LBG - FS2024 – Exercise 6

Author

Peter von Rohr

Problem 1: Numerator Relationship Matrix

The following pedigree is given

s_ped_data <- "https://charlotte-ngs.github.io/lbgfs2024/data/ped_num_rel_mat.csv"
tbl_pedigree <- readr::read_delim(file = s_ped_data, 
                                  col_types = readr::cols(Calf = readr::col_integer(),
                                                          Sire = readr::col_integer(),
                                                          Dam = readr::col_integer()))
knitr::kable(tbl_pedigree, booktabs = TRUE, longtable = TRUE)

The pedigree can be read from the file:

Compute the numerator relationship matrix $A$ for the given pedigree. Recall from the course notes that elements of matrix $A$ are computed differently for elements on the diagonal and for off-diagonal elements. In summary, we compute

diagonal element $(A)_{i i}$ as $(A)_{i i} = 1 + F_{i}$ where $F_{i} = 0.5 * (A)_{s d}$ where $s$ and $d$ are parents of $i$ .
off-diagonal element $(A)_{k i}$ as $(A)_{k i} = 0.5 * [(A)_{k s} + (A)_{k d}]$ where $s$ and $d$ are parents of $i$

Task

Use two nested loops over the rows and the columns of matrix $A$ to compute all the elements of matrix $A$ using the formulas given above. You can use the following steps to get to the solution

Read the pedigree from the given file
Add parents that do not appear as animals to the column of animals
Create an empty square matrix $A$ with as many rows and columns as we have animals in the pedigree.
Use the above formula to compute the diagonal element of the first row
Create a first loop to compute all the off-diagonal elements of the first row
Create a loop to repeat the computations over all rows.

Solution

Pedigree is read from the given file

R Code

# read pedigree

_webr_editor_3 = Object {code: null, options: Object, indicator: Ke}

Find animals that appear only as parents, start with sires

R Code

# find founder sires

_webr_editor_4 = Object {code: null, options: Object, indicator: Ke}

The same for the dams

R Code

# find founder dams

_webr_editor_5 = Object {code: null, options: Object, indicator: Ke}

Combining them into one vector and sorting them

R Code

# combine founders of both sexes

_webr_editor_6 = Object {code: null, options: Object, indicator: Ke}

Add pedigree records for founder animals at the top of the pedigree

R Code

# extend pedigree with founder records

_webr_editor_7 = Object {code: null, options: Object, indicator: Ke}

An empty matrix $A$ is initialized

R Code

# start with empty matrix A

_webr_editor_8 = Object {code: null, options: Object, indicator: Ke}

Start with the diagonal elemenent for animal 1

R Code

# diagonal for animal 1

_webr_editor_9 = Object {code: null, options: Object, indicator: Ke}

Offdiagonal elements belonging to animal 1

R Code

# offdiagonals in first row and first column

_webr_editor_10 = Object {code: null, options: Object, indicator: Ke}

The above two steps can be extended to all rows

R Code

# extension to all animals

_webr_editor_11 = Object {code: null, options: Object, indicator: Ke}

Problem 2: Verification

Use the function pedigreemm::getA() from package pedigreemm to verify your result from problem 1.

Solution

The pedigree is defined by

R Code

# define pedigree in pedigreemm based on extended version of pedigree

_webr_editor_12 = Object {code: null, options: Object, indicator: Ke}

The numerator relationship matrix is computed by

R Code

# compute numerator relationship matrix

_webr_editor_13 = Object {code: null, options: Object, indicator: Ke}

Check whether matrices are identical

R Code

# check whether matrices are the same

_webr_editor_14 = Object {code: null, options: Object, indicator: Ke}

Downloading webR