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A dense set of 5,000 SNPs on a 10-Mb region of human chromosome 20 has been typed on samples of African Americans, East Asians, and United Kingdom Caucasians. There are departures from Hardy-Weinberg equilibrium beyond the level at which markers are often discarded because of possible genotyping errors. The observation that markers showing such departures are often close together on the chromosome confirms the result that Hardy-Weinberg tests at two loci are correlated to an extent that depends on the linkage disequilibrium between those two markers. Linkage disequilibrium can be described by the composite linkage disequilibrium coefficient, the parameter that determines the behavior of case-control allelic tests of association. A useful preliminary investigation of datasets of this type is provided by counting the numbers of distinct multi-locus genotypes in windows of a few markers.

A moment-based method for estimating a measure of population diversity, theta or Wright's FST, is given for dominant markers such as amplified fragment length polymorphisms (AFLPs) or RAPDs in noninbred populations. Basic assumptions are that there is random mating, Hardy-Weinberg equilibrium, linkage equilibrium, no mutation from common ancestor and equally distant populations. It is based on the variances between and within populations of genotype frequencies, whereas previously moment methods for dominant markers have been indirect in that they have been based on first estimating allele frequencies and then using the variances of those frequencies. The use of genotype frequencies directly appears to be more robust. Approximate sampling errors of the estimates are given. Methods are extended to estimate genetic distances and their sampling errors. The AFLP data from samples of breeds of pig are used for illustration.

A moment estimator of, the coancestry coefficient for alleles within a population, was described by Weir & Cockerham in 1984 (100) and is still widely cited. The estimate is used by population geneticists to characterize population structure, by ecologists to estimate migration rates, by animal breeders to describe genetic variation, and by forensic scientists to quantify the strength of matching DNA profiles. This review extends the work of Weir & Cockerham by allowing different levels of coancestry for different populations, and by allowing non-zero coancestries between pairs of populations. All estimates are relative to the average value of theta between pairs of populations. Moment estimates for within- and between-population theta values are likely to have large sampling variances, although these may be reduced by combining information over loci. Variances also decrease with the numbers of alleles at a locus, and with the numbers of populations sampled. This review also extends the work of Weir & Cockerham by employing maximum likelihood methods under the assumption that allele frequencies follow the normal distribution over populations. For the case of equal theta values within populations and zero theta values between populations, the maximum likelihood estimate is the same as that given by Robertson & Hill in 1984 (70). The review concludes by relating functions of theta values to times of population divergence under a pure drift model.

Until recently, the estimation of the heritability of a trait has required knowledge of the pedigree within a population. In natural populations such knowledge is often unknown. Two techniques have been developed which use marker information to estimate heritabilities without reference to the exact nature of the relationships: a regression-based estimator that regresses phenotypic similarity for a pair of individuals against an estimate of their relationship and a likelihood-based estimator that maximizes the probability of the genotypic and phenotypic data given a known population structure. Computer simulation was used to compare the behaviour of these estimators. Bias in estimates of heritability decreased with increasing marker information, decreasing simulated heritability, increasing relatedness and increasing sample size. The techniques displayed reasonable tolerance to the percentage of missing data. The regression-based technique shows least average bias, but largest variance over simulations. Likelihood-based techniques show larger average bias, but smaller variances over estimates. A modified form of the likelihood technique, requiring fewer initial assumptions about population parameters, is presented. The modified form shows less bias in its estimates of heritability than the likelihood technique originally proposed.

Formulae were derived to predict genetic response under various selection schemes assuming an infinitesimal model. Account was taken of genetic drift, gametic (linkage) disequilibrium (Bulmer effect), inbreeding depression, common environmental variance, and both initial segregating variance within families (sigma AW02) and mutational (sigma M2) variance. The cumulative response to selection until generation t(CRt) can be approximated as [equation: see text] where Ne is the effective population size, sigma AW infinity 2 = Ne sigma M2 is the genetic variance within families at the steady state (or one-half the genic variance, which is unaffected by selection), and D is the inbreeding depression per unit of inbreeding. R0 is the selection response at generation 0 assuming preselection so that the linkage disequilibrium effect has stabilized. beta is the derivative of the logarithm of the asymptotic response with respect to the logarithm of the within-family genetic variance, i.e., their relative rate of change. R0 is the major determinant of the short term selection response, but sigma Me2 Ne and beta are also important for the long term. A selection method of high accuracy using family information gives a small Ne and will lead to a larger response in the short term and a smaller response in the long term, utilizing mutation less efficiently.

Until recently, attempts to map disease genes on the basis of population associations with linked markers have been based on expected values of linkage disequilibrium. These methods suffer from the large variances imposed on disequilibrium measures by the evolutionary process, but a more serious problem for many diseases is that they assume an equilibrium population. For diseases that arose only a few hundred generations ago, it is more appropriate to concentrate on the initial growth phase of the disease. We invoke a Poisson branching process for this early growth, and estimate the likelihood for the recombination fraction between marker and disease loci, on the basis of simulated disease populations. The limits of the resulting support intervals for the recombination fraction vary inversely with the age of the disease in generations. We illustrate the procedure with data on cystic fibrosis and diastrophic dysplasia, for which the method appears appropriate, and for Friedreich ataxia and Huntington disease, for which it does not. A valuable aspect of the method is the ability in some cases to compare likelihoods of the three orders for a disease locus and two linked marker loci.

Linkage disequilibrium, D, between a polymorphic disease and mapped markers can, in principle, be used to help find the map position of the disease gene. Likelihoods are therefore derived for the value of D conditional on the observed number of haplotypes in the sample and on the population parameter Nc, where N is the effective population size and c the recombination fraction between the disease and marker loci. The likelihood is computed explicitly for the case of two loci with heterozygote superiority and, more generally, by computer simulations assuming a steady state of constant population size and selective pressures or neutrality. It is found that the likelihood is, in general, not very dependent on the degree of selection at the loci and is very flat. This suggests that precise information on map position will not be obtained from estimates of linkage disequilibrium.

Nonrandom mating whereby parents are related is expected to cause a reduction in effective population size because their gene frequencies are correlated and this will increase the genetic drift. The published equation for the variance effective size, Ne, which includes the possibility of nonrandom mating, does not take into account such a correlation, however. Further, previous equations to predict effective sizes in populations with partial sib mating are shown to be different, but also incorrect. In this paper, a corrected form of these equations is derived and checked by stochastic simulation. For the case of stable census number, N, and equal progeny distributions for each sex, the equation is [formula: see text], where Sk2 is the variance of family size and alpha is the departure from Hardy-Weinberg proportions. For a Poisson distribution of family size (Sk2 = 2), it reduces to Ne = N/(1 + alpha), as when inbreeding is due to selfing. When nonrandom mating occurs because there is a specified system of partial inbreeding every generation, alpha can be substituted by Wright's FIS statistic, to give the effective size as a function of the proportion of inbred mates.

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 2 sigma i(Tii/Ni)/(k - 1), where Tii is an unbiased measure of the excess of the ith homozygote and Ni the number of the ith allele in the sample [frequency = Ni/(2N)]. No extra information is obtained from the Tij, where these are departures of numbers of heterozygotes from expectation. Alternatively, the best estimator can be computed from the Tij, ignoring the Tii. 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.

Expressions are derived for the response to directional selection for a quantitative trait that comes from fixation of new mutations in a finite population. For additive genes with a distribution of mutant gene effects symmetric about zero, the response from fixing mutations occurring in a single generation is 2 Ni sigma 2M/sigma, in which N is the effective population size, i is the selection intensity, sigma is the phenotypic standard deviation, and sigma 2M is the increment in variance in the generation immediately after occurrence of the mutations. This response is 2N times that immediately after occurrence of the mutations. With continuous mutation each generation, the asymptotic rate of response is also 2Ni sigma 2M/sigma and the asymptotic variance is independent of i. For completely dominant mutations with symmetric effects, the rates are Ni sigma 2M/sigma; and for recessive mutations the rates are proportional to (Ni)1/2. If the distribution of mutant gene effects, a, is not symmetric about zero, responses depend on the mean square of effects of mutations with positive effect, rather than on the variance of their effects. Rates of change in fitness and of traits correlated with fitness are also analyzed. It is argued that new mutations have contributed substantially to long-term responses in many laboratory selection experiments.

Analysis of linkage disequilibrium D among restriction sites or bases in DNA sequences, arising from mutations in finite populations, depends on a knowledge of the variance-covariance structure of measures such as D2 between different pairs of sites. This requires evaluation of the eighth moments of gene frequencies among two, three, and four loci, and the necessary methodology is derived here and results are computed. While primary emphasis is placed on disequilibrium arising from mutation or gene conversion, the methodology also allows for the joint effects of only drift and recombination. Numerical results confirm that squared linkage disequilibria can have high variances and covariances.