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( 4.10b)

Nothing in population-genetics theory tells us that should be independent of genotype. In fact, there is likely to be a different for each pair of alleles A_i and A_j. Since individual genotypes are usually rare, these values are inaccurately measured and ordinarily unknown. The best procedure is to use a conservative value of in Equations 4.10, knowing that the true individual values are likely to be smaller. Balding and Nichols (1994) extend Equations 4.10 to account for undetected bands. They also give an upper limit for homozygotes, analogous to the 2p rule. Their upper bound on the conditional probability is . We believe, however, that because Equation 4.10a is already conservative, this rule is usually unnecessary.

The value of has been estimated for several populations. As mentioned above, typical values for white and black populations are less than 0.01, usually about 0.002. Values for Hispanics are slightly higher, as expected because of the greater heterogeneity of this group, defined as it is mainly by linguistic criteria.

Table 4.9 gives numerical examples of calculations for three racial groups. using the data of Table 4.8. Two alternative assumptions are made: that the evidence profile is heterozygous (there are two clear bands) at all four loci, and that locus A has a single band at allele A₆. In this example, the three racial groups are very similar; if all are heterozygous or if the 2p rule is used for homozygotes, they are within a factor of 3. That will not always be true. If one locus is single-banded, the 2p rule makes a substantial difference in the calculation. With four multiallelic loci, such as VNTRs, most four-locus profiles will be heterozygous at all loci. (For example, if the heterozygosity per locus is 0.93, as it is for D2S44, the probability that all four loci will be heterozygous is about 0.75.)

If all loci are heterozygous, then assuming that the evidence DNA and the DNA from the suspect came from the same subpopulation, using Equations 4.10 has a fairly small effect on the calculations when . However, using a value of decreases the likelihood ratio (increases the match probability—see Chapter 5) by a factor of 10. If the A locus is homozygous, then Equation 4.1 a with the 2p rule is more conservative than Equation 4.10a with and very close to Formula 4.10a with .

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various statistical assumptions concerning the distribution of allele frequencies among the subpopulations. A more appropriate model would be a stepwise-mutation theory because VNTR lengths tend to change by small steps, but that has not been worked out. Even that would not be completely satisfactory unless one also takes migration, which may be more important than mutation, into account. When is small (< 0.02), the formulae derived from different models agree closely. Although the specific models are highly idealized, when different assumptions lead to similar results, it increases our confidence in the final formulae. The formulae given are from Balding and Nichols (1994), and were chosen because they are both simple to evaluate and accurate.