Testing Hardy-Weinberg equilibrium on allelic data from VNTR loci.

Several methods for testing independence of pairs of alleles in a population that are obtained from a VNTR locus are presented. We assume an exchangeable quasi-continuous distribution of the fragment lengths used to measure the allelic pairs. Bivariate-estimated quantiles computed from the quantiles of the entire data set are then utilized for testing independence. These methods have the advantage of being minimally susceptible to the criticism of (a) the inability of a technology to measure a few small-sized or rather large-sized fragments and (b) inadequate estimation of the homozygotic proportion.     

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.     

Diversity of ethnic and racial VNTR RFLP fixed-bin frequency distributions.

To examine the impact that intra- and interracial genetic diversities have on VNTR RFLP-fragment-size distributions, a multiracial (East Asian, African American, U.S. Southwest Hispanic, and European Caucasian) and multiethnic (Chinese, Japanese, Korean, and Vietnamese) database has been constructed for the following loci: D1S7, D2S44, D4S139, and D10S28. Homogeneity between samples was examined using the Komologorov-Smirnov two-sample test for RFLP fragment sizes and a log-likelihood test for fixed-bin frequencies with theoretical and Monte Carlo empirical significance levels. Small but significant differences between theoretical and empirical significance-level distributions were observed with both procedures, particularly with the latter. The significance levels of the two types of tests were poorly correlated. Statistically significant differences in fragment-size and fixed-bin distributions were found within and between races, with greater differences occurring between races. Cluster analysis and principal components analysis, using different similarity measures, did not support the hypothesis of greater intra- than interracial diversity, which suggests that ethnic variation can be conservatively estimated by racial variation.     

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.     

A note on goodness of fit of a population to Hardy-Weinberg structure.

The test of goodness of fit of a population to Hardy-Weinberg equilibrium given by Levene [2] and Haldane [1] is valid within a widely accepted recipe for testing goodness of fit of a composite hypothesis. The nature of the result of Cannings and Edwards [3] is described. The result was shown to be quite different than they claimed and, although possible of some interest, not relevant to the testing of goodness of fit to Hardy-Weinberg structure.     

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.     

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。     

An unconditional exact test for the Hardy-Weinberg equilibrium law: sample-space ordering using the Bayes factor

Much forensic inference based upon DNA evidence is made assuming that the Hardy-Weinberg equilibrium (HWE) is valid for the genetic loci being used. Several statistical tests to detect and measure deviation from HWE have been devised, each having advantages and limitations. The limitations become more obvious when testing for deviation within multiallelic DNA loci is attempted. Here we present an exact test for HWE in the biallelic case, based on the ratio of weighted likelihoods under the null and alternative hypotheses, the Bayes factor. This test does not depend on asymptotic results and minimizes a linear combination of type I and type II errors. By ordering the sample space using the Bayes factor, we also define a significance (evidence) index, P value, using the weighted likelihood under the null hypothesis. We compare it to the conditional exact test for the case of sample size n = 10. Using the idea under the method of chi(2) partition, the test is used sequentially to test equilibrium in the multiple allele case and then applied to two short tandem repeat loci, using a real Caucasian data bank, showing its usefulness.     

A note on the low-dimensional display of multivariate data using neural networks

A novel neural network technique has been proposed (Livingstone et al. J. Mol. Graphics 1991, 9, 115-118) which is useful for a low-dimensional display of multivariate data sets. The method makes use of the activity values of the hidden neurons in a trained three-layer feed-forward network to produce the low-dimensional display. It was claimed that in contrast to conventional techniques, such as principal components analysis or nonlinear mapping, this technique could be used also to reconstruct, from a given point in the low-dimensional display, the corresponding multivariate input vector via the completely known weight matrices of a suitably trained network. We show here that this claim is unjustified in this general form. When previously unknown, grossly different input vectors are presented to the trained network, they can occupy, for example, exactly the same point in the low-dimensional display which is occupied also by a given training vector, if certain linear relationships between the vector components are fulfilled. Thus, an infinite set of different linearly dependent input vectors is projected onto one single point in the low-dimensional display. Reconstruction of a multivariate vector, starting from this point in the low-dimensional display, is able to lead back to only one multivariate vector (in the example given, to the original training vector).     

Significant deviations from Hardy-Weinberg equilibrium caused by low levels of microsatellite genotyping errors

Microsatellite genotyping from samples with varying quality can result in an uneven distribution of errors. Previous studies reporting error rates have focused on estimating the effects of both randomly distributed and locus‐specific errors. Sample‐specific errors, however, can also significantly affect results in population studies despite a large sample size. From two studies including six microsatellite markers genotyped from 272 sperm whale DNA samples, and 33 microsatellites genotyped from 213 bowhead whales, we investigated the effects of sample‐ and locus‐specific errors on calculations of Hardy–Weinberg equilibrium. The results of a jackknife analysis in these two studies identified seven individuals that were highly influential on estimates of Hardy–Weinberg equilibrium for six different markers. In each case, the influential individual was homozygous for a rare allele. Our results demonstrate that Hardy–Weinberg P values are very sensitive to homozygosity in rare alleles for single individuals, and that > 50% of these cases involved genotype errors likely due to low sample quality. This raises the possibility that even small, normal levels of laboratory errors can result in an overestimate of the degree to which markers are out of Hardy–Weinberg equilibrium and hence overestimate population structure. To avoid such bias, we recommend routine identification of influential individuals and multiple replications of those samples.     

Quantification of the power of Hardy-Weinberg equilibrium testing to detect genotyping error

Deviation from Hardy-Weinberg equilibrium has become an accepted test for genotyping error. While it is generally considered that testing departures from Hardy-Weinberg equilibrium to detect genotyping error is not sensitive, little has been done to quantify this sensitivity. Therefore, we have examined various models of genotyping error, including error caused by neighboring SNPs that degrade the performance of genotyping assays. We then calculated the power of chi-square goodness-of-fit tests for deviation from Hardy-Weinberg equilibrium to detect such error. We have also examined the affects of neighboring SNPs on risk estimates in the setting of case-control association studies. We modeled the power of departure from Hardy-Weinberg equilibrium as a test to detect genotyping error and quantified the effect of genotyping error on disease risk estimates. Generally, genotyping error does not generate sufficient deviation from Hardy-Weinberg equilibrium to be detected. As expected, genotyping error due to neighboring SNPs attenuates risk estimates, often drastically. For the moment, the most widely accepted method of detecting genotyping error is to confirm genotypes by sequencing and/or genotyping via a separate method. While these methods are fairly reliable, they are also costly and time consuming.     

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.     

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.     

A new data analysis method to determine binding constants of small molecules to proteins using equilibrium analytical ultracentrifugation with absorption optics.

In principle, equilibrium analytical ultracentrifugation (AU) can be used to quantify the binding stoichiometry and affinity between small-molecule ligands and proteins in aqueous solution. We show here that heteromeric binding constants can be determined using a data-fitting procedure which utilizes a postfitting computation of the total amount of each component in the centrifuge cell. The method avoids overconstraining the fitting of the radial concentration profiles, but still permits unique binding constants to be determined using measurements at a single wavelength. The computational program is demonstrated by applying it to data obtained with mixtures of a 500-Da molecule and interleukin-2, a 16-kDa protein. The 1:1 binding stoichiometry and heteromeric dissociation constants (K(ab)) determined from centrifuge data at two different wavelengths are within the 4-9 microM range independently determined from a functional assay. Values for K(ab) have been obtained for ligands with affinities as weak as 500 microM. This AU method is applicable to compounds with significant UV absorbance (approximately 0.2) at concentrations within approximately 5- to 10-fold of their K(ab). The method, which has been incorporated into a user procedure for IgorPro (Wavemetrics, Oswego, OR), is included as supplementary material.