Applied Statistical Methods – Exercise 1

Problem 1: Linear Regression

Use the example dataset from the course notes which is used to demonstrate how to fit a regression of the response variable body weight (BW) on the predictor variable breast circumference (BC). The data is shown in the table below.

Dataset for Regression of Body Weight on Breast Circumference for ten Animals

Animal	Breast Circumference	Body Weight
1	176	471
2	177	463
3	178	481
4	179	470
5	179	496
6	180	491
7	181	518
8	182	511
9	183	510
10	184	541

Your Tasks

• Compute the regression coefficient using matrix computations. Use the function solve() in R to compute the inverse of a matrix.
• Verify your results using the function lm in R.

Your Solution

Please start your solution here, by completing the following R-code-chunks.

Problem 2: Breeding Values

During the lecture the computation of the breeding values for a given genotype was shown for a completely additive locus which means the genotypic value \(d\) of the heterozygous genotypes is \(0\). In this exercise, we want to compute the general solution for the breeding values of all three genotypes under a monogenic model. The term monogenic model is equivalent to a single-locus model.

We are given a single locus \(G\) with two alleles \(G_1\) and \(G_2\) which are closely linked to a QTL for a trait of interest. We assume that the population is in Hardy-Weinberg equilibrium at the given locus \(G\). It is important to note here, that the breeding values under this single-locus model are not the same as the direct genomic breeding values. In one of the following exercises, we will come back to this difference.

The allele frequencies are

Allele	Frequency
G1	p
G2	q

Allele \(G_1\) is the one with a positive effect on the trait of interest. The genotypic values are given in the following table.

Genotype	Value
G1G1	a
G1G2	d
G2G2	− a

Your Task

• Compute the breeding values for all three genotypes \(G_1G_1\), \(G_1G_2\) and \(G_2G_2\).
• Verify the results presented in the lecture by setting \(d=0\) in the breeding values you computed before.

Your Solution

Please start your solution here by first computing the breeding values for the three genotypes under a single-locus model. Then insert the numbers given in the problem description.

Problem 3: Linkage Between SNP and QTL

In a population of breeding animals, we are given a trait of interest which is determined by a QTL \(Q\) on chromosome \(1\). QTL \(Q\) is modelled as a bi-allelic QTL with alleles \(Q_1\) and \(Q_2\). Furthermore we have genotyped our population for two SNPs \(R\) and \(S\) with two alleles each. One of the SNPs is on chromosome \(1\) and is closely linked to \(Q\). The other SNP is on chromosome \(2\) and is unlinked. Figure @ref(fig:linkageqtlsnp) shows the situation in a diagram.

Based on the following small dataset, determine which of the two SNPs \(R\) and/or \(S\) is linked to QTL \(Q\).

Dataset showing linkage between SNP and QTL

SNP R	SNP S	Observation
R2R2	S1S1	23.17
R2R2	S2S2	-27.04
R1R2	S1S2	-2.79
R1R2	S2S2	-19.54
R1R2	S2S2	-24.05
R1R2	S1S1	25.84
R1R2	S1S2	-0.36
R1R1	S2S2	-23.34
R2R2	S1S2	1.38
R1R1	S1S2	-1.60
R1R2	S1S2	-2.97
R2R2	S1S2	-1.39

From the above table it might be difficult to decide which SNP is linked to the QTL. Plotting the data may help. Showing the observations as a function of the genotypes leads to Figure @ref(fig:problem2plot).

Your Tasks

• Determine which of the two SNPs \(R\) or \(S\) is closely linked to the QTL
• Estimate the value for \(a\) based on the data
• Try to fit a linear model through the genotypes that SNP which is linked to the QTL using the lm() function. The genotype data is available from

https://charlotte-ngs.github.io/gelasmss2021/data/asm_w02_ex01_p02_genodatafile.csv

Your Solution

Please start your solution here. First determine which of the two loci is linked by visually inspecting the given scatter-plots. Then estimate the marker effects based on the data. The marker effects can be obtained from the results of the linear model.