In technical terms it works as follows. To be able to estimate the breeding value we need to re-arrange the formula for calculating the regression coefficient (b):
b = cov(x,y)/var(x)
In the formula b is the regression coefficient, var(x) = variance of the phenotypic superiority, cov(x,y) = relation between the phenotypic superiority and the true breeding value: our estimated breeding values. So we can rearrange our formula to:
Cov(x,y) = b * var(x)
Var(EBV) = b * var(phenotypic superiority)
For individual animals we can translate this to:
EBV = b*phenotypic superiority
The better the phenotypic superiority is predicting the true breeding value of the animals, the more the cov(x,y) will resemble cov(y,y), which equals var(y) = variance of the true breeding values, but at the same time cov (x,x), the variance of the phenotypic observations. In other words: the more the regression coefficient will approach 1.
As a final step: in animal breeding we are interested in identifying the genetically superior animals. That is easier if we would express their EBV relative to an average animal. Any positive EBV would indicate a better than average animal. This is easier than just giving the unadjusted EBV. For example, if you would know that your animal has an EBV of 25, that is nice, but not very informative if you don’t know how the other animals score. It is much more informative if you would know that the average animal scores 23, so your animal scores +2. For convenience we, therefore, express the EBV relative to the average.
The phenotypic superiority can be calculated as (P- Pgemiddeld): the phenotypic information on the animal – the population average. Consequently, the true breeding value is also expressed as genetic superiority: (A-Agemiddeld) , and the EBV is an estimate of that.
The formula then becomes:
EBV = b * (P- Pgemiddeld)
This formula is the most basic formula to estimate the breeding value of an animal: it combines the phenotypic superiority of the animal and the regression coefficient of the genetic superiority on the phenotypic superiority.
Thus:
For estimating the breeding values of animals we try to find the best regression coefficient and the most informative phenotypic information so that our EBV comes as close to the TBV as possible.
EBV are expressed relative to the average animal to simplify identification of genetically superior animals.