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 relatedness between individuals in general populations with a focus on their use in conservation programs

Relatedness estimators are widely used in genetic studies, but effects of population structure on performance of estimators, criteria to evaluate estimators, and benefits of using such estimators in conservation programs have to date received little attention. In this article we present new estimators, based on the relationship between coancestry and molecular similarity between individuals, and compare them with existing estimators using Monte Carlo simulation of populations, either panmictic or structured. Estimators were evaluated using statistical criteria and a diversity criterion that minimized relatedness. Results show that ranking of estimators depends on the population structure. An existing estimator based on two-gene and four-gene coefficients of identity performs best in panmictic populations, whereas a new estimator based on coancestry performs best in structured populations. The number of marker alleles and loci did not affect ranking of estimators. Statistical criteria were insufficient to evaluate estimators for their use in conservation programs. The regression coefficient of pedigree relatedness on estimated relatedness (beta2) was substantially lower than unity for all estimators, causing overestimation of the diversity conserved. A simple correction to achieve beta2 = 1 improves both existing and new estimators. Using relatedness estimates with correction considerably increased diversity in structured populations, but did not do so or even decreased diversity in panmictic populations.     

Unbiased estimator for genetic drift and effective population size

Amounts of genetic drift and the effective size of populations can be estimated from observed temporal shifts in sample allele frequencies. Bias in this so-called temporal method has been noted in cases of small sample sizes and when allele frequencies are highly skewed. We characterize bias in commonly applied estimators under different sampling plans and propose an alternative estimator for genetic drift and effective size that weights alleles differently. Numerical evaluations of exact probability distributions and computer simulations verify that this new estimator yields unbiased estimates also when based on a modest number of alleles and loci. At the cost of a larger standard deviation, it thus eliminates the bias associated with earlier estimators. The new estimator should be particularly useful for microsatellite loci and panels of SNPs, representing a large number of alleles, many of which will occur at low frequencies.     

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.     

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.     

A simplified estimator of two and four gene relationship coefficients

Molecular marker data provide a means of circumventing the problem of not knowing the population structure of a natural population, as observed similarities between a pair's genotypes provide information on their genetic relationship. Numerous method-of-moment (MOM) estimators have been developed for estimating relationship coefficients using this information. Here, I present a simplified form of Wang's 2002 relationship estimator that is not dependent upon a previously required weighting scheme, thus improving the efficiency of the estimator when used with genuinely related pairs. The new estimator is compared against other estimators under a range of conditions, including situations where the parameter estimates are truncated to lie within the legitimate parameter space. The advantages of the new estimator are most notable for the two-gene coefficient of relatedness. Truncating the MOM estimators results in parameter estimates whose properties are similar to maximum likelihood estimates, with them having generally lower sampling variances, but being biased.     

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.     

Estimating relatedness and relationships using microsatellite loci with null alleles

Relatedness is often estimated from microsatellite genotypes that include null alleles. When null alleles are present, observed genotypes represent one of several possible true genotypes. If null alleles are detected, but analyses do not adjust for their presence (ie, observed genotypes are treated as true genotypes), then estimates of relatedness and relationship can be incorrect. The number of loci available in many wildlife studies is limited, and loci with null alleles are commonly a large proportion of data that cannot be discarded without substantial loss of power. To resolve this problem, we present a new approach for estimating relatedness and relationships from data sets that include null alleles. Once it is recognized that the probability of the observed genotypes is dependent on the probabilities of a limited number of possible true genotypes, the required adjustments are straightforward. The concept can be applied to any existing estimators of relatedness and relationships. We review established maximum likelihood estimators and apply the correction in that setting. In an application of the corrected method to data from striped hyenas, we demonstrate that correcting for the presence of null alleles affect results substantially. Finally, we use simulated data to confirm that this method works better than two common approaches, namely ignoring the presence of null alleles or discarding affected loci.     

Estimating pairwise relatedness from dominant genetic markers

Knowledge of the genetic relatedness between a pair of individuals is important in many research areas of quantitative genetics, conservation genetics, evolution and ecology. Many estimators have been developed to estimate such pairwise relatedness (r) using codominant markers, such as microsatellites and enzymes. In contrast, only two estimators are proposed to use dominant markers, such as random amplified polymorphic DNAs (RAPDs) and amplified fragment length polymorphisms (AFLPs), in relatedness inference. They are both biased estimators, and their statistical properties and robustness to the sampling errors in allele frequency have not been investigated. In this short paper, I propose two new pairwise relatedness estimators for dominant markers, and compare them in precision, accuracy and robustness to sampling with the two previous estimators using simulations. It was found that the new estimator based on the least squares approach is unbiased when allele frequencies are known or estimated from a sample without correcting for sampling effects. It has, however, a low precision and as a result, an intermediate overall performance among the four estimators in terms of the mean squared deviation (MSD) of estimates from actual values of r. The new estimator based on a similarity index is slightly biased but has generally the lowest MSD among the four estimators compared, regardless of the number of loci, type of actual relationships, allele frequencies known or estimated from samples. Simulations also show that the confidence intervals estimated by bootstrapping are appropriate for different estimators provided that the number of loci used in the estimation is not small.     

Methods to evaluate populations for alleles to improve an elite hybrid

Elite hybrids can be improved by the introgression of favorable alleles not already present in the hybrid. Our first objective was to evaluate several estimators derived from quantitative genetic theory that attempt to quantify the relative number of useful alleles in potential donor populations. Secondly, we wanted to evaluate two proposed ways of determining relatedness of donor populations to the parents of the elite hybrid. Two experiments, each consisting of 21 maize populations of known pedigree, were grown at three and four environments in Minnesota in 1991. Yield and plant height means were used to provide estimates of each of the following statistics: (1) LPLU, a minimally biased statistic, (2) UBND, the minimum estimate of an upper bound, (3) NI, the net improvement, (4) PTC, the predicted three-way cross, and (5) TCSC, the testcross of the populations. These statistics are biased estimators of the relative number of unique favorable alleles contained within a population compared to a reference elite hybrid. Based on rank correlations, all statistics except NI ranked populations similarly. The percent novel germplasm relative to the single cross to be improved was positively correlated with the estimates of favorable alleles except when NI was used as the estimator. The relationship estimators agreed with the genetic constitution of the donor populations. Strong positive correlations existed between diversity, based on the relationship rankings, and all the estimator rankings, except NI. Potential donor populations were effectively identified by LPLU, UBND, PTC, and TCSC. NI was not a good estimator of unique favorable alleles.     

Comparative evaluation of a new effective population size estimator based on approximate bayesian computation

We describe and evaluate a new estimator of the effective population size (N(e)), a critical parameter in evolutionary and conservation biology. This new "SummStat" N(e) estimator is based upon the use of summary statistics in an approximate Bayesian computation framework to infer N(e). Simulations of a Wright-Fisher population with known N(e) show that the SummStat estimator is useful across a realistic range of individuals and loci sampled, generations between samples, and N(e) values. We also address the paucity of information about the relative performance of N(e) estimators by comparing the SummStat estimator to two recently developed likelihood-based estimators and a traditional moment-based estimator. The SummStat estimator is the least biased of the four estimators compared. In 32 of 36 parameter combinations investigated using initial allele frequencies drawn from a Dirichlet distribution, it has the lowest bias. The relative mean square error (RMSE) of the SummStat estimator was generally intermediate to the others. All of the estimators had RMSE > 1 when small samples (n = 20, five loci) were collected a generation apart. In contrast, when samples were separated by three or more generations and N(e) < or = 50, the SummStat and likelihood-based estimators all had greatly reduced RMSE. Under the conditions simulated, SummStat confidence intervals were more conservative than the likelihood-based estimators and more likely to include true N(e). The greatest strength of the SummStat estimator is its flexible structure. This flexibility allows it to incorporate any potentially informative summary statistic from population genetic data.     

Estimation of quantitative genetic parameters using marker-inferred relatedness in Japanese flounder: a case study of upward bias

Marker-based methods for estimating heritability have been proposed as an effective means to study quantitative traits in long-lived organisms and natural populations. However, practical examinations to evaluate the usefulness and robustness of a regression method are limited. Using several quantitative traits of Japanese flounder Paralichthys olivaceus, the present study examined the influence of relatedness estimator and population structure on the estimation of heritability and genetic correlation under a regression method with 7 microsatellite loci. Significant heritability and genetic correlation were detected for several quantitative traits in 2 laboratory populations but not in a natural population. In the laboratory populations, upward bias in heritability appeared depending on the relatedness estimators and the populations. Upward bias in heritability increased with decreasing the actual variance of relatedness, suggesting that the estimates of heritability under the regression method tend to be overestimated due to the underestimation of the actual variance of relatedness. Therefore, relationship structure and precise estimation of relatedness are critical for applying this method.     

Estimation of genetically effective breeding numbers using a rejection algorithm approach

Polygynous mating results in nonrandom sampling of the adult male gamete pool in each generation, thereby increasing the rate of genetic drift. In principle, genetic paternity analysis can be used to infer the effective number of breeding males (Nebm). However, this requires genetic data from an exhaustive sample of candidate males. Here we describe a new approach to estimate Nebm using a rejection algorithm in association with three statistics: Euclidean distance between the frequency distributions of maternally and paternally inherited alleles, average number of paternally inherited alleles and average gene diversity of paternally inherited alleles. We quantify the relationship between these statistics and Nebm using an individual-based simulation model in which the male mating system varied continuously between random mating and extreme polygyny. We evaluate this method using genetic data from a natural population of highly polygynous fruit bats (Cynopterous sphinx). Using data in the form of mother-offspring genotypes, we demonstrate that estimates of Nebm are very similar to independent estimates based on a direct paternity analysis that included data on candidate males. Our method also permits an evaluation of uncertainty in estimates of Nebm and thus facilitates inferences about the mating system from genetic data. Finally, we investigate the sensitivity of our method to sample size, model assumptions, adult population size and the mating system. These analyses demonstrate that the rejection algorithm provides accurate estimates of Nebm across a broad range of demographic scenarios, except when the true Nebm is high.     

Estimating the total number of alleles using a sample coverage method.

Previously reported methods for estimating the number of different alleles at a single locus in a population have not described a useful general result. Using the number of alleles observed in a sample gives an underestimate for the true number of alleles. The similar problem of estimating the number of species in a population was first investigated in 1943. In this article we use the sample coverage method proposed by Chao and Lee in 1992 to estimate the number of alleles in a population when there are unequal allele frequencies. Simulation studies under the recurrent mutation model show that, for reasonable sample sizes, a significantly better estimate of the true number can be obtained than that using only the observed alleles. Results under the stepwise mutation model and infinite-allele model are presented. Possible applications include improving the characterization of the prior distribution for the allele frequencies, adjusting the estimates of genetic diversity, and estimating the range of microsatellite alleles.     

Estimating Relatedness in the Presence of Null Alleles

Studies of genetics and ecology often require estimates of relatedness coefficients based on genetic marker data. However, with the presence of null alleles, an observed genotype can represent one of several possible true genotypes. This results in biased estimates of relatedness. As the numbers of marker loci are often limited, loci with null alleles cannot be abandoned without substantial loss of statistical power. Here, we show how loci with null alleles can be incorporated into six estimators of relatedness (two novel). We evaluate the performance of various estimators before and after correction for null alleles. If the frequency of a null allele is <0.1, some estimators can be used directly without adjustment; if it is >0.5, the potency of estimation is too low and such a locus should be excluded. We make available a software package entitled PolyRelatedness v1.6, which enables researchers to optimize these estimators to best fit a particular data set.     

Colony sex ratios vary with queen number but not relatedness asymmetry in the ant Formica exsecta

Split-sex-ratio theory assumes that conflict over whether to produce predominately males or female reproductives (gynes) is won by the workers in haplodiploid insect societies and the outcome is determined by colony kin structure. Tests of the theory have the potential to provide support for kin-selection theory and evidence of social conflict. We use natural variation in kinship among polygynous (multiple-queen) colonies of the ant Formica exsecta to study the associations between sex ratios and the relatedness of workers to female versus male brood (relatedness asymmetry). The population showed split sex ratios with about 89% of the colonies producing only males, resulting in an extremely male-biased investment ratio in the population. We make two important points with our data. First, we show that queen number may affect sex ratio independently of relatedness asymmetry. Colonies producing only males had greater genetic effective queen number but did not have greater relatedness asymmetry from the perspective of the adult workers that rear the brood. This lack of a difference in relatedness asymmetry between colonies producing females and those producing only males was associated with a generally low relatedness between workers and brood. Second, studies that suggest support for the relatedness-asymmetry hypothesis based on indirect measures of relatedness asymmetry (e.g. queen number estimated from relatedness data taken from the brood only) should be considered with caution. We propose a new hypothesis that explains split sex ratios in polygynous social insects based on the value of producing replacement queens.     

The effects of habitat fragmentation on the social kin structure and mating system of the agile antechinus, Antechinus agilis

Habitat fragmentation is one of the major contributors to the loss of biodiversity worldwide. However, relatively little is known about its more immediate impacts on within-patch population processes such as social structure and mating systems, whose alteration may play an important role in extinction risk. We investigated the impacts of habitat fragmentation due to the establishment of an exotic softwood plantation on the social kin structure and breeding system of the Australian marsupial carnivore, Antechinus agilis. Restricted dispersal by males in fragmented habitat resulted in elevated relatedness among potential mates in populations in fragments, potentially increasing the risk of inbreeding. Antechinus agilis nests communally in tree hollows; these nests are important points for social contact between males and females in the mating season. In response to elevated relatedness among potential mates in fragmented habitat, A. agilis significantly avoided sharing nests with opposite-sex relatives in large fragment sites (but not in small ones, possibly due to limited nest locations and small population sizes). Because opposite-sex individuals shared nests randomly with respect to relatedness in unfragmented habitat, we interpreted the phenomenon in fragmented habitat as a precursor to inbreeding avoidance via mate choice. Despite evidence that female A. agilis at high inbreeding risk selected relatively unrelated mates, there was no overall increased avoidance of related mates by females in fragmented habitats compared to unfragmented habitats. Simulations indicated that only dispersal, and not nonrandom mating, contributed to inbreeding avoidance in either habitat context. However, habitat fragmentation did influence the mating system in that the degree of multiple paternity was reduced due to the reduction in population sizes and population connectivity. This, in turn, reduced the number of males available to females in the breeding season. This suggests that in addition to the obvious impacts of reduced recruitment, patch recolonization and increased genetic drift, the isolation of populations in habitat patches may cause changes in breeding behaviour that contribute to the negative impacts of habitat fragmentation.     

A new likelihood estimator and its comparison with moment estimators of individual genome-wide diversity

The inbreeding coefficient of an individual, F, is one of the central parameters in population genetics theory. It has found important applications in evolutionary biology, conservation and ecology, such as the study of inbreeding depression. In the absence of detailed and reliable pedigree records, researchers have developed quite a few estimators to estimate F or the genome-wide homozygosity from genetic marker data. The statistical properties and comparative performances of these metrics are rarely known, however, which impedes an informed choice of the most appropriate one in practical applications. In this investigation, I propose a new likelihood F estimator that makes efficient use of marker information and takes into account of allelic dropouts, null alleles and prior knowledge of inbreeding. I compare the likelihood estimator with three moment estimators of F and three metrics of genomic homozygosity (or heterozygosity) by analysing both simulated and empirical datasets. It is shown that the likelihood estimator invariably outperforms the other estimators and metrics across all datasets analysed. For a typical dataset in heterozygosity-fitness correlation studies involving 10-20 microsatellites and 50 individuals, the correlation between the likelihood estimator and F (the simulated true inbreeding coefficient) is about 8 ～ 35% higher than that between the moment estimators and F. A frequently applied metric, multilocus heterozygosity (MLH), and an F estimator based on the consideration of the proportion of alleles in homozygous conditions, [F R'), are shown to have particularly poor performances. The low correlation between MLH and fitness traits, which is widely observed in numerous empirical studies, might be partially caused by the adoption of this inefficient estimator of genomic inbreeding.     

Estimating relatedness in social groups

Genetic relatedness is a vital parameter in the evolution of social behaviour by kin selection. It can be easily estimated using genetic markers and calculating the genotypic correlation or regression of group members. Spatial gene frequency differentiation, due to population subdivision or isolation by distance, boosts the relatedness estimates. In such cases it may be useful to partition the estimate into components, the operational relatedness is normally that among individuals in social groups within the same subpopulation. Although it is straightforward to estimate the average relatedness in social groups, estimating values for specific individuals with the help of genetic markers is still problematic. Current estimators tend to give biased values and the sampling error is large. In spite of these shortcomings, studies of social behaviour combining relatedness and reproductive success are sorely needed.