[jibiki]

> two uncorrelated variables can't both have a correlation coefficient of .8.

Can you prove this? I think that the greatest possible correlation is sqrt(1/2) (around .7), but I have no idea how to prove it.

(Formally, the conjecture is as follows: if X,Y, and Z are random variables, and k is a real constant, with:

  corr(X,Y) = 0
  corr(X,Z) > k
  corr(Y,Z) > k

then k < sqrt(1/2).

It is indeed possible for corr(X,Z)=corr(Y,Z)=sqrt(1/2), here's an example. With sample space {a,b,c,d}, let X,Y, and Z be:

     a   b   c   d
  X: 1,  0,  0, -1
  Y: 0,  1, -1,  0
  Z: 1,  1, -1, -1

But I ran a program looking for cases with k > .7 and couldn't find anything.)

[tokenadult]

Thanks for doing the math. I appreciate the pursuit of curiosity here on HN. There is a different issue here to be curious about, and that is exactly what a calculation of broad heritability definitely predicts. It predicts a lot less than what many readers unfamiliar with genetics might guess. It happens that some of the leading authors on human behavioral genetics just wrote an article about what heritability does and does not mean

Johnson, Wendy; Turkheimer, Eric; Gottesman, Irving I.; Bouchard Jr., Thomas (2009). Beyond Heritability: Twin Studies in Behavioral Research. Current Directions in Psychological Science, 18, 4, 217-220.

http://www.ingentaconnect.com/content/bpl/cdir/2009/00000018...

(one online abstract)

http://www3.interscience.wiley.com/journal/122587149/abstrac...

(the main link to the article)

Alas, a peek behind the pay wall that was available the other day when I posted this article here on HN

http://news.ycombinator.com/item?id=838534

is now dead. But I have the full text of the article at hand, as I am currently attending a weekly journal club with some of the authors, and one key paragraph from the article must be read by anyone who draws conclusions from heritablity figures:

"Moreover, even highly heritable traits can be strongly manipulated by the environment, so heritability has little if anything to do with controllability. For example, height is on the order of 90% heritable, yet North and South Koreans, who come from the same genetic background, presently differ in average height by a full 6 inches (Pak, 2004; Schwekendiek, 2008)."

This simply reemphasizes a point that is familiar to anyone who has studied genetics carefully, namely that the pre-Mendelian concept of heritability says nothing about malleability, the degree to which a trait can be influenced by environmental variables.

Angoff, W. H. (1988). The nature-nurture debate, aptitudes, and group differences. American Psychologist, 43, 713-720.

Mange, A. & Mange, E. J. (1990). Genetics: Human Aspects.

Kaufman, Alan S. (1990). Assessing Adolescent and Adult Intelligence.

So the statement above that heritability somehow constrains the expression of IQ or of consequences of IQ such as occupational success is actually conceptually incorrect. But I appreciate you going to the effort of doing the math.

Another thought is that relating one correlation coefficient to another with linear algebra probably depends too heavily on applying linear tools to a not fully linear model. Increases of income are plainly linear and are on a ratio scale. (There is a zero point for income, and each dollar increase in income has the same magnitude anywhere along the scale.) But IQ test standard scores are at best ordinal scales, so it is already an abuse of mathematics to treat them as a linear variable, or to treat a figure derived from them as a linear variable.