Chapter 9 IBD notes
9.1 HE association
The classic \(h^2=\beta HLH\beta\)
In population-based design, the relatedness is measured in IBS
| \(h^2=\beta HLH\beta\) | \(h^2_d=d H_dL_dH_dd\) | |
| \(h^2_{SNP.A}=m\beta H\{\frac{ \sum_{k=1}^{m} v_k^Tv_k }{1^TP1}\} H \beta\) | \(h^2_{SNP.D}=md H_d\{\frac{ \sum_{k=1}^{m} v_{k,d}^T v_{k,d} }{1^TP_d1}\} H_d d\) | |
| \(\tilde{h}^2_{SNP.A}=m\beta H\{\frac{ \sum_{k=1}^{m}w_k v_{k}^Tv_k }{w^TPw}\} H \beta\) | \(\tilde{h}^2_{SNP.D}=md H_d\{\frac{ \sum_{k=1}^{m} w_{k,d}v_{k.d}^Tv_{k.d} }{w_{d}^TP_dw_{d}}\} H_d d\) | |
| \(h^2_{SNP.A_2}=m\beta_1 H_1\{\frac{ \sum_{k=1}^{m} v_{k_1}^Tv_{k_2} }{1^TP1}\} H_2 \beta_2\) | \(h^2_{SNP.D_2}=md_1 H_{d_1}\{\frac{ \sum_{k=1}^{m} v_{k,d_1}^Tv_{k,d_2} }{1^TP_d1}\} H_{d_2} d_2\) | |
| \(\tilde{h}^2_{SNP.A_2}=m\beta_1 H_1\{\frac{ \sum_{k=1}^{m}w_k v_{k,1}^Tv_{k,2} }{w^TPw}\} H_2 \beta_2\) | \(\tilde{h}^2_{SNP.D_2}=md_1 H_{d_1}\{\frac{ \sum_{k=1}^{m} w_{k,d}v_{k,d_1}^Tv_{k,d_2} }{w_{d}^TP_dw_{d}}\} H_{d_2} d_2\) |
In sibpair design, using IBD \[h^2_{fam}=m\beta H\{\frac{\sum_{k=1}^mz^T_kz_k}{\sum_{k_1=1}^m\sum_{k_2=1}^m (1-2c_{k_1k_2})^2}\}H\beta\]
\(z_k=[(1-2c_{k,1}), (1-2c_{k,2}), (1-2c_{k,3}),...,(1-2c_{k,l})]\)
9.2 IBD table
| IBD \((freq)\) | ||||||
|---|---|---|---|---|---|---|
| Mating type (\(freq\)) | Sib pair (\(freq\)) | \(\Omega\) | IBD=1 | IBD=\(\frac{1}{2}\) | IBD=0 | \(E(IBD)\) |
| \(AA , AA\) (\(p^4\)) | \(\color{red}{\{AA, AA\}}\) (1) | \(\color{red}{\frac{4q^2}{2pq}}\) | \(\color{red}{\frac{1}{4}}\) | \(\color{red}{\frac{1}{2}}\) | \(\color{red}{\frac{1}{4}}\) | \(\color{red}{\frac{1}{2}}\) |
| \(AA , Aa\) (\(4p^3q\)) | \(\color{red}{\{AA, AA\}}\) (\(\frac{1}{4}\)) | \(\color{red}{\frac{4q^2}{2pq}}\) | \(\color{red}{\frac{1}{2}}\) | \(\color{red}{\frac{1}{2}}\) | \(\color{red}{\frac{3}{4}}\) | |
| \(\{AA, Aa\}\) (\(\frac{1}{2}\)) | \(\frac{2q(q-p)}{2pq}\) | \(\frac{1}{2}\) | \(\frac{1}{2}\) | \(\frac{1}{4}\) | ||
| \(\color{green}{\{Aa, Aa\}}\) (\(\frac{1}{4}\)) | \(\color{green}{\frac{(q-p)^2}{2pq}}\) | \(\color{green}{\frac{1}{2}}\) | \(\color{green}{\frac{1}{2}}\) | \(\color{green}{\frac{3}{4}}\) | ||
| \(AA , aa\) (\(2p^2q^2\)) | \(\color{green}{\{Aa, Aa\}}\) (1) | \(\color{green}{\frac{(q-p)^2}{2pq}}\) | \(\color{green}{\frac{1}{4}}\) | \(\color{green}{\frac{1}{2}}\) | \(\color{green}{\frac{1}{4}}\) | \(\color{green}{\frac{1}{2}}\) |
| \(Aa , Aa\) (\(4p^2q^2\)) | \(\color{red}{\{AA, AA \}}\) (\(\frac{1}{16}\)) | \(\color{red}{\frac{4q^2}{2pq}}\) | \(\color{red}{1}\) | \(\color{red}{1}\) | ||
| \(\{AA, Aa \}\) (\(\frac{1}{4}\)) | \(\frac{2q(q-p)}{2pq}\) | \(1\) | \(\frac{1}{2}\) | |||
| \(\color{grey}{\{AA, aa\}}\) (\(\frac{1}{8}\)) | \(\color{grey}{\frac{-4pq}{2pq}}\) | \(\color{grey}{1}\) | \(\color{grey}{0}\) | |||
| \(\color{green}{\{Aa, Aa\}}\) (\(\frac{1}{4}\)) | \(\color{green}{\frac{(q-p)^2}{2pq}}\) | \(\color{green}{\frac{1}{2}}\) | \(\color{green}{\frac{1}{2}}\) | \(\color{green}{\frac{1}{2}}\) | ||
| \(\color{gold}{\{Aa, aa \}}\) (\(\frac{1}{4}\)) | \(\color{gold}{\frac{-2p(q-p)}{2pq}}\) | \(\color{gold}{1}\) | \(\color{gold}{\frac{1}{2}}\) | |||
| \(\color{blue}{\{aa, aa\}}\) (\(\frac{1}{16}\)) | \(\color{blue}{\frac{4p^2}{2pq}}\) | \(\color{blue}{1}\) | \(\color{blue}{1}\) | |||
| \(Aa, aa\) (\(4pq^3\)) | \(\color{green}{\{Aa, Aa\}}\) (\(\frac{1}{4}\)) | \(\color{green}{\frac{(q-p)^2}{2pq}}\) | \(\color{green}{\frac{1}{2}}\) | \(\color{green}{\frac{1}{2}}\) | \(\color{green}{\frac{3}{4}}\) | |
| \(\color{gold}{\{Aa, aa\}}\) (\(\frac{1}{2}\)) | \(\color{gold}{\frac{-2p(q-p)}{2pq}}\) | \(\color{gold}{\frac{1}{2}}\) | \(\color{gold}{\frac{1}{2}}\) | \(\color{gold}{\frac{1}{4}}\) | ||
| \(\color{blue}{\{aa, aa\}}\) (\(\frac{1}{4}\)) | \(\color{blue}{\frac{4p^2}{2pq}}\) | \(\color{blue}{\frac{1}{2}}\) | \(\color{blue}{\frac{1}{2}}\) | \(\color{blue}{\frac{3}{4}}\) | ||
| \(aa, aa\) (\(q^4\)) | \(\color{blue}{\{aa, aa\}}\) (1) | \(\color{blue}{\frac{4p^2}{2pq}}\) | \(\color{blue}{\frac{1}{4}}\) | \(\color{blue}{\frac{1}{2}}\) | \(\color{blue}{\frac{1}{4}}\) | \(\color{blue}{\frac{1}{2}}\) |
\(E(IBD)=\begin{matrix} \frac{1}{2}p^4&\\ +(\frac{3}{4}\frac{1}{4}+\frac{1}{4}\frac{1}{2}+\frac{3}{4}\frac{1}{4})4p^3q&\\ +\frac{1}{2}2p^2q^2&\\ +(1\cdot \frac{1}{16}+\frac{1}{2}\frac{1}{4}+0\cdot \frac{1}{8}+\frac{1}{2}\frac{1}{4}+\frac{1}{2}\frac{1}{4}+1\cdot \frac{1}{16})4p^2q^2&\\ +(\frac{3}{4}\frac{1}{4}+\frac{1}{4}\frac{1}{2}+\frac{3}{4}\frac{1}{4})4pq^3&\\ +\frac{1}{2}q^4 \end{matrix} =\frac{1}{2}\)
The variance of IBD is \(var(IBD)=\frac{p^4}{8} + \frac{p^3q}{2} + \frac{p^2q^2}{4} + \frac{3p^2q^2}{2} + \frac{pq^3}{2} + \frac{q^4}{8}=\frac{1}{8}+p^2q^2\).
However, if the IBD is known for sure, then \(var(IBD)=\frac{1}{8}\).
The expectation of \(\Omega\) is \(E(IBS) = 2pq[(p+\frac{q}{2})^2+(\frac{p}{2}+q)^2]-pq(p-q)^2-p^2q^2+\frac{(q-p)^2}{2}(1+pq) =\frac{1}{2}\)
The variance of \(\Omega\) is \(var(IBS) = E(\Omega^2)-E(\Omega)^2= \frac{p^6+7p^5q+19p^4q^2+26p^3q^3+19p^2q^4+7pq^5+q^6}{4pq} - \frac{1}{4}\)
| Mating type | Frequency | \(IBD \times IBS\) |
|---|---|---|
| \(AA \times AA\) | \(p^4\) | \(\frac{1}{2}\frac{4q^2}{2pq}p^4\) |
| \(aa \times aa\) | \(q^4\) | \(\frac{1}{2}\frac{4p^2}{2pq}q^4\) |
| \(AA \times aa\) | \(2p^2q^2\) | \(\frac{1}{2}\frac{(q-p)^2}{2pq}p^4\) |
| \(AA \times Aa\) | \(4p^3q\) | \(\{\frac{3}{16}\frac{4q^2}{2pq} + \frac{1}{8}\frac{2q(p-q)}{2pq} + \frac{3}{16}\frac{(q-p)^2}{2pq}\}4p^3q\) |
| \(Aa \times aa\) | \(4pq^3\) | \(\{\frac{3}{16}\frac{4q^2}{2pq} + \frac{1}{8}\frac{2p(p-q)}{2pq} + \frac{3}{16}\frac{(q-p)^2}{2pq}\}4pq^3\) |
| \(Aa \times Aa\) | \(4p^2q^2\) | \(\{\frac{1}{16}\frac{4q^2}{2pq} + \frac{1}{16}\frac{4p^2}{2pq}+ \frac{1}{8}\frac{2q(p-q)}{2pq}-\frac{1}{8}\frac{2p(p-q)}{2pq} + \frac{1}{8}\frac{(q-p)^2}{2pq}\}4p^2q^2\) |
The \(cov(IBD,IBS)=E(IBD\times IBS) - E(IBD)E(IBS)=\frac{3}{8}-\frac{1}{2}\frac{1}{2}=\frac{1}{8}\), and \(cor(IBD, IBS)=\frac{cov(IBS, IBD)}{\sqrt{var(IBD)var(IBS)}}=\frac{1}{\sqrt{8var(IBS)}}\).
It indicates that for sibpairs IBD and IBS are correlated for a locus.