Several interesting comments on this post. The puzzle is this: let X be the correlation between parental IQ and children's IQ. Let Y be the correlation between the child's IQ and the child's future earnings. Let Z be the correlation between parental earnings and the child's future earnings. Why is Z much higher than X times Y?

Steve Sailer writes,

Correlation isn't the same as effect size. In a complex system with many factors, a single factor can have a low correlation but still have a sizable effect

I think of effect size as the numerator in correlation. If you have a big effect size and a low correlation, then you must have a big denominator--in this case, a high variance of income. But that would tend to lower Z along with Y, so I think that the puzzle still stands.

John Thacker writes,

Correlation coefficients only measure the linear relationship, not any nonlinear relationship. It's quite possible that there is a nonlinear relationship. Ordinarily linear relationships dominate with two normally distributed r.v., so it's okay. However, if there's covariance between IQ and environment then the mathematical assumptions and simplifications fail, the chain of reasoning falls apart, and the conclusion does not hold.

But again it seems to me that nonlinearity ought to hurt Z as well as hurt Y. I have a hard time coming up with a story in which the relationship between IQ and income is nonlinear but the relationship between parental income and child's income is linear.

Bruce Charlton recommends a paper by Daniel Nettle. That paper says,

The data show clearly that general ability, an IQ-like score, is indeed a predictor of class mobility. Those reaching the professional class had a GA score about one and a half standard deviations higher than those reaching the unskilled class, regardless of their class of origin.
...The relationship of parental social class to attained social class was weak (6% of the variation). Most of this was mediated through General Ability. Partialling out GA differences left only 3% of the variation to be explained by direct effects of paternal class on occupational opportunities...

The vast majority of variation in both attained class and class mobility is not accounted for by IQ. ...Nonetheless, intelligence is the strongest single factor causing class mobility in contemporary societies that has been identified.

This is an interesting study, but it loses considerable precision by aggregating people into broad income classes. As I interpret the study, doing so brings Z down from something like .5 to something like .06

Personally, I think that the Bowles-Gintis-DeLong swindle comes from lowballing both X and Y. If you say that X is .2 and Y is .2, then X times Y is .04, which is close to zero as a correlation. If you say that X is .6 and Y is .6, then X times Y is .36, which is fairly big in terms of correlation.

I think that the correlation of parents' IQ with childrens' IQ is much higher than what you would get with random mating. I also think that this correlation has been getting higher in recent years, as more men choose women for their intelligence rather than for their obedience or domesticity. So I would expect X to be higher than the .2 that Bowles and Gintis appear to work with.

I also think that for measuring the impact of IQ on income, you have to be careful not to control for things that IQ can affect. Bowles and Gintis allow IQ to affect schooling. But it also may affect health, ability to defer gratification, marital stability, and other factors that affect income. By the time you are finished, you may wind up with a value for Y of .5 or higher.

Anyway, the overall point is that there is a large margin of error in estimating X times Y. If you were to report a 90 percent confidence interval for the product, rather than a specific number, my guess is that it would include values close to 0, but also values close to Z.

The puzzle is this: let X be the correlation between parental IQ and children's IQ. Let Y be the correlation between the child's IQ and the child's future earnings. Let Z be the correlation between parental earnings and the child's future earnings. Why is Z much higher than X times Y?Because you can't multiply correlations. They aren't transitive (unless, say, the r.v. are independent, in which case they're all zero). Let X be a normal random variable. Let Y = -X. Let Z be a normal random variable independent of X and Y.

The correlation of X and Z is 0. The correlation of Y and Z is 0. The correlation of X and Y is -1. 0 * 0 != -1 Note that this works no matter what the correlation of Z is with X and with Y. (It also works if X = Y)

So I would expect X to be higher than the .2 that Bowles and Gintis appear to work with.Their calculation has a few more steps.

.2 is actually the value that they use for the correlation between the two parents' IQs. They then determine that if the measured parent's IQ is x standard deviations from the mean, the average IQ of both parents is x*(1+.2)/2 = .6x standard deviations from the mean.

.5 is the value that they use for the correlation of parental IQ (both parents) to IQ. So they then assume that, given one parent with an IQ x standard deviations from the mean, the average IQ of the child is x*(.6)(.5) = .3x standard deviations from the mean. In other words, the correlation between one parent's IQ and the child's IQ is .3, what you call X.

They're assuming that Y is the correlation between IQ and earnings for both parent and child. Thus, they're wondering why Z is much higher than X times Y^2. (Correlate parental income to parental IQ, parental IQ to child IQ, child IQ to child income)

However, it doesn't work. Correlations don't work that way. The lay interpretation of correlation goes astray.

Consider that if this transitivity held, then you would have x*y = z, but also x*z = y and y*z = x by the same logic. That only works if all are 1 or all are zero.

It does work in certain cases, where you could claim that everything is coming from a joint normal distribution (a linear combination of normal distributions, or a "multivariate normal distribution"), and thus

Y = r_1X + A, where A is a normal r.v. uncorrelated with X.

Z = r_2Y + B, where B is a normal r.v. uncorrelated with Y.

It then follows that

Z = r_1*r_2X + r_2A + B

However, it's certainly not the case that B is necessarily uncorrelated with X, so we wouldn't know for sure that the correlation of Z with X would be r_1*r_2. It could easily be larger.

Even this case breaks down if we don't have a joint normal distribution. If there are any nonlinear effects, then we don't have one. They're assuming that the nonlinear effects are small, and that we have a joint normal, and that the effects independent of Y are also independent of X. You can design lab experiments to make that pretty much true, but I would be leery of doing so with survey data.

Hopefully these three comments explain my original point better. A PhD in probability should be good for something.

John, I'm not certain I followed your post, but Arnold is not questioning why correlation is not transitive. He is questioning why child's income is better predicted by parent income than by a (plausible if simple) story where heritable IQ drives income.

As to the statistics, I can't remember whether you can just multiply correlation coefficients or not. John, (I think) answering a different question, is suggesting it is not kosher.

I think what you really want to measure here is "which factor, if I knew it, would more precisely tell me the child's earnings: parent earnings? or parent IQ?"

Mathematically, we want to compare two conditional entropies:

H(ChildEarnings|ParentEarnings)

vs.

H(ChildEarnings|ParentIQ)

If the first one is less, then knowing the parent's earnings is more helpful, so society is "unfair."

However, I disagree with this line of reasoning because earnings may be well correlated with certain aptitudes (like "emotional intelligence") that are not captured by IQ.

To put it differently, IQ is not an ideal measure of aptitude. Earnings may even be a

bettermeasure of someone's aptitude than IQ, because we expect that high earners are not only smart but also socially well-adjusted, hard workers, etc.So, bottom line, we don't have a good enough measure of "aptitude independent of achievement". IQ doesn't fit the bill.

One problem is why stop with IQ? DeLong is interested in determining whether there are social effects that parents confer on their children that matter a lot for income. So why not separate it into heritable factors (like IQ to some extent) and non-heritable factors like social status that DeLong is concerned with? [This may be what Arnold means when he talks about other factors correlated with IQ.]

If you do that, then assuming that parents self-select for mates with similar genes (a not too overreaching assumption), and ignoring recessive genes (which of course you can't do), children's heritable factors are 100% correlated with their parents'. Accounting for recessive genes will lower that, but not to .2.

And I suspect that doing the same for the effect on income - widening it to include the effect of all heritable factors - would increase that coefficient a great deal as well.

Bowles and Gintis, along with co-author Melissa Osbourne, have a paper ("Determinants of Earnings: A Behavioral Approach") that delves into the issue.. my recollection is that they essentially find that personality and other non-cognitive aspects of human capital account for a lot more of the variation in earnings that does intelligence.

To maximize earnings, opt for patience, industry, and self-efficacy over braininess. Of course, since basic personality traits are probably significantly heritable...

Ironically, to me, this seems to weaken the social-democratic-Rawlsian argument, since it's always seemed more plausible to externalize one's raw intellectual horsepower as an accident of birth rather than one's personality.

There's a hierarchy of how easy it is to imagine oneself in the the original position:

1) Different wealth, health, strength, attractiveness is easy (or so we think)

2) Different raw generalized intelligence is harder

3) Different mix of types of intelligence is harder still (OK, Einstein, imagine not that you're just a bit dimmer, but imagine being really good at empathy and poetry, but totally unable to grasp anything analytically.)

4) Different value-system is very hard

5) Different personality traits is virtually impossible.

Its been a while since I have been here on this blog but I see the main subject still hasn't changed. First of all, in which imaginary world is social status 'non-heritable'?

Second of all, how can people claim to be economists, be involved with expensive universities, and still be totally non-plussed when it comes to 'earning power'? I have a very simple explanation. Many wealthy people spend most of their time and energy trying to justify their wealth to themselves and others. Whether its New Age crackpots convincing themselves that they are wealthy because they deserve it for being such 'good souls' or right-wingers trying to associate 'intelligence' with 'earning-power', its all the same, people refusing to face reality.

First of all, why do these idiot studies always try to correlate income with intelligence? Its because that information is easy to come by. Statistics on wealth don't exist in any accurate format. So they try to approximate income with wealth. But is that valid? It actually tries to correlate year to year reported income with wealth. Is that in any way useful information? Maybe over a person's 50 year adult lifetime, and not accurately, certainly not year to year.

So why mess around with statistical models based on bad data and false assumptions? There is absolutely nothing useful that is going to come out of it. Except maybe convincing oneself that he is wealthy because he is 'intelligent' and not because he was born into his social class.

Morgan, who asks for a wider search for causative factors to wealth, might enjoy Thomas J. Stanley's "The Millionnaire Mind." In an earlier book with Professor Danko, Stanley described "the millionnaire next door," and argued that it is seldom inheritance, or advanced degrees or even intelligence that builds fortunes in this country. Most of America's millionnaires are self-made, reveived no financial help to get started, and live modestly in middle class neighborhoods, drive used cars, and live below their means. I would emphasize the role of personal motivation--that comes from having little to begin with, and little help from others--that builds the self reliance and determination for an individual to lift himself by his own bootstraps. Benjamin Franklin, the son of a poor immigrant blacksmith, and who was apprenticed out at 14 years of age, served as a role model for America's rags to riches story. On my website I stress how it is common sense that builds success--for individuals as well as societies-- "The Radzewicz Rule" suggests that intelligence, at least in the soft sciences, does more harm than good. And EQ probably trumps IQ. Bill Greene

Correlation has extreme problems with nonlinearity. For example, suppose that X and Y are joint normal random variable. Now let W = exp(X) and Z = exp(Y). Then (W,Z) is a pair of bivariate lognormal random variables. In this case, the correlation between W and Z is strongly constrained. For example, if the standard deviation of X is 1 and the standard deviation of Y is 4, then the correlation between W and Z must lie in the interval [-0.000251, 0.01372]! In fact, a correlation of W and Z of 0.01372 indicates that X and Y are perfectly correlated, and completely functionally dependent on each other.

Correlation ONLY measures the linear relationship. Any time there is a nonlinear relationship between random variables, the correlation is much smaller, and extremely unpredictable effects can happen. You just can't multiple correlations.

He is questioning why child's income is better predicted by parent income than by a (plausible if simple) story where heritable IQ drives income.The problem is that the paper didn't, for example, compute the correlation between heritable IQ and income directly. They multiplied a bunch of correlations-- between parental IQ and parental income, between parental IQ and offspring IQ, and between offspring IQ and offspring income, and found that the multiple of the three correlations was much smaller than the direct correlation of parental income to child income. They then (incorrectly) claimed that this demonstrated that the portion of the correlation between incomes that was attributable to IQ was low.

This is bad statistics. It simply doesn't demonstrate their claim.

Arnold asked:

Why is Z much higher than X times Y?Because X times Y is not an accurate measure of Z except under some very strict assumptions, such as all variables being joint normal. Any nonlinearity, for example, causes extreme problems with that.