Definition and estimation of higher-order gene fixation indices

Fixation indices summarize the associations between genes that arise from the joint effects of inbreeding and selection. In this paper, fixation indices are derived for pairs, triplets and quadruplets of genes at a single multiallelic locus. The fixation indices are obtained by dividing cumulants by constants; the cumulants describe the statistical distribution of alleles and the constants are functions of gene frequency. The use of cumulants instead of moments is necessary only for four-gene indices, when the fourth cumulant is used. A second type of four-gene index is also required, and this index is based upon the covariation of second-order cumulants. At multiallelic loci, a large number of indices is possible. If alleles are selectively neutral, the number of indices is reduced and the relationship between gene identity and gene cumulants is shown.--Two-gene indices can always be estimated from genotypic frequency data at a single polymorphic locus. Three-gene indices are also estimable except when allele frequency equals one-half. Four-gene indices are not estimable unless selection is assumed to have an equal effect upon each allele (such as under selective neutrality) and the locus contains at least three alleles of unequal frequency. For diallelic or selected loci, an alternative four-gene fixation index is proposed. This index incorporates both types of four-gene associations but cannot be related to gene identity.