Inclusion of genetically identical animals to a numerator relationship matrix and modification of its inverse

In the field of animal breeding, estimation of genetic parameters and prediction of breeding values are routinely conducted by analyzing quantitative traits. Using an animal model and including the direct inverse of a numerator relationship matrix (NRM) into a mixed model has made these analyses possible. However, a method including a genetically identical animal (GIA) in NRM if genetic relationships between pairs of GIAs are not perfect, is still lacking. Here, we describe a method to incorporate GIAs into NRM using a K matrix in which diagonal elements are set to 1.0, off-diagonal elements between pairs of GIAs to (1-x) and the other elements to 0, where x is a constant less than 0.05. The inverse of the K matrix is then calculated directly by a simple formula. Thus, the inverse of the NRM is calculated by the products of the lower triangular matrix that identifies the parents of each individual, its transpose matrix, the inverse of the K matrix and the inverse of diagonal matrix D, in which the diagonal elements comprise a number of known parents and their inbreeding coefficients. The computing method is adaptable to the analysis of a data set including pairs of GIAs with imperfect relationships.