Power studies for the transmission/disequilibrium tests with multiple alleles.

Case-control studies compare marker-allele distributions in affected and unaffected individuals, and significant results suggest linkage but may simply reflect population structure. For markers with m alleles (m > or = 2), a McNemar-like statistic, I, estimates the level of population association between marker and disease loci. To test for linkage after significant case-control tests, within-family tests are performed. These operate on the contingency table, with i, jth element equal to the number of parents that transmit marker allele Mi and do not transmit marker allele Mi to an affected offspring. The dimension of the table is the number of alleles at the marker locus. Three test statistics have recently been proposed in the literature: Tc compares symmetric pairs of cells (i, j) and (j, i), Tm compares row and column totals for the same marker allele, and a likelihood ratio statistic Tl uses all the cells in the table. In addition, we consider a new statistic, Tmhet, that uses only the heterozygous parents and is approximately chi2 with (m - 1) df. We use a Monte Carlo test to guarantee valid tests and to demonstrate the inferiority of Tc and the equality of Tm and Tl in terms of power. The power of the Tmhet test is close but not always equal to the power of the Tm test. We also show that under the alternative hypothesis of linkage, Tm is approximately noncentral chi2 with (m - 1) df and noncentrality parameter 2NT(1 - 2theta)2I*, when data on single affecteds in NT families are used. If the disease has a low population frequency, then I* is estimated using the case-control statistic I. This offers a basis for choosing sample size, or choosing a marker system.     

Accounting for decay of linkage disequilibrium in haplotype inference and missing-data imputation

Although many algorithms exist for estimating haplotypes from genotype data, none of them take full account of both the decay of linkage disequilibrium (LD) with distance and the order and spacing of genotyped markers. Here, we describe an algorithm that does take these factors into account, using a flexible model for the decay of LD with distance that can handle both "blocklike" and "nonblocklike" patterns of LD. We compare the accuracy of this approach with a range of other available algorithms in three ways: for reconstruction of randomly paired, molecularly determined male X chromosome haplotypes; for reconstruction of haplotypes obtained from trios in an autosomal region; and for estimation of missing genotypes in 50 autosomal genes that have been completely resequenced in 24 African Americans and 23 individuals of European descent. For the autosomal data sets, our new approach clearly outperforms the best available methods, whereas its accuracy in inferring the X chromosome haplotypes is only slightly superior. For estimation of missing genotypes, our method performed slightly better when the two subsamples were combined than when they were analyzed separately, which illustrates its robustness to population stratification. Our method is implemented in the software package PHASE (v2.1.1), available from the Stephens Lab Web site.     

Two loci with two alleles: Linkage equilibrium and linkage disequilibrium can be simultaneously stable

A class of two-locus two-allele viability matrices is exhibited which has the property that, for a large range of recombination values, both linkage equilibrium and linkage disequilibrium are stable. The model is specified by five viabilities; the classical schemes previously analyzed involve at most four selection parameters.     

A simple correction for multiple testing for single-nucleotide polymorphisms in linkage disequilibrium with each other

In this report, we describe a simple correction for multiple testing of single-nucleotide polymorphisms (SNPs) in linkage disequilibrium (LD) with each other, on the basis of the spectral decomposition (SpD) of matrices of pairwise LD between SNPs. This method provides a useful alternative to more computationally intensive permutation tests. A user-friendly interface (SNPSpD) for performing this correction is available online (http://genepi.qimr.edu.au/general/daleN/SNPSpD/). Additionally, output from SNPSpD includes eigenvalues, principal-component coefficients, and factor "loadings" after varimax rotation, enabling the selection of a subset of SNPs that optimize the information in a genomic region.     

Volume measures for linkage disequilibrium

Background

A major QTL for fatness and growth, denoted FAT1, has previously been detected on pig chromosome 4q (SSC4q) using a Large White – wild boar intercross. Progeny that carried the wild boar allele at this locus had higher fat deposition, shorter length of carcass, and reduced growth. The position and the estimated effects of the FAT1 QTL for growth and fatness have been confirmed in a previous study. In order to narrow down the QTL interval we have traced the inheritance of the wild boar allele associated with high fat deposition through six additional backcross generations.

Results

Progeny-testing was used to determine the QTL genotype for 10 backcross sires being heterozygous for different parts of the broad FAT1 region. The statistical analysis revealed that five of the sires were segregating at the QTL, two were negative while the data for three sires were inconclusive. We could confirm the QTL effects on fatness/meat content traits but not for the growth traits implying that growth and fatness are controlled by distinct QTLs on chromosome 4. Two of the segregating sires showed highly significant QTL effects that were as large as previously observed in the F₂ generation. The estimates for the remaining three sires, which were all heterozygous for smaller fragments of the actual region, were markedly smaller. With the sample sizes used in the present study we cannot with great confidence determine whether these smaller effects in some sires are due to chance deviations, epistatic interactions or whether FAT1 is composed of two or more QTLs, each one with a smaller phenotypic effect. Under the assumption of a single locus, the critical region for FAT1 has been reduced to a 3.3 cM interval between the RXRG and SDHC loci.

Conclusion

We have further characterized the FAT1 QTL on pig chromosome 4 and refined its map position considerably, from a QTL interval of 70 cM to a maximum region of 20 cM and a probable region as small as 3.3 cM. The flanking markers for the small region are RXRG and SDHC and the orthologous region of FAT1 in the human genome is located on HSA1q23.3 and harbors approximately 20 genes. Our strategy to further refine the map position of this major QTL will be i) to type new markers in our pigs that are recombinant in the QTL interval and ii) to perform Identity-By-Descent (IBD) mapping across breeds that have been strongly selected for lean growth.     

Mapping multiple QTL using linkage disequilibrium and linkage analysis information and multitrait data

A multi-locus QTL mapping method is presented, which combines linkage and linkage disequilibrium (LD) information and uses multitrait data. The method assumed a putative QTL at the midpoint of each marker bracket. Whether the putative QTL had an effect or not was sampled using Markov chain Monte Carlo (MCMC) methods. The method was tested in dairy cattle data on chromosome 14 where the DGAT1 gene was known to be segregating. The DGAT1 gene was mapped to a region of 0.04 cM, and the effects of the gene were accurately estimated. The fitting of multiple QTL gave a much sharper indication of the QTL position than a single QTL model using multitrait data, probably because the multi-locus QTL mapping reduced the carry over effect of the large DGAT1 gene to adjacent putative QTL positions. This suggests that the method could detect secondary QTL that would, in single point analyses, remain hidden under the broad peak of the dominant QTL. However, no indications for a second QTL affecting dairy traits were found on chromosome 14.     

Exploring linkage disequilibrium

Linkage disequilibrium (LD, association of allelic states across loci) is poorly understood by many evolutionary biologists, but as technology for multilocus sampling improves, we ignore LD at our peril. If we sample variation at 10 loci in an organism with 20 chromosomes, we can reasonably treat them as 10 ‘independent witnesses’ of the evolutionary process. If instead, we sample variation at 1000 loci, many are bound to be close together on a chromosome. With only one or two crossovers per meiosis, associations between close neighbours decay so slowly that even LD created far in the past will not have dissipated, so we cannot treat the 1000 loci as independent witnesses (Barton 2011 ). This means that as marker density on genomes increases classic analyses assuming independent loci become mired in the problem of overconfidence: if 1000 independent witnesses are assumed, and that number should be much lower, any conclusion will be overconfident. This is of special concern because our literature suffers from a strong publication bias towards confident answers, even when they turn out to be wrong (Knowles 2008 ). In contrast, analyses that take into account associations across loci both control for overconfidence and can inform us about LD generating events far in the past, for example human/Neanderthal admixture (Fu et al. 2014 ). With increased marker density, biologists must increase their awareness of LD and, in this issue of Molecular Ecology Resources, Kemppainen et al. ( 2015 ) make software available that can only help in this process: LDna allows patterns of LD in a data set to be explored using tools borrowed from network analysis. This has great potential, but realizing that potential requires understanding LD.     

Highly variable patterns of linkage disequilibrium in multiple soybean populations

Prospects for utilizing whole-genome association analysis in autogamous plant populations appear promising due to the reported high levels of linkage disequilibrium (LD). To determine the optimal strategies for implementing association analysis in soybean (Glycine max L. Merr.), we analyzed the structure of LD in three regions of the genome varying in length from 336 to 574 kb. This analysis was conducted in four distinct groups of soybean germplasm: 26 accessions of the wild ancestor of soybean (Glycine soja Seib. et Zucc.); 52 Asian G. max Landraces, the immediate results of domestication from G. soja; 17 Asian Landrace introductions that became the ancestors of North American (N. Am.) cultivars, and 25 Elite Cultivars from N. Am. In G. soja, LD did not extend past 100 kb; however, in the three cultivated G. max groups, LD extended from 90 to 574 kb, likely due to the impacts of domestication and increased self-fertilization. The three genomic regions were highly variable relative to the extent of LD within the three cultivated soybean populations. G. soja appears to be ideal for fine mapping of genes, but due to the highly variable levels of LD in the Landraces and the Elite Cultivars, whole-genome association analysis in soybean may be more difficult than first anticipated.     

The trimmed-haplotype test for linkage disequilibrium

Single-marker linkage-disequilibrium (LD) methods cannot fully describe disequilibrium in an entire chromosomal region surrounding a disease allele. With the advent of myriad tightly linked microsatellite markers, we have an opportunity to extend LD analysis from single markers to multiple-marker haplotypes. Haplotype analysis has increased statistical power to disclose the presence of a disease locus in situations where it correctly reflects the historical process involved. For maximum efficiency, evidence of LD ought to come not just from a single haplotype, which may well be rare, but in addition from many similar haplotypes that could have descended from the same ancestral founder but have been trimmed in succeeding generations. We present such an analysis, called the "trimmed-haplotype method." We focus on chromosomal regions that are small enough that disequilibrium in significant portions of them may have been preserved in some pedigrees and yet that contain enough markers to minimize coincidental occurrence of the haplotype in the absence of a disease allele: perhaps regions 1-2 cM in length. In general, we could have no idea what haplotype an ancestral founder carried generations ago, nor do we usually have a precise chromosomal location for the disease-susceptibility locus. Therefore, we must search through all possible haplotypes surrounding multiple locations. Since such repeated testing obliterates the sampling distribution of the test, we employ bootstrap methods to calculate significance levels. Trimmed-haplotype analysis is performed on family data in which genotypes have been assembled into haplotypes. It can be applied either to conventional parent-affected-offspring triads or to multiplex pedigrees. We present a method for summarizing the LD evidence, in any pedigree, that can be employed in trimmed-haplotype analysis as well as in other methods.     

Homozygosity and linkage disequilibrium

We illustrate how homozygosity of haplotypes can be used to measure the level of disequilibrium between two or more markers. An excess of either homozygosity or heterozygosity signals a departure from the gametic phase equilibrium: We describe the specific form of dependence that is associated with high (low) homozygosity and derive various linkage disequilibrium measures. They feature a clear biological interpretation, can be used to construct tests, and are standardized to allow comparison across loci and populations. They are particularly advantageous to measure linkage disequilibrium between highly polymorphic markers.     

Decomposing multilocus linkage disequilibrium

We present a mathematically precise formulation of total linkage disequilibrium between multiple loci as the deviation from probabilistic independence and provide explicit formulas for all higher-order terms of linkage disequilibrium, thereby combining J. Dausset et al.'s 1978 definition of linkage disequilibrium with H. Geiringer's 1944 approach. We recursively decompose higher-order linkage disequilibrium terms into lower-order ones. Our greatest simplification comes from defining linkage disequilibrium at a single locus as allele frequency at that locus. At each level, decomposition of linkage disequilibrium is mathematically equivalent to number theoretic compositions of positive integers; i.e., we have converted a genetic decomposition into a mathematical decomposition.     

Towards linkage analysis with markers in linkage disequilibrium by graphical modelling

We review recent developments of MCMC integration methods for computations on graphical models for two applications in statistical genetics: modelling allelic association and pedigree based linkage analysis. We discuss and illustrate estimation of graphical models from haploid and diploid genotypes, and the importance of MCMC updating schemes beyond what is strictly necessary for irreducibility. We then outline an approach combining these methods to compute linkage statistics when alleles at the marker loci are in linkage disequilibrium. Other extensions suitable for analysis of SNP genotype data in pedigrees are also discussed and programs that implement these methods, and which are available from the author's web site, are described. We conclude with a discussion of how this still experimental approach might be further developed.     

Genotype determination for polymorphisms in linkage disequilibrium

Background  

Genome-wide association studies with single nucleotide polymorphisms (SNPs) show great promise to identify genetic determinants of complex human traits. In current analyses, genotype calling and imputation of missing genotypes are usually considered as two separated tasks. The genotypes of SNPs are first determined one at a time from allele signal intensities. Then the missing genotypes, i.e., no-calls caused by not perfectly separated signal clouds, are imputed based on the linkage disequilibrium (LD) between multiple SNPs. Although many statistical methods have been developed to improve either genotype calling or imputation of missing genotypes, treating the two steps independently can lead to loss of genetic information.     

On the formula for admixture linkage disequilibrium

Admixture linkage disequilibrium (ALD), a phenomenon created by gene flow between genetically distinct populations, has for some time been used as a tool in gene mapping. It is therefore important to analyze the pattern of ALD over generations. In this study we explore two models of admixture: the gradual admixture (GA) model, in which admixture occurs at a variable rate in every generation; and the immediate admixture (IA) model, a special case of the GA model, in which admixture occurs in a single generation. In the case of ALD, the well-known formula of linkage disequilibrium (Delta(t)=(1-r)t Delta(0)) is not applicable under these two models. We note the effect of a random mating population (RMP) to the gametic frequencies from the parental population to the offspring population, and provide the correct formula for ALD.     

Methods for linkage disequilibrium mapping in crops

Linkage disequilibrium (LD) mapping in plants detects and locates quantitative trait loci (QTL) by the strength of the correlation between a trait and a marker. It offers greater precision in QTL location than family-based linkage analysis and should therefore lead to more efficient marker-assisted selection, facilitate gene discovery and help to meet the challenge of connecting sequence diversity with heritable phenotypic differences. Unlike family-based linkage analysis, LD mapping does not require family or pedigree information and can be applied to a range of experimental and non-experimental populations. However, care must be taken during analysis to control for the increased rate of false positive results arising from population structure and variety interrelationships. In this review, we discuss how suitable the recently developed alternative methods of LD mapping are for crops.     

Estimating the timing of multiple admixture events using 3-locus linkage disequilibrium

Estimating admixture histories is crucial for understanding the genetic diversity we see in present-day populations. Allele frequency or phylogeny-based methods are excellent for inferring the existence of admixture or its proportions. However, to estimate admixture times, spatial information from admixed chromosomes of local ancestry or the decay of admixture linkage disequilibrium (ALD) is used. One popular method, implemented in the programs ALDER and ROLLOFF, uses two-locus ALD to infer the time of a single admixture event, but is only able to estimate the time of the most recent admixture event based on this summary statistic. To address this limitation, we derive analytical expressions for the expected ALD in a three-locus system and provide a new statistical method based on these results that is able to resolve more complicated admixture histories. Using simulations, we evaluate the performance of this method on a range of different admixture histories. As an example, we apply the method to the Colombian and Mexican samples from the 1000 Genomes project. The implementation of our method is available at https://github.com/Genomics-HSE/LaNeta.     

Simultaneous fine mapping of multiple closely linked quantitative trait Loci using combined linkage disequilibrium and linkage with a general pedigree

Within a small region (e.g., <10 cM), there can be multiple quantitative trait loci (QTL) underlying phenotypes of a trait. Simultaneous fine mapping of closely linked QTL needs an efficient tool to remove confounded shade effects among QTL within such a small region. We propose a variance component method using combined linkage disequilibrium (LD) and linkage information and a reversible jump Markov chain Monte Carlo (MCMC) sampling for model selection. QTL identity-by-descent (IBD) coefficients between individuals are estimated by a hybrid MCMC combining the random walk and the meiosis Gibbs sampler. These coefficients are used in a mixed linear model and an empirical Bayesian procedure combines residual maximum likelihood (REML) to estimate QTL effects and a reversible jump MCMC that samples the number of QTL and the posterior QTL intensities across the tested region. Note that two MCMC processes are used, i.e., an (internal) MCMC for IBD estimation and an (external) MCMC for model selection. In a simulation study, the use of the multiple-QTL model clearly removes the shade effects between three closely linked QTL located at 1.125, 3.875, and 7.875 cM across the region of 10 cM, using 40 markers at 0.25-cM intervals. It is shown that the use of combined LD and linkage information gives much more useful information compared to using linkage information alone for both single- and multiple-QTL analyses. When using a lower marker density (11 markers at 1-cM intervals), the signal of the second QTL can disappear. Extreme values of past effective size (resulting in extreme levels of LD) decrease the mapping accuracy.     

Linkage disequilibrium among multiple neutral alleles produced by mutation in finite population.

An analysis is undertaken for a finite random mating population of the linkage disequilibrium between two loci, at both of which all alleles are neutral, all mutant alleles differ from existing ones and several may be segregating at any time. Formulae are derived for the expected total squared disequilibrium, measured as the sum of squares of disequilibria between all pairs of alleles. The ratio of this quantity to the expected value of the product of the heterozygosities at the two loci is similar to that obtained previously by Ohta and Kimura for two nucleotide sites at each of which not more than two mutant types can segregate at any time.     

Positional cloning by linkage disequilibrium

Recently, metric linkage disequilibrium (LD) maps that assign an LD unit (LDU) location for each marker have been developed (Maniatis et al. 2002). Here we present a multiple pairwise method for positional cloning by LD within a composite likelihood framework and investigate the operating characteristics of maps in physical units (kb) and LDU for two bodies of data (Daly et al. 2001; Jeffreys et al. 2001) on which current ideas of blocks are based. False-negative indications of a disease locus (type II error) were examined by selecting one single-nucleotide polymorphism (SNP) at a time as causal and taking its allelic count (0, 1, or 2, for the three genotypes) as a pseudophenotype, Y. By use of regression and correlation, association between every pseudophenotype and the allelic count of each SNP locus (X) was based on an adaptation of the Malecot model, which includes a parameter for location of the putative gene. By expressing locations in kb or LDU, greater power for localization was observed when the LDU map was fitted. The efficiency of the kb map, relative to the LDU map, to describe LD varied from a maximum of 0.87 to a minimum of 0.36, with a mean of 0.62. False-positive indications of a disease locus (type I error) were examined by simulating an unlinked causal SNP and the allele count was used as a pseudophenotype. The type I error was in good agreement with Wald's likelihood theorem for both metrics and all models that were tested. Unlike tests that select only the most significant marker, haplotype, or haploset, these methods are robust to large numbers of markers in a candidate region. Contrary to predictions from tagging SNPs that retain haplotype diversity, the sample with smaller size but greater SNP density gave less error. The locations of causal SNPs were estimated with the same precision in blocks and steps, suggesting that block definition may be less useful than anticipated for mapping a causal SNP. These results provide a guide to efficient positional cloning by SNPs and a benchmark against which the power of positional cloning by haplotype-based alternatives may be measured.