Going beyond the First Moment in IQ Distributions
to jwai, Murray-Charles
Hello, Jonathan (if I may).
I saw your recent article via Twitter and posted a few comments on it. I commend you for looking at nonlinearities. I am an economist, so I don't know the literature, but I've long wondered about IQ effects that go beyond simple means. Finally, in the past 20 years, variance of the IQ distribution is getting talked about for male v. female at least, but there's so much to look at besides first moments of a distribution.
With IQ, in particular, since it is an artificial measure we should not expect to find linear effects. Going from 100 to 130 is not twice the addition to intelligence of going from 100 to 115--- it is just scaled to make it twice the number of standard deviations. So there is no good theory for why the effect of IQ on, say, income, should be double, even if the effect of intelligence on income is linear.
But it is still very much worthwhile to look at whether the effect of IQ on income is linear, since IQ is what we have to measure nad predict with.
I am critical of your methodology. Is your data easy to package up and send? I wonder about just doing a polynomial regression with an F-test for the x^2, x^3, etc. terms, or even just doing y = alpha + beta*x + gamma*x^2.
The problem with throwing out effects that have low "economic significance" as we call it in economics but high "statistical significance" is that in some contexts those seemingly tiny effects are still interesting and important. For example, suppose in your data there are 950 people with IQ of 100 and 50 with IQ of 120, and those two groups have incomes of 50K-80K and 90-100K. You would find IQ had very low R2, but very high statistical signifiance. It couldn't explain much variance or predict much, because almost all the variance consists of people with IQs of 100 earning different amounts of money, which you can't explain at all. But you can see that with different data, IQ would have extremely high R2.
Also, we wouldn't necessarily expect IQ to have a symmetric effect on outcomes. I just skimmed your article, but did you try separating out the bottom and the top ends? It might be that going from 100 to 145 has a linear effect, but going from 100 to 70 has a different linear effect. This would be particularly true of outcomes like ending up in prison, where very few high-IQ people do, so it is hard ot pick up effects accurately, but lots of low IQ people do.
Is that dubious-sound Simonschohn (2014) test for U-shapes any good? It's from a psychology journal, and not explained well, but deemed superior to just adding some quadratic terms and doing an F-test.
I was just suggesting on an econometrics thread that a good way to teach why R2 isn't everything is IQ regressions, where for, say, income,you might have very low R2 and be unable to predict individuals, but high t-test and be able to predict group means.
They do something odd. Rather than test of statistical signifiance of quadratic terms, they look to whether they add 1+% to R2. Thus, the quadratic term could be highly statistically significant, but they'd say it was unimportant.
The R2 thread was on Woodridge, who has an active metrics Twittering and wrote the standard text. He noted that its hard to keep undergrads from fixating on R2. I commented that Murray-Herrnstein obsesses about trying to explain that to ordinary people, e.g., re IQ-wages.