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.     

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.     

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.     

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.     

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.     

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.     

Alternatives to the Chi-Square Test of Homogeneity in 2×2 Tables and to Fisher's Exact Test

In comparing two independent binomial proportions by modified X² tests: Gart's modified likelihood-ratio, Overall and Starbuck's “tailored F”, Pearson's adjusted X²[=X² multiplied by (n - 1)/n] and its skewness-corrected version proposed by Berchtold, it was found that the last two have error probabilities that in the mean are close to the significance level.     

Deciphering diversity in populations of various linguistic and ethnic affiliations of different geographical regions of India: Analysis based on 15 microsatellite markers

The extent of genetic polymorphism at fifteen autosomal microsatellite markers in 54 ethnically, linguistically and geographically diverse human populations of India was studied to decipher intrapopulation diversity. The parameters used to quantify intrapopulation diversity were average allele diversity, average heterozygosity, allele range (base pairs), and number of alleles. Multilocus genotype frequencies calculated for selected populations were utilized for testing conformity with the assumption of Hardy-Weinberg equilibrium. The exact test values, after Bonferroni correction, showed significant deviation amongst Gowda (vWA, Penta E); Dhangar, Satnami and Gounder (D8S1179); Hmar (FGA); Kuki and Balti (vWA) groups. Relatively low number of alleles and allelic diversity (base-pairs size) had been observed in populations of central India as compared with southern and northern regions of the country. The communities of Indo-Caucasoid ethnic origin and Indo-European linguistic family (Kshatriya of Uttar Pradesh) showed highest allelic diversity, as well as rare alleles, not reported in any other Indian populations. Analysis based on average heterozygosity was also found to be lowest among the populations of central India (0.729) and highest among the populations from north (0.777) and west (0.784) regions of the country, having Indo-Caucasoid ethnic origin and Austro-Asiatic linguistic affiliation. The maximum power of discrimination (85%-89%) had been observed at loci FGA, Penta E, D18S51 and D21S11, suggested high intrapopulation diversity in India. Genetic diversity revealed by STR markers was consistent with the known demographic histories of populations. Thus, the present study clearly demonstrated that the intrapopulation diversity is not only present at the national level, but also within smaller geographical regions of the country. This is the first attempt to understand the extent of diversity within populations of India at such a large scale at genomic level.     

A Pooled Response Strategy for Combining Multiple Lines of Evidence to Quantitatively Estimate Impact

The impacts of sediment contaminants can be evaluated by different lines of evidence, including toxicity tests and ecological community studies. Responses from 10 different toxicity assays/tests were combined to arrive at a “site score.” We employed a relatively simple summary measure, pooled P-values where we quantify a potential decrement in response in a contaminated site relative to nominally clean reference sites. The response-specific P-values were defined relative to a “null” distribution of responses in reference sites, and were then pooled using standard meta-analytic methods. Ecological community data were also evaluated using an analogous strategy. A distribution of distances of the reference sites from thecentroid of the reference sites was obtained. The distance from each of the test sites from the centroid of the reference sites was then calculated, and the proportion of reference distances that exceed the test site difference was used to define an empirical P-value for that test site. A plot of the toxicity P-value versus the community P-value was used to identify sites based on both alteration in community structure and toxicity, that is, by weight-of-evidence. This approach provides a useful strategy for examining multiple lines of evidence that should be accessible to the broader scientific community. The use of a large collection of reference sites to empirically define P-values is appealing in that parametric distribution assumptions are avoided, although this does come at the cost of assuming the reference sites provide an appropriate comparison group for test sites.     

A Modified Exact Test for 2 × 2 Contingency Tables

A modified exact test is proposed for 2×2 contingency tables. This test, which is based on a less conservative definition of the concept of significance (STONE, 1969) is compared with a modified form of Pearson's X² test and with Tocher's randomized exact (UMPU) test. The sizes of the new test lie near the nominal 0.05 levels while those of the X² test usually exceed the nominal level, sometimes by a factor of 2 or more. The power of the modified test is usually close to that of the UMPU test.