Variance of actual inbreeding

Abstract

The variances of actual inbreeding and coancestry in terms of their corresponding identities by descent were studied for finite populations. For inbreeding at a single locus, the total variance sigma 2 equal F(1-F) (F is the inbreeding coefficient) is comprised of a component sigma 2w within populations and a component sigma 2b between replicate populations. These variances increase in time to a maximum at about 1.1Ne generations for sigma 2w, about 2.3Ne generations for sigma 2b, and about 1.4 NE generations for sigma 2, and decrease thereafter (Ne is effective population size). The ratio sigma 2b/sigma 2 is ever increasing to an asymptote in the range 0.4-0.5 depending on Ne and the mating system. For finite populations with variation in pedigree F's, there are contributions sigma 2wF within and sigma 2bF between populations. The component sigma 2bF is insignificant except for very small populations, and sigma 2wF is largest in the early generations and then decreases roughly as (1-F)2/KNe where K is formulated in terms of the mating strategy and the degree of avoidance of mating relatives. An additional degree of avoidance increases K by a factor of 4. In a large population at equilibrium with respect to mixed self and random mating, sigma 2wF accounts for one-half to two-thirds of sigma 2w. Bringing in more loci leads to decomposition of the total variance into four components whose values are affected by linkages among the loci. The relationships between these components and sigma 2w, sigma 2wF, and sigma 2bF are elaborated in terms of tight and loose linkage. The exact computations of sigma 2wF and sigma 2bF require the use of two locus descent measures without linkage. The variances of various averages of actual identities by descent, such as the proportions for individuals or populations, are formulated for a sample of individuals.