Deviations from Hardy-Weinberg Proportions: Sampling Variances and Use in Estimation of Inbreeding Coefficients

Abstract

An analysis is made of the distribution of deviations from Hardy-Weinberg proportions with k alleles and of estimates of inbreeding coefficients (f) obtained from these deviations.—If f is small, the best estimate of f in large samples is shown to be 2Σ _i(T_ii/N_i)/(k - 1), where T_ii is an unbiased measure of the excess of the ith homozygote and N_i the number of the ith allele in the sample [frequency = N_i/(2N)]. No extra information is obtained from the T_ij, where these are departures of numbers of heterozygotes from expectation. Alternatively, the best estimator can be computed from the T_ij, ignoring the T_ii. Also (1) the variance of the estimate of f equals 1/(N(k - 1)) when all individuals in the sample are unrelated, and the test for f = 0 with 1 d.f. is given by the ratio of the estimate to its standard error; (2) the variance is reduced if some alleles are rare; and (3) if the sample consists of full-sib families of size n, the variance is increased by a proportion (n - 1)/4 but is not increased by a half-sib relationship.—If f is not small, the structure of the population is of critical importance. (1) If the inbreeding is due to a proportion of inbred matings in an otherwise random-breeding population, f as determined from homozygote excess is the same for all genes and expressions are given for its sampling variance. (2) If the homozygote excess is due to population admixture, f is not the same for all genes. The above estimator is probably close to the best for all f values.

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