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Douglas_Knight comments on Open thread, December 7-13, 2015 - Less Wrong Discussion
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There are decades of studies of the heritability of IQ. Some of them measure H², which is full heritability and some of them measure h², "narrow sense heritability"; and some measure both. Narrow sense heritability is the linear part, a lower bound for the full broad sense heritability. A typical estimate of the nonlinear contribution is H²-h²=10%. In neither case do they make any assumptions about the genetic structure. Often they make assumptions about the relation between genes and environment, but they never assume linear genetics. Measuring h² is not assuming linearity, but measuring linearity.
This paper finds a lower bound for h² of 0.4 and 0.5 for crystallized and fluid intelligence, respectively, in childhood. I say lower bound because it only uses SNP data, not full genomes. It mentions earlier work giving a narrow sense heritability of 0.6 at that age. That earlier work probably has more problems disentangling genes from environment, but is unbiased given its assumptions.
The linked paper says:
If you have 3511 individuals and 549692 SNPs you won't find any nonlinear effects. 3511 observations of 549692 SNPs is already overfitted 3511 observations of 549692 * 549691 gene interactions is even more overfitted and I wouldn't expect that the four four principal components they calculate to find an existing needle in that haystack.
Apart from that it's worth noting that IQ is g fitted to a bell curve. You wouldn't expect a variable that you fit to a bell curve to behave fully linearly.
No, they didn't try to measure non-linear effects. Nor did they try to measure environment. That is all irrelevant to measuring linear effects, which was the main thing I wanted to convey. If you want to understand this, the key phrase is "narrow sense heritability." Try a textbook. Hell, try wikipedia.
That it did well on held-back data should convince you that you don't understand overfitting.
Actually, I would expect a bell curve transformation to be the most linear.
They didn't do well on the gene level:
Analyses of individual SNPs and genes did not result in any replicable genome-wide significant associationNo, the fact that you can calculate a linear model that predicts h_2 in a way that fits 0.4 or 0.5 of the variance doesn't mean that the underlying reality is structured in a way that gene's have linear effects.
To make a causal statement that genes work in a linear way the summarize statistic of is not enough.