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Normalize IQ to have variance 1. Consider a gene G with frequency p and effect E, in units of standard deviations. The frequency is easy to measure and eventually will be 20%. What we really want to know is the effect size. VG=p(1-p)E^2, so E=VG/p(1-p). Consider two hypotheses, that it explains 1% of the variance or that it explains 0.1% of the variance. In the first case, E^2=1%/(1/5 * 4/5)=1/16; E=0.25. In the second case, E=0.08, smaller by sqrt(10).

We take a sample of N people from the population, pN with the gene, (1-p)N without. We measure their IQs and average within each population. The standard error for these means is the reciprocal of the square root of the population, 1/sqrt(pN) or 1/sqrt((1-p)N). I’m just going to use the larger number, 1/sqrt(pN) and pretend that our measurement of the mean of the larger population is perfect. The smaller p, the more justified this approximation. It probably isn’t really justified at p=20%, so my confidence interval will be overconfident. In the worst case at p=50%, it will probably be half as big as it should, but we have plenty of room.

If the true effect is 1% of variance, what is the probability we will measure the effect as being only 0.1% of variance? It is the probability that our measurements of the mean are off by 0.25-.08=.17. That is .17*sqrt(pN) standard errors. The probability of such an error is e^-()^2. With N=100k, p=20%, e^-580, a pretty small p-value. For multiple comparison reasons, we multiply it by a million, but it is still zero.

What is the confidence interval? The 95% confidence interval is 2 standard errors. That is, for a p-value of 1-95%=0.05. But we want the divide the p-value by a million for multiple comparisons. A p-value of 1 in 20 million is about 5.3 standard errors. So if our corrected confidence interval is 0.08 +- .04 standard deviations, 0.6 to 1.8 IQ points. We’re just barely able to detect genes with a variance of 0.1%, but we can be very sure that we detect all genes with a variance of 1%. Our estimate of its strength might be too low, but it will be on the list of genes statistically significantly different from 0.

The important number is Esqrt(pN). Since VG=pE^2, the important number can be written as sqrt(VG*N). Thus the difficulty detecting genes depends on their variance, not really on frequency or effect size.

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I’m not actually aware of any test that quantifies certainty for the *absence* of differences

I didn’t say the absence of difference. The claim that the difference is not 0 is not falsifiable / subject to NHST. The claim that the difference is less than a fixed cutoff is testable.

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(guess) Is an intellectual sadist a person who likes proving people wrong and feeling intellectually superior, and an intellectual masochist someone who likes exploring all possible ways they might be wrong and misguided? Like teacher/student?

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no idea if this is actually true or not.

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Well, practically speaking, when the sample size is sufficiently high then absence of significance results indicates that no such results exist. But no, I’m not actually aware of any statistical method for quantifying certainty for something like that. (But like I said, if you singled out this one gene and went back into those GWAS’s and removed all corrections for multiple comparisons it should theoretically hit significance, if that’s what you meant)

>these studies supposedly had p<0.05 for genes accounting for 0.1% of the variance.

If I understand correctly, when p<.05, then their is a 5% chance that you would have gotten differences of greater magnitude by chance testing two equal random samples…that doesn't mean that you're allowed to infer the absence of differences based on the failure to hit the significance threshold.

I’m not actually aware of any test that quantifies certainty for the *absence* of differences, though I’m sure one exists and I would like to be pointed to it if so.

>Malacards doesn’t seem like a very reliable source. It’s just the medical literature, right?

You may be right about that.

>Of course you can’t reject the hypothesis that there exists an informative gene, somewhere out there, if you aren’t looking at the right region. But this SNP is on 23andMe v1, so it has probably been considered by every GWAS, ever.

Yes. My concern is that even when an informative SNP is in the GWAS, even if its on every GWAS which relates to that particular subject, you still need an insanely large sample size to successfully detect it because the sheer presence of so many other SNPs creating false positives requires stringent correction, and it would be easy to miss something.

I might be wrong, but *why*?

If we could just do a GWAS and immediately find these sorts of things, it seems that stuff would be progressing a lot faster than it currently is.

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As for the longevity thing: does that still hold once you correct for all the obvious confounders like income, tobacco use, social capital, and Adventism?

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