A proximal decomposition of inbreeding, coancestry and contributions

Wright's (1922) coalescence probabilistic method decomposes the average coancestry or the average inbreeding of the population of interest into contributions of ancestors who are directly responsible for coancestry. These ancestors are less numerous and younger than those given by Caballero & Toro's (2000) method. Then, Wright's method yields a more parsimonious representation of the origin of coancestry or inbreeding. Results of Wright's method, called proximal because ancestors are younger, can be obtained from the latter method using a simple matrix transformation. Detailed algorithms are presented. Both methods can be used for assessing the overall efficiency of a breeding scheme in balancing genetic gain and coancestry rate, yielding very similar results when generations are discrete.