Performing the exact test of Hardy-Weinberg proportion for multiple alleles.

The Hardy-Weinberg law plays an important role in the field of population genetics and often serves as a basis for genetic inference. Because of its importance, much attention has been devoted to tests of Hardy-Weinberg proportions (HWP) over the decades. It has long been recognized that large-sample goodness-of-fit tests can sometimes lead to spurious results when the sample size and/or some genotypic frequencies are small. Although a complete enumeration algorithm for the exact test has been proposed, it is not of practical use for loci with more than a few alleles due to the amount of computation required. We propose two algorithms to estimate the significance level for a test of HWP. The algorithms are easily applicable to loci with multiple alleles. Both are remarkably simple and computationally fast. Relative efficiency and merits of the two algorithms are compared. Guidelines regarding their usage are given. Numerical examples are given to illustrate the practicality of the algorithms.     

Properties of Statistical Tests of Neutrality for DNA Polymorphism Data

A class of statistical tests based on molecular polymorphism data is studied to determine size and power properties. The class includes TAJIMA''s D statistic as well as the D* and F* tests proposed by FU and LI. A new method of constructing critical values for these tests is described. Simulations indicate that TAJIMA''s test is generally most powerful against the alternative hypotheses of selective sweep, population bottleneck, and population subdivision, among tests within this class. However, even TAJIMA''s test can detect a selective sweep or bottleneck only if it has occurred within a specific interval of time in the recent past or population subdivision only when it has persisted for a very long time. For greatest power against the particular alternatives studied here, it is better to sequence more alleles than more sites.     

Testing for Hardy-Weinberg equilibrium in samples with related individuals

When the classical chi(2) goodness-of-fit test for Hardy-Weinberg (HW) equilibrium is used on samples with related individuals, the type I error can be greatly inflated. In particular the test is inappropriate in population isolates where the individuals are related through multiple lines of descent. In this article, we propose a new test for HW (the QL-HW test) suitable for any sample with related individuals, including large inbred pedigrees, provided that their genealogy is known. Performed conditional on the pedigree structure, the QL-HW test detects departures from HW that are not due to the genealogy. Because the computation of the QL-HW test becomes intractable for very polymorphic loci in large inbred pedigrees, a simpler alternative, the GCC-HW test, is also proposed. The statistical properties of the QL-HW and GCC-HW tests are studied through simulations considering a sample of independent nuclear families, a sample of extended outbred genealogies, and samples from the Hutterite population, a North American highly inbred isolate. Finally, the method is used to test a set of 143 biallelic markers spanning 82 genes in this latter population.     

An efficient method for stochastic simulation of biological populations in continuous time

We present an efficient algorithm for individual-based, stochastic simulation of biological populations in continuous time. A simple method for its implementation is given and it is compared to Gillespie's commonly used Direct Method. These two methods are proven to be exactly equivalent and, using a basic evolutionary model, it is demonstrated that the new algorithm can run thousands of times faster. Furthermore, while computational cost per event increases linearly with population size under the Direct Method, this cost is independent of population size under the new algorithm. We argue that this gain in efficiency opens up the possibility to explore a new class of models in population biology.     

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 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.     

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.     

Usefulness of molecular markers for detecting population bottlenecks via monitoring genetic change

It is important to detect population bottlenecks in threatened and managed species because bottlenecks can increase the risk of population extinction. Early detection is critical and can be facilitated by statistically powerful monitoring programs for detecting bottleneck-induced genetic change. We used Monte Carlo computer simulations to evaluate the power of the following tests for detecting genetic changes caused by a severe reduction in a population's effective size ( N _e): a test for loss of heterozygosity, two tests for loss of alleles, two tests for change in the distribution of allele frequencies, and a test for small N _e based on variance in allele frequencies (the 'variance test'). The variance test was most powerful; it provided an 85% probability of detecting a bottleneck of size N _e = 10 when monitoring five microsatellite loci and sampling 30 individuals both before and one generation after the bottleneck. The variance test was almost 10-times more powerful than a commonly used test for loss of heterozygosity, and it allowed for detection of bottlenecks before 5% of a population's heterozygosity had been lost. The second most powerful tests were generally the tests for loss of alleles. However, these tests had reduced power for detecting genetic bottlenecks caused by skewed sex ratios. We provide guidelines for the number of loci and individuals needed to achieve high-power tests when monitoring via the variance test. We also illustrate how the variance test performs when monitoring loci that have widely different allele frequency distributions as observed in five wild populations of mountain sheep ( Ovis canadensis ).     

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.     

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.     

A log-linear approach to case-parent-triad data: assessing effects of disease genes that act either directly or through maternal effects and that may be subject to parental imprinting.

We describe a log-linear method for analysis of case-parent-triad data, based on maximum likelihood with stratification on parental mating type. The method leads to estimates of association parameters, such as relative risks, for a single allele, and also to likelihood ratio chi2 tests (LRTs) of linkage disequilibrium. Hardy-Weinberg equilibrium need not be assumed. Our simulations suggest that the LRT has power similar to that of the chi2 "score" test proposed by Schaid and Sommer and that both can outperform the transmission/disequilibrium test (TDT), although the TDT can perform better under an additive model of inheritance. Because a restricted version of the LRT is asymptotically equivalent to the TDT, the proposed test can be regarded as a generalization of the TDT. The method that we describe generalizes easily to accommodate maternal effects on risk and, in fact, produces powerful and orthogonal tests of the contribution of fetal versus maternal genetic factors. We further generalize the model to allow for effects of parental imprinting. Imprinting effects can be fitted by a simple, iterative procedure that relies on the expectation-maximization algorithm and that uses standard statistical software for the maximization steps. Simulations reveal that LRT tests for detection of imprinting have very good operating characteristics. When a single allele is under study, the proposed method can yield powerful tests for detection of linkage disequilibrium and is applicable to a broader array of causal scenarios than is the TDT.     

Case-control studies of genetic markers: power and sample size approximations for Armitage's test for trend

The association of a candidate gene with disease can be efficiently evaluated by a case-control study in which allele frequencies are compared for diseased cases and unaffected controls. However, when the distribution of genotypes in the population deviates from Hardy-Weinberg proportions, the frequency of genotypes--rather than alleles--should be compared by the Armitage test for trend. We present formulas for power and sample size for studies that use Armitage's trend test. The formulas make no assumptions about Hardy-Weinberg equilibrium, but do assume random ascertainment of cases and controls, all of whom are independent of one another. We demonstrate the accuracy of the formulas by simulations.     

The size of the chi-square test for the Hardy-Weinberg law

Many scientific problems can be formulated in terms of a statistical model indexed by parameters, only some of which are of scientific interest and the other parameters, called nuisance parameters, are not of interest in themselves. For testing the Hardy-Weinberg law, a relation among genotype and allele probabilities is of interest and allele probabilities are of no interest and now nuisance parameters. In this paper we investigate how the size (the maximum of the type I error rate over the nuisance parameter space) of the chi-square test for the Hardy-Weinberg law is affected by the nuisance parameters. Whether the size is well controlled or not under the nominal level has been frequently investigated as basic components of statistical tests. The size represents the type I error rate at the worst case. We prove that the size is always greater than the nominal level as the sample size increases. Extensive computations show that the size of the chi-squared test (worst type I error rate over the nuisance parameter space) deviates more upwardly from the nominal level as the sample size gets larger. The value at which the maximum of the type I error rate was found moves closer to the edges of the the nuisance parameter space with increasing sample size. An exact test is recommended as an alternative when the type I error is inflated.     

Population growth enhances the mean fixation time of neutral mutations and the persistence of neutral variation

A fundamental result of population genetics states that a new mutation, at an unlinked neutral locus in a randomly mating diploid population, has a mean time of fixation of ～4N(e) generations, where N(e) is the effective population size. This result is based on an assumption of fixed population size, which does not universally hold in natural populations. Here, we analyze such neutral fixations in populations of changing size within the framework of the diffusion approximation. General expressions are derived for the mean and variance of the fixation time in changing populations. Some explicit results are given for two cases: (i) the effective population size undergoes a sudden change, representing a sudden population expansion or a sudden bottleneck; (ii) the effective population changes linearly for a limited period of time and then remains constant. Additionally, a lower bound for the mean time of fixation is obtained for an effective population size that increases with time, and this is applied to exponentially growing populations. The results obtained in this work show, among other things, that for populations that increase in size, the mean time of fixation can be enhanced, sometimes substantially so, over 4N(e,0) generations, where N(e,0) is the effective population size at the time the mutation arises. Such an enhancement is associated with (i) an increased probability of neutral polymorphism in a population and (ii) an enhanced persistence of high-frequency neutral variation, which is the variation most likely to be observed.     

Assessing population genetic structure via the maximisation of genetic distance

Background

The inference of the hidden structure of a population is an essential issue in population genetics. Recently, several methods have been proposed to infer population structure in population genetics.

Methods

In this study, a new method to infer the number of clusters and to assign individuals to the inferred populations is proposed. This approach does not make any assumption on Hardy-Weinberg and linkage equilibrium. The implemented criterion is the maximisation (via a simulated annealing algorithm) of the averaged genetic distance between a predefined number of clusters. The performance of this method is compared with two Bayesian approaches: STRUCTURE and BAPS, using simulated data and also a real human data set.

Results

The simulations show that with a reduced number of markers, BAPS overestimates the number of clusters and presents a reduced proportion of correct groupings. The accuracy of the new method is approximately the same as for STRUCTURE. Also, in Hardy-Weinberg and linkage disequilibrium cases, BAPS performs incorrectly. In these situations, STRUCTURE and the new method show an equivalent behaviour with respect to the number of inferred clusters, although the proportion of correct groupings is slightly better with the new method. Re-establishing equilibrium with the randomisation procedures improves the precision of the Bayesian approaches. All methods have a good precision for F_ST ≥ 0.03, but only STRUCTURE estimates the correct number of clusters for F_ST as low as 0.01. In situations with a high number of clusters or a more complex population structure, MGD performs better than STRUCTURE and BAPS. The results for a human data set analysed with the new method are congruent with the geographical regions previously found.

Conclusion

This new method used to infer the hidden structure in a population, based on the maximisation of the genetic distance and not taking into consideration any assumption about Hardy-Weinberg and linkage equilibrium, performs well under different simulated scenarios and with real data. Therefore, it could be a useful tool to determine genetically homogeneous groups, especially in those situations where the number of clusters is high, with complex population structure and where Hardy-Weinberg and/or linkage equilibrium are present.     

New Statistical Tests of Neutrality for DNA Samples from a Population

The purpose of this paper is to develop statistical tests of the neutral model of evolution against a class of alternative models with the common characteristic of having an excess of mutations that occurred a long time ago or a reduction of recent mutations compared to the neutral model. This class of population genetics models include models for structured populations, models with decreasing effective population size and models of selection and mutation balance. Four statistical tests were proposed in this paper for DNA samples from a population. Two of these tests, one new and another a modification of an existing test, are based on EWENS'' sampling formula, and the other two new tests make use of the frequencies of mutations of various classes. Using simulated samples and regression analyses, the critical values of these tests can be computed from regression equations. This approach for computing the critical values of a test was found to be appropriate and quite effective. We examined the powers of these four tests using simulated samples from structured populations, populations with linearly decreasing sizes and models of selection and mutation balance and found that they are more powerful than existing statistical tests of the neutral model of evolution.     

The Knox method and other tests for space-time interaction

The Knox method, as well as other tests for space-time interaction, are biased when there are geographical population shifts, i.e., when there are different percent population growths in different regions. In this paper, the size of the population shift bias is investigated for the Knox test, and it is shown that it can be a considerable problem. A Monte Carlo method for constructing unbiased space-time interaction tests is then presented and illustrated on the Knox test as well as for a combined Knox test. Practical implications are discussed in terms of the interpretation of past results and the design of future studies.     

A simple and robust TDT-type test against genotyping error with error rates varying across families

The transmission/disequilibrium test (TDT), a family based test of linkage and association, is a popular test for studies of complex inheritance, as it is nonparametric and robust against spurious conclusions induced by hidden genetic structure, such as stratification or admixture. However, the TDT may be biased by genotyping errors. Undetected genotyping errors may be contributing to an inflated type I error rate among reported TDT-derived associations. To adjust for bias, a popular approach is to assume a genotype error model for describing the pattern of errors and propose association tests using likelihood method. However, all model-based approaches tend to perform unsatisfactorily if the related genotyping error rates are not identical across all families. In this paper, we propose a TDT-type association test which is not only simple, robust against population stratification (and hence the assumption of Hardy-Weinberg equilibrium is not required), but also robust against genotyping error with error rates varying across families. Simulation studies confirm that the new test has very reasonable performance.     

LB3D: a protein three-dimensional substructure search program based on the lower bound of a root mean square deviation value

Searching for protein structure-function relationships using three-dimensional (3D) structural coordinates represents a fundamental approach for determining the function of proteins with unknown functions. Since protein structure databases are rapidly growing in size, the development of a fast search method to find similar protein substructures by comparison of protein 3D structures is essential. In this article, we present a novel protein 3D structure search method to find all substructures with root mean square deviations (RMSDs) to the query structure that are lower than a given threshold value. Our new algorithm runs in O(m + N/m(0.5)) time, after O(N log N) preprocessing, where N is the database size and m is the query length. The new method is 1.8-41.6 times faster than the practically best known O(N) algorithm, according to computational experiments using a huge database (i.e., >20,000,000 C-alpha coordinates).     

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.