Genuine Bayesian multiallelic significance test for the Hardy-Weinberg equilibrium law

Statistical tests that detect and measure deviation from the Hardy-Weinberg equilibrium (HWE) have been devised but are limited when testing for deviation at multiallelic DNA loci is attempted. Here we present the full Bayesian significance test (FBST) for the HWE. This test depends neither on asymptotic results nor on the number of possible alleles for the particular locus being evaluated. The FBST is based on the computation of an evidence index in favor of the HWE hypothesis. A great deal of forensic inference based on DNA evidence assumes that the HWE is valid for the genetic loci being used. We applied the FBST to genotypes obtained at several multiallelic short tandem repeat loci during routine parentage testing; the locus Penta E exemplifies those clearly in HWE while others such as D10S1214 and D19S253 do not appear to show this.     

A straightforward multiallelic significance test for the Hardy-Weinberg equilibrium law

Much forensic inference based upon DNA evidence is made assuming Hardy-Weinberg Equilibrium (HWE) for the genetic loci being used. Several statistical tests to detect and measure deviation from HWE have been devised, and their limitations become more obvious when testing for deviation within multiallelic DNA loci. The most popular methods-Chi-square and Likelihood-ratio tests-are based on asymptotic results and cannot guarantee a good performance in the presence of low frequency genotypes. Since the parameter space dimension increases at a quadratic rate on the number of alleles, some authors suggest applying sequential methods, where the multiallelic case is reformulated as a sequence of "biallelic" tests. However, in this approach it is not obvious how to assess the general evidence of the original hypothesis; nor is it clear how to establish the significance level for its acceptance/rejection. In this work, we introduce a straightforward method for the multiallelic HWE test, which overcomes the aforementioned issues of sequential methods. The core theory for the proposed method is given by the Full Bayesian Significance Test (FBST), an intuitive Bayesian approach which does not assign positive probabilities to zero measure sets when testing sharp hypotheses. We compare FBST performance to Chi-square, Likelihood-ratio and Markov chain tests, in three numerical experiments. The results suggest that FBST is a robust and high performance method for the HWE test, even in the presence of several alleles and small sample sizes.     

Distributions of Hardy-Weinberg equilibrium test statistics

It is well established that test statistics and P-values derived from discrete data, such as genetic markers, are also discrete. In most genetic applications, the null distribution for a discrete test statistic is approximated with a continuous distribution, but this approximation may not be reasonable. In some cases using the continuous approximation for the expected null distribution may cause truly null test statistics to appear nonnull. We explore the implications of using continuous distributions to approximate the discrete distributions of Hardy–Weinberg equilibrium test statistics and P-values. We derive exact P-value distributions under the null and alternative hypotheses, enabling a more accurate analysis than is possible with continuous approximations. We apply these methods to biological data and find that using continuous distribution theory with exact tests may underestimate the extent of Hardy–Weinberg disequilibrium in a sample. The implications may be most important for the widespread use of whole-genome case–control association studies and Hardy–Weinberg equilibrium (HWE) testing for data quality control.     

Exact tests of Hardy-Weinberg equilibrium and homogeneity of disequilibrium across strata

Detecting departures from Hardy-Weinberg equilibrium (HWE) of marker-genotype frequencies is a crucial first step in almost all human genetic analyses. When a sample is stratified by multiple ethnic groups, it is important to allow the marker-allele frequencies to differ over the strata. In this situation, it is common to test for HWE by using an exact test within each stratum and then using the minimum P value as a global test. This approach does not account for multiple testing, and, because it does not combine information over strata, it does not have optimal power. Several approximate methods to combine information over strata have been proposed, but most of them sum over strata a measure of departure from HWE; if the departures are in different directions, then summing can diminish the overall evidence of departure from HWE. An exact stratified test is more appealing because it uses the probability of genotype configurations across the strata as evidence for global departures from HWE. We developed an exact stratified test for HWE for diallelic markers, such as single-nucleotide polymorphisms (SNPs), and an exact test for homogeneity of Hardy-Weinberg disequilibrium. By applying our methods to data from Perlegen and HapMap--a combined total of more than five million SNP genotypes, with three to four strata and strata sizes ranging from 23 to 60 subjects--we illustrate that the exact stratified test provides more-robust and more-powerful results than those obtained by either the minimum of exact test P values over strata or approximate stratified tests that sum measures of departure from HWE. Hence, our new methods should be useful for samples composed of multiple ethnic groups.     

Correction method for null alleles in species with variable microsatellite flanking regions, a case study of the black-lipped pearl oyster Pinctada margaritifera

In bivalves, heterozygote deficiencies and departures from Hardy-Weinberg equilibrium (HWE) in microsatellite analysis are common and mainly attributed to inbreeding, genetic patchiness (Walhund effect), or null alleles. We checked for the occurrence of null alleles at 3 microsatellite loci in 3 populations of black-lipped pearl oyster, Pinctada margaritifera, using a step-by-step method to re-amplify homozygotes and null individuals with redesigned primer pair combinations. After amplification with original primer pairs, the 3 populations exhibited null alleles, absence of structure, and significant departure from HWE for all 3 loci due to heterozygote deficiencies. After 3 re-amplification steps, with modified primer sets, all loci were corrected for null alleles. Once corrected, all populations appeared at HWE, demonstrating that null alleles were responsible for the initial disequilibrium of the populations. Furthermore, analysis from corrected genotypes demonstrates significant genetic differentiation for one population from the other 2.     

A new method of testing for Hardy-Weinberg equilibrium and ordering populations

The assumption of Hardy-Weinberg equilibrium (HWE) among alleles in a nonevolving population is of fundamental importance in genetic studies. Deviation from HWE in a population usually indicates inbreeding, stratification and sometimes problems in genotyping. In populations of affected individuals, these deviations can also provide evidence for association. In this paper, we introduce a measure based on the Kullback-Leibler discrimination information function that quantifies the deviation from HWE in a population. We use this measure to order populations. We also propose a test for HWE based on an estimate of this measure. The test is a statistically consistent test of the null hypothesis for all alternatives and is very easy to implement. Our proposed test statistic is compared with an earlier, widely used, test. Finally, the use of the proposed new test is shown in an illustrative example.     

Significant evidence for linkage disequilibrium over a 5-cM region among Afrikaners

We explore the extent of deviations from Hardy-Weinberg equilibrium (HWE) at a marker locus and linkage disequilibrium (LD) between pairs of marker loci in the Afrikaner population of South Africa. DNA samples were used for genotyping of 23 loci on six chromosomes. The samples were collected from 91 healthy unrelated Afrikaner adults. Exact tests were used to determine evidence for deviations from HWE at a single marker locus or LD between pairs of marker loci. At the 0.05 level of significance, evidence was found for deviation from HWE at only one of the 23 loci. At the same level of significance, LD was found among 8 of the 34 intrachromosomal pairs of loci. On chromosome 21, there was evidence for LD (P = 0.02) between a pair of loci with a genetic distance of 5.51 cM. On chromosome 2, there was evidence for LD between a pair of loci with a genetic distance of 5.28 cM (P = 0.002) and a pair of loci with a genetic distance of 3.68 cM (P = 0.0004). Detailed analysis of LD for one locus pair indicated that only a few of all alleles participated in the LD and that strong LD was most often positive. Our findings indicate that Afrikaans-speaking Afrikaners represent one of those special populations deemed particularly suitable for disequilibrium mapping.     

An Investigation on the Allelic Chi‐Square Test Used in Genetic Association Studies

Case‐control studies are primary study designs used in genetic association studies. Sasieni (Biometrics 1997, 53, 1253–1261) pointed out that the allelic chi‐square test used in genetic association studies is invalid when Hardy‐Weinberg equilibrium (HWE) is violated in a combined population. It is important to know how much type I error rate is deviated from the nominal level under violated HWE. We examine bounds of type I error rate of the allelic chi‐square test. We also investigate power of the goodness‐of‐fit test for HWE which can be used as a guideline for selecting an appropriate test between the allelic chi‐square test and the modified allelic chi‐square test, the latter of which was proposed for cases of violated HWE. In small samples, power is not large enough to detect the Wright's inbreeding model of small values of inbreeding coefficient. Therefore, when the null hypothesis of HWE is barely accepted, the modified test should be considered as an alternative method. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on exact tests of Hardy-Weinberg equilibrium

Deviations from Hardy-Weinberg equilibrium (HWE) can indicate inbreeding, population stratification, and even problems in genotyping. In samples of affected individuals, these deviations can also provide evidence for association. Tests of HWE are commonly performed using a simple chi2 goodness-of-fit test. We show that this chi2 test can have inflated type I error rates, even in relatively large samples (e.g., samples of 1,000 individuals that include approximately 100 copies of the minor allele). On the basis of previous work, we describe exact tests of HWE together with efficient computational methods for their implementation. Our methods adequately control type I error in large and small samples and are computationally efficient. They have been implemented in freely available code that will be useful for quality assessment of genotype data and for the detection of genetic association or population stratification in very large data sets.     

Correlation-based inference for linkage disequilibrium with multiple alleles

The correlation between alleles at a pair of genetic loci is a measure of linkage disequilibrium. The square of the sample correlation multiplied by sample size provides the usual test statistic for the hypothesis of no disequilibrium for loci with two alleles and this relation has proved useful for study design and marker selection. Nevertheless, this relation holds only in a diallelic case, and an extension to multiple alleles has not been made. Here we introduce a similar statistic, R(2), which leads to a correlation-based test for loci with multiple alleles: for a pair of loci with k and m alleles, and a sample of n individuals, the approximate distribution of n(k - 1)(m - 1)/(km)R(2) under independence between loci is chi((k-1)(m-1))(2). One advantage of this statistic is that it can be interpreted as the total correlation between a pair of loci. When the phase of two-locus genotypes is known, the approach is equivalent to a test for the overall correlation between rows and columns in a contingency table. In the phase-known case, R(2) is the sum of the squared sample correlations for all km 2 x 2 subtables formed by collapsing to one allele vs. the rest at each locus. We examine the approximate distribution under the null of independence for R(2) and report its close agreement with the exact distribution obtained by permutation. The test for independence using R(2) is a strong competitor to approaches such as Pearson's chi square, Fisher's exact test, and a test based on Cressie and Read's power divergence statistic. We combine this approach with our previous composite-disequilibrium measures to address the case when the genotypic phase is unknown. Calculation of the new multiallele test statistic and its P-value is very simple and utilizes the approximate distribution of R(2). We provide a computer program that evaluates approximate as well as "exact" permutational P-values.     

Characterization of two New York City Jewish populations at six short tandem repeat loci

The Hasidic and non-Hasidic Jewish communities of New York City represent two subpopulations with long-documented histories of restrictive marriage patterns and a high degree of endogamy. As part of a continuing study into their genetic structure, allele frequencies were determined for the six tetrameric short tandem repeat (STR) loci: FESFPS, F13AO1, vWA, CSF1PO, TPOX, and THO1. All loci were tested for Hardy-Weinberg equilibrium (HWE) by three tests: chi-square analysis, Monte Carlo chi-square analysis. and the exact test. The non-Hasidic population failed to meet HWE at the F13A01, FESFPS, and CSF1PO loci by all three tests. The Hasidic population also failed to meet HWE at the same loci by some of the tests. Comparison of the Hasidic to the non-Hasidic population using an R x C contingency table demonstrated a similarity at only the vWA locus. Significant differences exist when comparing the two Jewish populations to a reference Caucasian population.     

Characterisation of 12 microsatellite loci in the Vietnamese commercial clam <Emphasis Type="Italic">Lutraria rhynchaena</Emphasis> Jonas 1844 (Heterodonta: Bivalvia: Mactridae) through next-generation sequencing

The marine clam Lutraria rhynchaena is gaining popularity as an aquaculture species in Asia. Lutraria populations are present in the wild throughout Vietnam and several stocks have been established and translocated for breeding and aquaculture grow-out purposes. In this study, we demonstrate the feasibility of utilising Illumina next-generation sequencing technology to streamline the identification and genotyping of microsatellite loci from this clam species. Based on an initial partial genome scan, 48 microsatellite markers with similar melting temperatures were identified and characterised. The 12 most suitable polymorphic loci were then genotyped using 51 individuals from a population in Quang Ninh Province, North Vietnam. Genetic variation was low (mean number of alleles per locus = 2.6; mean expected heterozygosity = 0.41). Two loci showed significant deviation from Hardy–Weinberg equilibrium (HWE) and the presence of null alleles, but there was no evidence of linkage disequilibrium among loci. Three additional populations were screened (n = 7–36) to test the geographic utility of the 12 loci, which revealed 100 % successful genotyping in two populations from central Vietnam (Nha Trang). However, a second population from north Vietnam (Co To) could not be successfully genotyped and morphological evidence and mitochondrial variation suggests that this population represents a cryptic species of Lutraria. Comparisons of the Qang Ninh and Nha Trang populations, excluding the 2 loci out of HWE, revealed statistically significant allelic variation at 4 loci. We reported the first microsatellite loci set for the marine clam Lutraria rhynchaena and demonstrated its potential in differentiating clam populations. Additionally, a cryptic species population of Lutraria rhynchaena was identified during initial loci development, underscoring the overlooked diversity of marine clam species in Vietnam and the need to genetically characterise population representatives prior to microsatellite development. The rapid identification and validation of microsatellite loci using next-generation sequencing technology warrant its integration into future microsatellite loci development for key aquaculture species in Vietnam and more generally, aquaculture countries in the South East Asia region.     

Powerful Haplotype-Based Hardy-Weinberg Equilibrium Tests for Tightly Linked Loci

Recently, there have been many case-control studies proposed to test for association between haplotypes and disease, which require the Hardy-Weinberg equilibrium (HWE) assumption of haplotype frequencies. As such, haplotype inference of unphased genotypes and development of haplotype-based HWE tests are crucial prior to fine mapping. The goodness-of-fit test is a frequently-used method to test for HWE for multiple tightly-linked loci. However, its degrees of freedom dramatically increase with the increase of the number of loci, which may lack the test power. Therefore, in this paper, to improve the test power for haplotype-based HWE, we first write out two likelihood functions of the observed data based on the Niu''s model (NM) and inbreeding model (IM), respectively, which can cause the departure from HWE. Then, we use two expectation-maximization algorithms and one expectation-conditional-maximization algorithm to estimate the model parameters under the HWE, IM and NM models, respectively. Finally, we propose the likelihood ratio tests LRT and LRT for haplotype-based HWE under the NM and IM models, respectively. We simulate the HWE, Niu''s, inbreeding and population stratification models to assess the validity and compare the performance of these two LRT tests. The simulation results show that both of the tests control the type I error rates well in testing for haplotype-based HWE. If the NM model is true, then LRT is more powerful. While, if the true model is the IM model, then LRT has better performance in power. Under the population stratification model, LRT is still more powerful. To this end, LRT is generally recommended. Application of the proposed methods to a rheumatoid arthritis data set further illustrates their utility for real data analysis.     

Testing Hardy–Weinberg Equilibrium and Homogeneity of Hardy–Weinberg Disequilibrium using Complex Survey Data

Summary For studies on population genetics, the use of representative random samples of the target population can avoid ascertainment bias. Genetic variation data from over a hundred genes were collected in a U.S. nationally representative sample in the Third National Health and Nutrition Examination Survey (NHANES III). Surveys such as the NHANES have complex stratified multistage cluster sample designs with sample weighting that can inflate variances and alter the expectations of test statistics. Thus, classical statistical tests of Hardy–Weinberg equilibrium (HWE) and homogeneity of HW disequilibrium (HHWD) for simple random samples are not suitable for data from complex samples. We propose using Wald tests for HWE and generalized score tests for HHWD that have been modified for complex samples. Monte Carlo simulation studies are used to investigate the finite sample properties of the proposed tests. Rao–Scott corrections applied to the tests were found to improve their type I error properties. Our methods are applied to the NHANES III genetic data for three loci involved in metabolizing lead in the body.     

Isolation and characterization of microsatellite markers for <Emphasis Type="Italic">Cedrela odorata</Emphasis> L. (Meliaceae), a high value neotropical tree

We describe 9 primers for amplification of microsatellite loci for the Neotropical tree Cedrela odorata L. (Meliaceae). Loci were isolated from an enriched library derived from a single DNA sample from a tree in Costa Rica. Levels of polymorphism were determined using samples from a large progeny trial. Across loci, the number of alleles ranged from 14 to 30. Observed heterozygosity levels ranged from 0.61 to 0.88. No linkage disequilibria were detected although some departures from Hardy-Weinberg equilibrium (HWE) were found, probably due to a Wahlund effect.     

A Bayes factor approach with informative prior for rare genetic variant analysis from next generation sequencing data

The discovery of rare genetic variants through next generation sequencing is a very challenging issue in the field of human genetics. We propose a novel region‐based statistical approach based on a Bayes Factor (BF) to assess evidence of association between a set of rare variants (RVs) located on the same genomic region and a disease outcome in the context of case‐control design. Marginal likelihoods are computed under the null and alternative hypotheses assuming a binomial distribution for the RV count in the region and a beta or mixture of Dirac and beta prior distribution for the probability of RV. We derive the theoretical null distribution of the BF under our prior setting and show that a Bayesian control of the false Discovery Rate can be obtained for genome‐wide inference. Informative priors are introduced using prior evidence of association from a Kolmogorov‐Smirnov test statistic. We use our simulation program, sim1000G, to generate RV data similar to the 1000 genomes sequencing project. Our simulation studies showed that the new BF statistic outperforms standard methods (SKAT, SKAT‐O, Burden test) in case‐control studies with moderate sample sizes and is equivalent to them under large sample size scenarios. Our real data application to a lung cancer case‐control study found enrichment for RVs in known and novel cancer genes. It also suggests that using the BF with informative prior improves the overall gene discovery compared to the BF with noninformative prior.     

Population genetics of the yellow fever mosquito in Trinidad: comparisons of amplified fragment length polymorphism (AFLP) and restriction fragment length polymorphism (RFLP) markers

Recent development of DNA markers provides powerful tools for population genetic analyses. Amplified fragment length polymorphism (AFLP) markers result from a polymerase chain reaction (PCR)-based DNA fingerprinting technique that can detect multiple restriction fragments in a single polyacrylamide gel, and thus are potentially useful for population genetic studies. Because AFLP markers have to be analysed as dominant loci in order to estimate population genetic diversity and genetic structure parameters, one must assume that dominant (amplified) alleles are identical in state, recessive (unamplified) alleles are identical in state, AFLP fragments segregate according to Mendelian expectations and that the genotypes of an AFLP locus are in Hardy-Weinberg equilibrium (HWE). The HWE assumption is untestable for natural populations using dominant markers. Restriction fragment length polymorphism (RFLP) markers segregate as codominant alleles, and can therefore be used to test the HWE assumption that is critical for analysing AFLP data. This study examined whether the dominant AFLP markers could provide accurate estimates of genetic variability for the Aedes aegypti mosquito populations of Trinidad, West Indies, by comparing genetic structure parameters using AFLP and RFLP markers. For AFLP markers, we tested a total of five primer combinations and scored 137 putative loci. For RFLP, we examined a total of eight mapped markers that provide a broad coverage of mosquito genome. The estimated average heterozygosity with AFLP markers was similar among the populations (0.39), and the observed average heterozygosity with RFLP markers varied from 0.44 to 0.58. The average FST (standardized among-population genetic variance) estimates were 0.033 for AFLP and 0.063 for RFLP markers. The genotypes at several RFLP loci were not in HWE, suggesting that the assumption critical for analysing AFLP data was invalid for some loci of the mosquito populations in Trinidad. Therefore, the results suggest that, compared with dominant molecular markers, codominant DNA markers provide better estimates of population genetic variability, and offer more statistical power for detecting population genetic structure.     

应用微卫星标DNA记对三个封闭群兔遗传分析

目的检测国内Et本大耳白兔、青紫蓝兔、新西兰白兔的遗传背景及遗传结构,为封闭群兔遗传检测方法建立和标准化提供基础资料。方法应用18个微卫星标记及荧光标记一半自动基因分型技术对三个群体95个个体进行Hardy．Weinberg检测,统计每个位点等位基因频率、杂合度、F值、遗传距离等信息。结果三个种群平均等位基因观测数为3．167、4．556、3．444,平均观测杂合度为0．444、0．5230、0．4976。18个位点平均多态信息含量（PIC）为0．410、0．549、0．470,日本大耳白兔6个位点HWE检验（P〈0．05）,显著偏离Hardy—Weinberg平衡,并在Satl3,INRCCDDV0088位点基因型完全纯和,新西兰白兔和青紫蓝兔分别有2个位点显著偏离Hardy—Weinberg平衡。三个种群遗传距离：青紫蓝兔与新西兰白兔遗传距离最近,为0．124,与日本大耳白兔遗传关系最远,为0．320;新西兰白兔与日本大耳白兔较远,为0：10。结论三个种群有各自不同遗传特征,遗传多样性较高,种群间分化明显。个别位点偏离遗传平衡,推测人工繁育过程中存在一定问题。     

Bioinformatic mining of EST-SSR loci in the Pacific oyster, Crassostrea gigas

A set of expressed sequence tag-simple sequence repeat (EST-SSR) markers of the Pacific oyster, Crassostrea gigas, was developed through bioinformatic mining of the GenBank public database. As of June 30, 2007, a total of 5132 EST sequences from GenBank were downloaded and screened for di-, tri- and tetra-nucleotide repeats, with criteria set at a minimum of 5, 4 and 4 repeats for the three categories of SSRs respectively. Seventeen polymorphic microsatellite markers were characterized. Allele numbers ranged from 3 to 10, and the observed and expected heterozygosity values varied from 0.125 to 0.770 and from 0.113 to 0.732 respectively. Eleven loci were at Hardy-Weinberg equilibrium (HWE); the other six loci showed significant departure from HWE (P < 0.01), suggesting possible presence of null alleles. Pairwise check of linkage disequilibrium (LD) indicated that 11 of 136 pairs of loci showed significant LD (P < 0.01), likely due to HWE present in single markers. Cross-species amplification was examined for five other Crassostrea species and reasonable results were obtained, promising usefulness of these markers in oyster genetics.     

Population admixture: detection by Hardy-Weinberg test and its quantitative effects on linkage-disequilibrium methods for localizing genes underlying complex traits

In association studies searching for genes underlying complex traits, the results are often inconsistent, and population admixture has been recognized qualitatively as one major potential cause. Hardy-Weinberg equilibrium (HWE) is often employed to test for population admixture; however, its power is generally unknown. Through analytical and simulation approaches, we quantify the power of the HWE test for population admixture and the effects of population admixture on increasing the type I error rate of association studies under various scenarios of population differentiation and admixture. We found that (1) the power of the HWE test for detecting population admixture is usually small; (2) population admixture seriously elevates type I error rate for detecting genes underlying complex traits, the extent of which depends on the degrees of population differentiation and admixture; (3) HWE testing for population admixture should be performed with random samples or only with controls at the candidate genes, or the test can be performed for combined samples of cases and controls at marker loci that are not linked to the disease; (4) testing HWE for population admixture generally reduces false positive association findings of genes underlying complex traits but the effect is small; and (5) with population admixture, a linkage disequilibrium method that employs cases only is more robust and yields many fewer false positive findings than conventional case-control analyses. Therefore, unless random samples are carefully selected from one homogeneous population, admixture is always a legitimate concern for positive findings in association studies except for the analyses that deliberately control population admixture.