A Two-Tailed P-Value for Fisher's Exact Test

A two-tailed P-value is presented for a significance test in two by two contingency tables. There is no extraneous quasi-observation such as is needed in the exact randomized uniformly most powerful unbiased (UMPU) test of the hypothesis of independence. The proposed P-value can never exceed unity and is always two-tailed, unlike other P-values proposed in the literature     

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.     

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.     

An Exact Unconditional test for the Hardy-Weinberg Equilibrium

An exact test based on the unconditional distribution of a test statistic for the Hardy-Weinberg equilibrium is introduced. This test is usually more powerful and requires less extensive tabulations compared with the (ordinary) exact conditional test. Tables of critical values are given for N = 5(1)100 and α =·10, ·05, ·01. Sample sizes required to attain a given power with the unconditional and conditional exact tests are provided.     

Hardy-Weinberg定律在专科教学中的课堂设计

Hardy-Weinberg定律是群体遗传学的第一理论基石,也是现代进化论、现代优生学和群体育种的理论基础,是遗传学教学中的重难点内容,但通过合理的教学设计可帮助学生全面理解、掌握并应用该定律,为后续学习奠定基础。     

A Two-Stage Test for Distinguishing Random,Pseudo-Random and Nonrandom Mating Populations

There is no such an implication that a population in Hardy-Weinberg equilibrium must have undergone random mating. Therefore, it is unequivocal that the usual tests for “Hardy-Weinberg equilibrium” are indeed tests for “random union of gametes” but not for “random mating”. In this paper, utilizing population characteristics expressed in equilibrium state (equilibrium or disequilibrium) and mating behavior (random or nonrandom), a two-stage testing procedure for distinguishing random, pseudo-random and nonrandom mating populations is proposed. At the first stage, a population is tested for Hardy-Weinberg equilibrium. If insignificant result (i.e., in equilibrium) is obtained, then to a second stage the population is further tested for mating behavior. Random mating-pairs data are needed here for analysis instead of random individuals for usual Hardy-Weinberg equilibrium tests. Since distinguishing the three types of mating populations depends on the combined results of two stages, the probability of correct determination of the two-stage tests is discussed by simulation studies.     

Large Hardy-Weinberg equilibrium deviations in the Daphnia longispina of Lake El Tobar

The population structure of Daphnia longispina in Lake El Tobar, Spain was studied by measuring variation at the aldehyde oxidase (AO), phosphoglucose isomerase (PGI) and phosphoglucose mutase (PGM) loci in each of 1337 individuals from four collections. In 9 of the 12 comparisons between observed allele frequencies and those expected by Hardy-Weinberg equilibrium there was an excess of heterozygotes. We found 27 of the potential number of 54 composite electromorphs (clones) based on the three allozymes. Clone diversities were rather high in all collections. Three clones reached frequencies of over 25% and different clones were dominant in each of the four collections. Strong temporal variation was found in the genetic structure of this Daphnia population. This variation was driven by changes in the relative frequencies of the component clones in the lake rather than by a recruitment of novel clones into the population.We conclude with a consideration of the role of models relating allele and genotype frequencies in populations of cyclical parthenogens. Because the breeding system of these populations infrequently involves recombination between clones, models such as the Hardy-Weinberg have limited value in providing meaningful measures of population structure.     

关于最大信息熵原理与群体遗传平衡一致性的探讨

汪小龙等建立了用最大信息熵原理推导一个基因座上群体遗传平衡的统一数学模型,并给出了模型的最大值解,此解正是Hardy-Weinberg平衡定律所给出的基因型频率。这说明当群体基因型信息熵最大时,群体基因型频率不再变化,达到平衡状态,从而证明了最大信息熵原理与Hardy-Weinberg平衡定律具有一致性,同时指出这一结论可以推广至有迁移、突变、选择、遗传漂变、近亲交配的群体以及多个基因座情形。概括地说就是：最大信息熵原理与群体遗传平衡具有一致性。但是,他们仅仅证明了最大信息熵原理与一个基因座上Hardy-Weinberg平衡定律具有一致性,本文在这个范围内将其推广至多个基因座,且每一个基因座均为复等位基因情形。至于最大信息熵原理是否与其它的群体遗传平衡具有一致性,他们的结论仅仅是猜想,并未严格推导。事实上,要想将这种一致性推广到迁移、突变、随机漂变和近亲交配等群体,则不见得正确。      

On Nonrandom Mating Systems for Attaining Hardy-Weinberg Equilibrium

LI (1988) showed that random mating is a sufficient, not a necessary condition for the Hardy-Weinberg principle. A nonrandom mating population that behaves like a random mating population is thus called a ‘pseudo-random mating population’ by him. The pseudo-random mating system studied by him has been focused on those populations in which the parental generation is in Hardy-Weinberg proportions. In other words, the mating type frequency deviations from random mating for each parental genotype add up to zero. In this article these restrictions are relaxed and new pseudo-random mating systems that immediately yield Hardy-Weinberg offspring are also obtained. This is possible because reciprocal crosses have identical segregation probabilities for an autosomal locus, and the manipulation of the combined frequency of reciprocal crosses does not change the gene frequency of the population. A comparison of these new patterns with that of Li is given in the Discussion.     

基因型错误对数量性状精细定位的哈迪-温伯格不平衡指数的效应

使用紧密相邻的标记位点且与标记基因频率无关的哈迪-温伯格不平衡(HWD)指数被用来对数量性状位点(QTL)进行精细定位.本文讨论了当存在基因型错误时HWD指数的性质.文章指出,当存在基因型错误时,对于在群体的标记基因频率已知的情形使用的两个HWD指数尽管受基因型错误的影响但仍然有效;而仅仅极端样本的标记基因频率已知的情形下使用的两个HWD指数同时与基因型错误和标记基因频率有关.计算机模拟表明,仅仅极端样本的标记基因频率已知的情形下使用的两个HWD指数在精细定位时会产生偏差,不适宜作精细定位.     

Representation of nonepistatic selection models and analysis of multilocus Hardy-Weinberg equilibrium configurations

Summary The paper develops conditions for the existence and the stability of central equilibria emanating from selection recombination interaction with generalized nonepistatic selection forms operating in multilocus multiallele systems. The selection structure admits a natural representation as simple sums of Kronecker products based on a common set of marginal selection components. A flexible parametrization of the recombination process is introduced leading to a canonical derivation of the transformation equations connecting gamete frequency states over successive generations. Conditions for the existence and stability of multilocus Hardy-Weinberg (H.W.) type equilibria are elaborated for the classical nonepistatic models (multiplicative and additive viability effects across loci) as well as for generalized nonepistatic selection expressions. It is established that the range of recombination distributions maintaining a stable H.W. polymorphic equilibrium is confined to loose linkage in the pure multiplicative case, but is not restricted in the additive model. In the bisexual case we ascertain for the generalized nonepistatic model the stability conditions of a common H.W. polymorphism.This paper was supported in part by NIH Grant GM 10452-14 and NSF Grant MCS 75-23608.     

Exact tests for association between alleles at arbitrary numbers of loci

Associations between allelic frequencies, within and between loci, can be tested for with an exact test. The probability of the set of multi-locus genotypes in a sample, conditional on the allelic counts, is calculated from multinomial theory under the hypothesis of no association. Alleles are then permuted and the conditional probability calculated for the permuted genotypic array. The proportion of arrays no more probable than the original sample provides the significance level for the test. An algorithm is provided for counting genotypes efficiently in the arrays, and the powers of the test presented for various kinds of association. The powers for the case when associations are generated by admixture of several populations suggest that exact tests are capable of detecting levels of association that would affect forensic calculations to a significant extent.     

A Note on a Continuity Correction for Testing in Three-Dimensional Configural Frequency Analysis

A continuity correction is proposed for the test statistics of HABERMAN (1978) and LEHMACHER (1981) for identifying overfrequented (or underfrequented) cells in the three-dimensional Configural Frequency Analysis (CFA). Its quality is shown by comparison with the test based on the exact distribution of the cell frequencies.     

On S.N. Bernstein's derivation of Mendel's Law and 'rediscovery' of the Hardy-Weinberg distribution

Around 1923 the soon-to-be famous Soviet mathematician and probabilist Sergei N. Bernstein started to construct an axiomatic foundation of a theory of heredity. He began from the premise of stationarity (constancy of type proportions) from the first generation of offspring. This led him to derive the Mendelian coefficients of heredity. It appears that he had no direct influence on the subsequent development of population genetics. A basic assumption of Bernstein was that parents coupled randomly to produce offspring. This paper shows that a simple model of non-random mating, which nevertheless embodies a feature of the Hardy-Weinberg Law, can produce Mendelian coefficients of heredity while maintaining the population distribution. How W. Johannsen's monograph influenced Bernstein is discussed.     

Small Sample Properties of the Mantel-Haenszel Test Statistic for Density Follow-Up Studies

For the analysis of combinations of 2×2 non-contingency tables as obtained from density follow-up studies (relating a number of events to a number of person-years of follow-up) an analogue of the Mantel-Haenszel test for 2×2 contingency tables is widely used. In this paper the small sample properties of this test, both with and without continuity correction, are evaluated. Also the improvement of the test-statistic by using the first four cumulants via the Edgeworth expansion was studied. Results on continuity correction agree with similar studies on the Mantel-Haenszel statistic for 2×2 contingency tables: Continuity correction gives a p-value which approximates the exact p-value better than the p-value obtained without this correction; both the exact test and its approximations show considerable conservatism in small samples; the uncorrected Mantel-Haenszel test statistic gives a p-value that agrees more with the nominal significance level, but can be anti-conservative. The p-value based on the first four cumulants gives a better approximation of the exact p-value than the continuity corrected test, especially when the distribution has marked skewness.     

两个基因座位的遗传平衡原理

由于许多教科书中关于连锁平衡的介绍,多是引用结论或是推导不太严谨,使学生在学习群体遗传学时对理解连锁平衡的原理感到困难。本文从遗传平衡的基本条件出发,通过较严谨的数学推导,介绍了两个基因座的连锁平衡条件、平衡过程等原理,供教师和学生在群体遗传学教学中参考。Abstract: Because linkage equilibrium is introduced by directly quoting the conclusions or imprecise mathematical reasoning in most of textbooks, many students are puzzled with the problem of linkage equilibrium when they learn population genetics. Based on the radical conditions of genetic equilibrium, the principle of linkage equilibrium condition and process, for two gene loci is introduced by precise mathematical reasoning. The article may provide reference to teachers and students in the teaching and learning of population genetics.     

A Novel Test for Independence Derived from an Exact Distribution of ith Nearest Neighbours

Dependence measures and tests for independence have recently attracted a lot of attention, because they are the cornerstone of algorithms for network inference in probabilistic graphical models. Pearson''s product moment correlation coefficient is still by far the most widely used statistic yet it is largely constrained to detecting linear relationships. In this work we provide an exact formula for the th nearest neighbor distance distribution of rank-transformed data. Based on that, we propose two novel tests for independence. An implementation of these tests, together with a general benchmark framework for independence testing, are freely available as a CRAN software package (http://cran.r-project.org/web/packages/knnIndep). In this paper we have benchmarked Pearson''s correlation, Hoeffding''s , dcor, Kraskov''s estimator for mutual information, maximal information criterion and our two tests. We conclude that no particular method is generally superior to all other methods. However, dcor and Hoeffding''s are the most powerful tests for many different types of dependence.