Maximum likelihood estimation of recombination rates from population data

We describe a method for co-estimating r = C/mu (where C is the per-site recombination rate and mu is the per-site neutral mutation rate) and Theta = 4N(e)mu (where N(e) is the effective population size) from a population sample of molecular data. The technique is Metropolis-Hastings sampling: we explore a large number of possible reconstructions of the recombinant genealogy, weighting according to their posterior probability with regard to the data and working values of the parameters. Different relative rates of recombination at different locations can be accommodated if they are known from external evidence, but the algorithm cannot itself estimate rate differences. The estimates of Theta are accurate and apparently unbiased for a wide range of parameter values. However, when both Theta and r are relatively low, very long sequences are needed to estimate r accurately, and the estimates tend to be biased upward. We apply this method to data from the human lipoprotein lipase locus.     

Correcting Coalescent Analyses for Panel-Based SNP Ascertainment

Single-nucleotide polymorphism (SNP) data are routinely obtained by sequencing a region of interest in a small panel, constructing a chip with probes specific to sites found to vary in the panel, and using the chip to assay subsequent samples. The size of the chip is often reduced by removing low-frequency alleles from the set of SNPs. Using coalescent estimation of the scaled population size parameter, Θ, as a test case, we demonstrate the loss of information inherent in this procedure and develop corrections for coalescent analysis of SNPs obtained via a panel. We show that more accurate Θ-estimates can be recovered if the panel size is known, but at considerable computational cost as the panel individuals must be explicitly modeled in the analysis. We extend this technique to apply to the case where rare alleles have been omitted from the SNP panel. We find that when appropriate corrections for panel ascertainment and rare-allele omission are used, the biases introduced by ascertainment are largely correctable, but recovered estimates are less accurate than would be obtained with fully sequenced data. This method is then applied to recombinant multiple population data to investigate the effects of recombination and migration on the estimate of Θ.     

Estimation of population parameters and recombination rates from single nucleotide polymorphisms

Some general likelihood and Bayesian methods for analyzing single nucleotide polymorphisms (SNPs) are presented. First, an efficient method for estimating demographic parameters from SNPs in linkage equilibrium is derived. The method is applied in the estimation of growth rates of a human population based on 37 SNP loci. It is demonstrated how ascertainment biases, due to biased sampling of loci, can be avoided, at least in some cases, by appropriate conditioning when calculating the likelihood function. Second, a Markov chain Monte Carlo (MCMC) method for analyzing linked SNPs is developed. This method can be used for Bayesian and likelihood inference on linked SNPs. The utility of the method is illustrated by estimating recombination rates in a human data set containing 17 SNPs and 60 individuals. Both methods are based on assumptions of low mutation rates.     

Correcting for ascertainment biases when analyzing SNP data: applications to the estimation of linkage disequilibrium

As large-scale sequencing efforts turn from single genome sequencing to polymorphism discovery, single nucleotide polymorphisms (SNPs) are becoming an increasingly important class of population genetic data. But because of the ascertainment biases introduced by many methods of SNP discovery, most SNP data cannot be analyzed using classical population genetic methods. Statistical methods must instead be developed that can explicitly take into account each method of SNP discovery. Here we review some of the current methods for analyzing SNPs and derive sampling distributions for single SNPs and pairs of SNPs for some common SNP discovery schemes. We also show that the ascertainment scheme has a large effect on the estimation of linkage disequilibrium and recombination, and describe some methods of correcting for ascertainment biases when estimating recombination rates from SNP data.     

Precision and accuracy of divergence time estimates from STR and SNPSTR variation

Inference of intraspecific population divergence patterns typically requires genetic data for molecular markers with relatively high mutation rates. Microsatellites, or short tandem repeat (STR) polymorphisms, have proven informative in many such investigations. These markers are characterized, however, by high levels of homoplasy and varying mutational properties, often leading to inaccurate inference of population divergence. A SNPSTR is a genetic system that consists of an STR polymorphism closely linked (typically < 500 bp) to one or more single-nucleotide polymorphisms (SNPs). SNPSTR systems are characterized by lower levels of homoplasy than are STR loci. Divergence time estimates based on STR variation (on the derived SNP allele background) should, therefore, be more accurate and precise. We use coalescent-based simulations in the context of several models of demographic history to compare divergence time estimates based on SNPSTR haplotype frequencies and STR allele frequencies. We demonstrate that estimates of divergence time based on STR variation on the background of a derived SNP allele are more accurate (3% to 7% bias for SNPSTR versus 11% to 20% bias for STR) and more precise than STR-based estimates, conditional on a recent SNP mutation. These results hold even for models involving complex demographic scenarios with gene flow, population expansion, and population bottlenecks. Varying the timing of the mutation event generating the SNP revealed that estimates of divergence time are sensitive to SNP age, with more recent SNPs giving more accurate and precise estimates of divergence time. However, varying both mutational properties of STR loci and SNP age demonstrated that multiple independent SNPSTR systems provide less biased estimates of divergence time. Furthermore, the combination of estimates based separately on STR and SNPSTR variation provides insight into the age of the derived SNP alleles. In light of our simulations, we interpret estimates from data for human populations.     

A dimension reduction approach for modeling multi-locus interaction in case-control studies

Studying one locus or one single nucleotide polymorphism (SNP) at a time may not be sufficient to understand complex diseases because they are unlikely to result from the effect of only one SNP. Each SNP alone may have little or no effect on the risk of the disease, but together they may increase the risk substantially. Analyses focusing on individual SNPs ignore the possibility of interaction among SNPs. In this paper, we propose a parsimonious model to assess the joint effect of a group of SNPs in a case-control study. The model implements a data reduction strategy within a likelihood framework and uses a test to assess the statistical significance of the effect of the group of SNPs on the binary trait. The primary advantage of the proposed approach is that the dimension reduction technique produces a test statistic with degrees of freedom significantly lower than a multiple logistic regression with only main effects of the SNPs, and our parsimonious model can incorporate the possibility of interaction among the SNPs. Moreover, the proposed approach estimates the direction of association of each SNP with the disease and provides an estimate of the average effect of the group of SNPs positively and negatively associated with the disease in the given SNP set. We illustrate the proposed model on simulated and real data, and compare its performance with a few other existing approaches. Our proposed approach appeared to outperform the other approaches for independent SNPs in our simulation studies.     

Maximum likelihood estimation of linkage disequilibrium in half-sib families

Maximum likelihood methods for the estimation of linkage disequilibrium between biallelic DNA-markers in half-sib families (half-sib method) are developed for single and multifamily situations. Monte Carlo computer simulations were carried out for a variety of scenarios regarding sire genotypes, linkage disequilibrium, recombination fraction, family size, and number of families. A double heterozygote sire was simulated with recombination fraction of 0.00, linkage disequilibrium among dams of δ=0.10, and alleles at both markers segregating at intermediate frequencies for a family size of 500. The average estimates of δ were 0.17, 0.25, and 0.10 for Excoffier and Slatkin (1995), maternal informative haplotypes, and the half-sib method, respectively. A multifamily EM algorithm was tested at intermediate frequencies by computer simulation. The range of the absolute difference between estimated and simulated δ was between 0.000 and 0.008. A cattle half-sib family was genotyped with the Illumina 50K BeadChip. There were 314,730 SNP pairs for which the sire was a homo-heterozygote with average estimates of r2 of 0.115, 0.067, and 0.111 for half-sib, Excoffier and Slatkin (1995), and maternal informative haplotypes methods, respectively. There were 208,872 SNP pairs for which the sire was double heterozygote with average estimates of r2 across the genome of 0.100, 0.267, and 0.925 for half-sib, Excoffier and Slatkin (1995), and maternal informative haplotypes methods, respectively. Genome analyses for all possible sire genotypes with 829,042 tests showed that ignoring half-sib family structure leads to upward biased estimates of linkage disequilibrium. Published inferences on population structure and evolution of cattle should be revisited after accommodating existing half-sib family structure in the estimation of linkage disequilibrium.     

SNP-based heritability estimation using a Bayesian approach

Heritability is a central element in quantitative genetics. New molecular markers to assess genetic variance and heritability are continually under development. The availability of molecular single nucleotide polymorphism (SNP) markers can be applied for estimation of variance components and heritability on population, where relationship information is unknown. In this study, we evaluated the capabilities of two Bayesian genomic models to estimate heritability in simulated populations. The populations comprised different family structures of either no or a limited number of relatives, a single quantitative trait, and with one of two densities of SNP markers. All individuals were both genotyped and phenotyped. Results illustrated that the two models were capable of estimating heritability, when true heritability was 0.15 or higher and populations had a sample size of 400 or higher. For heritabilities of 0.05, all models had difficulties in estimating the true heritability. The two Bayesian models were compared with a restricted maximum likelihood (REML) approach using a genomic relationship matrix. The comparison showed that the Bayesian approaches performed equally well as the REML approach. Differences in family structure were in general not found to influence the estimation of the heritability. For the sample sizes used in this study, a 10-fold increase of SNP density did not improve precision estimates compared with set-ups with a less dense distribution of SNPs. The methods used in this study showed that it was possible to estimate heritabilities on the basis of SNPs in animals with direct measurements. This conclusion is valuable in cases when quantitative traits are either difficult or expensive to measure.     

Reconstituting the frequency spectrum of ascertained single-nucleotide polymorphism data

Most of the available SNP data have eluded valid population genetic analysis because most population genetical methods do not correctly accommodate the special discovery process used to identify SNPs. Most of the available SNP data have allele frequency distributions that are biased by the ascertainment protocol. We here show how this problem can be corrected by obtaining maximum-likelihood estimates of the true allele frequency distribution. In simple cases, the ML estimate of the true allele frequency distribution can be obtained analytically, but in other cases computational methods based on numerical optimization or the EM algorithm must be used. We illustrate the new correction method by analyzing some previously published SNP data from the SNP Consortium. Appropriate treatment of SNP ascertainment is vital to our ability to make correct inferences from the data of the International HapMap Project.     

Haplotype estimation from fuzzy genotypes using penalized likelihood

The Composite Link Model is a generalization of the generalized linear model in which expected values of observed counts are constructed as a sum of generalized linear components. When combined with penalized likelihood, it provides a powerful and elegant way to estimate haplotype probabilities from observed genotypes. Uncertain ("fuzzy") genotypes, like those resulting from AFLP scores, can be handled by adding an extra layer to the model. We describe the model and the estimation algorithm. We apply it to a data set of accurate human single nucleotide polymorphism (SNP) and to a data set of fuzzy tomato AFLP scores.     

Joint modeling of linkage and association: identifying SNPs responsible for a linkage signal

Once genetic linkage has been identified for a complex disease, the next step is often association analysis, in which single-nucleotide polymorphisms (SNPs) within the linkage region are genotyped and tested for association with the disease. If a SNP shows evidence of association, it is useful to know whether the linkage result can be explained, in part or in full, by the candidate SNP. We propose a novel approach that quantifies the degree of linkage disequilibrium (LD) between the candidate SNP and the putative disease locus through joint modeling of linkage and association. We describe a simple likelihood of the marker data conditional on the trait data for a sample of affected sib pairs, with disease penetrances and disease-SNP haplotype frequencies as parameters. We estimate model parameters by maximum likelihood and propose two likelihood-ratio tests to characterize the relationship of the candidate SNP and the disease locus. The first test assesses whether the candidate SNP and the disease locus are in linkage equilibrium so that the SNP plays no causal role in the linkage signal. The second test assesses whether the candidate SNP and the disease locus are in complete LD so that the SNP or a marker in complete LD with it may account fully for the linkage signal. Our method also yields a genetic model that includes parameter estimates for disease-SNP haplotype frequencies and the degree of disease-SNP LD. Our method provides a new tool for detecting linkage and association and can be extended to study designs that include unaffected family members.     

Estimating individuals’ genetic and non-genetic effects underlying infectious disease transmission from temporal epidemic data

Individuals differ widely in their contribution to the spread of infection within and across populations. Three key epidemiological host traits affect infectious disease spread: susceptibility (propensity to acquire infection), infectivity (propensity to transmit infection to others) and recoverability (propensity to recover quickly). Interventions aiming to reduce disease spread may target improvement in any one of these traits, but the necessary statistical methods for obtaining risk estimates are lacking. In this paper we introduce a novel software tool called SIRE (standing for “Susceptibility, Infectivity and Recoverability Estimation”), which allows for the first time simultaneous estimation of the genetic effect of a single nucleotide polymorphism (SNP), as well as non-genetic influences on these three unobservable host traits. SIRE implements a flexible Bayesian algorithm which accommodates a wide range of disease surveillance data comprising any combination of recorded individual infection and/or recovery times, or disease diagnostic test results. Different genetic and non-genetic regulations and data scenarios (representing realistic recording schemes) were simulated to validate SIRE and to assess their impact on the precision, accuracy and bias of parameter estimates. This analysis revealed that with few exceptions, SIRE provides unbiased, accurate parameter estimates associated with all three host traits. For most scenarios, SNP effects associated with recoverability can be estimated with highest precision, followed by susceptibility. For infectivity, many epidemics with few individuals give substantially more statistical power to identify SNP effects than the reverse. Importantly, precise estimates of SNP and other effects could be obtained even in the case of incomplete, censored and relatively infrequent measurements of individuals’ infection or survival status, albeit requiring more individuals to yield equivalent precision. SIRE represents a new tool for analysing a wide range of experimental and field disease data with the aim of discovering and validating SNPs and other factors controlling infectious disease transmission.     

Comparison of REML and Gibbs sampling estimates of multi-trait genetic parameters in Scots pine

Multi-trait (co)variance estimation is an important topic in plant and animal breeding. In this study we compare estimates obtained with restricted maximum likelihood (REML) and Bayesian Gibbs sampling of simulated data and of three traits (diameter, height and branch angle) from a 26-year-old partial diallel progeny test of Scots pine (Pinus sylvestris L.). Based on the results from the simulated data we can conclude that the REML estimates are accurate but the mode of posterior distributions from the Gibbs sampling can be overestimated depending on the level of the heritability. The mean and median of the posteriors were considerably higher than the expected values of the heritabilities. The confidence intervals calculated with the delta method were biased downwardly. The highest probablity density (HPD) interval provides a better interval estimate, but could be slightly biased at the lower level. Similar differences between REML and Gibbs sampling estimates were found for the Scots pine data. We conclude that further simulation studies are needed in order to evaluate the effect of different priors on (co)variance components in the genetic individual model.     

Robustness of coalescent estimators to between-lineage mutation rate variation

Data from HIV and from human neoplastic cells can show substantial between-lineage mutation rate variation even within a single population. Such variation may affect estimators of population quantities such as Theta = 4N(e)mu. Using simulated DNA data, I measured the effect of rate variation on recovery of Theta by the summary-statistic estimator of Watterson (Watterson GA. 1975. On the number of segregating sites in genetical systems without recombination. Theor Popul Biol. 7:256-276) and the coalescent maximum likelihood algorithm LAMARC (Kuhner MK. 2006. LAMARC 2.0: maximum likelihood and Bayesian estimation of population parameters. Bioinformatics. Advance Access doi: 10.1093/bioinformatics/btk051). Watterson's estimator showed a downward bias, as expected, with high values of Theta. LAMARC's mean estimate was accurate for all tested values of Theta and rate variation except for a downward bias when rate variation was maximal (i.e., the slow rate was zero). LAMARC had consistently narrower confidence intervals (CIs) than Watterson's estimator. Both methods tended to reject the truth too often when rate variation was 8x or greater and independent among branches, as well as when variation was 4x or greater and correlated among branches. In the case of Watterson's estimate, this excess rejection was fully attributable to variation among genealogies in the amount of total branch length associated with the fast and slow rates. However, in the case of LAMARC, some excess rejection was still observed even when between-genealogy variation was taken into account. Both estimators are robust to modest rate variation; however, their use should be coupled with a statistical test to rule out extreme rate variation as the resulting CIs may not be reliable.     

Inferring population parameters from single-feature polymorphism data

This article is concerned with a statistical modeling procedure to call single-feature polymorphisms from microarray experiments. We use this new type of polymorphism data to estimate the mutation and recombination parameters in a population. The mutation parameter can be estimated via the number of single-feature polymorphisms called in the sample. For the recombination parameter, a two-feature sampling distribution is derived in a way analogous to that for the two-locus sampling distribution with SNP data. The approximate-likelihood approach using the two-feature sampling distribution is examined and found to work well. A coalescent simulation study is used to investigate the accuracy and robustness of our method. Our approach allows the utilization of single-feature polymorphism data for inference in population genetics.     

Genomic Selection Using Low-Density Marker Panels

Genomic selection (GS) using high-density single-nucleotide polymorphisms (SNPs) is promising to improve response to selection in populations that are under artificial selection. High-density SNP genotyping of all selection candidates each generation, however, may not be cost effective. Smaller panels with SNPs that show strong associations with phenotype can be used, but this may require separate SNPs for each trait and each population. As an alternative, we propose to use a panel of evenly spaced low-density SNPs across the genome to estimate genome-assisted breeding values of selection candidates in pedigreed populations. The principle of this approach is to utilize cosegregation information from low-density SNPs to track effects of high-density SNP alleles within families. Simulations were used to analyze the loss of accuracy of estimated breeding values from using evenly spaced and selected SNP panels compared to using all high-density SNPs in a Bayesian analysis. Forward stepwise selection and a Bayesian approach were used to select SNPs. Loss of accuracy was nearly independent of the number of simulated quantitative trait loci (QTL) with evenly spaced SNPs, but increased with number of QTL for the selected SNP panels. Loss of accuracy with evenly spaced SNPs increased steadily over generations but was constant when the smaller number individuals that are selected for breeding each generation were also genotyped using the high-density SNP panel. With equal numbers of low-density SNPs, panels with SNPs selected on the basis of the Bayesian approach had the smallest loss in accuracy for a single trait, but a panel with evenly spaced SNPs at 10 cM was only slightly worse, whereas a panel with SNPs selected by forward stepwise selection was inferior. Panels with evenly spaced SNPs can, however, be used across traits and populations and their performance is independent of the number of QTL affecting the trait and of the methods used to estimate effects in the training data and are, therefore, preferred for broad applications in pedigreed populations under artificial selection.     

DAIRRy-BLUP: A High-Performance Computing Approach to Genomic Prediction

In genomic prediction, common analysis methods rely on a linear mixed-model framework to estimate SNP marker effects and breeding values of animals or plants. Ridge regression–best linear unbiased prediction (RR-BLUP) is based on the assumptions that SNP marker effects are normally distributed, are uncorrelated, and have equal variances. We propose DAIRRy-BLUP, a parallel, Distributed-memory RR-BLUP implementation, based on single-trait observations (y), that uses the Average Information algorithm for restricted maximum-likelihood estimation of the variance components. The goal of DAIRRy-BLUP is to enable the analysis of large-scale data sets to provide more accurate estimates of marker effects and breeding values. A distributed-memory framework is required since the dimensionality of the problem, determined by the number of SNP markers, can become too large to be analyzed by a single computing node. Initial results show that DAIRRy-BLUP enables the analysis of very large-scale data sets (up to 1,000,000 individuals and 360,000 SNPs) and indicate that increasing the number of phenotypic and genotypic records has a more significant effect on the prediction accuracy than increasing the density of SNP arrays.     

Estimation of allele frequency in pooled DNA by using PCR–RFLP combined with microchip electrophoresis

Biallelic marker, most commonly single nucleotide polymorphism (SNP), is widely utilized in genetic association analysis, which can be speeded up by estimating allele frequency in pooled DNA instead of individual genotyping. Several methods have shown high accuracy and precision for allele frequency estimation in pools. Here, we explored PCR restriction fragment length polymorphism (PCR–RFLP) combined with microchip electrophoresis as a possible strategy for allele frequency estimation in DNA pools. We have used the commercial available Agilent 2100 microchip electrophoresis analysis system for quantifying the enzymatically digested DNA fragments and the fluorescence intensities to estimate the allele frequencies in the DNA pools. In this study, we have estimated the allele frequencies of five SNPs in a DNA pool composed of 141 previously genotyped health controls and a DNA pool composed of 96 previously genotyped gastric cancer patients with a frequency representation of 10–90% for the variant allele. Our studies show that accurate, quantitative data on allele frequencies, suitable for investigating the association of SNPs with complex disorders, can be estimated from pooled DNA samples by using this assay. This approach, being independent of the number of samples, promises to drastically reduce the labor and cost of genotyping in the initial association analysis.     

Selection at linked sites in the partial selfer Caenorhabditis elegans

Natural selection can produce a correlation between local recombination rates and levels of neutral DNA polymorphism as a consequence of genetic hitchhiking and background selection. Theory suggests that selection at linked sites should affect patterns of neutral variation in partially selfing populations more dramatically than in outcrossing populations. However, empirical investigations of selection at linked sites have focused primarily on outcrossing species. To assess the potential role of selection as a determinant of neutral polymorphism in the context of partial self-fertilization, we conducted a multivariate analysis of single-nucleotide polymorphism (SNP) density throughout the genome of the nematode Caenorhabditis elegans. We based the analysis on a published SNP data set and partitioned the genome into windows to calculate SNP densities, recombination rates, and gene densities across all six chromosomes. Our analyses identify a strong, positive correlation between recombination rate and neutral polymorphism (as estimated by noncoding SNP density) across the genome of C. elegans. Furthermore, we find that levels of neutral polymorphism are lower in gene-dense regions than in gene-poor regions in some analyses. Analyses incorporating local estimates of divergence between C. elegans and C. briggsae indicate that a mutational explanation alone is unlikely to explain the observed patterns. Consequently, we interpret these findings as evidence that natural selection shapes genome-wide patterns of neutral polymorphism in C. elegans. Our study provides the first demonstration of such an effect in a partially selfing animal. Explicit models of genetic hitchhiking and background selection can each adequately describe the relationship between recombination rate and SNP density, but only when they incorporate selfing rate. Clarification of the relative roles of genetic hitchhiking and background selection in C. elegans awaits the development of specific theoretical predictions that account for partial self-fertilization and biased sex ratios.     

Using molecular markers to map multiple quantitative trait loci: models for backcross,recombinant inbred,and doubled haploid progeny

Summary To maximize parameter estimation efficiency and statistical power and to estimate epistasis, the parameters of multiple quantitative trait loci (QTLs) must be simultaneously estimated. If multiple QTL affect a trait, then estimates of means of QTL genotypes from individual locus models are statistically biased. In this paper, I describe methods for estimating means of QTL genotypes and recombination frequencies between marker and quantitative trait loci using multilocus backcross, doubled haploid, recombinant inbred, and testcross progeny models. Expected values of marker genotype means were defined using no double or multiple crossover frequencies and flanking markers for linked and unlinked quantitative trait loci. The expected values for a particular model comprise a system of nonlinear equations that can be solved using an interative algorithm, e.g., the Gauss-Newton algorithm. The solutions are maximum likelihood estimates when the errors are normally distributed. A linear model for estimating the parameters of unlinked quantitative trait loci was found by transforming the nonlinear model. Recombination frequency estimators were defined using this linear model. Certain means of linked QTLs are less efficiently estimated than means of unlinked QTLs.