Estimated Breeding Value
When calculating EBV (estimated breeding value) for an animal, we usually want to combine information from various sources. We have results from the animal itself, but also from its relatives. However, EBV is always for one trait only.
The formula for an EBV is
 = b1x1 + b2x2 + ... bnxn
where  is the EBV, b1 is the regression coefficient for trait 1 and x1 is the result for trait 1. Seems simple, doesn't it? Now all we need to do is calculate the b values. The key to the b values is to remember where dealing with several bits of information at once, and every "bit" is actually an equation
where  is the EBV, y1 is the animal's own result in trait 1, μ is the population mean result in trait 1, a1 is the additive genetic effects contributing to the trait and e denotes the environmental factors contributing to the trait. Simply put: an EBV consists of genetics and environmental factors.
So, the b's must fulfill equations for traits 1 ... n at the same time. Instead of a group of equations we use matrices to calculate the b's. Only then can we continue to calculating the actual selection index. The matrix notation for calculating b's is
 = (y1 - μ) = a1 + e1
where  is the EBV, y1 is the animal's own result in trait 1, μ is the population mean result in trait 1, a1 is the additive genetic effects contributing to the trait and e denotes the environmental factors contributing to the trait. Simply put: an EBV consists of genetics and environmental factors.
So, the b's must fulfill equations for traits 1 ... n at the same time. Instead of a group of equations we use matrices to calculate the b's. Only then can we continue to calculating the actual selection index. The matrix notation for calculating b's is
b = P-1G
Here P and G are matrices. P includes the variances and covariances between phenotypic results. The marking P-1 simply means that the transpose of the matrix P is used in the calculation. G is another matrix, which links the information sources to true breeding values. In the G matrix Ai denotes the true breeding value of animal i. Info1 and Info2 below are different information sources, for example animals 1 and 2.
You might remember that we never know the actual breeding value A, so we always work with the estimated BV, denoted as Â. However, with enough information and correct calculations it is assumed that A = Â. The trick in matrix G is to consider the genetic relationships between the information sources, here animals 1 and 2. If Info 1 is the animal itself, so that the source for info 1 = i, then Cov(Info 1, Ai) = Var(Ai). More generally,
So now we can calculate the EBV for one trait. What if we want to combine several traits into one number? Then we need a selection index. It works like EBV, but combines information from several sources and several traits into one.
A selection index can either be optimal or common. The difference is in the coefficients: optimal coefficients minimize the variance between true breeding values and estimated breeding values. In a common selection index the coefficient b is said to be "any b0", but in the optimal index b = P-1Cv. The optimal index considers covariances, and breeding accuracies impact the b coefficients.
Here we can see a new matrix, C. C is used if the measured traits are not the same as the traits to be improved. For example, we might measure weight and thickness of back fat, but we want to improve weights and percentage of lean meat. Now we need the matrix C, which relates to the other matrices as shown in the picture below.
If we want to include money to the calculations, we get a total breeding value. Money is used in breeding values to give economical weights to each trait. This weight is entirely decided by animal breeders, and based on what they think is most important. Economical weight isn't linked to genetics or phenotype in any way. It is simply a way to put the traits into some order of importance.Often economical values is derived from actual profits or costs regarding the trait in question. The weight is currency per 1 unit of increase/decrease in the trait, for example euros per +- 1 kg of meat or dollars per +- 1 weaned piglet. The economical value can be used to ompare the costs and profits between different breeding schemes.
For example:
We have two schemes for pig breeding. One scheme gives us - 0,5 piglets per sow, but 10 kg more meat since the surviving piglets are heavier. The other scheme gives + 0,7 piglets but -6 kg meat. Let us assume that 1 kg of meat is +10 euros and 1 piglet is 15 euros.
Now the first scheme yields (-0,5 * 15) + (10*10) = 92,5 euros, and the second scheme (0,7 * 15) + (-6 * 10) = -49,5 euros. With these exaggerated numbers it is easy to see which scheme would be more profitable for the producer.
Economic weight can also be used when restricting a selection index. We may want to improve one trait, but leave another trait untouched. In that case the economic value of the trait, which is not allowed to change, is set to 0,
Using indices and economic breeding values we can calculate a total breeding value for an animal. The formulas are
H = total breeding value, estimated using the index I
v = economical weights of traits
g = breeding values of traits
I = selection index
b = regression coefficients for traits
x = vector of observations [result1 result2 resultn]
P = covariances and variances between observations
G = covariances and variances between traits to be improved and measured traits.
Cameron, N. D. Selection Indices and Prediction of Genetic Merit in Animal Breeding. CAB International, USA. ISBN-13: 978-085199-169-6
GenUp-software for playing with genetics: http://www-personal.une.edu.au/~bkinghor/genup.htm
You might remember that we never know the actual breeding value A, so we always work with the estimated BV, denoted as Â. However, with enough information and correct calculations it is assumed that A = Â. The trick in matrix G is to consider the genetic relationships between the information sources, here animals 1 and 2. If Info 1 is the animal itself, so that the source for info 1 = i, then Cov(Info 1, Ai) = Var(Ai). More generally,
Cov(Info x, Ay) = a(x,y) * Var(A)
and
Var(A) = h2 * s.d. (P)
and
Var(A) = h2 * s.d. (P)
where a(x,y) is the coefficient of genetic relationship between animals x and y. If x and y are full siblings, their coefficient of genetic relationship is 0,5, and Cov(Info x, Ay) = 0,5 Var(A). If they're half-sibs, it's Cov(Info x, Ay) = 0,25 Var(A) and so on. s.d. (P) is the standard deviation of the trait P, or the trait for which we are calculating the breeding value for. Standard deviation of P is the square root of the variance of P.
Selection index and economic breeding value
So now we can calculate the EBV for one trait. What if we want to combine several traits into one number? Then we need a selection index. It works like EBV, but combines information from several sources and several traits into one.
A selection index can either be optimal or common. The difference is in the coefficients: optimal coefficients minimize the variance between true breeding values and estimated breeding values. In a common selection index the coefficient b is said to be "any b0", but in the optimal index b = P-1Cv. The optimal index considers covariances, and breeding accuracies impact the b coefficients.
Here we can see a new matrix, C. C is used if the measured traits are not the same as the traits to be improved. For example, we might measure weight and thickness of back fat, but we want to improve weights and percentage of lean meat. Now we need the matrix C, which relates to the other matrices as shown in the picture below.
If we want to include money to the calculations, we get a total breeding value. Money is used in breeding values to give economical weights to each trait. This weight is entirely decided by animal breeders, and based on what they think is most important. Economical weight isn't linked to genetics or phenotype in any way. It is simply a way to put the traits into some order of importance.Often economical values is derived from actual profits or costs regarding the trait in question. The weight is currency per 1 unit of increase/decrease in the trait, for example euros per +- 1 kg of meat or dollars per +- 1 weaned piglet. The economical value can be used to ompare the costs and profits between different breeding schemes.
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We have two schemes for pig breeding. One scheme gives us - 0,5 piglets per sow, but 10 kg more meat since the surviving piglets are heavier. The other scheme gives + 0,7 piglets but -6 kg meat. Let us assume that 1 kg of meat is +10 euros and 1 piglet is 15 euros.
Now the first scheme yields (-0,5 * 15) + (10*10) = 92,5 euros, and the second scheme (0,7 * 15) + (-6 * 10) = -49,5 euros. With these exaggerated numbers it is easy to see which scheme would be more profitable for the producer.
Economic weight can also be used when restricting a selection index. We may want to improve one trait, but leave another trait untouched. In that case the economic value of the trait, which is not allowed to change, is set to 0,
Total breeding value
Using indices and economic breeding values we can calculate a total breeding value for an animal. The formulas are
H = v'g
I = b'x where b = P-1Gv
H = total breeding value, estimated using the index I
v = economical weights of traits
g = breeding values of traits
I = selection index
b = regression coefficients for traits
x = vector of observations [result1 result2 resultn]
P = covariances and variances between observations
G = covariances and variances between traits to be improved and measured traits.
Additional information and sources
Mrode, R. A. Linear Models for the Prediction of Animal Breeding Values, 2nd edition. CABI Publishing, USA. ISBN-13: 978-085199-000-2Cameron, N. D. Selection Indices and Prediction of Genetic Merit in Animal Breeding. CAB International, USA. ISBN-13: 978-085199-169-6
GenUp-software for playing with genetics: http://www-personal.une.edu.au/~bkinghor/genup.htm
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