Imaginings of a Livestock Geneticist
This section describes how to generate the inverse of A (A-1) based on simple addition rules Below are the rules Henderson (1976) generated to calculate A-1 and more importantly every element doesn't need to be stored in memory. In summary all you need to generate A-1 is a list of sires and dams on an individual along with the the diagonal elements of L (i.e. inbreeding coefficients).
The bii value refers to (D-1)2 and its computation is outlined in a
previous section . Lastly, if you assume the population isn't inbred then the bi values will always be a particular value depending on whether both, 1 or
none of the parents are known. Look back at a previous section and you will notice how those values were obtained.
After playing around with simple pedigrees you will begin to see where the values to add to particular elements within the matrix come from. I will leave that up
to you!!
Similar to previous methods, the pedigree has to be sorted so that parents come before progeny. Due to the pedigree being sorted, the recursive
method has a very simple form. Look at the 'Recursive Method to Create G Inverse" sections R Code to see the impact of using a pedigree that isn't sorted. Lastly,
if animals are numbered from 1 to the very last animal then the sire and dam values can be used to index where the respective elements are located within the algorithm.
Algorithms to sort and renumber a pedigree is outlined in the following section.
The following pedigree file from Henderson (1976) can be utilized with the R code above. The columns are animal, sire and dam and the pedigree is already ordered so that parents come before progeny. Lastly the animals go from 1 to the total number of animals. Outlined below is what A-1 looks like at the end of each iteration of the for loop.
