LBG - FS2024 – Exercise 3

Author

Peter von Rohr

Problem 1: Parent Offspring Breeding Values

As shown in the course notes, the breeding value $u_{i}$ of animal $i$ can be decomposed into the average of the parent breeding values plus a mendelian sampling term ( $m_{i}$ ). This means

$u_{i} = \frac{1}{2} u_{s} + \frac{1}{2} u_{d} + m_{i}$

where animal $i$ has parents $s$ and $d$ . The mendelian sampling term $m_{i}$ is the deviation of the single breeding value $u_{i}$ from the parent average breeding value. Because $m_{i}$ is modelled as a deviation, it follows that for a large number ( $N$ ) of offspring from parents $s$ and $d$ , the average over all mendelian sampling terms must be $0$ .

Your Task

Show that the average mendelian sampling term over a large number of offspring is $0$ using a single locus model for the following cases.

Case 1: Homozygous and Heterozygous Parents

Parent $s$ with genotype $G_{1} G_{1}$ and parent $d$ with genotype $G_{1} G_{2}$

Case 2: Homozygous and Heterozygous Parents

Parent $s$ with genotype $G_{2} G_{2}$ and parent $d$ with genotype $G_{1} G_{2}$

Case 3: Heterozygous Parents

Both parents $s$ and $d$ have genotype $G_{1} G_{2}$

Your Solution

For each of the following cases compute the parent average of breeding values.
Compute the difference between the breeding values of every possible offspring and the parent average
Compute the average over all mendelian sampling terms