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)

Similarity Between Individuals {#similaritybetweenindividuals}

At the genetic level there are two different kinds of similarity

1. Identity by descent (IBD)
2. Identity by state

IBD versus IBS

#rmdhelp::use_odg_graphic(ps_path = "odg/ibdvsibs.odg")
knitr::include_graphics(path = "odg/ibdvsibs.png")

Numerator Relationship Matrix

• probability of IBD alleles in two individuals: coancestry or coefficient of kinship
• additive genetic relationship between two individuals is twice their coancestry
• matrix containing all additive genetic relationships in a population is called numerator relationship matrix ($A$)
• $A$ is symmetric and contains on
  • diagonal: $(A)_{ii} = (1 + F_i)$
  • off-diagonal: $(A)_{ij} = cov(u_i, u_j) / \sigma_u^2$ (with $i \ne j$)

Recursive Computation of $A$

• If both parents $s$ and $d$ of animal $i$ are known then

  • the diagonal element $(A){ii}$ corresponds to: $(A){ii} = 1 + F_i = 1 + {1\over 2} (A)_{sd}$ and
  • the offdiagonal element $(A){ji}$ is computed as: $(A){ji} = {1\over 2} ((A){js} + (A){jd})$
  • because $A$ is symmetric $(A){ji} = (A){ij}$

• If only one parent $s$ is known and assumed unrelated to the mate

  • $(A)_{ii} = 1$
  • $(A){ij} = (A){ji} = {1\over 2} ((A)_{js}$

• If both parents are unknown

  • $(A)_{ii} = 1$
  • $(A){ij} = (A){ji} = 0$

Example

#rmdhelp::use_odg_graphic(ps_path = "odg/pedexamplegraph.odg")
knitr::include_graphics(path = "odg/pedexamplegraph.png")

Tabular Representation of Pedigree

suppressPackageStartupMessages( library(pedigreemm) )
n_nr_ani_ped <- 6
n_nr_parent <- 2
tbl_ped <- tibble::tibble(Calf = c((n_nr_parent+1):n_nr_ani_ped),
                             Sire = c(1, 1, 4, 5),
                             Dam  = c(2, NA, 3, 2))
ped <- pedigree(sire = c(rep(NA, n_nr_parent), tbl_ped$Sire), dam = c(rep(NA, n_nr_parent), tbl_ped$Dam), label = as.character(1:n_nr_ani_ped))
matA <- as.matrix(getA(ped = ped))
matAinv <- as.matrix(getAInv(ped = ped))
ped

knitr::kable(tbl_ped,
             booktabs = TRUE,
             longtable = TRUE,
             caption = "Example Pedigree To Compute Additive Genetic Relationship Matrix")

Stepwise Computation of $A$

• Start by extending pedigree with animals that do not have parents
• Order animals, such that parents before progeny

tbl_ped_extended <- tibble::tibble(Animal = c(1:n_nr_ani_ped),
                                      Sire = c(NA, NA, 1, 1, 4, 5),
                                      Dam  = c(NA, NA, 2, NA, 3, 2))

knitr::kable(tbl_ped_extended,
             booktabs = TRUE,
             longtable = TRUE)

Initialize With Empty Matrix $A$

• Dimensions of $A$: number of rows and number of columns equal to the number of animals
• Our example: $r n_nr_ani_ped \times r n_nr_ani_ped$

matA_empty <- matrix(NA, nrow = n_nr_ani_ped, ncol = n_nr_ani_ped)
### # display
cat("$$\n")
cat("A = \\left[")
cat(paste(rmdhelp::sConvertMatrixToLaTexArray(pmatAMatrix = matA_empty), collapse = "\n"), "\n")
cat("\\right]\n")
cat("$$\n")

First Diagonal Element

• Compute first element $(A)_{11} = 1 + F_1$
• Animal $1$ has both parents unknown $\rightarrow$ $F_1 = 0$

matA_empty[1,1] <- matA[1,1]
### # display
cat("$$\n")
cat("A = \\left[")
cat(paste(rmdhelp::sConvertMatrixToLaTexArray(pmatAMatrix = matA_empty), collapse = "\n"), "\n")
cat("\\right]\n")
cat("$$\n")

Off-diagonal Elements

• Assume animal $i$ has parents $s$ and $d$
• $(A){ji} = {1\over 2} ((A){js} + (A)_{jd})$

First Row of $A$

matA_empty[1,] <- matA[1,]
### # display
cat("$$\n")
cat("A = \\left[")
cat(paste(rmdhelp::sConvertMatrixToLaTexArray(pmatAMatrix = matA_empty), collapse = "\n"), "\n")
cat("\\right]\n")
cat("$$\n")

Use Symmetry of $A$

• Copy first row into first column

matA_empty[,1] <- matA[,1]
### # display
cat("$$\n")
cat("A = \\left[")
cat(paste(rmdhelp::sConvertMatrixToLaTexArray(pmatAMatrix = matA_empty), collapse = "\n"), "\n")
cat("\\right]\n")
cat("$$\n")

Remaining Elements of $A$

• Continue with rows and columns $2$ to $r n_nr_ani_ped$ using the same recipe

Final Result

### # display
cat("$$\n")
cat("A = \\left[")
cat(paste(rmdhelp::sConvertMatrixToLaTexArray(pmatAMatrix = matA, pnDigits = 4), collapse = "\n"), "\n")
cat("\\right]\n")
cat("$$\n")

The Inverse Numerator Relationship Matrix {#inversenumeratorrelationshipmatrix}

• Recap: Henderson's mixed model equations depend on four matrices

• Design matrix $X$ for the fixed effects

• Design matrix $Z$ for the random effects
• The inverse variance-covariance matrix $R^{-1}$ for the residuals $e$ and
• The inverse variance-covariance matrix $G^{-1}$ for the random breeding values $a$.

Animal Model

• Breeding values of all individuals as random effects
• Variance-Covariance matrix $G$ corresponds to variance-covariance matrix of breeding values

$$G = A * \sigma_u^2$$

• We need: $G^{-1}$

$$G^{-1} = A^{-1} * {1\over \sigma_u^2}$$

Need For Efficient Computation of $A{-1}$

• In practical livestock breeding evaluations $A$ is very large
• Dimensions of $A$ can be $10^7 \times 10^7$
• Explicit general inversion not possible
• Special structure of $A^{-1}$ leads to efficient computation

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