A maximum-likelihood estimation of pairwise relatedness for autopolyploids

Relatedness between individuals is central to ecological genetics. Multiple methods are available to quantify relatedness from molecular data, including method-of-moment and maximum-likelihood estimators. We describe a maximum-likelihood estimator for autopolyploids, and quantify its statistical performance under a range of biologically relevant conditions. The statistical performances of five additional polyploid estimators of relatedness were also quantified under identical conditions. When comparing truncated estimators, the maximum-likelihood estimator exhibited lower root mean square error under some conditions and was more biased for non-relatives, especially when the number of alleles per loci was low. However, even under these conditions, this bias was reduced to be statistically insignificant with more robust genetic sampling. We also considered ambiguity in polyploid heterozygote genotyping and developed a weighting methodology for candidate genotypes. The statistical performances of three polyploid estimators under both ideal and actual conditions (including inbreeding and double reduction) were compared. The software package POLYRELATEDNESS is available to perform this estimation and supports a maximum ploidy of eight.     

Unbiased relatedness estimation in structured populations

Knowledge of the genetic relatedness between individuals is important in many research areas in quantitative genetics, conservation genetics, forensics, evolution, and ecology. In the absence of pedigree records, relatedness can be estimated from genetic marker data using a number of estimators. These estimators, however, make the critical assumption of a large random mating population without genetic structures. The assumption is frequently violated in the real world where geographic/social structures or nonrandom mating usually lead to genetic structures. In this study, I investigated two approaches to the estimation of relatedness between a pair of individuals from a subpopulation due to recent common ancestors (i.e., relatedness is defined and measured with the current focal subpopulation as reference). The indirect approach uses the allele frequencies of the entire population with and without accounting for the population structure, and the direct approach uses the allele frequencies of the current focal subpopulation. I found by simulations that currently widely applied relatedness estimators are upwardly biased under the indirect approach, but can be modified to become unbiased and more accurate by using Wright's F(st) to account for population structures. However, the modified unbiased estimators under the indirect approach are clearly inferior to the unmodified original estimators under the direct approach, even when small samples are used in estimating both allele frequencies and relatedness.     

A pairwise relatedness estimator for polyploids

Studies in genetics and ecology often require estimates of relatedness coefficients based on genetic marker data. Many diploid estimators have been developed using either method‐of‐moments or maximum‐likelihood estimates. However, there are no relatedness estimators for polyploids. The development of a moment estimator for polyploids with polysomic inheritance, which simultaneously incorporates the two‐gene relatedness coefficient and various ‘higher‐order’ coefficients, is described here. The performance of the estimator is compared to other estimators under a variety of conditions. When using a small number of loci, the estimator is biased because of an increase in ill‐conditioned matrices. However, the estimator becomes asymptotically unbiased with large numbers of loci. The ambiguity of polyploid heterozygotes (when balanced heterozygotes cannot be distinguished from unbalanced heterozygotes) is also considered; as with low numbers of loci, genotype ambiguity leads to bias. A software, PolyRelatedness , implementing this method and supporting a maximum ploidy of 8 is provided.     

Multilocus estimation of pairwise relatedness with dominant markers

Estimators for pairwise relatedness designed for dominant markers are derived, based on a genetic model that accounts for the full structure of pairwise relatedness between two individuals at a diploid locus with dominance. They jointly estimate 'relatedness' and 'fraternity', in which case the estimators are inherently multilocus, as at least two loci of differing gene frequency are required. Extensions to cases of zero fraternity and isolation by distance (inbreeding) are also examined. Properties of estimators are examined by simulation and compared to the estimator of Hardy. The most statistical power for pairwise relatedness occurs when roughly half of individuals are the recessive phenotype. Estimation procedures are implemented in the computer program mark.     

TECHNICAL ADVANCES: A maximum-likelihood relatedness estimator allowing for negative relatedness values

Previously reported maximum-likelihood pairwise relatedness (r) estimator of Thompson and Milligan (M) was extended to allow for negative r estimates under the regression interpretation of r. This was achieved by establishing the equivalency of the likelihoods used in the kinship program and the likelihoods of Thompson. The new maximum-likelihood (ML) estimator was evaluated by Monte Carlo simulations. It was found that the new ML estimator became unbiased significantly faster compared to the original M estimator when the amount of genotype information was increased. The effects of allele frequency estimation errors on the new and existing relatedness estimators were also considered.     

A comparison of microsatellite-based pairwise relatedness estimators

Studies of inbreeding depression or kin selection require knowledge of relatedness between individuals. If pedigree information is lacking, one has to rely on genotypic information to infer relatedness. In this study we investigated the performance (absolute and relative) of 10 marker-based relatedness estimators using allele frequencies at microsatellite loci obtained from natural populations of two bird species and one mammal species. Using Monte Carlo simulations we show that many factors affect the performance of estimators and that different sets of loci promote the use of different estimators: in general, there is no single best-performing estimator. The use of locus-specific weights turns out to greatly improve the performance of estimators when marker loci are used that differ strongly in allele frequency distribution. Microsatellite-based estimates are expected to explain between 25 and 79% of variation in true relatedness depending on the microsatellite dataset and on the population composition (i.e. the frequency distribution of relationship in the population). We recommend performing Monte Carlo simulations to decide which estimator to use in studies of pairwise relatedness.     

A PCA-based method for ancestral informative markers selection in structured populations

Identification of population structure can help trace population histories and identify disease genes. Structured association (SA) is a commonly used approach for population structure identification and association mapping. A major issue with SA is that its performance greatly depends on the informativeness and the numbers of ancestral informative markers (AIMs). Present major AIM selection methods mostly require prior individual ancestry information, which is usually not available or uncertain in practice. To address this potential weakness, we herein develop a novel approach for AIM selection based on principle component analysis (PCA), which does not require prior ancestry information of study subjects. Our simulation and real genetic data analysis results suggest that, with equivalent AIMs, PCA-based selected AIMs can significantly increase the accuracy of inferred individual ancestries compared with traditionally randomly selected AIMs. Our method can easily be applied to whole genome data to select a set of highly informative AIMs in population structure, which can then be used to identify potential population structure and correct possible statistical biases caused by population stratification.     

An estimator for pairwise relatedness using molecular markers

I propose a new estimator for jointly estimating two-gene and four-gene coefficients of relatedness between individuals from an outbreeding population with data on codominant genetic markers and compare it, by Monte Carlo simulations, to previous ones in precision and accuracy for different distributions of population allele frequencies, numbers of alleles per locus, actual relationships, sample sizes, and proportions of relatives included in samples. In contrast to several previous estimators, the new estimator is well behaved and applies to any number of alleles per locus and any allele frequency distribution. The estimates for two- and four-gene coefficients of relatedness from the new estimator are unbiased irrespective of the sample size and have sampling variances decreasing consistently with an increasing number of alleles per locus to the minimum asymptotic values determined by the variation in identity-by-descent among loci per se, regardless of the actual relationship. The new estimator is also robust for small sample sizes and for unknown relatives being included in samples for estimating allele frequencies. Compared to previous estimators, the new one is generally advantageous, especially for highly polymorphic loci and/or small sample sizes.     

Estimating the correlation of pairwise relatedness along chromosomes

The 'spatial' pattern of the correlation of pairwise relatedness among loci within a chromosome is an important aspect for an insight into genomic evolution in natural populations. In this article, a statistical genetic method is presented for estimating the correlation of pairwise relatedness among linked loci. The probabilities of identity-in-state (IIS) are related to the probabilities of identity-by-descent (IBS) for the two- and three-loci cases. By decomposing the joint probabilities of two- or three-loci IBD, the probability of pairwise relatedness at a single locus and its correlation among linked loci can be simultaneously estimated. To provide effective statistical methods for estimation, weighted least square (LS) and maximum likelihood (ML) methods are evaluated through extensive Monte Carlo simulations. Results show that the ML method gives a better performance than the weighted LS method with haploid genotypic data. However, there are no significant differences between the two methods when two- or three-loci diploid genotypic data are employed. Compared with the optimal size for haploid genotypic data, a smaller optimal sample size is predicted with diploid genotypic data.     

A method for the estimation of parameters for natural stage-structured populations

A relatively simple method is proposed for the estimation of parameters of stage-structured populations from sample data for situation where (a) unit time survival rates may vary with time, and (b) the distribution of entry times to stage 1 is too complicated to be fitted with a simple parametric model such as a normal or gamma distribution. The key aspects of this model are that the entry time distribution is approximated by an exponential function withp parameters, the unit time survival rates in stages are approximated by anr parameter exponential polynomial in the stage number, and the durations of stages are assumed to be the same for all individuals. The new method is applied to four Zooplankton data sets, with parametric bootstrapping used to assess the bias and variation in estimates. It is concluded that good estimates of demographic parameters from stagefrequency data from natural populations will usually only be possible if extra information such as the durations of stages is known.     

Informativeness of genetic markers for pairwise relationship and relatedness inference

Measuring the information content of markers in relationship/relatedness inferences is important in selecting highly informative markers to attain a given statistical power with the minimal genotyping effort. Using information-theoretic principles, I introduce the informativeness for relationship (I(R)) and the informativeness for relatedness (I(r)) to measure the amount of information provided by markers in inferring pairwise relationships (R) and relatedness (r), respectively. I also propose a fast and accurate algorithm to calculate the power (PW(R)) of a set of markers in differentiating two candidate relationships, and the reciprocal of the mean squared deviations of relatedness estimates (RMSD) to measure the amount of information of markers actually used by an estimator in estimating relatedness. All of the four measurements (I(R), I(r), PW(R), RMSD) apply to dominant and codominant markers, haploid and diploid individuals, and take into account of mutations and typing errors in data. The statistical properties of the four measurements and their relationships are investigated analytically and are examined by applying these methods to simulated and empirical data.     

A parametric model for estimation of dispersal patterns applied to five passerine spatially structured populations

Natal dispersal capture-recapture data from five fragmented populations of house sparrows, great tits, and blue tits were analyzed using maximum likelihood methods. A new two-parametric distribution was constructed that includes four distributions previously used as special cases in the literature. Dispersal standard deviations were estimated at 22.9 km for the house sparrows and ranged from 0.66 to 4.4 km for the tits. Female great tits and blue tits dispersed consistently further than males. Estimates of the shape parameter of the dispersal distribution ranged from 0.66 to 2.27, indicating strong to moderately leptokurtic dispersal displacements. There were significant effects of density on local immigration rates and a consistent tendency for immigration rates to depend underproportionally on local densities. Potential implications of the shape of the dispersal distribution for the spread of invading organisms were investigated and compared with previous results. It is shown that the wave speed, for a given dispersal standard deviation, depends only to some extent on leptokurtosis, provided that the intrinsic growth rate of the population is moderate or small. In estimating the dispersal standard deviation, however, incorrect assumptions about the degree of leptokurtosis can lead to a large bias in estimation and predictions.     

A diffusion model for geographically structured populations

Summary A diffusion model is derived for the evolution of a diploid monoecious population under the influence of migration, mutation, selection, and random genetic drift. The population occupies an unbounded linear habitat; migration is independent of genotype, symmetric, and homogeneous. The treatment is restricted to a single diallelic locus without dominance. With the customary diffusion hypotheses for migration and the assumption that the mutation rates, selection coefficient, variance of the migrational displacement, and reciprocal of the population density are all small and of the same order of magnitude, a boundary value problem is deduced for the mean gene frequency and the covariance between the gene frequencies at any two points in the habitat. Supported by the National Science Foundation (Grant No. DEB77-21494).     

A simple persistence condition for structured populations

The fundamental question in both basic and applied population biology of whether a species will increase in numbers is often investigated by finding the population growth rate as the largest eigenvalue of a deterministic matrix model. For a population classified only by age, and not stage or size, a simpler biologically interpretable condition can be used, namely whether R ₀, the mean number of offspring per newborn, is greater than one. However, for the many populations not easily described using only age classes, stage-structured models must be used for which there is currently no quantity like R ₀. We determine analogous quantities that must be greater than one for persistence of a general structured population model that have a similar useful biological interpretation. Our approach can be used immediately to determine the magnitude of changes and interactions that would either allow population persistence or would ensure control of an undesirable species.     

Genetic relatedness in viscous populations

Summary Hamilton's inclusive fitness rule shows that the evolution of altruism is facilitated by high genetic relatedness of altruists to their beneficiaries. But the evolution of altruism is inhibited when the beneficiaries are also close competitors of the altruist, as will often be true in structured or viscous populations. However, Hamilton's rule still gives the correct condition for the evolution of altruism if relatedness is measured with respect to the local competitive neighbourhood.     

Maximum-likelihood estimation of relatedness

Relatedness between individuals is central to many studies in genetics and population biology. A variety of estimators have been developed to enable molecular marker data to quantify relatedness. Despite this, no effort has been given to characterize the traditional maximum-likelihood estimator in relation to the remainder. This article quantifies its statistical performance under a range of biologically relevant sampling conditions. Under the same range of conditions, the statistical performance of five other commonly used estimators of relatedness is quantified. Comparison among these estimators indicates that the traditional maximum-likelihood estimator exhibits a lower standard error under essentially all conditions. Only for very large amounts of genetic information do most of the other estimators approach the likelihood estimator. However, the likelihood estimator is more biased than any of the others, especially when the amount of genetic information is low or the actual relationship being estimated is near the boundary of the parameter space. Even under these conditions, the amount of bias can be greatly reduced, potentially to biologically irrelevant levels, with suitable genetic sampling. Additionally, the likelihood estimator generally exhibits the lowest root mean-square error, an indication that the bias in fact is quite small. Alternative estimators restricted to yield only biologically interpretable estimates exhibit lower standard errors and greater bias than do unrestricted ones, but generally do not improve over the maximum-likelihood estimator and in some cases exhibit even greater bias. Although some nonlikelihood estimators exhibit better performance with respect to specific metrics under some conditions, none approach the high level of performance exhibited by the likelihood estimator across all conditions and all metrics of performance.     

Selection for recombination in structured populations

In finite populations, linkage disequilibria generated by the interaction of drift and directional selection (Hill-Robertson effect) can select for sex and recombination, even in the absence of epistasis. Previous models of this process predict very little advantage to recombination in large panmictic populations. In this article we demonstrate that substantial levels of linkage disequilibria can accumulate by drift in the presence of selection in populations of any size, provided that the population is subdivided. We quantify (i) the linkage disequilibrium produced by the interaction of drift and selection during the selective sweep of beneficial alleles at two loci in a subdivided population and (ii) the selection for recombination generated by these disequilibria. We show that, in a population subdivided into n demes of large size N, both the disequilibrium and the selection for recombination are equivalent to that expected in a single population of a size intermediate between the size of each deme (N) and the total size (nN), depending on the rate of migration among demes, m. We also show by simulations that, with small demes, the selection for recombination is stronger than both that expected in an unstructured population (m = 1 - 1/n) and that expected in a set of isolated demes (m = 0). Indeed, migration maintains polymorphisms that would otherwise be lost rapidly from small demes, while population structure maintains enough local stochasticity to generate linkage disequilibria. These effects are also strong enough to overcome the twofold cost of sex under strong selection when sex is initially rare. Overall, our results show that the stochastic theories of the evolution of sex apply to a much broader range of conditions than previously expected.     

Genetic linkage in the estimation of pairwise relationship

 New types of markers, such as RAPDs, microsatellite markers, AFLPs, and SNPs provide the opportunity to obtain information on individuals at multiple genetic loci across the genome. This increase in the number of marker loci has provided enhanced opportunities for statistical analysis of the genetic consequences of genealogical relationship among individuals. In place of the classical models, we can now investigate empirical multilocus segregation patterns. Linkage among loci decreases the precision of relationship estimation but permits additional dimensions of genome sharing to be explored. In this paper we consider the effect of linkage on the pattern of genome sharing among relatives who share (on average) 25% of their dipolid genomes using the empirical meioses giving rise to 58 gametophytes from a single maternal plant of the species Pinus taeda (loblolly pine). The genome sharing among relatives is quantified in terms of the linkage map of the markers. Received: 26 November 1997 / Accepted: 3 March 1998