Extensions to pedigree analysis. V. Optimal calculation of Mendelian likelihoods

Mendelian likelihoods are computed from human pedigree data for purposes of gene mapping, risk prediction in genetic counseling, and hypothesis testing in genetic epidemiology. The Mendelian likelihood of an extended pedigree can be written as a sum of products, the sum ranging over all possible genotypic combinations for the individuals in the pedigree. Exclusion of genotypes incompatible with the phenotypic information and pedigree structure reduces the ranges of summation and simplifies the likelihood calculation. To evaluate the likelihood with the fewest possible arithmetic operations requires carrying out the summations over one individual at a time and the intervening multiplications in some appropriate order. Each such removal of an individual reduces the likelihood evaluation to another evaluation of the same numerical form. Greedy-type algorithms are suggested for determining an order in which the summations and multiplications may be carried out. The greedy methods are fast and appear to generate good removal sequences. They are shown to work well when applied to a large, complex pedigree.