class: center, middle, inverse, title-slide # Resemblance between relatives ### Jinliang Yang ### Nov. 11th, 2019 --- # Announcements ### Next Exam date: __Nov. 15th (F) at 7:30am__ --- # Parent-offspring ### From Breeding value - Parent genotypic value: `\(G = A + D\)`. - Offspring (half the breeding value of the parents) : `\(G=1/2A\)` `\begin{align*} Cov(P, O) & = Cov(A + D, 1/2A) \\ & = 1/2Cov(A, A) + 1/2Cov(A, D) = \frac{1}{2}\sigma_A^2 \\ \end{align*}` Because `\(Cov(A, D) = 0\)`. -- #### Assumptions 1. Diploidy 2. Autosomal loci 3. Linkage equilibrium 4. No maternal effects 5. No GxE interactions 6. No selection --- # Parent-offspring | Genotype | Freq | Breeding Value | Dominance Deviation | Genotypic value | Offspring ( `\(\mu_G\)` ) | | :-------: | :-------: | :-----------: | :-------: | :-------: | :-------: | :-------: | | `\(A_1A_1\)` | `\(p^2\)` | `\(2q\alpha\)` | `\(-2q^2d\)` | `\(2q(\alpha - qd)\)` | ? | | `\(A_1A_2\)` | `\(2pq\)` | `\((q-p)\alpha\)` | `\(2pqd\)` | `\((q-p)\alpha + 2pqd\)` | ? | | `\(A_2A_2\)` | `\(q^2\)` | `\(-2p\alpha\)` | `\(-2p^2d\)` | `\(-2p(\alpha + pd)\)` | ? | --- # Parent-offspring | Genotype | Freq | Breeding Value | Dominance Deviation | Genotypic value | Offspring ( `\(\mu_G\)` ) | | :-------: | :-------: | :-----------: | :-------: | :-------: | :-------: | :-------: | | `\(A_1A_1\)` | `\(p^2\)` | `\(2q\alpha\)` | `\(-2q^2d\)` | `\(2q(\alpha - qd)\)` | `\(q\alpha\)` | | `\(A_1A_2\)` | `\(2pq\)` | `\((q-p)\alpha\)` | `\(2pqd\)` | `\((q-p)\alpha + 2pqd\)` | `\(1/2(q-p)\alpha\)` | | `\(A_2A_2\)` | `\(q^2\)` | `\(-2p\alpha\)` | `\(-2p^2d\)` | `\(-2p(\alpha + pd)\)` | `\(-p\alpha\)` | -- $$ `\begin{aligned} Cov(X, Y) & = E(XY) - E(X)E(Y) \\ \end{aligned}` $$ where, $$ `\begin{aligned} E(XY) = \sum_i \sum_j x_i y_j Pr(X = x_i, Y = y_j) \end{aligned}` $$ -- $$ `\begin{aligned} E(PO) = & p^2 \times 2q(\alpha-qd) \times q\alpha + 2pq \times ((q-p)\alpha+2pqd) \times 1/2(q-p)\alpha \\ & + q^2 \times (-2p(\alpha+pd)) \times (-p\alpha) \\ = & [2p^2q^2\alpha^2 - 2p^2q^3d\alpha] + [pq\alpha^2(q^2-2pq+p^2) + 2p^2q^2d\alpha(q-p)] \\ & + [2p^2q^2\alpha^2 + 2p^3q^2d\alpha] \\ = & pq\alpha^2(2pq + q^2 -2pq +p^2 + 2pq) + 2p^2q^2d\alpha(-q+q-p+p) \\ = & pq\alpha^2 \end{aligned}` $$ --- # Parent-offspring | Genotype | Freq | Breeding Value | Dominance Deviation | Genotypic value | Offspring ( `\(\mu_G\)` ) | | :-------: | :-------: | :-----------: | :-------: | :-------: | :-------: | :-------: | | `\(A_1A_1\)` | `\(p^2\)` | `\(2q\alpha\)` | `\(-2q^2d\)` | `\(2q(\alpha - qd)\)` | `\(q\alpha\)` | | `\(A_1A_2\)` | `\(2pq\)` | `\((q-p)\alpha\)` | `\(2pqd\)` | `\((q-p)\alpha + 2pqd\)` | `\(1/2(q-p)\alpha\)` | | `\(A_2A_2\)` | `\(q^2\)` | `\(-2p\alpha\)` | `\(-2p^2d\)` | `\(-2p(\alpha + pd)\)` | `\(-p\alpha\)` | $$ `\begin{aligned} Cov(P, O) & = E(PO) - E(P)E(O) \\ \end{aligned}` $$ -- $$ `\begin{aligned} E(PO) = & pq\alpha^2 \\ E(O) = & 0 \end{aligned}` $$ -- Therefore, $$ `\begin{aligned} Cov(P, O) & = E(PO) - E(P)E(O) \\ & = pq\alpha^2 \\ & = 1/2 V_A \end{aligned}` $$ Because `\(V_A = 2pq\alpha^2\)`. --- # Offspring and mid-parent The covariance of the mean of the offsrping and the mean of both parents (commonly called the __mid-parent__) Let `\(P\)` and `\(P'\)` be the values of the two parents, therefore `\(\bar{P}=1/2(P + P')\)` -- $$ `\begin{aligned} Cov(\bar{P}, O) & = Cov(1/2(P + P'), O) \\ & = 1/2(Cov(P,O) + Cov(P', O))\\ & = 1/2 V_A \end{aligned}` $$ If `\(P\)` and `\(P'\)` have the same variance. See _P149 of F&M_ for the algebraic reduction. --- # What is the genetic covariance for half sibs? <div align="center"> <img src="hs-covar.png" height=400> </div> --- # Genetic covariance for half sibs? `\begin{align*} Cov(1/2A, 1/2A) & = 1/4V_A\\ \end{align*}` -- | Genotype | Freq | Breeding Value | Dominance Deviation | Genotypic value | Offspring ( `\(\mu_G\)` ) | | :-------: | :-------: | :-----------: | :-------: | :-------: | :-------: | :-------: | | `\(A_1A_1\)` | `\(p^2\)` | `\(2q\alpha\)` | `\(-2q^2d\)` | `\(2q(\alpha - qd)\)` | `\(q\alpha\)` | | `\(A_1A_2\)` | `\(2pq\)` | `\((q-p)\alpha\)` | `\(2pqd\)` | `\((q-p)\alpha + 2pqd\)` | `\(1/2(q-p)\alpha\)` | | `\(A_2A_2\)` | `\(q^2\)` | `\(-2p\alpha\)` | `\(-2p^2d\)` | `\(-2p(\alpha + pd)\)` | `\(-p\alpha\)` | -- `\begin{align*} Cov_{HS} = & p^2(q\alpha)^2 + 2pq \times 1/4(q-p)^2\alpha^2 + q^2 \times p^2\alpha^2 \\ =& pq\alpha^2[pq + 1/2(q-p)^2 + pq] \\ =& pq\alpha^2[1/2(p+q)^2] \\ =& 1/2pq\alpha^2 \\ = & 1/4V_A \end{align*}` --- # Genetic covariance for full sibs? ### Additive genetic covariance - Genotypic value for full sibs: `\begin{align*} G_{o1} & = \frac{1}{2}A + \frac{1}{2}A'\\ G_{o2} & = \frac{1}{2}A + \frac{1}{2}A'\\ \end{align*}` -- - The genetic covariance for full sibs: `\begin{align*} Cov(G_{o1}, G_{o2}) = & Cov(\frac{1}{2}A + \frac{1}{2}A', \frac{1}{2}A + \frac{1}{2}A') \\ = & Var(\frac{1}{2}(A + A')) \\ = & \frac{1}{4}(\sigma_A^2 + \sigma^2_{A'}) \\ = & \frac{1}{2}\sigma_A^2 \\ \end{align*}` --- # Genetic covariance for full sibs? <div align="center"> <img src="fs_covar.png" height=200> </div> -- - Among the progeny, only __four possible genotypes__, each with a frequency of 1/4. - `\(A_1A_3\)`, `\(A_1A_4\)`, `\(A_2A_3\)`, `\(A_2A_4\)` -- - Let the first sib has any of these genotypes. The 2nd has the same genotype is 1/4. - Thus one-quarter of all sib-pairs have the same genotype and consequently the same dominance deviation, `\(D\)`. -- - Thus, the cross-product of the dominance deviation is `\(\sigma_D^2\)`, times frequency 1/4 = `\(1/4\sigma_D^2\)` --- # Genetic covariance for full sibs? ### The genetic covariance of full sibs is therefore, `\begin{align*} Cov_{FS} = \frac{1}{2}\sigma_A^2 + \frac{1}{4}\sigma_D^2 \end{align*}` -- ### Covariance of half sibs `\begin{align*} Cov_{HS} = \frac{1}{4}\sigma_A^2 \end{align*}` -- In priciple, dominance variance can be calculated using full sibs and half sibs: `\begin{align*} Cov_{FS} - 2Cov_{HS} = & \frac{1}{2}\sigma_A^2 + \frac{1}{4}\sigma_D^2 - 2 \times \frac{1}{4}\sigma_A^2 \\ = & \frac{1}{4}\sigma_D^2 \end{align*}` --- # Twins ### Monozygotic (identical) twins $$ Cov_{MZ} = V_G $$ -- ### Dizygotic (fraternal) twins - Not identical twins - = Full sibs --- # Summary | Relationship | | r (of `\(\sigma^2_A\)`) | u (of `\(\sigma^2_D\)`) | | :-------: | :-------: | :-----------: | :-------: | :-------: | | Identical twins | | 1 | 1| | First degree | Parent-offspring | 1/2 | 0 | | Second degree | Half sibs | 1/4 | 0 | | | Full sibs | 1/2 | 1/4 |