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one to determine the chance of such a random match. Let the frequency of an allele in the matching haplotype be pi where i indicates the locus. At each locus the chance of matching at random is simply the probability that an individual is either homozygous (p2) or heterozygous (2pi(1 - p)) for the allele in question. Thus, the total probability of a random match for one locus is:

under the assumptions of random mating and panmixia. Assuming that all of the loci used in a parentage analysis are independent, the probability of a random match for all loci in a given haplotype is the product of the locus by locus frequency of a random match, or:

P(multilocus random match)

loci

where n indicates chain multiplication over all loci.

Returning to our C. alia example, we can calculate the chances of a random match for each of the paternal haplotypes. The haplotypes, allele frequencies (see Table 4.3), probability of a random match at each locus, and probability of a random match at all five loci are given in Table 4.4. Focus first on the haplotype for tree 1946. Given that allele 32 7 at locus A has an observed frequency of 0.2703 in the population of candidate parents (which is an estimate of the allele frequency in the entire population), the chance of any genotype having one copy of this allele is (0.2703)2 + 2(0.2703)(1 - 0.2703) = 0.4675. We therefore expect 46.75% of individuals in the population to have a genotype with either one or two copies of the 327 allele. This is the same as the probability that an individual taken at random from the population (and not necessarily included in the sample of candidate parents) could provide the correct haplo-type to be included as a possible father of seed 989 1-1 in Table 4.2. The chances of a random match at each of the five loci is calculated in the same fashion. We see that a genotype that would compliment the known parent's haplotype and explain the seed genotype is expected to occur between about 2 and 47% of the time for any single locus. When these probabilities are combined across all five loci the expected frequency of a random match becomes very small. As shown in Table 4.4, the expected frequency of a random match at all five loci is between 44 in 1000 and 66 in 1,000,000 genotypes under the assumption of random mating. This is a demonstration of the general principle that the ability to distinguish true parentage from apparent parentage due to random matches depends on both the allele frequencies at each locus as well as the total number of loci available. Random matches become less likely as allele frequencies decrease and as the number of independent loci increases.

Table 4.4 The chance of a random match for the included fathers in Table 4.2. The probability of a random match at each locus is p2 + 2p;(1 - ). The combined probability of a random match for all loci in the haplotype is the product of the probabilities of a random match at each independent locus. Paternal haplotype data are treated as missing (0) for the purposes of probability calculations when progeny genotype data are missing. In the cases where the paternal haplotype has multiple possible alleles at some loci, the highest probability of a chance match is given. The allele frequencies for each locus are given in Table 4.3.

Microsatellite haplotype

- P(multilocus

Included father A B C D E random match)

1946 (seed 1-1)

0 0