In charlotte-ngs/lbgfs2020: Livestock Breeding and Genomics (Fall 2020)

knitr::opts_chunk$set(echo = FALSE, results = "asis")
knitr::knit_hooks$set(hook_convert_odg = rmdhelp::hook_convert_odg)

So Far ...

• Prediction of Breeding Values for one trait

$\rightarrow$ univariate analyses

• In Livestock Breeding, populations are improved with respect to several traits

$\rightarrow$ multi-trait or multiple trait

• Different selection strategies and different approaches of how data is analysed are possible

Multiple Trait Selection

• Selection index theory provides a tool for optimal integration of different sources of information
• But still other strategies are applied
  • Tandem selection
  • Selection based on independent thresholds

Tandem Selection

• Improve one trait at the time until they all reach a certain threshold
• Problem: For traits which are not improved
  • only correlated selection responses
  • can be negative
• Populations with long generation intervals, response per year is very small

Independent Selection Thresholds

• Applied before selection index
• Define selection thresholds in each of the traits
• Select animals as parents which are above thresholds for all traits

Example

### # selection thresolods
n_milk_sel_thr <- 6900
n_prot_sel_thr <- 3.5
### # mean and sd of traits
n_nr_trait <- 2
n_milk_mean <- 6800
n_milk_sd <- 600
n_prot_mean <- 3.4
n_prot_sd <- .2
n_corr <- -.4
### # variance - covariance matrix
n_cov <- n_corr * n_milk_sd * n_prot_sd
mat_varcov <- matrix(c(n_milk_sd^2, n_cov,
                       n_cov,       n_prot_sd^2), 
                     nrow = n_nr_trait, 
                     ncol = n_nr_trait, 
                     byrow = TRUE)

### # generate observations
n_nr_obs <- 50
### # cholesky decomposition of varcov
mat_varcov_chol <- chol(mat_varcov)
### # generate independent observations
set.seed(5432)
mat_unrel_obs <- matrix(c(rnorm(n_nr_obs),rnorm(n_nr_obs)), nrow = n_nr_trait, byrow = TRUE)
mat_obs <- crossprod(mat_unrel_obs, mat_varcov_chol) + matrix(c(n_milk_mean, n_prot_mean), nrow=n_nr_obs, ncol=n_nr_trait, byrow = TRUE)

### # convert data matrix into a tibble that is later used for plotting
tbl_milk_prot <- tibble::as_tibble(mat_obs)
colnames(tbl_milk_prot) <- c("Milk", "Protein")
### # define the colours of the threshold lines
s_col_milk_thr <- "red"
s_col_prot_thr <- "blue"
### # use ggplot2 to do the plot
library(ggplot2)
milk_prot_plot <- qplot(Milk, Protein, data=tbl_milk_prot, geom="point", 
                         xlab = "Milk Yield", ylab = "Protein Content")
milk_prot_plot <- milk_prot_plot + 
  geom_hline(yintercept = n_prot_sel_thr, colour = s_col_prot_thr) +
  geom_vline(xintercept = n_milk_sel_thr, colour = s_col_milk_thr)
print(milk_prot_plot)

Pros and Cons

• Selection response in all traits
• Thresholds often set to only positive predicted breeding values in all traits

$\rightarrow$ exclusion of very many animals and reduction in genetic variability

• Genetic relationships between traits ignored

$\rightarrow$ genetic gain will not be as expected

1. Differences in the economic relevance ignored.

$\rightarrow$ threshold in all traits above positive predicted breeding values emphasizes traits with high heritability

Aggregate Genotype

• Define the set of important traits for which population should be improved
• Determine economic values $w$ for these traits
• Aggregate genotype $H$ follows as

$$H = w^Tu$$

Selection Index

• Use index $I$ to estimate $H$ where $I$ is a linear combination of information sources

$$I = b^T\hat{u}$$

• Index weights $b$ are determined using selection index theory as

$$b = P^{-1}Gw$$

• Information sources are predicted breeding values
• If traits in $u$ and $\hat{u}$ are the same and $\hat{u}$ were estimated using BLUP, then $b = w$

Implementations

• First possible implementation

  • Do univariate predictions of breeding values using BLUP animal model
  • Combine $\hat{u}$ with appropriate $b$-values

• Imrprovement

  • get $\hat{u}$ from multivariate analysis

Multivariate Analysis

• Given two traits with univariate models

$$y_1 = X_1 \beta_1 + Z_1u_1 + e_1$$ $$y_2 = X_2 \beta_2 + Z_2u_2 + e_2$$

• Combine both univariate models by stacking one on top of the other, resulting in

$$\left[ \begin{array}{c} y_1 \ y_2 \end{array} \right] = \left[ \begin{array}{lr} X_1 & 0 \ 0 & X_2 \end{array} \right] \left[ \begin{array}{c} \beta_1 \ \beta_2 \end{array} \right] + \left[ \begin{array}{lr} Z_1 & 0 \ 0 & Z_2 \end{array} \right] \left[ \begin{array}{c} u_1 \ u_2 \end{array} \right] + \left[ \begin{array}{c} e_1 \ e_2 \end{array} \right] $$

Multivariate Model

$$\left[ \begin{array}{c} y_1 \ y_2 \end{array} \right] = \left[ \begin{array}{lr} X_1 & 0 \ 0 & X_2 \end{array} \right] \left[ \begin{array}{c} \beta_1 \ \beta_2 \end{array} \right] + \left[ \begin{array}{lr} Z_1 & 0 \ 0 & Z_2 \end{array} \right] \left[ \begin{array}{c} u_1 \ u_2 \end{array} \right] + \left[ \begin{array}{c} e_1 \ e_2 \end{array} \right] $$

can be written as

$$y = X \beta + Zu + e$$

with $y = \left[ \begin{array}{c} y_1 \ y_2 \end{array} \right]$, $\beta = \left[ \begin{array}{c} \beta_1 \ \beta_2 \end{array} \right]$, $u = \left[ \begin{array}{c} u_1 \ u_2 \end{array} \right]$, $e = \left[ \begin{array}{c} e_1 \ e_2 \end{array} \right]$

\vspace{2ex} $X = \left[ \begin{array}{lr} X_1 & 0 \ 0 & X_2 \end{array} \right]$, $Z = \left[ \begin{array}{lr} Z_1 & 0 \ 0 & Z_2 \end{array} \right]$

Multivariate Variance-Covariance Matrices

$$G_0 = \left[ \begin{array}{lr} \sigma_{g_{1}}^2 & \sigma_{g1,g2} \ \sigma_{g1,g2} & \sigma_{g_{2}}^2 \end{array} \right] = \left[ \begin{array}{lr} g_{11} & g_{12} \ g_{21} & g_{22} \end{array} \right] $$

$$var(u) = var\left[\begin{array}{c} u_1 \ u_2 \end{array} \right]

\left[ \begin{array}{lr} g_{11}A & g_{12}A \ g_{21}A & g_{22}A \end{array} \right] = G_0 \otimes A = G $$

$$R_0 = \left[ \begin{array}{lr} r_{11} & r_{12} \ r_{21} & r_{22} \end{array} \right]$$

$$ R = var(e) = var\left[\begin{array}{c} e_1 \ e_2 \end{array} \right] =
\left[ \begin{array}{lr} r_{11}I_n & r_{12}I_n \ r_{21}I_n & r_{22}I_n \end{array} \right] = R_0 \otimes I_n $$

Solutions

• Mixed Model Equations

$$ \left[ \begin{array}{lr} X^TR^{-1}X & X^TR^{-1}Z \ Z^TR^{-1}X & Z^TR^{-1}Z + G^{-1} \end{array} \right] \left[ \begin{array}{c} \hat{\beta} \ \hat{u} \end{array} \right] = \left[ \begin{array}{c} X^TR^{-1}y \ Z^TR^{-1}y \end{array} \right] $$

Advantages

• some traits have lower heritability than others
• environmental correlations exist between traits measured on the same animal
• some traits are available only a subset of all animals
• some traits were used for a first round of selection
• accuracies are higher in multivariate analyses

charlotte-ngs/lbgfs2020 documentation built on Dec. 20, 2020, 5:39 p.m.