Inbred pairs of individuals can share two, three, four or two pairs of alleles identical by descent (ibd) at a locus. Although there are 15 possible ibd states, it is generally sufficient to combine these into nine states with probabilities adding to one, or eight summary states with eight probabilities. For SNP data, some authors claim only five states have identifiable probabilities, while other authors give estimates for probabilities of nine states. In any event the higher-order ibd probabilities for a target pair of individuals cannot be estimated by single-SNP methods unless allele probabilities are known in the population to which ibd refers, and this is unlikely to be the case. Nor is it appropriate to use sample allele frequencies in place of allele probabilities because powers of these frequencies have expectations that depend on the inbreeding and relatedness of all individuals in the study sample.
A remedy is to estimate within-population measures of ibd. Instead of estimating the probability an indivdual carries two ibd alleles, for example, that probability can be estimated relative to the average coancestry of pairs of individuals in the study sample: this is the individual-specific version of Wright's FIS. It is FIS rather than FIT that is estimable with data from a single population. We work with observed proportions of allele pairs that match within or between individuals, and employ ratios of functions of these proportions for which the unknown allele probabilities cancel out of the expected values. With a large number of SNPs, these methods give good estimates of within-population inbreeding and kinship coefficients for pairs of individuals. We extend this approach for within-population estimators of functions of ibd probabiliities for three, four or two pairs of alleles for pairs of inbred individuals.
As an example, we estimate the probability that there is no ibd among any three alleles from two individuals, two alleles from one and one allele from the other, relative to the probability there is no ibd among any three alleles taken randomly, one from each of three individuals in the study sample. This can be compared to the corresponding quantity for two alleles to establish the extent of ibd at sets of three alleles. These measures are needed to express the trait covariance for pairs of inbred relatives.
The advanatage of working with allele-matching, or allele-sharing, is that the estimates are rank-invariant: one pair of individuals has its ibd probabilities less than those for another pair, for example, regardless of the scope of the study including those individuals. Invariance is not guaranteed for estimators that make explicxit use of sample allele frequencies.