text: user15062: I am trying to understand the derivations from Sullivan et al. (2015) in the Supplementary Material. There, it is mentioned in the first page that the least squares estimate of the j-th SNP effect, considering the polygenicity linear model $φ=Xβ + ε$, is $\hat{β_j} := X^T_j φ/N$ . Normally, the least squares solution of $\hat{\beta}$ is $(X^TX)^{-1}X^T\phi$. By comparing the two equations with each other, this means that, in this case, $X^TX=N I_{M\times M}$. However, LD is assumed, therefore $E(X_{ij}X_{ik})\neq 0$, so the covariance matrix cannot be diagonal. Could someone explain to me how the derivation for the beta estimate was performed, please? What am I missing? I checked that the same thing is mentioned in more than one sources such as here.
  
  user15286: Not entirely clear on what you're asking but the beta estimates in LDSC come from GWAS summary statistics. As I understand it, they aren't "derived" so much as just set equivalent to what they linear model defines them as, which is their additive effects on the phenotype as implied by the model. So for each SNP j, it's effect size (beta) is whatever difference one copy of the allele makes on the phenotype (phi), divided by N, the number of SNPs.
  
original_text: 15062: <p>I am trying to understand the derivations from <a href="https://www.nature.com/articles/ng.3211" rel="nofollow noreferrer">Sullivan et al. (2015)</a> in the <a href="https://static-content.springer.com/esm/art%3A10.1038%2Fng.3211/MediaObjects/41588_2015_BFng3211_MOESM39_ESM.pdf" rel="nofollow noreferrer">Supplementary Material</a>. There, it is mentioned in the first page that the least squares estimate of the j-th SNP effect, considering the polygenicity linear model <span class="math-container">$φ=Xβ + ε$</span>, is <span class="math-container">$\hat{β_j} := X^T_j φ/N$</span> . Normally, the least squares solution of <span class="math-container">$\hat{\beta}$</span> is <span class="math-container">$(X^TX)^{-1}X^T\phi$</span>. By comparing the two equations with each other, this means that, in this case, <span class="math-container">$X^TX=N I_{M\times M}$</span>. However, LD is assumed, therefore <span class="math-container">$E(X_{ij}X_{ik})\neq 0$</span>, so the covariance matrix cannot be diagonal. Could someone explain to me how the derivation for the beta estimate was performed, please? What am I missing? I checked that the same thing is mentioned in more than one sources such as <a href="https://static-content.springer.com/esm/art%3A10.1038%2Fng.3404/MediaObjects/41588_2015_BFng3404_MOESM34_ESM.pdf" rel="nofollow noreferrer">here</a>.</p>
   15286: <p>Not entirely clear on what you're asking but the beta estimates in LDSC come from GWAS summary statistics. As I understand it, they aren't &quot;derived&quot; so much as just set equivalent to what they linear model defines them as, which is their additive effects on the phenotype as implied by the model. So for each SNP j, it's effect size (beta) is whatever difference one copy of the allele makes on the phenotype (phi), divided by N, the number of SNPs.</p>
  