Genetic Evaluation - Notebook 4

Problem 1: Prediction Of Breeding Values Using A Sire Model

The dataset for this exercise is available from

https://charlotte-ngs.github.io/gelasmss2021/data/gel_sire_ex04_p01.csv

to predict breeding values using a sire model. The sire model is a mixed linear effects model where the sire effects are random effects. This leads to the following model

\[\begin{equation} y = Xb + Zs +e (\#eq:gel-ex3-siremodel) \end{equation}\]

For the random effects, we have to specify the expected values and the variance-covariance matrices. Because the random errors \(e\) and the sire effects \(s\) are deviations from a common mean and hence their expected values are defined as

\[\begin{align} E\left[ e \right] = 0 \notag \\ E\left[ s \right] = 0 \notag \\ E\left[ y \right] = Xb \notag \end{align}\]

The expected value \(E\left[ y \right]\) is computed using the defined expected values for \(e\) and \(s\) and from the model @ref(eq:gel-ex3-siremodel).

The random error terms \(e_i\) are uncorrelated and hence the variance-covariance matrix \(var(e)\) is given by

\[\begin{equation} var(e) = I * \sigma_e^2 \notag \end{equation}\]

In the case where the sires are unrelated, the sire effects are also uncorrelated and the variance-covariance matrix \(var(s)\) corresponds to

\[\begin{equation} var(s) = I * \sigma_s^2 \notag \end{equation}\]

The values for \(\sigma_e^2\) and \(\sigma_s^2\) are taken from the results of the variance components estimation from last weeks exercise. The variance-covariance matrix of the observations \(y\) can be computed as

\[\begin{equation} var(y) = ZZ^T * \sigma_s^2 + I * \sigma_e^2 \notag \end{equation}\]

Hints

• Use the package pedigreemm to predict the breeding values for all the sires.
• The function ranef() can be used to extract realised values of random effects.

Your Solution

Start by reading the data from the given URL

# read data with readr::read_csv2()

Convert all fixed effects into factors

# use as.factor() for the conversion

Create pedigree

# use pedigreemm::pedigree() to create the pedigree

Analyse the data

# use require(pedigreemm);pedigreemm() to do the analysis

Extract predicted breeding values

# use function raneff() to extract solutions

Problem 2: Compare Offspring Of Sires

For the purpose of livestockbreeding the realised values themselves are not so interesting. But for the selection decision require a ranking of the sires according to their breeding values. Find the ranking of the sires according to their breeding values.

According to the definition of breeding value, it corresponds to the deviation of the offspring from the population mean. Hence the offspring of the sire with the best breeding value should on average be better than the offspring of the sire with the worst breeding value. Verify the difference between the average phenotypic values of the offsprings of the sires with the best and the worst predicted breeding values.

Your Solution

Group the average of the offpsring performances by the sires.

# grouping using dplyr::group_by() and dplyr::summarise()

Compare average of offspring performance of sire with best and worst breeding value

# use dplyr::filter() to extract offspring value for sire with best and worst predicted breeding value

Problem 3: Predict Breeding Values Using Animal Model

As in Exercise 2, we are using the full dataset to predicted breeding values with an animal model. The computations for the solution of this Problem will have a very long runtime. That is why the solution is only sketched and not explicitly computed.

The data is available from https://charlotte-ngs.github.io/gelasmss2021/data/gel_bp_ex04_p03.csv.

Your Solution

Use the same steps as in the solution for Problem 1, but use an animal model instead of a sire model. The difference between a sire model and an animal model is mainly reflected by the pedigree.

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Latest Changes: 2021-05-07 10:59:41 (pvr)