HIGHER ORDER PROBABILITY FUNCTIONS OF IDENTITY OF ALLELES BY DESCENT
A R R I S (1964) provided a parameterization of nine probabilities for the fifteen states of identity of allelic genes of inbred relatives, and with these developed the joint frequencies and covariances of the relatives.He also showed how to find the probabilities when one relative is not an ancestor of the other.At about the same time (1965) GILLOIS organized various probability functions for the states of identity and, although not involving explicit expressions of the joint frequencies, found the covariances of inbred relatives, which he later (1966) extended to multiple alleles.DENNISTON (1967) produced a system of path counting for arriving at the probabilities, and extended his system to include two linked loci.In this work all two-, three-and four-gene probability functions for a single locus are interrelated for all situations, me, two, three, or four individuals in which the gene states of identity can arise.The probability functions in conjunction with gene frequencies provide the joint genotypic frequencies.An algorithm is developed for the systematic computation of all the probability functions from pedigrees.For regular systems of mating the algorithm need be applied only once between successive generations and is illustrated for full sib mating and a finite monoecious population.As mentioned before, the probability functions were used to find the covariances of inbred relatives.They can be used to compute exact probabilities of fixation and the time to fixation in finite populations, but which requires an extension of the set of functions to the order of the number of genes in the population.Most important, however, they provide models for the analyses of gene frequencies, models which clearly define parameters that are estimable and hypotheses that are testable.These models may be applied to the analyses of frequencies of individuals and groupings of individuals for which the pedigree relationships are known or, more important, for which the relationships are unknown-the usual situation in natural populations.