Equivalence by descent: pedigree analysis with inbreeding and gametic phase disequilibrium

In the presence of gametic phase disequilibrium and inbreeding, multiple locus genotype frequencies cannot be written solely in terms of identity by descent (IBD) probabilities. Following Cockerham & Weir (1973) we introduce the concept of 'equivalence by descent' (EBD), an extension of the concept of IBD to include non-allelic genes. Two genes are said to be EBD if they derive ultimately from the same founding gamete of a pedigree. Allelic genes that are EBD are also IBD. For two loci 11 EBD probabilities, the 'J-coefficients,' are required and for three loci 117 J-coefficients are required to write genotype probabilities. It is shown how the 117 J-coefficients for three loci can be reduced to a basic set of 37. Computer programs, written in the algebraic programming language, MAPLE, are described which are capable of calculating the two- and three-locus J-coefficients for any pedigree, subject only to size limitations. The MAPLE packages are available from the author upon request.