Kinship coefficient (${\varphi }_{XY}$)

The kinship coefficient ${\varphi }_{XY}$ is the probability that one randomly selected allele copy from each of $X,Y$ is identical by descent. For a noninbred individual, ${\varphi }_{XX}=1/2$, and more generally ${\varphi }_{XX}=(1+F_X)/2$. For distinct individuals with parents $M_X,P_X$, Mendelian segregation gives ${\varphi }_{XY}=[{\varphi }_{M_{X}Y}+{\varphi }_{P_{X}Y}]/2$. The offspring's inbreeding coefficient equals the parents' kinship coefficient.

// Target: biology.bigb

For $N$ individuals in monogamous families of exactly two offspring, let ${\Phi }_G$ be mean pairwise kinship coefficient, $F_G$ mean inbreeding and ${\Phi }_{m,G}$ mean kinship of mates. A random distinct offspring pair are siblings with probability $1/(N-1)$. Accounting separately for within-family and cross-family pairs gives the displayed recurrence, while $F_{G+1}={\Phi }_{m,G}$. Random mating makes ${\Phi }_{m,G}={\Phi }_G$. Sibling avoidance changes the mating-pair mean; it does not create unrelated founders or stop eventual drift to ancestral identity in a closed finite population.

For otherwise outbred opposite-sex first cousins, the probability that the mother's randomly transmitted X copy matches the father's X by descent depends on which of their parents are siblings. It is zero when the male cousin is related through his father, $1/8$ when the female cousin is related through her father and the male through his mother, and $3/16$ when their mothers are sisters. Daughters can be X-autozygous; sons have one X and a Y and are not homozygous for a paired X chromosome. Recombination and pseudoautosomal regions require a locus-based formulation instead of whole-chromosome blocks.