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Recent modeling studies interested in runs of homozygosity (ROH) and identity by descent (IBD) have sought to connect these properties of genomic sharing to pairwise coalescence times. Here, we examine a variety of features of pairwise coalescence times in models that consider consanguinity. In particular, we extend a recent diploid analysis of mean coalescence times for lineage pairs within and between individuals in a consanguineous population to derive the variance of coalescence times, studying its dependence on the frequency of consanguinity and the kinship coefficient of consanguineous relationships. We also introduce a separation-of-time-scales approach that treats consanguinity models analogously to mathematically similar phenomena such as partial selfing, using this approach to obtain coalescence-time distributions. This approach shows that the consanguinity model behaves similarly to a standard coalescent, scaling population size by a factor 1 − 3c, wherecrepresents the kinship coefficient of a randomly chosen mating pair. It provides the explanation for an earlier result describing mean coalescence time in the consanguinity model in terms ofc. The results extend the potential to make predictions about ROH and IBD in relation to demographic parameters of diploid populations.

Modeling covariance structure based on genetic similarity between pairs of relatives plays an important role in evolutionary, quantitative and statistical genetics. Historically, genetic similarity between individuals has been quantified from pedigrees via the probability that randomly chosen homologous alleles between individuals are identical by descent (IBD). At present, however, many genetic analyses rely on molecular markers, with realized measures of genomic similarity replacing IBD-based expected similarities. Animal and plant breeders, for example, now employ marker-based genomic relationship matrices between individuals in prediction models and in estimation of genome-based heritability coefficients. Phenotypes convey information about genetic similarity as well. For instance, if phenotypic values are at least partially the result of the action of quantitative trait loci, one would expect the former to inform about the latter, as in genome-wide association studies. Statistically, a non-trivial conditional distribution of unknown genetic similarities, given phenotypes, is to be expected. A Bayesian formalism is presented here that applies to whole-genome regression methods where some genetic similarity matrix, e.g., a genomic relationship matrix, can be defined. Our Bayesian approach, based on phenotypes and markers, converts prior (markers only) expected similarity into trait-specific posterior similarity. A simulation illustrates situations under which effective Bayesian learning from phenotypes occurs. Pinus and wheat data sets were used to demonstrate applicability of the concept in practice. The methodology applies to a wide class of Bayesian linear regression models, it extends to the multiple-trait domain, and can also be used to develop phenotype-guided similarity kernels in prediction problems.

Principal component analysis (PCA) is widely used in genome-wide association studies (GWAS), and the principal component axes often represent perpendicular gradients in geographic space. The explanation of PCA results is of major interest for geneticists to understand fundamental demographic parameters. Here, we provide an interpretation of PCA based on relatedness measures, which are described by the probability that sets of genes are identical-by-descent (IBD). An approximately linear transformation between ancestral proportions (AP) of individuals with multiple ancestries and their projections onto the principal components is found.In addition, a new method of eigenanalysis “EIGMIX” is proposed to estimate individual ancestries. EIGMIX is a method of moments with computational efficiency suitable for millions of SNP data, and it is not subject to the assumption of linkage equilibrium. With the assumptions of multiple ancestries and their surrogate ancestral samples, EIGMIX is able to infer ancestral proportions (APs) of individuals. The methods were applied to the SNP data from the HapMap Phase 3 project and the Human Genome Diversity Panel. The APs of individuals inferred by EIGMIX are consistent with the findings of the program ADMIXTURE.In conclusion, EIGMIX can be used to detect population structure and estimate genome-wide ancestral proportions with a relatively high accuracy.

Contrary to what is often assumed in population genetics, independently segregating loci do not have completely independent ancestries, since all loci are inherited through a single, shared population pedigree. Previous work has shown that the non-independence between gene genealogies of independently segregating loci created by the population pedigree is weak in panmictic populations, and predictions made from standard coalescent theory are accurate for populations that are at least moderately sized. Here, we investigate patterns of coalescence in pedigrees of structured populations. We find that the pedigree creates deviations away from the predictions of the structured coalescent that persist on a longer timescale than in the case of panmictic populations. Nevertheless, we find that the structured coalescent provides a reasonable approximation for the coalescent process in structured population pedigrees so long as migration events are moderately frequent and there are no migration events in the recent pedigree of the sample. When there are migration events in the recent sample pedigree, we find that distributions of coalescence in the sample can be modeled as a mixture of distributions from different initial sample configurations. We use this observation to motivate a maximum-likelihood approach for inferring migration rates and mutation rates jointly with features of the pedigree such as recent migrant ancestry and recent relatedness. Using simulation, we show that our inference framework accurately recovers long-term migration rates in the presence of recent migration events in the sample pedigree.

The coancestry coefficient, also known as the population structure parameter, is of great interest in population genetics. It can be thought of as the intraclass correlation of pairs of alleles within populations and it can serve as a measure of genetic distance between populations. For a general class of evolutionary models it determines the distribution of allele frequencies among populations. Under more restrictive models it can be regarded as the probability of identity by descent of any pair of alleles at a locus within a random mating population. In this paper we review estimation procedures that use the method of moments or are maximum likelihood under the assumption of normally distributed allele frequencies. We then consider the problem of testing hypotheses about this parameter. In addition to parametric and non-parametric bootstrap tests we present an asymptotically-distributed chi-square test. This test reduces to the contingency-table test for equal sample sizes across populations. Our new test appears to be more powerful than previous tests, especially for loci with multiple alleles. We apply our methods to HapMap SNP data to confirm that the coancestry coefficient for humans is strictly positive.

In this paper, we present a unified mathematical model for linkage analysis that allows for inbreeding among founders in all families. The identical by descent (IBD) configuration of each pedigree is modeled as a Markov process containing two parameters; the inverse inbreeding and kinship coefficient and a rate parameter proportional to the inverse expected length of chromosome segments shared IBD by two different founder haplotypes. We use hidden Markov models and define a forward-backward algorithm for computing the conditional IBD-distribution given marker data, thereby extending the multipoint method of Lander and Green [1987. Construction of multilocus genetic maps in humans, Proc. Natl. Acad. Sci. USA 84, 2363-2367] to situations where founders are inbred. Our methodology is valid for arbitrary pedigree structures. Simulation and theoretical approximations for nonparametric linkage (NPL) analysis based on affected sib pairs reveal that NPL scores are inflated and type 1 errors increased when the inbreeding coefficient or rate parameter is underestimated. When the parents are genotyped, we present a general way of modifying the score function to drastically reduce this effect.

In this paper, we show how to construct the genealogy of a sample of genes for a large class of models with selection and mutation. Each gene corresponds to a single locus at which there is no recombination. The genealogy of the sample is embedded in a graph which we call the ancestral selection graph . This graph contains all the information about the ancestry; it is the analogue of Kingman's coalescent process which arises in the case with no selection. The ancestral selection graph can be easily simulated and we outline an algorithm for simulating samples. The main goal is to analyze the ancestral selection graph and to compare it to Kingman's coalescent process. In the case of no mutation, we find that the distribution of the time to the most recent common ancestor does not depend on the selection coefficient and hence is the same as in the neutral case. When the mutation rate is positive, we give a procedure for computing the probability that two individuals in a sample are identical by descent and the Laplace transform of the time to the most recent common ancestor of a sample of two individuals; we evaluate the first two terms of their respective power series in terms of the selection coefficient. The probability of identity by descent depends on both the selection coefficient and the mutation rate and is different from the analogous expression in the neutral case. The Laplace transform does not have a linear correction term in the selection coefficient. We also provide a recursion formula that can be used to approximate the probability of a given sample by simulating backwards along the sample paths of the ancestral selection graph, a technique developed by Griffiths and Tavare (1994).

The distribution of the proportion of genome shared identical by descent (IBD) by c half-sibs is calculated for a general c . With data availability the consideration of the chromosome as a whole rather than discrete loci becomes of increasing practical importance. Assuming the genomic continuum model, which allows for recombination, the crossover process in a chromosome pedigree is viewed as a continuous-time Markov random walk on the vertices of a hypercube with time parameter map distance along the chromosome. The desired distribution corresponds to the distribution of sojourn times of the process in a small set of vertices. The sojourn times are well approximated via the Poisson clumping heuristic. It is shown explicitly that there is no “equivalent” number of independently segregating loci that will yield the same results as with the genomic continuum model. Results are given for the human genome.

We investigate the probabilities of identity-by-descent at three loci in order to find a signature which differentiates between the two types of crossing over events: recombination and gene conversion. We use a Markov chain to model coalescence, recombination, gene conversion and mutation in a sample of size two. Using numerical analysis, we calculate the total probability of identity-by-descent at the three loci, and partition these probabilities based on a partial ordering of coalescent events at the three loci. We use these results to compute the probabilities of four different patterns of conditional identity and non-identity at the three loci under recombination and gene conversion. Although recombination and gene conversion do make different predictions, the differences are not likely to be useful in distinguishing between them using three locus patterns between pairs of DNA sequences. This implies that measures of genetic identity in larger samples will be needed to distinguish between gene conversion and recombination.

An algorithm to predict the level of identity by descent simultaneously at multiple loci is presented, which can in principle be extended to any number of loci. The model assumes a random mating population, with random association of haplotypes. The relationship is shown between coefficients of multi-locus identity or non-identity by descent and moments of multi-locus linkage disequilibrium. Thus, these moments can be computed from the multilocus identity or, using algorithms derived previously to predict the disequilibria moments, vice-versa. The results can be applied to predict multi-locus identity in, for example, gene mapping.

For a model of diallelic loci with arbitrary epistasis, Barton and Turelli [2004. Effects of genetic drift on variance components under a general model of epistasis. Evolution 58, 2111-2132] gave results for variances among and within replicate lines obtained by inbreeding without selection. Here, we discuss the relation between their population genetic methods and classical quantitative genetic arguments. In particular, we consider the case of no dominance using classical identity by descent arguments, which generalizes their results from two alleles to multiple alleles. To clarify the connections between the alternative methods, we obtain the same results using an intermediate method, which explicitly identifies the statistical effects of sets of loci. We also discuss the effects of population bottlenecks on covariances among relatives.

Linkage disequilibrium (LD) reflects coinheritance of an ancestral segment by chromosomes in a population. To begin to understand the effects of population history on the extent of LD, we model the length of a tract of identity-by-descent (IBD) between two chromosomes in a finite, random mating population. The variance of an IBD tract is large: a model described by (Genet. Res. Cambridge 35 (1980) 131) underestimates this variance. Using Fisher's concept of junctions, we predict the mean length of an IBD tract, given the age of the population and the population sizes over time. We derive results also for subdivided populations, given times of subdivision events and sizes of the resulting subpopulations. The model demonstrates that population growth and subdivision strongly affect the expected length of an IBD tract in small populations. These effects are less dramatic in large populations.

Important methods for calculating likelihoods of genealogical relationships and mapping genes are based on hidden Markov models for the process of identity by descent along chromosomes. The computational time for the algorithms depends critically on the size of the statespace of the hidden Markov model. We describe the maximal grouping together of states of the model to reduce the size of the statespace. This grouping is based on pedigree symmetries. We also present an efficient algorithm for finding the maximal grouping.

This paper reviews linkage analysis methods for detecting loci associated with quantitative traits in humans. All such methods are based on the underlying principle that family members who have similar trait values should have higher than expected levels of sharing of genetic material (identity by descent) near the genes that influence those traits. A number of different statistical methods for testing that association between shared trait values and shared identity by descent have been developed over the past 30 or more years. These different types of tests are reviewed here, with emphasis on their theory and derivations. Robustness and power are also discussed.

The Hasemann–Elston method of linkage detection is based on the probabilities of a sib pair having 0, 1, or 2 alleles identical by descent (IBD) at a marker and a trait locus. These probabilities form a 3×3 matrix. Here, the characteristic values and characteristic vectors of this matrix were used to clarify the structure of the equations and to simplify calculations. As examples, the regression coefficients were derived for three genetic systems: a trait and a marker, two epistatic traits and two markers, and one trait locus and two markers. The last model was studied under the assumption of no crossover interference, the expression for allele IBD sharing at a trait locus was derived as a function of allele IBD sharing at two marker loci, and the regression is shown to be non-linear.

The sampling distribution of neutral alleles under a stochastic birth, death, and immigration (BDI) process, proposed as a model of fluctuating island populations, is studied by analytical methods. A new result is presented for the distribution of allele types in a sample of N individuals from an island when the allele frequencies among immigrants are constant. The sampling distribution of allele types depends on the sample size N, the array of allele frequencies among immigrants p, and the parameter theta=phi/lambda, where phi is the immigration rate and lambda is the individual birth rate. The sampling distribution of alleles does not depend on time or population size, and no "genetic equilibrium" assumption is therefore needed to apply the model to natural populations. The moments of the sampling distribution of allele types are used to calculate the expectation and variance of a sample identity by descent estimate (fN 0) within islands under the BDI model.

An analysis is made of the variation among individuals in finite populations of the proportion of their genes which are identical by descent. There are two causes of this variability: variation in pedigree among individuals, and linkage which causes whole blocks of genes to be identical or nonidentical by descent. The variation between and within populations is analyzed in detail for several mating systems: monoecious populations with and without random selfing, and dioecious populations with and without a hierarchical mating structure. Transition matrices for two-locus descent measures are given for each system. Total variability is obtained by integrating these measures over the distribution of map distances over whole chromosomes. Approximate methods are also developed for unlinked loci. Unless populations have a very small effective size (Ne) there is little variation in inbreeding between populations. For unlinked loci, the coefficient of variation in nonidentity within populations approaches about 1 √3N e for random selfing, 1 √6N e for monogamous matings and 1 √12N e for monoecy with selfing excluded or dioecy with random pairing. If there is no association between map distance and initial heterozygosity or effect on quantitative traits, the coefficient of variation in mean heterozygosity over the genome is related to that in nonidentity, and the additional variation in a quantitative trait due to dominant genes equals the product of the square of the initial inbreeding depression and the squared coefficient of variation of nonidentity.

Attention is drawn to errors common in the derivation of forms for the genotypic covariance of noninbred relatives from a Hardy-Weinberg population of diploids. A synthesis of Fisher's least-squares method of partitioning the genotypic variance and Malecot's probability method of expressing kinship, yields a general form. For one locus, the form is (P ss + P sd + P ds + P dd ) 1 2 σ a 2 + (P ss P dd + P sd P ds ) σa d 2 , where σa2 is the additive genetic variance, αd2 is the variance of dominance deviations, pij is the probability that parental gamete i is identical by descent to parental gamete j, i = s, d indexes the parents of one relative, and j = s, d indexes those of the other. The form provides a framework for obtaining the covariance of relatives from an equilibrium population with linkage.

The processes of gene identity by descent and of allelic identities (or likenesses) between genes have been previously studied under a wide variety of migration and subdivision models of population evolution. Since the two processes follow probabilistically parallel paths, there has been a tendency to consider the two processes as equivalent, and to equate estimates of correlation of allele frequency with those of gene identity by descent. The adequacy of this procedure must depend upon the evolutionary history. Here, by distinguishing the processes and considering them jointly as a population evolves in time, we demonstrate the relationship between them in a simple situation, thus providing a basis for study of the extent to which observed allelic similarity between populations is a reflection of underlying gene identity.

A formal mathematical framework is presented for the study of linkage in man and the concept of chromosome pedigree is defined for both autosomes and X chromosomes. It is shown that, assuming no interference, all the crossover processes in the pedigree may be viewed jointly as a continuous-time Markov random walk on the vertices of a hypercube, the time parameter being map distance along the chromosome. The event that two individuals have a segment of chromosome in common, thus proving them to be related, corresponds to the random walk hitting a particular set of vertices. The probability of this happening is calculated for various types of relationship, making use of the symmetry of the situation to partition the vertices into a very much smaller number of orbits and render the computation manageable. The probability that an individual with n children passes on all his or her genes to them is also calculated in this way.

In considering populations of finite size, there are two approaches to studying the correlation of genotype frequencies at two linked loci. The first is based on disequilibrium parameters, and the second on identity-by-descent methods. The basic connection between the two approaches may be stated in the following way. The expected square of the correlation of gene frequencies at two loci r2 = D2pApapBpb, is approximately equal to Q, the probability that, given two genes at one locus which are identical by descent, then the two genes at the second locus will be identical by descent through the same pathways. Using this relationship it is shown that for a monoecious population of effective size Ne with no selection, the expected value of r2 will tend to approximately 1(1 + 4Nec), where c is the recombination frequency, the rate of approach being given by (1 − 12Ne)(1 − c)2. Computer simulation has shown that this formula holds with reasonable accuracy, and that it is also quite accurate in populations with heterozygote advantage stabilizing gene frequencies. It is suggested that the measurement of linkage disequilibrium in natural populations might thus be used to give information about the effective population size. The identity-by-descent approach is extended to derive the distribution of the length of homozygous chromosome segment surrounding a locus which is identical by descent. The mean such length is approximately 12Ne(log Ne − 1). It is suggested that a general theory of stability of the genotype in small populations might be based on parameters such as this.

The genetic variability in a subdivided population of finite size was investigated in terms of the effective number of alleles maintained. The population structures used in the investigation are the island model and the stepping stone model. For these models, exact formulas or approximation formulas which give the probability that two randomly chosen homologous genes, one from each of two specified colonies (or both from the same colony) are identical by descent were obtained. It is shown that the genetic variability in the whole population with stepping stone structure is known if the mutation rate and the variability in a single colony is known, without knowing the migration rate. The validity of the analyses was checked by simulation.

In a haploid population of constant finite size, in the absence of mutation and selection, the probability that there will be less than i different alleles in the population in the next generation, given that there are i alleles in the population at present, tends asymptotically with time to 1 − Gii, where Gii is the probability that i randomly chosen haploid individuals are descended from exactly i individuals in the previous generation. When i is 2, this proves that the rate of increase of identity by descent equals the asymptotic rate of fixation of the whole population. The formula is used to calculate rates of loss of multiple alleles under several different models of population reproduction.

Motivated by problems in conservation biology we study genetic dynamics in structured populations of diploid organisms (monoecious or dioecious). Our analysis provides an analytical framework that unifies substantial parts of previous work in terms of exact identity by descent (IBD) and identity by state (IBS) recursions. We provide exact conditions under which two structured haploid and diploid populations are equivalent, and some sufficient conditions under which a dioecious diploid population can be treated as a monoecious diploid one. The IBD recursions are used for computing local and metapopulation inbreeding and coancestry effective population sizes and for predictions of several types of fixation indices over different time horizons.

Probability of identity by type is studied for regular systems of inbreeding in the presence of mutation. Analytic results are presented for half-sib mating, first cousin mating, and half nth cousin mating under both infinite allele and two allele (back mutation) models. Reasonable rates of mutation do not provide significantly different results from probability of identity by descent in the absence of mutation. Homozygosity is higher under half-sib mating than under first cousin mating, but the expected number of copies of a gene in the population is higher under first cousin mating than under half-sib mating.

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.

Relationship inference in a population is of interest for many areas of research from anthropology to genetics. It is possible to directly infer the relationship between the two individuals in a couple from their genetic data or to indirectly infer it from the genetic data of one of their offspring. For this reason, one can wonder if it is more advantageous to sample couples or single individuals to study relationships of couples in a population. Indeed, sampling two individuals is more informative than sampling one as we are looking at four haplotypes instead of two, but it also doubles the cost of the study and is a more complex sampling scheme. To answer this question, we performed simulations of 1000 trios from 10 different relationships using real human haplotypes to have realistic genome-wide genetic data. Then, we compared the genome sharing coefficients and the relationship inference obtained from either a pair of individuals or one of their offspring using both single-point and multi-point approaches. We observed that for relationships closer than 1st cousin, pairs of individuals were more informative than one of their offspring for relationship inference, and kinship coefficients obtained from single-point methods gave more accurate or equivalent genome sharing estimations. For more remote relationships, offspring were more informative for relationship inference, and inbreeding coefficients obtained from multi-point methods gave more accurate genome sharing estimations. In conclusion, relationship inference on a parental pair or on one of their offspring provides complementary information. When possible, sampling trios should be encouraged as it could allow spanning a wider range of potential relationships.