Inverse Numerator Relationship Matrix

Problem 1: Inverse Numerator Relationship Matrix

During the lecture the method of computing the inverse numerator relationship matrix \(A^{-1}\) directly was introduced. The computation is based on the LDL-decomposition. As a result, we can write

\[A^{-1} = (L^T)^{-1} \cdot D^{-1} \cdot L^{-1}\] where \(L^{-1} = I-P\), and \(D^{-1}\) is a diagonal matrix with \((D^{-1})_{ii} * \sigma_u^{-2} = var(m_i)^{-1}\).

Tasks

• Use the example pedigree given below and compute the matrices \(L^{-1}\) and \(D^{-1}\) to compute \(A^{-1}\)
• Verify your result using the function getAinv() from package pedigreemm.

Pedigree

nr_animal <- 6
tbl_pedigree <- tibble::tibble(Calf = c(1:nr_animal),
                               Sire = c(NA, NA, NA, 1 ,3, 4),
                               Dam = c(NA, NA, NA, 2, 2, 5))
tbl_pedigree

Solution

Problem 2: Rules

The following diagram helps to illustrate the rules for constructing \(A^{-1}\)

Tasks

• Go through the list of animals in the pedigree and write down the contributions that are made to the different elements of matrix \(A^{-1}\)
• Based on the different contributions, try to come up with some general rules

Solution