In charlotte-ngs/lbgfs2021: Livestock Breeding and Genomics (ETHZ) - Fall Semester 2021

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Generic Relation for Single Observation

• Data: parent $i$ with $k$ offspring performances
• Variance according to genetic model

$$var(y) = var(u) + var(e) = \sigma_y^2 = \sigma_u^2 + \sigma_e^2$$ * Genetic model of observation for offspring $k$

$$y_k = \mu + u_k + e_k = \mu + {1\over 2}(u_i + u_d) + m_k + e_k$$ where $i$ and $d$ are parents of offspring $k$

Variance for Single Observation

The phenotypic variance $\sigma_y^2$ is computed by $var(y_k)$

$$\sigma_y^2 = var(y_k) = var(\mu + {1\over 2}(u_i + u_d) + m_k + e_k)$$

$$ = var(\mu) + var({1\over 2}u_i) + var({1\over 2}u_d) + var(m_k) + var(e_k)$$

$$= {1\over 4}var(u_i) + {1\over 4} var(u_d) + var(m_k) + var(e_k)$$

Permanent and Non-Permanent Variance Components

• Permanent refers to what is constant across all offsprings ($1 \ldots k$) of parent $i$ which is only ${1\over 4}var(u_i)$
• Define $t$ as the ratio between the parmanent and the total variance, hence

$$t = \frac{{1\over 4}var(u_i)}{var(y_k)} = \frac{h^2}{4}$$

• Consequently

$${1\over 4}var(u_i) = t * \sigma_y^2$$

$$(1-t)\sigma_y^2 = {1\over 4} var(u_d) + var(m_k) + var(e_k) $$

Parent $i$

• Observations of $n$ offspring of parent $i$

$$\bar{y_i} = {1\over n}\sum_{k=1}^n y_k $$

• Assume $n$ offspring are half-sibs

$$\bar{y_i} = {1\over n}\sum_{k=1}^n y_k = {1\over n}\sum_{k=1}^n \left[ \mu + {1\over 2}(u_i + u_{d,k}) + m_k + e_k \right] $$

charlotte-ngs/lbgfs2021 documentation built on Dec. 19, 2021, 3:01 p.m.