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individual alleles to estimate the frequency of a given profile. That approach necessitates some assumptions about the mating structure of the population, and that is where population genetics comes in.1

Allele and Genotype Proportions

It is conventional in genetics to designate each gene or marker locus with a letter and each allele at that locus with a subscript numeral. So, A10 designates the tenth allele at locus A, B5 the fifth allele at locus B, and so on. When we want a statement to apply to any of the alleles of a given locus, we use a literal subscript, such as i or j. We designate the frequencies (it is customary to use the word frequency for relative frequency, meaning proportion) of alleles with the letter p and a corresponding subscript. Thus, the frequency of allele A3 is p3 and of allele Ai is pi. The sum of all the pi values is 1 because it includes all the possibilities. Symbolically, if S stands for summation, Spi = 1.

At the DQA locus, discussed in Chapter 2, six alleles are customarily used in forensic analysis (Table 4.1). For example, allele D1.1 (designated as 1.1 in the table), has a proportion of 0.150, or 15.0%, in the black population; this was computed from the proportions in the right-hand portion of the table. The first six genotypes include the 1.1 allele (the top one has two copies) and adding their frequencies—0.036 + (0.076 + 0.009 + 0.036 + 0.027 + 0.080)/2—yields 0.150. The division by 2 is because in heterozygotes only half the alleles are D1.1·

Random Mating and Hardy-Weinberg Proportions

In the simplest population structure, mates are chosen at random. Clearly, the population of the United States does not mate at random; a person from Oregon is more likely to mate with another from Oregon than with one from Florida. Furthermore, people often choose mates according to physical and behavioral attributes, such as height and personality. But they do not choose each other according to the markers used for forensic studies, such as VNTRs and STRs. Rather, the proportion of matings between people with two marker genotypes is determined by their frequencies in the mating population. If the allele frequencies in Oregon and Florida are the same as those in the nation as a whole, then the proportions of genotypes in the two states will be the same as those for the United States, even though the population of the whole country clearly does not mate at random.

We use random mating to refer to choice of mates independently of genotype at the relevant loci and independently of ancestry. The expected proportions with

1An elementary exposition of population genetics is found in Hartl and Clark (1989). A more advanced text, with discussion of many of the formulae used here, is Nei (1987). Practical details of estimation and analysis are given by Weir (1990). See also Weir (1995a).

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