Properties of the multiallelic trend test

Disease genes can be mapped on the basis of associations between genetic markers and disease status, with the case-control design having the advantage of not requiring individuals from different generations. When the marker loci have multiple alleles, there has been debate on whether the power of tests for association increases or decreases. We show here that the multiple-allele version of Armitage's trend test has increased power over the two-allele version under the requirement of equifrequent alleles, but not in general. The trend test has the advantage of remaining valid even when the sampled population is not in Hardy-Weinberg equilibrium. A departure from Hardy-Weinberg means that association tests depend on gametic and nongametic linkage disequilibrium between marker and disease loci, and we illustrate the magnitude of these effects with simulated data.     

Testing Hardy-Weinberg proportions in a frequency-matched case-control genetic association study

In case-control genetic association studies, cases are subjects with the disease and controls are subjects without the disease. At the time of case-control data collection, information about secondary phenotypes is also collected. In addition to studies of primary diseases, there has been some interest in studying genetic variants associated with secondary phenotypes. In genetic association studies, the deviation from Hardy-Weinberg proportion (HWP) of each genetic marker is assessed as an initial quality check to identify questionable genotypes. Generally, HWP tests are performed based on the controls for the primary disease or secondary phenotype. However, when the disease or phenotype of interest is common, the controls do not represent the general population. Therefore, using only controls for testing HWP can result in a highly inflated type I error rate for the disease- and/or phenotype-associated variants. Recently, two approaches, the likelihood ratio test (LRT) approach and the mixture HWP (mHWP) exact test were proposed for testing HWP in samples from case-control studies. Here, we show that these two approaches result in inflated type I error rates and could lead to the removal from further analysis of potential causal genetic variants associated with the primary disease and/or secondary phenotype when the study of primary disease is frequency-matched on the secondary phenotype. Therefore, we proposed alternative approaches, which extend the LRT and mHWP approaches, for assessing HWP that account for frequency matching. The goal was to maintain more (possible causative) single-nucleotide polymorphisms in the sample for further analysis. Our simulation results showed that both extended approaches could control type I error probabilities. We also applied the proposed approaches to test HWP for SNPs from a genome-wide association study of lung cancer that was frequency-matched on smoking status and found that the proposed approaches can keep more genetic variants for association studies.     

An incomplete enumeration algorithm for an exact test of Hardy–Weinberg proportions with multiple alleles

Testing of Hardy–Weinberg proportions (HWP) with asymptotic goodness-of-fit tests is problematic when the contingency table of observed genotype counts has sparse cells or the sample size is low, and exact procedures are to be preferred. Exact p-values can be (1) calculated via computational demanding enumeration methods or (2) approximated via simulation methods. Our objective was to develop a new algorithm for exact tests of HWP with multiple alleles on the basis of conditional probabilities of genotype arrays, which is faster than existing algorithms. We derived an algorithm for calculating the exact permutation significance value without enumerating all genotype arrays having the same allele counts as the observed one. The algorithm can be used for testing HWP by (1) summation of the conditional probabilities of occurrence of genotype arrays with smaller probability than the observed one, and (2) comparison of the sum with a nominal Type I error rate α. Application to published experimental data from seven maize populations showed that the exact test is computationally feasible and reduces the number of enumerated genotype count matrices about 30% compared with previously published algorithms.     

A clarification of the Hardy-Weinberg law

C. C. Li showed that Hardy-Weinberg proportions (HWP) can be maintained in a large population by nonrandom mating as well as random mating. In particular he gave the mating matrix for the symmetric case in the most general form possible. Thus Li showed that, once HWP are attained, the same proportions can be maintained by what he called pseudorandom mating. This article shows that, starting from any genotypic distribution at a single locus with two alleles, the same in each sex, HWP can be reached in one round of nonrandom mating with no change in allele frequency. In the model that demonstrates this fact, random mating is represented by a single point in a continuum of nonrandom possibilities.     

Polymorphic microsatellite DNA markers in the mangrove crab Ucides cordatus (Brachyura: Ocypodidae)

The isolation and characterization of the first polymorphic microsatellite markers for the mangrove crab Ucides cordatus are described. The number of alleles at each locus ranged from three to 25, mean of nine alleles, in 46 crabs captured in two Brazilian mangroves. The markers averaged high levels of observed (0.709 ± 0.183) and expected (0.716 ± 0.170) heterozygosities. Departures from Hardy-Weinberg equilibrium were observed at two loci. Linkage disequilibrium tests were not significant and no evidence of null alleles was detected. All these microsatellite loci are expected to be useful in estimating fine-scale population processes of this valuable mangrove species currently subjected to excessive fishing efforts.     

Testing for association on the X chromosome

The problem of testing for genotype-phenotype association with loci on the X chromosome in mixed-sex samples has received surprisingly little attention. A simple test can be constructed by counting alleles, with males contributing a single allele and females 2. This approach does assume not only Hardy-Weinberg equilibrium in the population from which the study subjects are sampled but also, perhaps, an unrealistic alternative hypothesis. This paper proposes 1 and 2 degree-of-freedom tests for association which do not assume Hardy-Weinberg equilibrium and which treat males as homozygous females. The proposed method remains valid when phenotype varies between sexes, provided the allele frequency does not, and avoids the loss of power resulting from stratification by sex in such circumstances.     

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.     

The variance for the disequilibrium coefficient in the individual Hardy-Weinberg test

It is of interest to detect the deviation from Hardy-Weinberg proportion (HWP) for a particular heterozygote. Hernández and Weir (1989, Biometrics 45, 53-70) suggested a disequilibrium coefficient approach and proposed a 1-d.f. chi2 test. This note derives the appropriate variance under the null hypothesis for performing this individual HWP test. Examples of applying the test to human genetic disease data are presented.     

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.     

鄱阳湖日本沼虾15个群体遗传多样性的微卫星分析

利用徽卫星分子标记研究鄱阳湖15个日本沼虾(Macrobrachium nipponense野生群体的遗传多样性.结果表明:10个微卫星位点呈高度多态性,共检测到129个多态性位点;每个群体中至少有5个位点显著偏离Hardy-Weinberg平衡,表现出杂合不足;15个日本沼虾群体均表现出较高的遗传多样性,其中大坝外群体和星池群体遗传多样性较高,万户和湖口群体相对较低.瓶颈效应和突变-漂移平衡分析表明,鄱阳湖日本沼虾群体近期没有经历过瓶颈效应,群体数量也没有下降.群体间F统计量及AMOVA分析显示,鄱阳湖日本沼虾群体存在极显著的遗传分化程度(FST=0.04709,P＜O.01).综上所述,鄱阳湖日本沼虾群体具有丰富的遗传多样性,可作为日本沼虾选育的基础群.     

Hardy-Weinberg equilibria in random mating populations

The structure of multiloci random mating populations is examined. Sufficient conditions for the existence of stable local Hardy-Weinberg equilibria for n loci and an arbitrary number of alleles per locus, are then derived for specified situations under the assumption of multiplicative gene action between loci. It is shown that a stable Hardy-Weinberg equilibrium can not be a local maximum of the mean fitness function with multiplicative gene action between loci. The stability of Hardy-Weinberg type border points and the condition for the increase of newly introduced genes are topics on which some n-loci results are also obtained for an arbitrary number of alleles per locus in systems that allow Hardy-Weinberg equilibria.     

New polymorphic microsatellite loci for population studies in the European anchovy, Engraulis encrasicolus (L.)

Eleven microsatellite loci were developed in the European anchovy, Engraulis encrasicolus, and tested in samples from two geographically distant populations (Atlantic and Mediterranean Sea). Number of alleles ranged from eight to 28 and observed heterozygosity from 0.440 to 0.920. There was no evidence of linkage disequilibrium, although two loci are indeed linked. All loci were in Hardy-Weinberg equilibrium, except for one locus in the Atlantic and two loci in the Mediterranean sample. These three loci plus two more showed evidence for null alleles.     

A disequilibrium coefficient approach to Hardy-Weinberg testing

A comparison was made of various tests for Hardy-Weinberg equilibrium, with emphasis on methods for multiple alleles. For an overall test of deviations from equilibrium, the classical chi-square goodness-of-fit test generally performs well, with continuity corrections needed for extreme gene frequencies or extreme departures from equilibrium. For small samples, probability tests are preferable and for multiple alleles these probability tests may be performed on a sample of all possible sets of genotypic frequencies having a fixed set of sample gene frequencies. Numerical work showed that the continuity-corrected chi-square was the most conservative test procedure, and the uncorrected chi-square the least conservative. With multiple alleles, a better appreciation of the nature of departures from equilibrium is given by the use of disequilibrium coefficients, defined for each heterozygote as the difference between observed and expected frequencies. Likelihood-ratio tests can be used to test each of these coefficients individually but a satisfactory procedure is to divide the squared estimate of each coefficient by its estimated variance and regard the ratio as a single-degree-of-freedom chi-square. Numerical studies confirmed the validity of this approach, which has the great advantage of not requiring solutions of nonlinear equations.     

Pseudo-Random Mating Populations. in Celebration of the 80th Anniversary of the Hardy-Weinberg Law

That random mating leads to Hardy-Weinberg distribution of genotypes is well known. This report is to show that, if the deviations from random mating are of a certain pattern, the offspring generation will also be in Hardy-Weinberg proportions. This brings out the fact that random mating is a sufficient condition, not a necessary one, for the attainment of the Hardy-Weinberg proportions. Such nonrandom-mating populations are tentatively said to be pseudo-random mating. Pseudo-random-mating populations exist for both autosomal and sex-linked systems with two or multiple alleles. This report covers the basic case of a two-allele autosomal locus in detail, but the possible extension to two loci and cytonuclear systems have also been mentioned in discussion.     

Characteristics of 11 polymorphic microsatellite markers in the red imported fire ant, Solenopsis invicta Buren

We have characterized 11 polymorphic microsatellite loci in the invasive ant Solenopsis invicta. Primer pairs were evaluated on fire ants collected from monogyne mounds in Lauderdale County, Mississippi. The observed and effective number of alleles ranged from two to six and from 1.31 to 2.64, respectively. The observed and expected heterozygosity values ranged from 0.1613 to 0.7826 and from 0.1491 to 0.6242, respectively. The polymorphism information content of the microsatellites ranged from 0.1482 to 0.6208. Probability tests indicated significant deviations from the Hardy-Weinberg equilibrium at three loci. Pairwise tests did not detect linkage disequilibrium between any pair of loci.     

Isolation and characterization of microsatellites in the harlequin ladybird, Harmonia axyridis (Coleoptera, Coccinellidae), and cross-species amplification within the family Coccinellidae

A total of 18 microsatellite DNA loci were isolated and characterized from the harlequin ladybird, Harmonia axyridis (Coleoptera: Coccinellidae). We optimized a multiplex panel consisting of two polymerase chain reactions, allowing the genotyping of all loci. The number of alleles and heterozygosity observed at each locus ranged from 1 to 12 and from 0 to 100%, respectively. After Bonferroni correction for multiple tests, none of the loci deviated significantly from Hardy-Weinberg equilibrium and there was no indication of significant linkage disequilibrium among pairs of loci. Successful cross-species amplification was obtained for only three of the seven tested species of Coccinellidae.     

Continuous selective models

Neglecting age-structure, but taking into account matings with differential fertility in Mendelian reproduction, continuous selective models are formulated for a single locus with an arbitrary number of alleles, with or without distinguishing the sexes, and for two alleles at each of two loci in a monoecious population. In each case, without restricting the mating system, differential equations are derived for the genotypic frequencies, and the validity of the customary Malthusian-parameter differential equations for the gametic frequencies is established. Particular attention is devoted to the conditions for Hardy-Weinberg proportions under random mating. For multiple alleles at a single locus in a monoecious population, exact solutions are obtained for the following three Hardy-Weinberg models: gametic selection, no dominance, and the same selective effect for all alleles but one. The last scheme includes, as special cases, a completely dominant or recessive distinguished allele, and arbitrary selection with only two alleles. Two single-locus assortative mating patterns are analyzed for a monoecious organism using the general formalism. One of these has an arbitrary number of alleles, all the genotypes being distinguishable, while the other involves two alleles, one of which is completely dominant to the other.     

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.     

新疆维吾尔族HLA-Cw基因座遗传多态性调查 The Study of HLA-Cw Polymorphism in Uygur Population

采用序列特异性寡核苷酸探针杂交技术（PCR-SSOP）对146位新疆维吾尔族无关个体HLA-Cw基因座进行基因分型,研究该民族HLA-Cw基因座遗传多态性,建立新疆维吾尔族HLA-Cw基因频率数据库。检出18种等位基因,基因频率分布在0.0069~02460,其中HLA-Cw*04、07、08、14基因频率比较高,基因频率分别为02460、0.1151、0.1010、0.1202,共占新疆维吾尔族可检出等位基因的58.23%,PCR-SSOP分型技术使新疆维吾尔族HLA-Cw基因座空白基因频率降至0.0064。经χ2检验,基因型分布符合Hardy-Weinberg平衡定律。建立民族HLA-Cw基因座基因频率数据库,为临床器官移植配型、人类学、法医学提供重要的群体遗传学资料。 Abstract:The HLA-Cw loci polymorphism in Uygur population was investigated using the PCR- sequence specific oligonucleotide probe (SSOP) method,and the genetic database on the distribution of gene frequency of the HLA-Cw loci was established.From 146 individuals of Uygur population,18 HLA-Cw alleles were detected.The gene frequency was from 0.0069 to 0.2460.The four most common alleles were HLA-Cw*04(24.60%)、07(11.51%)、08(1010%)、14(12.02%),and they covered 58.23% of total alleles detected from Uygur population.We have made a survey of HLA-Cw alleles frequencies in a Uygur population,with blank frequency being lowered to 0.0064.The distribution of genotype frequencies met the law of Hardy-Weinberg equilibrium by hi-square test.The frequency data can be used in forensic and paternity tests to estimate the frequency of a DNA profile in the Uygur population,transplant matching and anthropology.     

Identification, Inheritance, and Variation of Microsatellite Markers in the Black Scallop Mimachlamys varia

Five polymorphic microsatellite loci were identified in the black scallop Mimachlamys varia after construction of a genomic library enriched for (GT)n. To examine the transmission pattern of microsatellite alleles, several families were created and genotypes scored for three loci. The expected Mendelian ratios were found in 12 of 14 segregations examined. Unexpected segregations may be explained by a genotyping error (allelic dropout), given that when a specific allele was treated as dominant, the phenotypic ratios conformed to Mendelian expectations. The five loci were also examined in two samples from the Spanish coast. The two localities displayed similar mean values for the number of alleles per locus (7.2-8.4), allelic richness (7.2-7.9), and observed (0.389-0.484) and expected heterozygosity (0.545-0.618). Significant Hardy-Weinberg deviations were observed at three loci, with heterozygote deficiency occurring in all cases. Global multilocus θ value and allele frequencies at one locus revealed significant differentiation between the two localities.