don't tell the Chinese, they'll start doing it to everyone.

As you get more intelligent there are less and less people in the same cohort, so you chances of fiinding someone of the same iq must be lower.

Pretty much. With an IQ of 100, you have 83% of the population with an IQ within 15 points of yours. At 115, it drops to just under half. At 130, only about 15% of the population is within 15 points of you. And at 145 the pool shrinks to 2% or so.

Maybe, but given the way modern society sorts us into intellectual classes during the relevant years of our lives, I wonder how high you have to get to run into this effect. I mean, the average person at Harvard Law School is probably +3 sigmas, but they are also hanging around in a pool of other people of comparable intelligence. Perhaps only 1% of the population are your peers, but you don't have much trouble meeting them.

I'd be curious to see this controlled for age at marriage (or even better time since achieving highest level of educational attainment). My hunch is that you are correct about people who marry within small groups of like individuals; but it may just be that Tyler's comment, "furthermore higher-IQ individuals may marry later in life, put more effort into choosing, and encounter a wider variety of potential partners," is offsetting that self-cohorting impact.

This one works for me. The pool gets smaller and IQ is not the sole suitability criterion or proxy, certainly not for males.

There are two questions about assortative mating. How do different mating models affect the income moments of the couple and given further assumptions about heritability of traits how does it affect the trait distributions over generations. They might imply different metrics.

High IQ men (or women) are more successful, richer etc. Ergo they have more choice to select a mate they want.

Most men (and women) fall for attractive mates. Conventional physical attractiveness is a trait uncorrelated, or even anti-correlated with high-IQ in today's knowledge economy.

End result: High IQ male / female pairs with a spouse of substantially lower IQ.


This. Physically beautiful people (especially women) can easily marry "up" to higher intelligence, higher income spouses.

Supposedly, the most beautiful woman in the world once proposed marriage to the most intelligent man. "Why our children would be the most beautiful and intelligent offspring around," she proposed. The intelligent man declined, saying, "what if the children have my looks and your IQ?"

Lies. I would never be so rude to a beautiful woman.

I just don't meet enough beautiful women at the restaurants I frequent...

Poor Leonard--all that intelligence sure doesn't make him verbally smooth....

False symmetry.

IQ and attractiveness are positively correlated. A large reason is because both IQ and attractiveness are positively correlated with good health/ good genetics.

A simple (simplistic?) explanation could be that for from the extreme high end of the distribution there's simply not much quantity to select from at that IQ. Probability dictates that a spouse will probably be from the lower, more densely populated bulge of the IQ distribution.

It's like most BMWs that were involved in a crash, had crashed into much cheaper cars.

Well the whole premise of "assortive mating" is that BMWs almost always crash into other BMWs, so it seems relevant.

And this study can be seen to support a certain drag of regression to the mean on that assortive mating. It's a flippant remark that hides a deeper truth, I have always liked the saying that "Hot is how crazy stays in the gene pool". Now, I don't think that's true for intelligence, but it does note the papering-over effect of physical attractiveness. And as I'm sure has been noted, women are often attracted to a man noticeably more intelligent than they are, but the reverse is not true unless the woman is also very attractive. My explanation has always been that I am attracted to the physical, but I can't stand to stay with a woman without a certain modicum of intelligence, which while well above the average, is well below my own.

Income variation in the tails is surely (roughly) correlated with IQ, but moving into the high income tail involves an explicit or implicit commitment from a spouse to earn less, so as to improve geographic mobility and time-of-work-day inflexibility for that higher earning spouse. With children, one of the two spouses has to have the work flexibility to get the kids to dentists, doctors, events, etc., and that sort of commitment leads to lower income.

First let's determine what it is that IQ actually measures.

And why is it suddenly a credible number again? Humans are idiots.

First let's find out what g, the gravitational constant, actually measures. Physicists are idiots.

My pet theory is nothingness loves company.

First let's determine what centigrade actually measures.

And why are thermometers suddenly a credible way of measuring again? Meteorologists are idiots.

I'm personally a fairly strong believer in the utility of IQ testing, but if you think the strength of IQ as a measure parallels the deep utility and fundamental relevance of Temperature, you are probably sadly uninformed about both Physics and Cognitive Measurements.

I am in the same boat as Rahul. I just think Wondering wanted to pile on.

As long as IQ correlates reasonably well with what we care about (intelligence, which we can sort of recognize when we see it even if we can't perfectly define it), it will help us to answer questions about assortive mating by intelligence. Though imperfections in the test (like having a weaker correlation with intelligence at higher or lower scores) can complicate things a lot.

mike and Wondering - really?

G *defines* a relationship between mass, distance, and force. All of these are easily measurable parameters of a system. G thus defines the function of a fundamental physical force.

Centigrade (or temperature) measures the average kinetic energy of particles in a system. Many physical properties, such as conductivity, and thermal voltage (i.e. thermal noise) stem from temperature.

IQ, on the other hand, attempts to measure a range of cognitive functions using a one-dimensional quantity. At best there is a positive correlation between (some) cognitive function and IQ. However, there are plenty of individuals that are brilliant along some axis (for example, pandering) while being average or perhaps below average along other measures. IQ is perhaps as sophisticated a cognitive measure as Aristotle's Five Elemental theory was for modeling matter.

"At best there is a positive correlation between (some) cognitive function and IQ."

I'd replace your "some" by "a lot of". @Mike and @Wondering erred in the direction of overselling IQ, whereas you are underestimating IQ's utility, I suspect.

"IQ is perhaps as sophisticated a cognitive measure as Aristotle’s Five Elemental theory was for modeling matter."

I don't agree. IQ is a lot more sophisticated than what you are making it out to be. With all its faults, it might still be the best single measure of cognitive ability we have.

Many non-physical properties stem from IQ. How is this so radically different from temperature?

As far as we can tell, IQ test scores represent the influence of basic primary cognitive abilities (like memory, reaction speed, basic spatial skills, etc.), as determined mainly by genetic and random variance, over sets of tasks with which most people in a culture tend to be roughly familiar or unfamiliar (either because they are saturated by them, or they're completely novel).

It would probably give more predictive accuracy if each of these basic primary cognitive abilities could be broken out separately and sophisticated models built on them (and that's why smart psychometricians are interested in doing so), but they tend to be both positively correlated (probably because "good" mutations and random events tend in general to be "good" globally across the brain) and are equally useful for predicting the same outcomes, so its not so lossy to have either a total IQ or a g score for most purposes (even though IQ and g are probably further from neurologically "real" than a pure measure of working memory might be).

It's certainly not the sole god variable for predicting individual and group variation (or to select individuals or groups for certain functions), but is useful.

The full paper is available here: https://dl.dropboxusercontent.com/u/182368464/2013-brant.pdf

Note that they discuss assortative mating (AM) not for its own sake but because AM is a potential confound that could invalidate the main finding of the paper, which is that "individuals with high IQ show high environmental influence on IQ into adolescence (resembling younger children), whereas individuals with low IQ show high heritability of IQ in adolescence (resembling adults)." AM could invalidate this finding because unmodeled AM can decrease heritability coefficients and increase shared family environment coefficients in twin studies.

However, they show that there is less AM among higher-IQ parents, which means that the finding of lower heritability and higher environmentality in high-IQ adolescents is not an aftefact of greater AM among high-IQ parents. In the context of this study, it doesn't matter why higher-IQ parents show less AM.

That's not the full paper and it doesn't say anymore about assortive mating than Tyler quoted.

Does anyone have an ungated copy of the supplementary materials?

As I said, they're not interested in AR per se, but here's what they say about it in the supplementary materials:

Assortative mating in parents of the LTS twins: We tested for patterns of assortative mating for IQ in the parents of the LTS twins. Assortative mating increases between-family variability and therefore manifests as c2 in the twin design. If assortative mating was higher among higher IQ parents, then this would be a potential explanation for the increased c2 seen in higher IQ individuals at age 16.

Full-scale WISC-R IQ scores were assessed in parents at the time of intake of the family into either the Twin Infant Project (TIP) or the Longitudinal Twin Study (between 3 and 14 months post partum; the two studies were amalgamated to construct the current LTS). In the event that this information was given twice during this period, the responses were averaged within parent. If information was available from just one parent, this value was used alone. In total, data was available for 400 fathers and 447 mothers, for a total of 399 families with complete parental information. The mean IQ score for mothers was 104.91 (sd = 12.37) and for fathers 107.50 (sd = 12.89). The correlation between parental scores was r(399) = .388, p < .001. This demonstrates a moderate amount of assortative mating for IQ in the parents of the LTS twins, which could account for some of the variance attributed to c2. In order to assess whether the extent of assortative mating was different depending on the ability level of the parents, we correlated mean parental IQ and absolute difference between parental scores. The average difference was 11.26 points (sd = 8.68). The correlation between mean parental score and the this difference score was r(399) = .155, p = .002. This suggests that assortative mating was significantly less strong as mean parental IQ score increased. For this reason, patterns of assortative mating cannot account for our results.

If IQ is normally distributed, and income has more of a pareto distribution, the analogy does not hold up very well.

+1 Income is also Parento distributed for the kids who inherit.

Bill, that is the funniest joke I have ever heard on MR. Can I use that?

Yeah, I pointed this out. But income is not really pareto throughout. It's lognormal for the bottom 97 to 99% and approximately power for the rest.

Isn't logarithmic a power?


It's worth keeping in mind that IQ is an ordinal measure of cognitive ability forced into a normal curve and dressed up like a cardinal measure.

Since we don't have a good cardinal measure of cognitive ability, we can't say for sure that cognitive ability is actually normally distributed.

Which is to say that an IQ point is not a fixed value. The difference in cognitive ability between an IQ of 100 and an IQ of 110 is unlikely to be the same as the difference between 110 and 120, or between 120 and 130, All we know is that a fifteen-point difference means that two individuals are separated by one standard deviation in the rank ordering of cognitive ability. Keeping in mind, of course, that there is likely some measurement error.

Actually, we can say that IQ is *NOT* normally distributed. By definition of the Gaussian distribution, any quantity that cannot take on a negative number is not Gaussian. A lot more people need to study Aitchison's book on lognormal distributions.

That being said, often the normal is close enough to the empirical distribution to be useful for modeling and theoretically simpler to interpret, so we model lots of things that way. Which works until, say, you try and use a normal where you should be using a Lorentzian and the referee points it out...

What rules out that there's someone out there with a -3 IQ?

This confuses me. IQ is normalized so 100 is always the average, right? 100 is arbitrary though. You could assume a 0 mean and then say people with an IQ of 90 actually have an IQ of -10. Is this really an issue?

WTF? That paper has 13 co-authors .... I wonder who the free-riders are ...

Probably the ones with the higher IQs.

The paragraph on author contributions says:

Author Contributions
A. M. Brant developed the study concept, performed analyses,
and wrote the manuscript under the supervision of J. K. Hewitt.
Y. Munakata provided theoretical input to inform interpretation
and critical revisions. Testing and longitudinal data collection
was directed by D. I. Boomsma, J. C. DeFries, J. K. Hewitt,
M. McGue, N. G. Martin, S. A. Petrill, R. Plomin, S. J. Wadsworth,
and M. J. Wright. J. C. DeFries, M. C. Keller, and C. M. A.
Haworth aided in data analysis.

Why would I pay attention to authors who wrote that para so unintelligently?

You can replicate this in simulation even if you assume strictly ordinal sorting. Assume a population of 100 men and 100 women with a trait (such as IQ) drawn from Z. Then have them pair by ordinal sorting. The result is that men close to the median have are very similar to their wives, those in the tails can be non-trivially different (~ .2 sigmas). The effect would go away if you assumed infinite population, but in some ways small population is a reasonable approximation of search costs.
local popsize 100
set obs `popsize'
gen x=rnormal()
sort x
save _x.dta, replace
set obs `popsize'
gen y=rnormal()
sort y
merge 1:1 _n using _x /*sort order merge*/
gen diff=abs(x-y)
corr diff x</pre

Output from one draw:
. corr diff x

| diff x
diff | 1.0000
x | 0.0603 1.0000

The thing is, IQ is an inexact proxy for attractive traits.....in men. It is not perfect, but IQ is fairly well correlated to educational attainment, which is in turn correlated to earnings potential. Earnings potential is an attractive trait in a man, a non-issue in a woman. I mean, all else being equal, I guess rich is more attractive than poor, but guys just don't sit around and dream of their perfect girl's career and what sort of cash she'll be pulling down. So in an "assortive mating" scenario, one sees two forces in slight tension, but not a lot. The first is smart people tend to be around other smart people (in college, say), and tend to like people who are like them (basic rule of humanity). Then you have the slightly lower - IQ females who want to attach to a high-earning-potential mate. They compete with the smarter girls, and the more attractive of them may well be successful, because men are more attracted to physical traits and emotional kindness than raw earning potential.

I mean, all else being equal, I guess rich is more attractive than poor, but guys just don’t sit around and dream of their perfect girl’s career and what sort of cash she’ll be pulling down.

You aren't married, are you? Let's just say that if you're planning a serious long-term relationship with any chance of success, you're probably looking at a *lot* of characteristics, among which is earning power. Certainly it will have a lot more impact on your life's course (for good or ill) than her physical beauty.

Obviously her physical beauty is very important to you. Not just because you'll be sleeping with her and all, but she will also pass on her good looks to her kids.

One might want to steer clear of a bimbo, but it seems the best mate is a smart good looking woman who cultivated her smarts in a way that makes her a better companion and mother rather then whatever it is the market demands (often corporate drudgery).

I'm sorry, but I doubt physical beauty with the 10th-90th percentile is going to make a lot of difference to my children's lives. Looking at perhaps 100 people I knew in college, I see no correlation between physical beauty and happiness 30 years later. (Okay, I actually see an anti-correlation at the high end, but I'll call it zero.)

There's no doubt that companionship and parental potential are valuable, but there are precious few one-income families in the middle class, so earning potential is important (at least when I observe that same ~100 people 10-30 years after college).

"guys just don’t sit around and dream of their perfect girl’s career and what sort of cash she’ll be pulling down"

Speak for yourself, buddy.

Wait, never mind, carry on.

On the other hand, you might want smart kids, and you might want to be able to talk to the woman you're sharing your life with about something other than what's on TV.

Obviously, the implied model is this:

15. What is your IQ?

Under 85: -1 point
85 to 110: 0 points
110 to 130: +1 point
130 to 145: 0 points
over 145: -1 point

Any such model needs to account for the fact that hamsters and fruit flies mate assortatively all the time.

I'm not sure it makes sense to look at the average behavior of women and men here, the two are encoded with somewhat different dominant reproductive strategies. I suspect women seek higher IQ significantly more than men.

Perhaps the study breaks this down, I'll have to read the whole thing later.

Two possible explanations that haven't been mentioned by other commenters:

1. Highly intelligent people being oddballs. Women will put up with an oddball husband if he's smart and a good provider. Men have less incentive to put up with the weirdness. Thus a lot of highly intelligent, mismatched men.

2. Very intelligent women are browbeaten into becoming careerists. Careerist women often don't marry. Nearly all careerist men do. Same result.

Since there is already a disproportion of men to women on the right end of the bell curve, either of these would enhance the effect.

On the oddballishness of the high IQ

There's a bit of a mismatch between IQ and our cultural idea of intelligence.

IQ mostly represents the quality of basic cognitive functions, as captured through a set of tests where training functionally generally doesn't contribute to society wide variance (either on individual tests, or once the tests are averaged out).

Whereas what we think of intelligence as being seems to be usually having a lot of unconventional (but obviously sound) insights and a mastery of a body of arcane and unusual knowledge.

We obtain this intelligence through the kind of high quality of basic cognitive functions measured by IQ tests, but also by having wholly unconventional (or unusually broad) interests and fixations (e.g. being obsessed with maths, or just a plain infovore like for'ex, Tyler Cowen), which aren't very correlated with IQ.

These unusual patterns of interest probably are typical of "oddballs", while high quality of basic cognitive functions aren't, really.

(Unusual patterns of interest are not easy to describe as a single dimensional metric that makes a lot of sense - the Openness to Experience Big 5 Personality trait gets close, but is not really as crisp and useful as IQ, and has strong differences according to question battery, while IQ has strong agreement between batteries).

So we would expect "intelligent" (and "Open to Experience") people to tend to have some oddballishness about them, but the high IQ generally would not.

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