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The theory may become more clear with an example. In dairy cattle breeding, for example, a young bull only has a breeding value for milk production based on that of his parents. He may have a very high EBV, the accuracy of that estimate is low. His father may have an EBV with high accuracy of 90% (larger group of daughters with performance records), his mother only has her own performance and maybe some information of relatives, so her accuracy will be around 35%. The accuracy of the breeding value of a son or a daughter will be equal to √(0.25*reliability of the sire + 0.25* reliability of the dam). Thus the accuracy of the breeding value of the young bull will be √(0.25*90 0.81 + 0.25*35 = 31.25%0.1225) = √0.233125= 48%. Important reason why it is so much lower than the average of this parents is because of the Mendelian sampling: you know he inherited half of his genetics from his father and half from his mother. But you don’t know which half. He may have inherited the best halves of each, resulting in a better than expected young bull. But he may also have inherited the worst half of both parents, resulting in very disappointing performance of his daughters. Because you will have to wait for his daughters to produce milk before you get an impression of how good his genetics really are. And you need many daughters before you can be 90% sure (accuracy of 90%). Despite this insecurity, the young bull with the highest EBV still is expected to be the best bull. This example was with dairy cattle but of course exactly the same story could be told for any other type of animal. The main message is that even though you know that the parents are good (high accuracy of high EBV), still the offspring may perform different than expected because of the effect of the Mendelian sampling component.

 

Thus:

The accuracy of the EBV of young offspring is not equal to the average of that of their parents because of the relatively large influence of Mendelian sampling: breeding remains gambling until you have insufficient information to estimate the EBV accurately.