Body: <p>I'm an interested layman reading the paper <a href="https://arxiv.org/pdf/1408.3421.pdf" rel="nofollow noreferrer">https://arxiv.org/pdf/1408.3421.pdf</a> (Hsu, Stephen, On the genetic architecture of intelligence and other quantitative traits, 2014). I'm trying to understand Hsu's claim (my possibly incomplete/faulty interpretation; for his words see pages 13 and 16):</p>
  <ol>
  <li>Alleles affecting a quantitative phenotype trait that has been strongly selected for (such as height or intelligence for humans) will have their frequencies pushed close to 0 or 1 by evolution.</li>
  <li>If there are two possible alleles at each of <span class="math-container">$N$</span> sites contributing to the quantitative trait, say with each minor allele having frequency <span class="math-container">$p$</span>, then linear contributions of minor alleles would number <span class="math-container">$O(p)$</span> whereas non-linear effects would number <span class="math-container">$O(p^2)$</span>. By the previous point, <span class="math-container">$p$</span> is small, so the linear terms dominate the total contribution.</li>
  </ol>
  <p>Two things confuse me with this:</p>
  <ol>
  <li>Might not the larger number of non-linear interactions (at least <span class="math-container">$\binom N2)$</span> vs <span class="math-container">$N$</span>) compensate for the extra factor <span class="math-container">$p$</span>?</li>
  <li>Shouldn't we expect alleles only involved in non-linear contributions to experience less selective pressure, hence making the <span class="math-container">$p$</span> for such minor alleles larger than for those contributing linearly, again compensating for the differing orders in <span class="math-container">$p$</span>?</li>
  </ol>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2021-09-12T20:42:56.640Z
Comments.Text: Hoo boy. This is complicated. First, Hsu is a somewhat controversial character, many (most?) geneticists think of him as a scientific racist; his own students are trying to get him fired: https://statenews.com/article/2020/06/michigan-state-geu-calls-to-remove-vp-research-administrator-stephen-hsu?ct=content_open&cv=cbox_latest. That said, additive vs. epistatic effects is an active and complex debate. Your points 1 and 2 are reasonable and argued about frequently. Review articles arguing each way: https://pubmed.ncbi.nlm.nih.gov/24161664/, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4639398/
Comments.UserDisplayName: Maximilian Press
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2021-09-13T21:06:38.623Z
Comments.Text: Thanks, those links are helpful!  I guess the argument that intelligence/height heredity being twice as high for monozygotic twins as for dizygtotic twins implies linearity seems convincing enough to me, although I don't get the more general theoretical argument put forward to explain why this should be the case.
Comments.UserDisplayName: keraC
Tags: <gwas>
Title: Reason for linear effects dominating the difference in a quantitative trait between individual genomes
ThreadText: # Reason for linear effects dominating the difference in a quantitative trait between individual genomes
  
  (Asked by: username_0 on 2021-09-11)
  
  I'm an interested layman reading the paper <https://arxiv.org/pdf/1408.3421.pdf> (Hsu, Stephen, On the genetic architecture of intelligence and other quantitative traits, 2014). I'm trying to understand Hsu's claim (my possibly incomplete/faulty interpretation; for his words see pages 13 and 16):
  
  1. Alleles affecting a quantitative phenotype trait that has been strongly selected for (such as height or intelligence for humans) will have their frequencies pushed close to 0 or 1 by evolution.
  2. If there are two possible alleles at each of $N$ sites contributing to the quantitative trait, say with each minor allele having frequency $p$, then linear contributions of minor alleles would number $O(p)$ whereas non-linear effects would number $O(p^2)$. By the previous point, $p$ is small, so the linear terms dominate the total contribution.
  
  Two things confuse me with this:
  
  1. Might not the larger number of non-linear interactions (at least $\binom N2)$ vs $N$) compensate for the extra factor $p$?
  2. Shouldn't we expect alleles only involved in non-linear contributions to experience less selective pressure, hence making the $p$ for such minor alleles larger than for those contributing linearly, again compensating for the differing orders in $p$?
  
  **Tags:** gwas
  
  > **Comments:**
  > **username_1**: Hoo boy. This is complicated. First, Hsu is a somewhat controversial character, many (most?) geneticists think of him as a scientific racist; his own students are trying to get him fired: https://statenews.com/article/2020/06/michigan-state-geu-calls-to-remove-vp-research-administrator-stephen-hsu?ct=content_open&cv=cbox_latest. That said, additive vs. epistatic effects is an active and complex debate. Your points 1 and 2 are reasonable and argued about frequently. Review articles arguing each way: https://pubmed.ncbi.nlm.nih.gov/24161664/, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4639398/  
  > **username_0**: Thanks, those links are helpful!  I guess the argument that intelligence/height heredity being twice as high for monozygotic twins as for dizygtotic twins implies linearity seems convincing enough to me, although I don't get the more general theoretical argument put forward to explain why this should be the case.  
  
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  ## 0 answers
  
  