The Paternity Index, Population Heterogeneity, and the Product Rule

Recent controversy regarding the validity of the product rule in forensic DNA identification cases raises the possibility of living in a “product rule free” world. We have examined methods to determine PI’s across genetic loci without using the product rule. In analogy with HLA, collection of haplotypes from family studies would permit such calculations from tables, but fixed bins and very large numbers of haplotypes would be required. Following a suggestion of C. Brenner that a “product rule free” PI can be obtained directly from multilocus VNTR phenotype tables, we have devised a simple method that relies only on counting: i) Determine, from direct counting, the proportion of individuals in the multilocus phenotype data base that match at all loci tested. This yields the “exclusion probability” term of the PI [J. Morris. 3rd Intl Symp on Hum Ident 1992, Promega p. 177–190, 1993]. ii) For each of the randomly matching phenotypes, determine from direct counting the relative “fit” to the mother-child pair compared to the phenotype of the tested man. The average relative “fit” gives the “goodness of fit” term of the PI. iii) Multiplication of the two terms yields the directly observed “gamete count” PI. For example, suppose in a paternity case tested at 3 VNTR loci, the tested man, heterozygous at all loci, matches the single paternal allele at each locus, and that search of the three locus data base yields 43 of 6146 non-fathers who also match. The exclusion probability PI is 6146/43 = 143. Of the tested man’s eight equiprobable gametes, only one contains the paternal marker at each locus, yielding 1/8 as the numerator of the “goodness of fit” term. If the 43 matching non-fathers consist of 38 who match as well as the tested man and 5 who are apparent homozygotes at one of the three loci, the denominator of the “goodness of fit” term would be 1/43 (38•1/8 + 5•2/8), yielding a “gamete count” PI of 6146/48 = 128.