In charlotte-ngs/asmss2022: Applied Statistical Methods in Animal Science - Spring Semester 2022

knitr::opts_chunk$set(echo = FALSE)

Animal Model

• Extension of sire model
• Include dams in pedigree
• Predict breeding values for all animals in the pedigree

Dataset

sigma_p2 <- 60
sigma_s2 <- 5
sigma_u2 <- 4 * sigma_s2
sigma_e2 <- sigma_p2 - sigma_u2
tbl_animal_model <- tibble::tibble(Animal = c(4:8),
                                 Sire   = c(1,3,1,4,3),
                                 Dam    = c(NA, 2, 2, 5, 6),
                                 Sex    = c("M","F","F","M","M"),
                                 WWG    = c(4.5, 2.9, 3.9, 3.5, 5.0))
n_nr_founder <- min(tbl_animal_model$Animal) - 1
n_nr_animal <- max(tbl_animal_model$Animal)
n_nr_rec <- nrow(tbl_animal_model)

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

where NA stands for unknown

The Model

\begin{equation} \mathbf{y} = \mathbf{Xb} + \mathbf{Zu} + \mathbf{e} \notag \end{equation}

• Random effects $\mathbf{u}$ and $\mathbf{e}$

$$E(\mathbf{e}) = \mathbf{0}$$ $$var(\mathbf{e}) = \mathbf{I} * \sigma_e^2$$ $$E(\mathbf{u}) = \mathbf{0}$$ $$var(\mathbf{u}) = \mathbf{A} * \sigma_u^2$$

with numerator relationship matrix $\mathbf{A}$

Numerator Relationship Matrix $\mathbf{A}$

• Diagonal elements

\begin{equation} (\mathbf{A})_{ii} = 1 + F_i \notag \end{equation}

where $F_i$ is the inbreeding coefficient

\begin{equation} F_i = {1 \over 2} * (\mathbf{A})_{sd} \notag \end{equation}

• Off-diagonal elements

\begin{equation} cov(u_i, u_j) = (\mathbf{A})_{ij} * \sigma_u^2 \notag \end{equation}

Example

library(pedigreemm)
ped_ani <- pedigree(sire = c(rep(NA, n_nr_founder),
                             1,3,1,4,3),
                    dam  = c(rep(NA, n_nr_founder),
                             NA,2,2,5,6),
                    label = as.character(1:n_nr_animal))
mat_A <- as.matrix(getA(ped = ped_ani))

cat(paste0(rmdhelp::bmatrix(pmat = mat_A, ps_name = "A", ps_env = "$$"), collapse = "\n"), "\n")

Solution

• pedigreemm cannot handle such small datasets with only one observation per animal
• Mixed model equations

\begin{equation} \left[ \begin{array}{cc} X^TX & X^TZ \ Z^TX & Z^TZ + \lambda * A^{-1} \end{array} \right] \left[ \begin{array}{c} \hat{b} \ \hat{u} \end{array} \right] = \left[ \begin{array}{c} X^Ty \ Z^Ty \end{array} \right] \notag \end{equation}

with $\lambda = \sigma_e^2 / \sigma_u^2$

Genomic BlUP

1. Marker effect models (MEM): Linear mixed effects models with marker effects as random effects
2. Breeding-value based models (BVM): Genomic breeding values as random effects

Marker Effect Models

• Model \begin{equation} y = 1_n \mu + Wq + e \notag \end{equation}

• Solution

\begin{equation} \left[ \begin{array}{cc} 1_n^T1_n & 1_n^TW \ W^T1_n & W^TW + \lambda_q * I \end{array} \right] \left[ \begin{array}{c} \hat{\mu} \ \hat{q} \end{array} \right] = \left[ \begin{array}{c} 1_n^Ty \ W^Ty \end{array} \right] \notag \end{equation}

with $\lambda_q = \sigma_e^2 / \sigma_q^2$.

Breeding Value Models

• Model

\begin{equation} y = Xb + Zg + e \notag \end{equation}

• Solution

\begin{equation} \left[ \begin{array}{cc} X^TX & X^TZ \ Z^TX & Z^TZ + \lambda_g * G^{-1} \end{array} \right] \left[ \begin{array}{c} \hat{b} \ \hat{g} \end{array} \right] = \left[ \begin{array}{c} X^Ty \ Z^Ty \end{array} \right] \notag \end{equation}

with $\lambda_g = \sigma_e^2 / \sigma_g^2$.

Genomic Relationship Matrix

\begin{equation} g = U \cdot q \notag \end{equation}

with $U = W - P$ and $P$ has columns $2p_j-1$ with $p_j$ being the frequency of the positive allele at locus $j$.

\begin{equation} var(g) = G * \sigma_g^2 \notag \end{equation}

\begin{equation} var(g) = UU^T * \sigma_q^2 \notag \end{equation}

\begin{equation} \sigma_g^2 = 2 \sum_{j=1}^m p_j(1-p_j)\sigma_q^2 \notag \end{equation}

Genomic Relationship Matrix II

\begin{equation} var(g) = G * \sigma_g^2 = UU^T\sigma_q^2 \notag \end{equation}

\begin{equation} G = \frac{UU^T}{2 \sum_{j=1}^m p_j(1-p_j)} \notag \end{equation}

charlotte-ngs/asmss2022 documentation built on June 7, 2022, 1:33 p.m.