class: center, middle, inverse, title-slide # Resemblance between relatives ### Jinliang Yang ### Oct. 22nd, 2018 --- # Half Sibs <div align="center"> <img src="https://i.imgur.com/hOtOWLB.png" height=400> </div> With an experimental design including family structure, we can partition phenotypic variance into that __among families__ and that __within families__. --- # Between and within group variance A key concept in the __analysis of variance (ANOVA)__ is that the variance between groups is equal to the covariance within groups. -- Let `\begin{align*} y_{ij}= \mu + b_i + e_{ij} \end{align*}` Where `\(b_i\)` is the group effect and `\(e_{ij}\)` is the residual. -- The covariance between two individuals in the same group ( `\(y_{ij}\)` and `\(y_{ik}\)` ) is `\begin{align*} Cov(y_{ij}, y_{ik}) & = Cov(\mu + b_i + e_{ij}, \mu + b_i + e_{ik}) \\ & = Cov(b_i, b_i) \\ & =Var(b) \\ \end{align*}` -- As similarity within families increases, variation among families increases. --- # Intraclass correlation The proportion of between-family variance to the total variance is called the __intraclass correlation__, usually expressed with the symbol __t__. -- Thus, if we have the variance partitioned into the between families ( `\(\sigma^2_B\)` ) and within familes ( `\(\sigma^2_W\)` ), and these sum to the phenotypic variance ( `\(\sigma_P^2=\sigma_B^2 + \sigma_W^2\)` ), then the intraclass correlation is: `\begin{align*} t =\frac{\sigma_B^2}{\sigma_B^2 + \sigma_W^2} \\ \end{align*}` -- ### Assumptions 1. Diploidy 2. Autosomal loci 3. Linkage equilibrium 4. No maternal effects 5. No GxE interactions 6. No selection --- # General framework for calculating resemblance The resemblance (covariance) between relatives is a function of the __coefficient of co-ancestry__ between individuals and __the componenets of `\(\sigma^2_G\)`__ underlying trait variation. -- Alleles shared between relatives that are __identical by descent (IBD)__ contribute to the covariance between relatives. For example, if `\(\sigma_A^2 = 0\)`, meaning no variation in breeding values exists in the population for the trait of interest, then shared average allelic effects will not contribute to resemblance. -- The more closely related a group of individual is, the greater the probability they share alleles that are IBD, and therefore the greater the covariance. --- # General framework The resemblance (covariance) between relatives is a function of the __coefficient of co-ancestry__ between individuals and __the componenets of `\(\sigma^2_G\)`__ underlying trait variation. -- If allele `\(x_i\)` carried by individual X is IBD to allele `\(y_k\)` in Y, then the covariance due to this allele is: `\begin{align*} Cov(\alpha_i, \alpha_k) & = E[(\alpha_i - \mu_{\alpha})(\alpha_k - \mu_{\alpha})] \\ & = E[(\alpha_i - \mu_{\alpha})^2] \\ & = \sigma_{\alpha}^2 \end{align*}` Because `\(\alpha_i = \alpha_k\)` if alleles `\(x_i\)` and `\(y_k\)` are IBD. --- # Covariance due to additive genetic covariance The additive genetic covariance between relatives is generated through individuals sharing average effects of alleles. -- Alleles in individuals `\(X (x_ix_j)\)` and `\(Y (y_ky_l)\)` can be IBD through four possible events: -- `\begin{align*} & x_i \equiv y_k \\ & x_i \equiv y_l \\ & x_j \equiv y_k \\ & x_j \equiv y_l \\ \end{align*}` -- `\begin{align*} Cov_\alpha(X, Y) = & P(x_i \equiv y_k)Cov(\alpha_i, \alpha_k) + P(x_i \equiv y_l)Cov(\alpha_i, \alpha_l) \\ & + P(x_j \equiv y_k)Cov(\alpha_j, \alpha_k) + P(x_j \equiv y_l)Cov(\alpha_j, \alpha_l) \\ = & 4f_{XY}\sigma_\alpha^2 \\ = & 2f_{XY}\sigma_A^2 \\ \end{align*}` Because `\(\sigma_A^2 = \sigma_{\alpha_i}^2 + \sigma_{\alpha_j}^2 = 2\sigma_\alpha^2\)` and `\(\alpha_i = \alpha_k\)` when alleles `\(i\)` and `\(k\)` are IBD. --- # Parent-offspring Recall that the coefficient of co-ancestry between a non-inbred parent and non-inbred offspring is 1/4. Thus, the coefficient for `\(\sigma_A^2\)` is 1/2. Consider the convariance between parent (P) and offspring (O), if we assume the parents are unrelated, then all offspring can't share a common dominance deviation ( `\(\sigma^2_D\)` =0 ). Therefore, `\begin{align*} Cov(P, O) = \frac{1}{2}\sigma_A^2 \end{align*}` -- ### From Breeding value - Parent genotypic value: `\(G = A + D\)`. - Offspring (half the breeding value of the parents from Ch.7) : `\(G=1/2A\)` -- `\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 `\(Cov(A, D) = 0\)` from Ch.8.