sl/w03_gen_eval/lbgfs2020_slides_w03_gen-eval.md

In charlotte-ngs/lbgfs2020: Livestock Breeding and Genomics (Fall 2020)

Genetic Evaluation

Peter von Rohr 02 Oktober 2020

Consequences of Definition of Breeding Value

• Based on the average of a large number of offspring, because
  • offspring inherit a random sample of parental alleles
  • average over a large number of offspring reduces sampling effect
• The breeding value is defined as a deviation from the population mean
  • population mean depends on allele frequencies which are specific for each population
  • hence breeding values can only be compared within one population.
• Because the breeding value is defined as a deviation its expected value of the breeding value is (0)

The Basic Model

Re-arranging Terms

New Model

Infinitesimal Model

• Central Limit Theorem for (u_i) and (e_{ij}) lead to multivariate normal distributions with
  • (E(u) = 0) and (E(e) = 0) and
  • Known variances and co-variances
  • No co-variances between (u_i) and (e_{ij})
• (\mu) is assumed to be constant for a given evaluation
• Phenotypic observation (y_{ij}) is the sum of two normally distributed random variables, therefore
  • (y_{ij}) also follows a multivariate normal distribution
  • (E(y) = \mu)

Central Limit Theorem

Decomposition of Breeding Value

Basic Principle of Predicting Breeding Values

Breeding values are predicted according to the following two steps.

1. Observations corrected for the appropriate mean performance values of animals under the same conditions
   • conditions are described by the effects captured in (\mu_i).
2. The corrected observations are weighted by a certain factor
   • factor reflects the amount of information available for prediction

Animal’s Own Performance - Single Record

• one phenotypic observation per animal
• search for prediction (\hat{u_i}) of the breeding value (u_i) of animal (i)
• assume (u_i) and (y_i) known for a certain population

(\rightarrow) plot

Plot (u) against (y)

Regression

• red line denotes regression line from (u_i) onto (y_i)
• because phenotypes have genetic basis (\rightarrow) connection between (u_i) and (y_i)
• measure for connection: regression coefficient (b)
• new model can be interpreted as regression

Regression Coefficient

where (h^2) is called heritability

Prediction

• Given a new (y_i), what would be the predicted (u_i)?
• Use regression line and compute (\hat{u_i})

Accuracy

• Measured as correlation between true breeding value (u) and selection criterion (y)

Response To Selection

Repeated Records

• Additional component of variation

• Predicted breeding value

Regression Coefficient

Putting Results together

Progeny Records

where

Covariance and Variance

with (t = \sigma_u^2 / 4)

Intra-Class (t)

• Progeny mean

• Variance

with (cov(.) = 0), (t = var({1 \over 2} u_i) / var(y) = h^2/4)

Results

with (k=\frac{4-h^2}{h^2}).

charlotte-ngs/lbgfs2020 documentation built on Dec. 20, 2020, 5:39 p.m.