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 σ 2 = F(1 − F) ( F is the inbreeding coefficient) is comprised of a component σ w 2 within populations and a component σ b 2 between replicate populations. These variances increase in time to a maximum at about 1.1 N e generations for σ w 2, about 2.3 N e generations for σ b 2, and about 1.4 N e generations for σ 2, and decrease thereafter ( N e is effective population size). The ratio σ b 2 σ 2 is ever increasing to an asymptote in the range 0.4-0.5 depending on N e and the mating system. For finite populations with variation in pedigree F's, there are contributions σ w F 2 within and σ b F 2 between populations. The component σ b F 2 is insignificant except for very small populations, and σ w F 2 is largest in the early generations and then decreases roughly as (1 − F) 2 KN e 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, σ w F 2 accounts for onehalf to two-thirds of σ w 2. Bringing in more loci leads to the decomposition of the total variance into four components whose values are affected by linkages among the loci. The relationships between these components and σ w 2, σ b 2, σ w F 2, and σ b F 2, are elaborated in terms of tight and loose linkage. The exact computations of σ w F 2 and σ b F 2 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.