ASCI 896 Statistical Genomics

One locus to the infinitesimal model - 1

Assumptions:

• Hardy–Weinberg equilibrium (HWE)
• linkage equilibrium (LE)
• additive gene action

\[ \begin{align*} y_i &= u_i + e_i \\ y_i &= W_{i1}a_1 + W_{i2}a_2 + \cdots W_{ik}a_k + e_i \end{align*} \] Variance of \( \mathbf{u} \) is \[ \begin{align*} Var(\mathbf{u}) &= \sum^K_{k=1} Var(W_k a_k) \\ &= \sum^K_{k=1} Var(W_k) a_k^2 \\ &= \sum^K_{k=1} 2p_k(1-p_k) a_k^2 \end{align*} \]

Assumptions:

• Hardy–Weinberg equilibrium (HWE)
• linkage disequilibrium (LD)
• additive gene action \[ \begin{align*} Var(\mathbf{u}) &= \sum^K_{k=1} 2p_k(1 - p_k) a_k^2 + 2 \sum^K_{k=1}\sum^{K}_{l=k+1}Cov(W_{ik}, W_{il})a_ka_l \\ &= \sum^K_{k=1} 2p_k(1-p_k) a_k^2 + 2 \sum^K_{k=1}\sum^{K}_{l=k+1}2\rho_{kl} \sqrt{p_k(1-p_k)p_l(1 - p_l)} a_ka_l \\ &= \sum^K_{k=1} 2p_k(1-p_k) a_k^2 + 2 \sum^K_{k=1}\sum^{K}_{l=k+1} 2D_{kl} a_k a_l \end{align*} \] where \( \rho_{kl} \) is the correlation between allelic counts between loci \( k \) and \( l \) and \( D_{kl} \) is the LD between loci \( k \) and \( l \)

Assumptions:

• Hardy–Weinberg equilibrium (HWE)
• linkage equilibrium (LE)
• additive gene action

\[ \begin{align*} h^2_g &= \frac{\sum^K_{k=1}2p_k(1-p_k) a_k^2 }{\sum^K_{k=1}2p_k(1-p_k) a_k^2 + \sigma^2_e} \end{align*} \] Assumptions:

• Hardy–Weinberg equilibrium (HWE)
• linkage disequilibrium (LD)
• additive gene action

\[ \begin{align*} h^2_g &= \frac{\sum^K_{k=1} 2p_k(1-p_k) a_k^2 + 2 \sum^K_{k=1}\sum^{K}_{l=k+1} 2 \rho_{kl} \sqrt{p_k(1-p_k)p_l(1 - p_l)} a_k a_l }{\sum^K_{k=1} 2p_k(1-p_k) a_k^2 + 2 \sum^K_{k=1}\sum^{K}_{l=k+1} 2 \rho_{kl} \sqrt{p_k(1-p_k)p_l(1 - p_l)} a_k a_l + \sigma^2_e} \end{align*} \]