Biostatistical evaluation of HLA-A, B findings in cases with suspected recombination

In a given case with two or more siblings the bloodgroup findings may yield a high plausibility of paternity for the alleged father of the children whilst the HLA-A,B characteristics may exhibit an exclusion of fatherhood of one or other of the siblings. But this "exclusion" may disappear assuming an HLA-A,B recombination of a father's (or mother's) gamete. In such cases simple calculations (i.e., by hand) can provide the serostatistical characteristic L = Y/X (after Essen-M?ller) for the given HLA-A,B constellation. In two-child cases with HLA-A,B findings for the putative father, the two pedigrees for the counterhypothesis Y are Yg1 and Yg2. The one holds for the case where the putative father is the father of child 1 and the other for the case where he is the father of child 2; in each case an unknown man is the father of the other child. The likelihood ratio is calculated as follows: L = Y/X = 1/r.p(3,3).p(3,4).p(4,4) + p(4,3)]/p(3,3).p(4,4) + p(3,4).p(4,3)], where p(3,3) and p(4,4) are the frequencies of the putative father's HLA-A,B haplotypes in the pedigree YH1, p(3,4) and p(4,3) those in pedigree YH2, and where r is the recombination rate. In three-child cases and others in which the putative father is deceased and HLA results from legitimate children or siblings are available, and in which fatherhood is only possible if a recombinant haplotype interferes, only one counter-hypothesis exists. In these cases, the likelihood ratio is represented by L = Y/X = 2/r.p(HR).2I, where p(HR) means the frequency of the recombinant HLA-A,B haplotype, r the recombination rate and I(0, 1 or 2) the "difference in determined types."