Mathematical properties of the r2 measure of linkage disequilibrium

Statistics for linkage disequilibrium (LD), the non-random association of alleles at two loci, depend on the frequencies of the alleles at the loci under consideration. Here, we examine the r(2) measure of LD and its mathematical relationship to allele frequencies, quantifying the constraints on its maximum value. Assuming independent uniform distributions for the allele frequencies of two biallelic loci, we find that the mean maximum value of r(2) is approximately 0.43051, and that r(2) can exceed a threshold of 4/5 in only approximately 14.232% of the allele frequency space. If one locus is assumed to have known allele frequencies--the situation in an association study in which LD between a known marker locus and an unknown trait locus is of interest--we find that the mean maximum value of r(2) is greatest when the known locus has a minor allele frequency of approximately 0.30131. We find that in 1/4 of the space of allowed values of minor allele frequencies and haplotype frequencies at a pair of loci, the unconstrained maximum r(2) allowing for the possibility of recombination between the loci exceeds the constrained maximum assuming that no recombination has occurred. Finally, we use r(max)(2) to examine the connection between r(2) and the D(') measure of linkage disequilibrium, finding that r(2)/r(max)(2)=D('2) for approximately 72.683% of the space of allowed values of (p(a),p(b),p(ab)). Our results concerning the properties of r(2) have the potential to inform the interpretation of unusual LD behavior and to assist in the design of LD-based association-mapping studies.     

Examining the effect of linkage disequilibrium on multipoint linkage analysis

Most linkage programs assume linkage equilibrium among multiple linked markers. This assumption may lead to bias for tightly linked markers where strong linkage disequilibrium (LD) exists. We used simulated data from Genetic Analysis Workshop 14 to examine the possible effect of LD on multipoint linkage analysis. Single-nucleotide polymorphism packets from a non-disease-related region that was generated with LD were used for both model-free and parametric linkage analyses. Results showed that high LD among markers can induce false-positive evidence of linkage for affected sib-pair analysis when parental data are missing. Bias can be eliminated with parental data and can be reduced when additional markers not in LD are included in the analyses.     

GOLD--graphical overview of linkage disequilibrium

SUMMARY: We describe a software package that provides a graphical summary of linkage disequilibrium in human genetic data. It allows for the analysis of family data and is well suited to the analysis of dense genetic maps. AVAILABILITY: http://www.well.ox.ac.uk/asthma/GOLD CONTACT: goncalo@well.ox.ac.uk     

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.     

Bias of allele-sharing linkage statistics in the presence of intermarker linkage disequilibrium

Current genome-wide linkage-mapping single-nucleotide polymorphism (SNP) panels with densities of 0.3 cM are likely to have increased intermarker linkage disequilibrium (LD) compared to 5-cM microsatellite panels. The resulting difference in haplotype frequencies versus that predicted may affect multipoint linkage analysis with ungenotyped founders; a common haplotype may be assumed to be rare, leading to inflation of identical-by-descent (IBD) allele-sharing estimates and evidence for linkage. Using data simulated for the Genetic Analysis Workshop 14, we assessed bias in allele-sharing measures and nonparametric linkage (NPL all) and Kong and Cox LOD (KC-LOD) scores in a targeted analysis of regions with and without LD and with and without genes. Using over 100 replicates, we found that if founders were not genotyped, multipoint IBD estimates and delta parameters were modestly inflated and NPL all and KC-LOD scores were biased upwards in the region with LD and no gene; rather than centering on the null, the mean NPL all and KC-LOD scores were 0.51 +/- 0.91 and 0.19 +/- 0.38, respectively. Reduction of LD by dropping markers reduced this upward bias. These trends were not seen in the non-LD region with no gene. In regions with genes (with and without LD), a slight loss in power with dropping markers was suggested. These results indicate that LD should be considered in dense scans; removal of markers in LD may reduce false-positive results although information may also be lost. Methods to address LD in a high-throughput manner are needed for efficient, robust genomic scans with dense SNPs.     

Extent of linkage disequilibrium in chicken

Many of the economically important traits in chicken are multifactorial and governed by multiple genes located at different quantitative trait loci (QTLs). The optimal marker density to identify these QTLs in linkage and association studies is largely determined by the extent of linkage disequilibrium (LD) around them. In this study, we investigated the extent of LD on two chromosomes in a white layer and two broiler chicken breeds. Pairwise levels of LD were calculated for 33 and 36 markers on chromosomes 10 and 28, respectively. We found that useful LD (i.e. an r(2) value higher than 0.3) in Nutreco chicken breed E5 (inbred) can extend to around 1 cM on chromosomes 10 and 28, although in a second region on chromosome 28 it extends to about 2.5 cM. The extent in breed Nutreco E3 (outbred) was very short in chromosome 10 (15 kb) but very much larger on chromosome 28, particularly in one region of depressed heterozygosity. The layer breed E2 (inbred) showed an extent of useful LD up to 4 cM on chromosome 10; the extent on chromosome 28 could not be assessed due to an erratic pattern of LD on that chromosome, although in one region LD appears to be in the order of 0.8 cM. This indicates that there may be very large differences in patterns of LD between different chicken breeds and different genomic regions.     

A genealogical interpretation of linkage disequilibrium

The degree of association between alleles at different loci, or linkage disequilibrium, is widely used to infer details of evolutionary processes. Here I explore how associations between alleles relate to properties of the underlying genealogy of sequences. Under the neutral, infinite-sites assumption I show that there is a direct correspondence between the covariance in coalescence times at different parts of the genome and the degree of linkage disequilibrium. These covariances can be calculated exactly under the standard neutral model and by Monte Carlo simulation under different demographic models. I show that the effects of population growth, population bottlenecks, and population structure on linkage disequilibrium can be described through their effects on the covariance in coalescence times.     

Patterns of linkage disequilibrium in the human genome

Particular alleles at neighbouring loci tend to be co-inherited. For tightly linked loci, this might lead to associations between alleles in the population a property known as linkage disequilibrium (LD). LD has recently become the focus of intense study in the hope that it might facilitate the mapping of complex disease loci through whole-genome association studies. This approach depends crucially on the patterns of LD in the human genome. In this review, we draw on empirical studies in humans and Drosophila, as well as simulation studies, to assess the current state of knowledge about patterns of LD, and consider the implications for the use of LD as a mapping tool.     

Evaluation of linkage disequilibrium measures between multi-allelic markers as predictors of linkage disequilibrium between markers and QTL

Effectiveness of marker-assisted selection (MAS) and quantitative trait loci (QTL) mapping using population-wide linkage disequilibrium (LD) between markers and QTL depends on the extent of LD and how it declines with distance in a population. Because marker-QTL LD cannot be observed directly, the objective of this study was to evaluate alternative measures of observable LD between multi-allelic markers as predictors of usable LD of multi-allelic markers with presumed biallelic QTL. Observable LD between marker pairs was evaluated using eight existing measures and one new measure. These consisted of two pooled and standardized measures of LD between pairs of alleles at two markers based on Lewontin's LD measure, two pooled measures of squared correlations between alleles, one standardized measure using Hardy-Weinberg heterozygosities, and four measures based on the chi-square statistic for testing for association between alleles at two loci. In simulated populations with a range of LD generated by drift and a range of marker polymorphism, marker-marker LD measured by a standardized chi-square statistic (denoted chi(2')) was found to be the best predictor of useable marker-QTL LD for a group of multi-allelic markers. Estimates of the level and decline of marker-marker LD with distance obtained from chi(2') were linearly and highly correlated with usable LD of those markers with QTL across population structures and marker polymorphism. Corresponding relationships were poorer for the other marker-marker LD measures. Therefore, when LD is generated by drift, chi(2') is recommended to quantify the amount and extent of usable LD in a population for QTL mapping and MAS based on multi-allelic markers.     

Evaluation of linkage disequilibrium measures between multi-allelic markers as predictors of linkage disequilibrium between single nucleotide polymorphisms

Effectiveness of marker-assisted selection (MAS) and quantitative trait locus (QTL) mapping using population-wide linkage disequilibrium (LD) between markers and QTLs depends on the extent of LD and how it declines with distance between markers and QTLs in a population. Marker-QTL LD can be predicted from LD between markers. Our previous work evaluated LD measures between multi-allelic markers as predictors of usable LD of multi-allelic markers with QTLs. Since single nucleotide polymorphisms (SNPs) are the current marker of choice for high-density genotyping and LD-mapping of QTLs, the objective of this study was to use LD between multi-allelic markers to predict LD among biallelic SNPs or between SNPs and QTLs. Observable LD between multi-allelic markers was evaluated using nine measures. These included two pooled and standardized measures of LD between pairs of alleles at two markers based on Lewontin's LD measure, two pooled measures of squared correlations between alleles, one standardized measure using Hardy-Weinberg heterozygosities, and four measures based on the chi-square statistic for testing for association between alleles at two loci. The standardized chi-square measure that best predicted usable LD between multi-allelic markers and QTLs, based on our previous work, overestimated usable SNP-SNP or SNP-QTL LD. Instead, three other measures were found to be good predictors of usable SNP-SNP or SNP-QTL LD when LD is generated by drift. Therefore, the LD measure between multi-allelic markers that is best for predicting usable LD in a population depends on the type of markers (i.e. multi-allelic or biallelic) that will eventually be used for QTL mapping or MAS.     

Human prehistory: the message from linkage disequilibrium

The vast amount of information being gathered about human DNA sequence variation raises the question of what these data can tell us about events in our past. A new way has been found by which patterns of linkage disequilibrium can be used to detect the effects of natural selection in human prehistory.     

Population genomics: linkage disequilibrium holds the key

Current efforts to find disease-causing genes depend on patterns of linkage disequilibrium in human populations. Recent work has shown that linkage disequilibrium can extend over much larger genomic regions than expected, and that the patterns of linkage disequilibrium can differ markedly among populations.     

Expected behavior of conditional linkage disequilibrium.

The ubiquitousness of RFLPs in the human genome has greatly helped the mapping of human disease genes, and it has been suggested that population measures of association between disease and marker loci could help with this mapping. For rare diseases, random samples are taken from within disease genotypes in order to obtain reasonable sample sizes, but this sampling strategy requires a modification of the usual measures of association. We present theoretical predictions for the mean and variance of such a modified measure, under the assumption that the disease gene is maintained at a constant low frequency in the population. The coefficient of variation of this modified measure is large enough that caution is needed in using the measure to locate disease genes, and, furthermore, the coefficient of variation cannot be made arbitrarily small by increasing sample size. The modified association measure is calculated for recently published data on cystic fibrosis.     

The effect of linkage disequilibrium on linkage analysis of incomplete pedigrees

Dense SNP maps can be highly informative for linkage studies. But when parental genotypes are missing, multipoint linkage scores can be inflated in regions with substantial marker-marker linkage disequilibrium (LD). Such regions were observed in the Affymetrix SNP genotypes for the Genetic Analysis Workshop 14 (GAW14) Collaborative Study on the Genetics of Alcoholism (COGA) dataset, providing an opportunity to test a novel simulation strategy for studying this problem. First, an inheritance vector (with or without linkage present) is simulated for each replicate, i.e., locations of recombinations and transmission of parental chromosomes are determined for each meiosis. Then, two sets of founder haplotypes are superimposed onto the inheritance vector: one set that is inferred from the actual data and which contains the pattern of LD; and one set created by randomly selecting parental alleles based on the known allele frequencies, with no correlation (LD) between markers. Applying this strategy to a map of 176 SNPs (66 Mb of chromosome 7) for 100 replicates of 116 sibling pairs, significant inflation of multipoint linkage scores was observed in regions of high LD when parental genotypes were set to missing, with no linkage present. Similar inflation was observed in analyses of the COGA data for these affected sib pairs with parental genotypes set to missing, but not after reducing the marker map until r2 between any pair of markers was 相似文献   

Haplotype diversity and the block structure of linkage disequilibrium

Several recent studies indicate that patterns of linkage disequilibrium in the human genome cannot be reconciled with a uniform distribution of recombination events, but crossovers appear to be localized in short hot-spots that separate longer stretches of DNA. Markers within these low-recombination blocks show increased levels of linkage disequilibrium and very low haplotype diversity. This could simplify study of the genetic basis of complex diseases if causal variants are common.     

Linkage detection adaptive to linkage disequilibrium: the disequilibrium maximum-likelihood-binomial test for affected-sibship data

It has been demonstrated in the literature that the transmission/disequilibrium test (TDT) has higher power than the affected-sib-pair (ASP) mean test when linkage disequilibrium (LD) is strong but that the mean test has higher power when LD is weak. Thus, for ASP data, it seems clear that the TDT should be used when LD is strong but that the mean test or other linkage tests should be used when LD is weak or absent. However, in practice, it may be difficult to follow such a guideline, because the extent of LD is often unknown. Even with a highly dense genetic-marker map, in which some markers should be located near the disease-predisposing mutation, strong LD is not inevitable. Besides the genetic distance, LD is also affected by many factors, such as the allelic heterogeneity at the disease locus, the initial LD, the allelic frequencies at both disease locus and marker locus, and the age of the mutation. Therefore, it is of interest to develop methods that are adaptive to the extent of LD. In this report, we propose a disequilibrium maximum-binomial-likelihood (DMLB) test that incorporates LD in the maximum-binomial-likelihood (MLB) test. Examination of the corresponding score statistics shows that this method adaptively combines two sources of information: (a) the identity-by-descent (IBD) sharing score, which is informative for linkage regardless of the existence of LD, and (b) the contrast between allele-specific IBD sharing score, which is informative for linkage only in the presence of LD. For ASP data, the proposed test has higher power than either the TDT or the mean test when the extent of LD ranges from moderate to strong. Only when LD is very weak or absent is the DMLB slightly less powerful than the mean test; in such cases, the TDT has essentially no power to detect linkage. Therefore, the DMLB test is an interesting approach to linkage detection when the extent of LD is unknown.     

A global survey of haplotype frequencies and linkage disequilibrium at the DRD2 locus

A four-site haplotype system at the dopamine D2 receptor locus (DRD2) has been studied in a global sample of 28 distinct populations. The haplotype system spans about 25 kb, encompassing the coding region of the gene. The four individual markers include three TaqI restriction site polymorphisms (RSPs) – TaqI “A”, “B”, and “D” sites – and one dinucleotide short tandem repeat polymorphism (STRP). All four of the marker systems are polymorphic in all regions of the world and in most individual populations. The haplotype system shows the highest average heterozygosity in Africa, a slightly lower average heterozygosity in Europe, and the lowest average heterozygosities in East Asia and the Americas. Across all populations, 20 of the 48 possible haplotypes reached a frequency of at least 5% in at least one population sample. However, no single population had more than six haplotypes reaching that frequency. In general, African populations had more haplotypes present in each population and more haplotypes occurring at a frequency of at least 5% in that population. Permutation tests for significance of overall disequilibrium (all sites considered simultaneously) were highly significant (P<0.001) in all 28 populations. Except for three African samples, the pairwise disequilibrium between the outermost RSP markers, TaqI “B” and “A”, was highly significant with D’ values greater than 0.8; in two of those exceptions the RSP marker was not polymorphic. Except for those same two African populations, the 16-repeat allele at the STRP also showed highly significant disequilibrium with the TaqI “B” site in all populations, with D’ values usually greater than 0.7. Only four haplotypes account for more than 70% of all chromosomes in virtually all non-African populations, and two of those haplotypes account for more than 70% of all chromosomes in most East Asian and Amerindian populations. A new measure of the amount of overall disequilibrium shows least disequilibrium in African populations, somewhat more in European populations, and the greatest amount in East Asian and Amerindian populations. This pattern seems best explained by random genetic drift with low levels of recombination, a low mutation rate at the STRP, and essentially no recurrent mutation at the RSP sites, all in conjunction with an “Out of Africa” model for recent human evolution. Received: 14 January 1998 / Accepted 19 March 1998