Understanding heritability

CIP wrote:

Interesting (or not) as that story may be, it starts becoming irrelevant if genes linked to IQ are found. If there aren't any, then the statistical studies showing such heritability must be flawed.

Let's go over heritability again, in case Cosma Shalizi's essay was not understood.

1. The observable characteristics of an organism constitute its phenotype. 
2. In a population of organisms that lives in some range of environmental conditions, we observe variations in their phenotypes.
3. In a simple model, we attribute the variations in a population's phenotypes to genetic variation and environmental variation.  Heritability is the fraction of the variance of a phenotype attribute in the population in that set of environments that can be attributed to genetic variation.
   
   (Variance is the average squared difference from the mean.  E.g., if my population pf n organisms had heights h1, h2, h3, ..., hn,  then the mean hmean =  (h1+h2+...+hn)/n.
   The variance is then  hvariance =  ((h1-hmean)^2 + (h2-hmean^2) + ...+ (hn-hmean)^2 )/n.
   
   If height was determined by genes and environment in a simple additive way,
   h = g + e   (i.e., h1 = g1 + e1; h2 = g2 + e2, etc., g being the component of height determined by genes, and e is the component of height determined by the environment.  Then there is a gmean, emean, gvariance and evariance, and if genes and environment vary independently, then a little arithmetic and the definitions above will show that hvariance = gvariance + evariance.
   
   Heritability is  just gvariance/hvariance.
4. Heritability does not tell you how much an attribute under genetic control.
5. To borrow an example from Cosma Shalizi: Having two eyes is entirely genetically determined.  The genetic variation for having one or three eyes is virtually zero, any such mutations are rapidly weeded out by natural selection.  The environment (disease, accident, etc.) causes phenotypes to lose one or two eyes, and is thus the only source of variation.  Thus the heritability of having two eyes is zero, though two eyes is a genetically determined feature.   This is because all the variation in the number of eyes arises from the environment.
6. Suppose we don't know what specific genes are responsible for number of eyes, but discover them. We haven't learned anything new about heritability.
7. Likewise with genes for intelligence.  Of course,  if we discover the genes that govern IQ, it may be possible to do a from-first-principles study of genetic variation versus IQ variation.  It will still be only the simple model which neglects things like gene-environment interactions.
8. While Cosma Shalizi mentions it, on the subject of gene-environment interactions,  let's go to someone CIP trusts more,  Steve Hsu (Hsu is on CIP's blog roll.) Childhood Socio-Economic Status amplifies genetic effects on adult IQ, we are told.   Hsu quotes Alison Gopnik from the Wall Street Journal:

... When psychologists first started studying twins, they found identical twins much more likely to have similar IQs than fraternal ones. They concluded that IQ was highly "heritable"—that is, due to genetic differences. But those were all high SES twins. Erik Turkheimer of the University of Virginia and his colleagues discovered that the picture was very different for poor, low-SES twins. For these children, there was very little difference between identical and fraternal twins: IQ was hardly heritable at all. Differences in the environment, like whether you lucked out with a good teacher, seemed to be much more important. In the new study, the Bates team found this was even true when those children grew up. IQ was much less heritable for people who had grown up poor. This might seem paradoxical: After all, your DNA stays the same no matter how you are raised....

9. Note for the above: identical twins have identical genes; fraternal twins share only half their genes.  Hence a heritability estimate is possible.  But such twin studies took place only with very limited environmental variation.  (Roughly speaking, all were in the same privileged middle/upper class.)  What the quote from Gopnik is saying is that the heritability of intelligence is greatly reduced when the population being measured is subject to the full range of environments that humans are subjected to - most of the variation then arises from environmental variation, not from genetic variation.
   
   Or in other words, the twin studies by their circumstances, have limited variation in environment.  Therefore they show a high heritability.
10. Another way of looking at gene-environment effects.  Suppose you have a population with variation.  The variation is due to genetic differences and environmental differences, (but not necessarily expressible as a simple additive model.).  Suppose you could take that population and put it in a common, uniform environment. The variation they show could then be all attributed to genes.  Heritability would be the variance in this common, uniform environment divided by the variance of the population in the original situation.
    
    But what should this common, uniform environment be?  I could pick one of the many original environments.  If there are gene-environment interactions, then the variation (due to genes)would depend on which environment I picked to be the common, uniform environment.   Which is essentially the thrust of the quote on twin studies.
    
    
11. I'd also record my inexpert opinion here, that if IQ is strongly selected for in evolution, then IQ variation due to genetic variation should be expected to be low, i.e., heritability should be low.  We see this in the example of the two-eyed phenotype - the heritability is zero.
    A high heritability for IQ means, in my opinion, that IQ was only weakly selected for in evolution.
12. I don't think the IQ hereditarians (the authors of the Bell Curve and anyone who thinks that that book is great) have decided which way they want the world to work.  Ideally, they want IQ to have high heritability, so that they can argue (within their simple additive model) that social programs won't work - we can't induce much improvement in IQ by improving the environment in which people live.   But they want IQ to have been strongly selected for during human evolution, because they believe IQ is strongly determinative of life outcomes, including presumably reproductive success (this last at least until the Industrial Revolution.  The changes in fertility post-industrial revolution dismay the IQ hereditarians - the "smart" people have fewer children.) .  Strong selection for IQ fits their Social Darwinist inclinations.  But for that,  IQ must have low heritability.

PS: for another example of gene-environment interactions, I quote Shalizi:

To see why gene-environment interactions matter, consider one of the best-established links between genetic variations and intelligence, phenylketonuria. This is a recessive genetic disease which interferes with the normal metabolism of the amino acid phenylalanine. If someone with one of the defective forms of the gene for phenylalanine hydroxylase consumes too much dietary phenylalanine, it leads, among other problems, to serious mental retardation. Under suitable diets low in phenylalanine, however, they grow up mentally normal. Assigning shares of this effect to the genes and to the environment is exactly as sensible as trying to say how much of the fact that a car can go is due to its having an engine and how much is due to their being fuel in the tank. The best the usual biometric model could do here would be to predict that having the gene always reduced intelligence, as did consuming phenylalanine (which would be bad news for makers of artificial sweeteners); the fact that it's the combination, and only the combination, which is a problem would be missed, and the predicted size of the effect would be badly wrong. (The situation is similar for, say, hypothyroidism and a lack, rather than an excess, of iodine, but the genetics there are messier.) So while everyone piously says that genes and environments interact in development, they typically use models which assume that they do so only in trivial ways, and hope that any actual interactions are small enough to be treated as noise.

PPS: on why heritability says little about the malleability of a trait, Cosma Shalizi again (emphasis added)

Does a trait's heritability tells us anything about its malleability, about how easy it is to change the trait with environmental manipulations? The answer is "no, of course not", even assuming (1) the basic biometric model holds, and (2) we are talking about true heritability and not biased-to-nonsensical estimated heritabilities.

It's banging on an often-sounded drum, but it's worth doing because it makes the point clearly: height is heritable, and estimates for the population of developed countries put the heritability around 0.8. Moreover, tall people tend to be at something of a reproductive advantage. Applying the standard formulas for response to selection, we straightforwardly predict that average height should increase. If we select a population without a lot of immigration or emigration to mess this up, say 20th century Norway, we find that that's true: the average height of Norwegian men increased by about 10 centimeters over the century. But that's much more than selection can account for. Doing things by discrete generations, rather than in continuous time, height grew by 2.5 centimeters per generation. (The conclusion is not substantially altered by going to continuous time.) If the heritability of height is 0.8, for this change to be due entirely to selection, the average Norwegian parent must have been 3 centimeters taller than the average Norwegian. This, needless to say, was not how it happened; the change was almost entirely environmental. The moral is that highly heritable traits with an indubitable genetic basis can be highly responsive to changes in environment (such as nutrition, disease, environmental influences on hormone levels, etc.).

Conversely, the very low heritability of eye number does not tell us that it is easy to increase how many eyes someone has by exercise, education and training, manipulating diet, manipulating ambient light, trepanation, etc.