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SUMMARYAn infinite population practising a constant amount of selfing and random mating is studied. The effects of the mating system on two linked loci with an arbitrary number of neutral alleles are determined. Expressions are obtained for the two-locus descent measure, and hence genotypic frequencies and disequilibria functions, in any generation. The nature of the equilibrium population is deduced. The special cases of pure selfing or pure random mating and completely linked or completely unlinked loci are considered separately.

For human complex traits, non-additive genetic variation has been invoked to explain "missing heritability,'' but its discovery is often neglected in genome-wide association studies. Here we propose a method of using SNP data to partition and estimate the proportion of phenotypic variance attributed to additive and dominance genetic variation at all SNPs (h(SNP)(2) and delta(2)(SNP)) in unrelated individuals based on an orthogonal model where the estimate of h(SNP)(2) is independent of that of delta(2)(SNP). With this method, we analyzed 79 quantitative traits in 6,715 unrelated European Americans. The estimate of delta(2)(SNP) averaged across all the 79 quantitative traits was 0.03, approximately a fifth of that for additive variation (average h(SNP)(2) = 0.15). There were a few traits that showed substantial estimates of delta(2)(SNP), none of which were replicated in a larger sample of 11,965 individuals. We further performed genome-wide association analyses of the 79 quantitative traits and detected SNPs with genome-wide significant dominance effects only at the ABO locus for factor VIII and von Willebrand factor. All these results suggest that dominance variation at common SNPs explains only a small fraction of phenotypic variation for human complex traits and contributes little to the missing narrow-sense heritability problem.

SummaryWe review and extend a recent suggestion that fine-scale localization of a disease-susceptibility locus for a complex disease be done on the basis of deviations from Hardy-Weinberg equilibrium among affected individuals. This deviation is driven by linkage disequilibrium between disease and marker loci in the whole population and requires a heterogeneous genetic basis for the disease. A finding of marker-locus Hardy-Weinberg disequilibrium therefore implies disease heterogeneity and marker-disease linkage disequilibrium. Although a lack of departure of Hardy-Weinberg disequilibrium at marker loci implies that disease susceptibility–weighted linkage disequilibria are zero, given disease heterogeneity, it does not follow that the usual measures of linkage disequilibrium are zero. For disease-susceptibility loci with more than two alleles, therefore, care is needed in the drawing of inferences from marker Hardy-Weinberg disequilibria.

The statistical assessment of gene-frequency data on protein polymorphisms in natural populations remains a contentious issue. Here we formulate a test of whether polymorphisms detected by electrophoresis are in accordance with the stepwise, or charge-state, model of mutation in finite populations in the absence of selection. First, estimates of the model parameters are derived by minimizing chi-square deviations of the observed frequencies of genotypes with alleles (0,1,2…) units apart from their theoretical expected values. Then the remaining deviation is tested under the null hypothesis of neutrality. The procedure was found to be conservative for false rejections in simulation data. We applied the test to Ayala and Tracey 's data on 27 allozymic loci in six populations of Drosophila willistoni. About one-quarter of polymorphic loci showed significant departure from the neutral theory predictions in virtually all populations. A further quarter showed significant departure in some populations. The remaining data showed an acceptable fit to the charge state model. A predominating mode of selection was selection against alleles associated with extreme electrophoretic mobilities. The advantageous properties and the difficulties of the procedure are discussed.

The analysis of gene frequencies for a nested structure of genes within individuals, individuals within subpopulations, and subpopulations within populations is considered. Alternative parameterizations are provided by measures of correlation and of identity by descent, but the latter parameters provide more flexibility. The effects of population size, mating system, mutation, and migration can be incorporated into transition equations for identity measures and the structure of equilibrium populations can be determined; the procedures are illustrated for a finite island model. With parameters defined before estimation procedures are developed, problems of estimates depending on the numbers of sampled subpopulations are avoided, while the descent measures also avoid the approximations found in other treatments.

A general model for the evolution of pathogen populations on mixtures or multilines is developed. This model is used to extend previous analyses of the effects of the widespread cultivation of multilines on the evolution of virulence in obligate parasites to mixtures of lines carrying different numbers of resistance genes. It is concluded that the composition of an equilibrium pathogen population growing on a multiline may vary within wide limits and the prinicipal determinant of its composition is the number of components in the multiline and the resistance genes they carry. Other factors of importance are (i) the relative contribution made by each host class (with different numbers of resistance genes) to the pathogen spore pool each generation; (ii) the levels of ‘stabilizing selection’ against unnecessary virulence genes; and (iii) the way in which unnecessary genes for virulence combine to reduce pathogen fitness. Conditions for the fixation of avirulent biotypes in the pathogen population and the evolution of a pathogen superrace are given for multilines of various compositions.

The variances of actual inbreeding and coancestry in terms of their corresponding identities by descent were studied for finite populations. For inbreeding at a single locus, the total variance sigma 2 equal F(1-F) (F is the inbreeding coefficient) is comprised of a component sigma 2w within populations and a component sigma 2b between replicate populations. These variances increase in time to a maximum at about 1.1Ne generations for sigma 2w, about 2.3Ne generations for sigma 2b, and about 1.4 NE generations for sigma 2, and decrease thereafter (Ne is effective population size). The ratio sigma 2b/sigma 2 is ever increasing to an asymptote in the range 0.4-0.5 depending on Ne and the mating system. For finite populations with variation in pedigree F's, there are contributions sigma 2wF within and sigma 2bF between populations. The component sigma 2bF is insignificant except for very small populations, and sigma 2wF is largest in the early generations and then decreases roughly as (1-F)2/KNe where K is formulated in terms of the mating strategy and the degree of avoidance of mating relatives. An additional degree of avoidance increases K by a factor of 4. In a large population at equilibrium with respect to mixed self and random mating, sigma 2wF accounts for one-half to two-thirds of sigma 2w. Bringing in more loci leads to decomposition of the total variance into four components whose values are affected by linkages among the loci. The relationships between these components and sigma 2w, sigma 2wF, and sigma 2bF are elaborated in terms of tight and loose linkage. The exact computations of sigma 2wF and sigma 2bF require the use of two locus descent measures without linkage. The variances of various averages of actual identities by descent, such as the proportions for individuals or populations, are formulated for a sample of individuals.

Measurement of linkage disequilibrium involves two sampling processes. First, there is the sampling of gametes in the population to form successive generations, and this generates disequilibrium dependent on the effective population size (Ne) and the mating structure. Second, there is sampling of a finite number (n) of individuals to estimate the population disequilibrium. ——Two-locus descent measures are used to describe the mating system and are transformed to disequilibrium moments at the final sampling. Approximate eigenvectors for the transition matrix of descent measures are used to obtain formulae for the variance of the observed disequilibria as a function of Ne, mating structure, n, and linkage or recombination parameter.—The variance of disequilibrium is the same for monoecious populations with or without random selfing and for dioecious populations with random pairing for each progeny. With monogamy, the variance is slightly higher, the proportional difference being greater for unlinked loci.

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.

Transition equations are established for descent measures for pairs of loci in finite randomly mating monoecious populations. The two special cases of equal chance gamete formation and two gametes per parent are considered in detail. The descent measures allow genotypic frequencies to be found but are used mainly to evaluate three quadrigenic moments, including the variance of the linkage disequilibrium. Numerical properties of these moments are compared to previously reported values.

For loci with multiple alleles, hypotheses about linkage disequilibrium may be tested on the complete set of gametic data, or on various collapsed sets of data. Collapsing data into a few alleles at each locus can change the power of the tests, as implied in a recent paper by Zouros, Golding and Mackay (1977). We show that the nature of such changes can be found from properties of the noncentral chi-square distribution, and that the magnitude and direction of these changes depend on the levels of linkage disequilibria, allelic frequencies and degrees of freedom.

Nine laboratory populations of D. melanogaster were surveyed by starch gel electrophoresis for variation at 17 enzyme loci. A single-fly extract could be assayed for all 17 enzymes, so that the data consist of 17-locus genotypes.— Pairwise linkage disequilibria were estimated from the multilocus genotypic frequencies, using both BURROWS' and HILL'S methods. Large amounts of link-age disequilibrium were found, in contrast to the results reported for natural populations.—Knowledge of the approximate sizes of these populations was used to compare the observed heterozygosities and linkage disequilibria with predictions of the neutral allele hypothesis. The relatively large amount of linkage disequilibrium is consistent with the small sizes of the populations. However, the levels of heterozygosity in at least some populations suggest that some mechanism has been operating to retard the rate of decay by random drift. Several examples of significant deviation from Hardy-Weinberg frequencies and the large amount of linkage disequilibnim present in these populations indicate that a likely mechanism is selective effects associated with neutral alleles because of linkage disequilibrium with selected loci (e.g., "associative overdominance"). The results are therefore consistent with both neutralist, and selectionist hypotheses, but suggest the importance of considering linkage disequilibrium between neutral and selected loci when attempting to explain the dynamics of enzyme polymorphisms.

For a population undergoing recurrent selection, a method is presented for determining the average inbreeding coefficients at the end of each breeding cycle. The coefficients are derived in terms of probability measures that genes are identical by descent. For the one-locus case, two digametic measures are defined and employed in the derivation of a recurrence formula for the inbreeding coefficient. Two further classes of measures, trigametic and quadrigametic, are required for transition from one cycle to the previous one to allow the calculation of the inbreeding function for the two-locus case. Numerical values of the average probability of double identity by descent for populations with various imposed assumptions are listed to illustrate the effects of linkage and population size on the accrual of inbreeding and hence of homozygosity.

We discuss the statistical significance of local similarities found between DNA sequences, and illustrate the procedure with reference to the Queen and Korn algorithm. If the longest similarity found for two sequences has length L, this length is said to be significant at the 5% level if there is a probability of no more than 0.05 of finding a length of L or greater between a pair of sequences consisting of randomly chosen bases with the same overall base frequencies. The distribution of longest lengths is related to that of lengths from any particular pair of starting positions on the two sequences. For our implementation of the Queen and Korn algorithm, this latter distribution is constructed by combining the five different blocks of bases that may be added to extend a similarity. A table is given to assess the significance of longest similarities in sequences of length up to 1000 bases. Quite long similarities are expected to occur by chance alone. The critical values we calculate for assessing significance are preferable to expected numbers of similarities used by some commercial computer packages.

For any four genes, two at each of two loci, in a population, a 15 component descent measure has been introduced. These components are the probabilities of the 15 possible arrangements on a set of initial gametes of those genes of which the four of interest are copies. Since identity by descent of genes is equivalent to their being copies of a single gene on an initial gamete, descent measures have inbreeding coefficients as special cases. The individual descent measure, defined for four genes on two uniting gametes can be evaluated for any pedigree by means of an algorithm developed here. If initial gametic frequencies are specified, descent measures allow genotypic frequencies and disequilibria functions at one and two loci to be found. The procedures are illustrated for selfing and for sib mating. Several applications of the descent measures are discussed.

Statistical tests for Hardy–Weinberg equilibrium have been an important tool for detecting genotyping errors in the past, and remain important in the quality control of next generation sequence data. In this paper, we analyze complete chromosomes of the 1000 genomes project by using exact test procedures for autosomal and X-chromosomal variants. We find that the rate of disequilibrium largely exceeds what might be expected by chance alone for all chromosomes. Observed disequilibrium is, in about 60% of the cases, due to heterozygote excess. We suggest that most excess disequilibrium can be explained by sequencing problems, and hypothesize mechanisms that can explain exceptional heterozygosities. We report higher rates of disequilibrium for the MHC region on chromosome 6, regions flanking centromeres and p-arms of acrocentric chromosomes. We also detected long-range haplotypes and areas with incidental high disequilibrium. We report disequilibrium to be related to read depth, with variants having extreme read depths being more likely to be out of equilibrium. Disequilibrium rates were found to be 11 times higher in segmental duplications and simple tandem repeat regions. The variants with significant disequilibrium are seen to be concentrated in these areas. For next generation sequence data, Hardy–Weinberg disequilibrium seems to be a major indicator for copy number variation. Electronic supplementary material The online version of this article (doi:10.1007/s00439-017-1786-7) contains supplementary material, which is available to authorized users.

Modern forensic DNA profiles are constructed using microsatellites, short tandem repeats of 2–5 bases. In the absence of genetic data on a crime-specific subpopulation, one tool for evaluating profile evidence is the match probability. The match probability is the conditional probability that a random person would have the profile of interest given that the suspect has it and that these people are different members of the same subpopulation. One issue in evaluating the match probability is population differentiation, which can induce coancestry among subpopulation members. Forensic assessments that ignore coancestry typically overstate the strength of evidence against the suspect. Theory has been developed to account for coancestry; assumptions include a steady-state population and a mutation model in which the allelic state after a mutation event is independent of the prior state. Under these assumptions, the joint allelic probabilities within a subpopulation may be approximated by the moments of a Dirichlet distribution. We investigate the adequacy of this approximation for profiled loci that mutate according to a generalized stepwise model. Simulations suggest that the Dirichlet theory can still overstate the evidence against a suspect with a common microsatellite genotype. However, Dirichlet-based estimators were less biased than the product-rule estimator, which ignores coancestry.

Standard statistical tests for equality of allele frequencies in males and females and tests for Hardy-Weinberg equilibrium are tightly linked by their assumptions. Tests for equality of allele frequencies assume Hardy-Weinberg equilibrium, whereas the usual chi-square or exact test for Hardy-Weinberg equilibrium assume equality of allele frequencies in the sexes. In this paper, we propose ways to break this interdependence in assumptions of the two tests by proposing an omnibus exact test that can test both hypotheses jointly, as well as a likelihood ratio approach that permits these phenomena to be tested both jointly and separately. The tests are illustrated with data from the 1000 Genomes project.

Three different models, a two-way factorial model for familiarity, an orthogonalizing transform of this model to a diallel model, and a bio model more representative of the biological situation, are interrelated in terms of their components of variance and covariance. It is clarified that there are five components that can be reckoned with in the analysis of reciprocal crosses, including distinct maternal and paternal variances. Estimation of the components and tests of hypotheses concerning them are outlined for two types of mating designs with reciprocals. One deisgn involves a factorial mating design between two distinct sets of parents or parental lines and the other a diallel of all crosses from a single set of parents or parental lines. Both designs provide the same types of information and similar tests of hypotheses. At least some parts of the analyses corresponding to the factorial model are required to separate the maternal and paternal variances. A least squares partitioning of the sums of squares according to the diallel model, but with expectations expressed in terms of the bio model, provides most of the tests of hypotheses of interest. Worked examples are given.

We consider covariances of all parent and first-generation relatives from outcrossing or self-fertilization in a parent population that is in equilibrium with respect to these processes. The results, which are for any number of alleles and loci with additive and dominance effects, are phrased in terms of six quadratic genetic components whose coefficients are given by descent measures for equilibrium populations. Because of the variation in the inbreeding coefficients for this system of mating, the expressions include joint contributions of loci to the variances and covariances of relatives. By inclusion of the full complement of relatives, all quadratic components can be estimated. The findings of Ghai (1982, Biometrics 38, 87-92) for compound functions of the covariances with two alleles at a single locus are analyzed in terms of the more general model.

Existing theory for inferences about linkage disequilibrium is restricted to a measure defined on gametic frequencies. Unless gametic frequencies are directly observable, they are inferred from genotypic frequencies under the assumption of random union of gametes. Primary emphasis in this paper is given to genotypic data, and disequilibrium coefficients are defined for all subsets of two or more of the four genes, two at each of two loci, carried by an individual. Linkage disequilibrium coefficients are defined for genes within and between gametes, and methods of estimating and testing these coefficients are given for gametic data. For genotypic data, when coupling and repulsion double heterozygotes cannot be distinguished. Burrows' composite measure of linkage disequilibrium is discussed. In particular, the estimate for this measure and hypothesis tests based on it are compared to the usual maximum likelihood estimate of gametic linkage disequilibrium, and corresponding likelihood ratio or contingency chi-square tests. General use of the composite measure, whether or not random union of gametes is an appropriate assumption, is recommended. Attention is given to small samples, where the non-normality of gene frequencies will have greatest effect on methods of inference based on normal theory. Even tools such as Fisher's z-transformation for the correlation of gene frequencies are found to perform quite satisfactorily.

Statistical Analysis of DNA Sequence Data. Edited by B. S. Weir. New York, Basel, Marcel Dekker, 1983. ix, 255 p. SFr 120. (Statistics: Textbooks and Monographs, Vol. 47.)

A comparison was made of various tests for Hardy-Weinberg equilibrium, with emphasis on methods for multiple alleles. For an overall test of deviations from equilibrium, the classical chi-square goodness-of-fit test generally performs well, with continuity corrections needed for extreme gene frequencies or extreme departures from equilibrium. For small samples, probability tests are preferable and for multiple alleles these probability tests may be performed on a sample of all possible sets of genotypic frequencies having a fixed set of sample gene frequencies. Numerical work showed that the continuity-corrected chi-square was the most conservative test procedure, and the uncorrected chi-square the least conservative. With multiple alleles, a better appreciation of the nature of departures from equilibrium is given by the use of disequilibrium coefficients, defined for each heterozygote as the difference between observed and expected frequencies. Likelihood-ratio tests can be used to test each of these coefficients individually but a satisfactory procedure is to divide the squared estimate of each coefficient by its estimated variance and regard the ratio as a single-degree-of-freedom chi-square. Numerical studies confirmed the validity of this approach, which has the great advantage of not requiring solutions of nonlinear equations.

A clarification is given of the difference between a multiplicative model (Haber, 1984, Biometrics 40, 189-198) and an additive model (Weir, 1979, Biometrics 35, 235-254) for association between genes at two loci. Although the parametrization of Haber contains both intergametic and intragametic linkage disequilibrium terms, there are problems in separating these terms when the two types of double heterozygote cannot be distinguished. An alternative formulation that contains the sum and product of these terms also contains measures of association between three or four genes at two loci, and a reexamination of data discussed by Haber shows that the inclusion of trigenic disequilibria can significantly improve the fit of the model.