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Subpopulation Theory

We can deal with a structured population by using a theory that is very similar to that of inbreeding. We shall reserve the symbol F for inbreeding caused by a specified degree of relationship of the parents, such as cousins. The symbol is sometimes used in forensic science, so we employ it to designate the effects of population subdivision. The following formulae, which are analogous to those for inbreeding, define a parameter _ij for each genotype A_iA_j. These formulae do not require that the subpopulations mate at random or even that they be distinct.

(4.4a)
(4.4b)

In general, the parameters may be positive or negative. However, substituting the inequalities P_{ii £}  p_i and P_ij £ 1 into equations 4.4a and 4.4b, respectively, demonstrates that for every i and j.

Let f₀ denote the actual homozygosity in the entire population, and let h₀ = 1 - f₀ denote the corresponding heterozygosity. If the population were divided into distinct subpopulations and mating were random within each subpopulation, we would designate f_s and h_s by f_s and h_s, respectively. If mating were random within the entire population, these quantities would become f_T and h_T, respectively.

The average of the parameters _ij over all genotypes is precisely Wright's (1951) fixation index F_IT:

(4.5)

For an elementary explanation of Equation 4.5 for equal subpopulation numbers, see Hartl and Clark (1989, p 293); Nei (1987, p 162) presents a more detailed treatment. We also provide an alternative and more general derivation (Appendix 4A).

It is clear that is a composite quantity, averaged over all genotypes, whereas Equations 4.4 involve and for individual genotypes. In general, may be positive or negative, but . However, if the local populations are mating at random or if there is local inbreeding, then the true value of is positive. In empirical data, if statistical uncertainties are taken into account, is almost always positive or very small. For selectively neutral loci, population values of for particular genotypes may be negative only temporarily, except in highly unusual situations. Of course, point estimates from samples, which are quite inaccurate, may be negative even when the true value is positive (Weir and Cockerham 1984; Nei 1987; Chakraborty and Danker-Hopfe 1991).

Most of the forensic literature posits distinct subpopulations in HW proportions. In that case, comparison of Equations 4.4 with Equations 4.3 shows that _ij and _ij are given by

(4.6a)