3.1 Population genetics for GRM

3.1.1 Unrelated GRM - \(m_e\)

\[ m_e=1/\text{var}(\text{GRM}_{\text{offdiag}}) \]

3.1.2 Sib GRM - genome length (Morgan)

The genome length measured in Morgan can be put in the following:

\[ m_{e.\text{IBS}}=\text{var}(\text{GRM}_{\text{sib}})=\frac{1}{16} \]

We can depict the distribution of IBS across all families and calculate the \(\frac{1}{\text{var}(\text{GRM}_{\text{sib}})}\)

3.1.3 Brothers, Sisiters, Bro-Sis pairs, Bro pairs + Sis pairs

We can separate the families based on the gender and depict the distribution of IBS across the whole genome:

Moreover, we can calculate IBS across different chromosomes,

and the length of each chromosome:

3.1.4 IBS for each locus

The variance IBS score (\(g_i\)) for \(i^{th}\) locus between a pair sib is \(\text{var}(g_i)=\theta^2r_{ii}^2+r_{ii}^2\). We calculated \(\text{var}(g_i)\) across all loci with minor allel frequency greater than 0.05 and number of genotyped families greater than 500, and the density of \(\text{var}(g_i)\) can be shown below:

The vertical blue line is x=1.25

3.1.5 IBD correlation for sibs

In contrast, for IBD for a pair of sib is \((1-2c_{ij})^2\), in which \(c_{ij}\) is the recombination faction between a pair of loci \(i\) and \(j\).

3.2 Estimation for heritability (\(h^2_{\text{SNP}}\))

3.2.1 unrelated \(h^2\) - MOM

This part was calculated by Guoan using 280K individuals across 81 quantitative traits.

3.2.2 Sib \(h^2\) - IBS-linkage

Two scenarios:

  • SingleSib: randomly selected one individual from each family and calculate \(h^2\) using HE-reg method implemented in GCTA

  • Sib: Calculate \(h^2\) (cross-product or squared difference) using HE-reg based on family IBS

Results

  • Sib vs Unrelated

  • Sib vs SingleSib

  • SingleSib vs Unrelated

3.3 GWAS

3.3.1 Unrelated GWAS 280K unrelated UKB

\(y_{\text{self}}=a+bx+e\)

NCP and smpling variance, statistical power.

3.3.2 Single-sib GWAS, 18K unrelated UKB

\(y_{\text{self}}=a+bx+e\)

3.3.3 Alternative sib GWAS, 18K

\(y_{\text{sib}}=a+bx+e\), here \(y_{\text{sib}}\) means the phenotype of the other sib.

We can now check the results between Single-sib and Alternative sib:

Height

  • Miami

+ Chisq-statistic scatter plot

+ Chisq-statistic scatter plot

  • Beta scatter plot

BMI

  • Miami

  • Chisq-statistic scatter plot

+ Chisq-statistic scatter plot

  • Beta scatter plot

AgeDiabetes

  • Miami

  • Chisq-statistic scatter plot

  • Chisq-statistic scatter plot

  • Beta scatter plot

4 Discrepancy between GWAS and linkage (IBS-linkage)

I have completed the comparison between IBS GWAS based on square difference and single-sib GWAS:

  • Height

  • BMI

  • AgeDiabetes

between IBS GWAS based on cross product and single-sib GWAS:

  • Height

  • BMI

  • AgeDiabetes