An exact test of the Hardy-Weinberg law.

An exact distribution of a finite sample drawn from an infinite population in Hardy-Weinberg Equilibrium is described for k-alleles. Accordingly, an exact test of the law is presented and compared with two x2-tests for two and three alleles. For two alleles, it is shown that the "classical" c2-test is very adequate for sample sizes as small as ten. For three alleles, it is shown that a simpler formulation based on Leven's distribution approximates the exact test of this paper rather closely. However, it is recommended that researchers continue to employ the standard x2-test for all sample sizes and abide by it if the corresponding probability value is not "too close" to the critical level; otherwise, an exact test should be used.     

Hierarchical analyses of genetic differentiation in a hybrid zone of Sorex araneus (Insectivora: Soricidae)

Microsatellites are used to unravel the fine-scale genetic structure of a hybrid zone between chromosome races Valais and Cordon of the common shrew ( Sorex araneus ) located in the French Alps. A total of 269 individuals collected between 1992 and 1995 was typed for seven microsatellite loci. A modified version of the classical multiple correspondence analysis is carried out. This analysis clearly shows the dichotomy between the two races. Several approaches are used to study genetic structuring. Gene flow is clearly reduced between these chromosome races and is estimated at one migrant every two generations using R -statistics and one migrant per generation using F -statistics. Hierarchical F - and R -statistics are compared and their efficiency to detect inter- and intraracial patterns of divergence is discussed. Within-race genetic structuring is significant, but remains weak. F _ST displays similar values on both sides of the hybrid zone, although no environmental barriers are found on the Cordon side, whereas the Valais side is divided by several mountain rivers. We introduce the exact G -test to microsatellite data which proved to be a powerful test to detect genetic differentiation within as well as among races. The genetic background of karyotypic hybrids was compared with the genetic background of pure parental forms using a CRT–MCA. Our results indicate that, without knowledge of the karyotypes, we would not have been able to distinguish these hybrids from karyotypically pure samples.     

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     

Testing for group effect in a 2 x k heteroscedastic ANOVA model

It is natural to want to relax the assumption of homoscedasticity and Gaussian error in ANOVA models. For a two-way ANOVA model with 2 x k cells, one can derive tests of main effect for the factor with two levels (referred to as group) without assuming homoscedasticity or Gaussian error. Empirical likelihood can be used to derive testing procedures. An approximate empirical likelihood ratio test (AELRT) is derived for the test of group main effect. To approximate the distributions of the test statistics under the null hypothesis, simulation from the approximate empirical maximum likelihood estimate (AEMLE) restricted by the null hypothesis is used. The homoscedastic ANOVA F -test and a Box-type approximation to the distribution of the heteroscedastic ANOVA F -test are compared to the AELRT in level and power. The AELRT procedure is shown by simulation to have appropriate type I error control (although possibly conservative) when the distribution of the test statistics are approximated by simulation from the constrained AEMLE. The methodology is motivated and illustrated by an analysis of folate levels in the blood among two alcohol intake groups while accounting for gender.     

Self-reported confidence, attitudes and skills in practical procedures among medical students: questionnaire study

The aim of our study was to investigate the influence of gender, loss of academic year(s), confidence and attitudes of students on the clinical experience gained by undergraduate education. The survey was conducted during 2004 and 2005 in a sample of 182 students of the 5th and the 6th year at J.J. Strossmayer University School of Medicine in Osijek. The participants were grouped and matched according to their gender, regular studying, the number of time(s) student has performed certain practical medical procedure and the self-confidence arisen by performing one. Furthermore, participants were grouped and compared due to their own assessment of their own practical and theoretical medical knowledge, courses which provide them the least and oppositely--the most practical medical knowledge and their attitude toward current medical faculty curriculum on clinical courses as well as the possibilities of improving them. Fisher's exact test and chi2-test were used to estimate statistical differences between the groups and the parameters in research, while coefficient of contingency was introduced with the aim of defining their correlation. The results showed statistically significant differences between male students who performed more practical medical procedures than female (p < 0.001), non-repeaters performed medical procedures more often than repeaters (p < 0.001, C = 0.658) while repeaters thought higher of their theoretical knowledge than non-repeaters (p < 0.005). Data analysis showed statistically significant correlation between clinical experience and the level of confidence (C = 0.944). This study confirmed influence of male gender, regular studying, better opinion about one's own practical skills and higher confidence in one's own work on greater number of clinical skills performed during undergraduate education.     

A statistically valid alternative to the TDT

OBJECTIVE: To present an alternative linkage test to the transmission/disequilibrium test (TDT) which is conservative under the null hypothesis and generally more powerful under alternatives. METHODS: The exact distribution of the TDT is examined under both the null hypothesis and relevant alternatives. The TDT is rewritten in an alternate form based on the contributions from each of the three relevant parental mating types. This makes it possible to show that a particular term in the estimate is an exact tie and thus to rewrite the estimate without this term and to replace the multinomial 'variance estimate' of Spielman et al. [Am J Hum Genet 1993;52:506-516] by the binomial variance. RESULTS: The resulting test is shown to be a stratified McNemar test (SMN). The significance level attained by the SMN is shown to be conservative when compared to the asymptotic chi(2) distribution, while the TDT often exceeds the nominal level alpha. Under alternatives, the proposed test is shown to be typically more powerful than the TDT. CONCLUSION: The properties of the TDT as a statistical test have never been fully investigated. The proposed test replaces the heuristically motivated TDT by a formally derived test, which is also computationally simple.     

Haplotype-sharing analysis using Mantel statistics for combined genetic effects

We applied a new approach based on Mantel statistics to analyze the Genetic Analysis Workshop 14 simulated data with prior knowledge of the answers. The method was developed in order to improve the power of a haplotype sharing analysis for gene mapping in complex disease. The new statistic correlates genetic similarity and phenotypic similarity across pairs of haplotypes from case-control studies. The genetic similarity is measured as the shared length between haplotype pairs around a genetic marker. The phenotypic similarity is measured as the mean corrected cross-product based on the respective phenotypes. Cases with phenotype P1 and unrelated controls were drawn from the population of Danacaa. Power to detect main effects was compared to the X2-test for association based on 3-marker haplotypes and a global permutation test for haplotype association to test for main effects. Power to detect gene x gene interaction was compared to unconditional logistic regression. The results suggest that the Mantel statistics might be more powerful than alternative tests.     

Exact sample size needed to detect dependence in 2 x 2 x 2 tables

In the Georgia Centenarian Study (Poon et al., Exceptional Longevity, 2006), centenarian cases and young controls are classified according to three categories (age, ethnic origin, and single nucleotide polymorphisms [SNPs] of candidate longevity genes), where each factor has two possible levels. Here we provide methodologies to determine the minimum sample size needed to detect dependence in 2 x 2 x 2 tables based on Fisher's exact test evaluated exactly or by Markov chain Monte Carlo (MCMC), assuming only the case total L and the control total N are known. While our MCMC method uses serial computing, parallel computing techniques are employed to solve the exact sample size problem. These tools will allow researchers to design efficient sampling strategies and to select informative SNPs. We apply our tools to 2 x 2 x 2 tables obtained from a pilot study of the Georgia Centenarians Study, and the sample size results provided important information for the subsequent major study. A comparison between the results of an exact method and those of a MCMC method showed that the MCMC method studied needed much less computation time on average (10.16 times faster on average for situations examined with S.E. = 2.60), but its sample size results were only valid as a rule for larger sample sizes (in the hundreds).     

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.     

Genomic control for association studies: a semiparametric test to detect excess-haplotype sharing

Individuals who share a disease mutation from a common ancestor often share alleles at genetic markers adjacent to the mutation, even if the common ancestor is remote. The alleles at these adjacent markers, called the haplotype, can be visualized as a string of realizations of random variables, which may be dependent when individuals are related in some fashion. Ideally, for a sample of individuals all having the same (genetic) disease, this dependence-measured as haplotype-sharing-will be greater in the vicinity of disease genes than in other regions of the genome. In this paper we present a semiparametric test for haplotype-sharing. We begin by developing a model assuming that the ancestral haplotype is known and thus the extent of haplotype-sharing from a common ancestor can be determined unambiguously. The amount of overlap at markers far from the disease is treated as a random variable with an unknown distribution F, which we estimate non-parametrically. Overlap of markers surrounding disease genes are modeled as a mixture pF(x - theta) + (1 - p)F(x), in which p is the fraction of subjects with the disease mutation. Testing for a disease gene then amounts to testing whether p = 0. Next we drop the assumption that the ancestral haplotype is known. To detect excess clustering of haplotypes, we measure the pairwise overlap of a set of haplotypes. As in the simpler scenario, this distribution is modeled as a location-shift mixture. To test the hypothesis we construct a score test with a simple limiting distribution.     

A robust measure of HIV-1 population turnover within chronically infected individuals

A simple nonparameteric test for population structure was applied to temporally spaced samples of HIV-1 sequences from the gag-pol region within two chronically infected individuals. The results show that temporal structure can be detected for samples separated by about 22 months or more. The performance of the method, which was originally proposed to detect geographic structure, was tested for temporally spaced samples using neutral coalescent simulations. Simulations showed that the method is robust to variation in samples sizes and mutation rates, to the presence/absence of recombination, and that the power to detect temporal structure is high. By comparing levels of temporal structure in simulations to the levels observed in real data, we estimate the effective intra-individual population size of HIV-1 to be between 10(3) and 10(4) viruses, which is in agreement with some previous estimates. Using this estimate and a simple measure of sequence diversity, we estimate an effective neutral mutation rate of about 5 x 10(-6) per site per generation in the gag-pol region. The definition and interpretation of estimates of such "effective" population parameters are discussed.     

Parameter Estimation and Quantitative Parametric Linkage Analysis with GENEHUNTER-QMOD

Objective: We present a parametric method for linkage analysis of quantitative phenotypes. The method provides a test for linkage as well as an estimate of different phenotype parameters. We have implemented our new method in the program GENEHUNTER-QMOD and evaluated its properties by performing simulations. Methods: The phenotype is modeled as a normally distributed variable, with a separate distribution for each genotype. Parameter estimates are obtained by maximizing the LOD score over the normal distribution parameters with a gradient-based optimization called PGRAD method. Results: The PGRAD method has lower power to detect linkage than the variance components analysis (VCA) in case of a normal distribution and small pedigrees. However, it outperforms the VCA and Haseman-Elston regression for extended pedigrees, nonrandomly ascertained data and non-normally distributed phenotypes. Here, the higher power even goes along with conservativeness, while the VCA has an inflated type I error. Parameter estimation tends to underestimate residual variances but performs better for expectation values of the phenotype distributions. Conclusion: With GENEHUNTER-QMOD, a powerful new tool is provided to explicitly model quantitative phenotypes in the context of linkage analysis. It is freely available at http://www.helmholtz-muenchen.de/genepi/downloads.     

Phylogenetically nested comparisons for testing correlates of species richness: a simulation study of continuous variables

Abstract.— Explaining the uneven distribution of species among lineages is one of the oldest questions in evolution. Proposed correlations between biological traits and species diversity are routinely tested by making comparisons between phylogenetic sister clades. Several recent studies have used nested sister-clade comparisons to test hypotheses linking continuously varying traits, such as body size, with diversity. Evaluating the findings of these studies is complicated because they differ in the index of species richness difference used, the way in which trait differences were treated, and the statistical tests employed. In this paper, we use simulations to compare the performance of four species richness indices, two choices about the branch lengths used to estimate trait values for internal nodes and two statistical tests under a range of models of clade growth and character evolution. All four indices returned appropriate Type I error rates when the assumptions of the method were met and when branch lengths were set proportional to time. Only two of the indices were robust to the different evolutionary models and to different choices of branch lengths and statistical tests. These robust indices had comparable power under one nonnull scenario. Regression through the origin was consistently more powerful than the t -test, and the choice of branch lengths exerts a strong effect on both the validity and power. In the light of our simulations, we re-evaluate the findings of those who have previously used nested comparisons in the context of species richness. We provide a set of simple guidelines to maximize the performance of phylogenetically nested comparisons in tests of putative correlates of species richness.     

Experimental designs and statistical methods for mapping quantitative trait loci underlying triploid endosperm traits without maternal genetic variation

Many endosperm traits are related to grain quality in cereal crops. Endosperm traits are mainly controlled by the endosperm genome but may be affected by the maternal genome. Studies have shown that maternal genotypic variation could greatly influence the estimation of the direct effects of quantitative trait loci (QTLs) underlying endosperm traits. In this paper, we propose methods of interval mapping of endosperm QTLs using seeds of F2 or BC1 (an equal mixture of F1 x P1 and F1 x P2 with F1 as the female parent) derived from a cross between 2 pure lines (P1 x P2). The most significant advantage of our experimental designs is that the maternal effects do not contribute to the genetic variation of endosperm traits and therefore the direct effects of endosperm QTLs can be estimated without the influence of maternal effects. In addition, the experimental designs can greatly reduce environmental variation because a few F1 plants grown in a small block of field will produce sufficient F2 or BC1 seeds for endosperm QTL analysis. Simulation studies show that the methods can efficiently detect endosperm QTLs and unbiasedly estimate their positions and effects. The BC1 design is better than the F2 design.     

An exact test for detecting inconsistency in readers interpretation over time in screening mammograms

Radiologists' interpretation on screening mammograms is measured by accuracy indices such as sensitivity and specificity. The hypothesis that radiologists' interpretation on screening mammograms is constant across time can be tested by measuring overdispersion. However, small sample sizes are problematic for the accuracy of asymptotic approaches. In this article, we propose an exact conditional distribution for testing overdispersion of the binomial assumption that is assumed for the accuracy indices. An exact p -value can be defined from the developed distribution. We also describe an algorithm for computing this exact test. This proposed method is applied to data from a study in reading screening mammograms in a population of US radiologists (Beam et al., 2003). The exact method is compared analytically with a currently available method based on large sample approximations.     

MASKER: improved solvent-excluded molecular surface area estimations using Boolean masks

A fast algorithm for computing the solvent-accessible molecular surface area (SAS) using Boolean masks [Le Grand,S.M. and Merz,K.M.J. (1993). J. Comput. Chem., 14, 349-352) has been modified to estimate the solvent-excluded molecular surface area (SES), including contact, toroidal and re-entrant surface components. Numerical estimates of arc lengths of intersecting atomic SAS are used to estimate the toroidal surface and intersections between those arcs are used to estimate the re-entrant surface area. The new method is compared with an exact analytical method. Boolean molecular surface areas are continuous and pairwise differentiable and should be useful for molecular dynamics simulations, especially as the basis for an implicit solvent model.     

A test for genetic association that incorporates information about deviation from Hardy-Weinberg proportions in cases

For assessment of genetic association between single-nucleotide polymorphisms (SNPs) and disease status, the logistic-regression model or generalized linear model is typically employed. However, testing for deviation from Hardy-Weinberg proportion in a patient group could be another approach for genetic-association studies. The Hardy-Weinberg proportion is one of the most important principles in population genetics. Deviation from Hardy-Weinberg proportion among cases (patients) could provide additional evidence for the association between SNPs and diseases. To develop a more powerful statistical test for genetic-association studies, we combined evidence about deviation from Hardy-Weinberg proportion in case subjects and standard regression approaches that use case and control subjects. In this paper, we propose two approaches for combining such information: the mean-based tail-strength measure and the median-based tail-strength measure. These measures integrate logistic regression and Hardy-Weinberg-proportion tests for the study of the association between a binary disease outcome and an SNP on the basis of case- and control-subject data. For both mean-based and median-based tail-strength measures, we derived exact formulas to compute p values. We also developed an approach for obtaining empirical p values with the use of a resampling procedure. Results from simulation studies and real-disease studies demonstrate that the proposed approach is more powerful than the traditional logistic-regression model. The type I error probabilities of our approach were also well controlled.     

Exact unconditional inference for risk ratio in a correlated 2 x 2 table with structural zero

In this article, we consider small-sample statistical inference for rate ratio (RR) in a correlated 2 x 2 table with a structural zero in one of the off-diagonal cells. Existing Wald's test statistic and logarithmic transformation test statistic will be adopted for this purpose. Hypothesis testing and confidence interval construction based on large-sample theory will be reviewed first. We then propose reliable small-sample exact unconditional procedures for hypothesis testing and confidence interval construction. We present empirical results to evince the better confidence interval performance of our proposed exact unconditional procedures over the traditional large-sample procedures in small-sample designs. Unlike the findings given in Lui (1998, Biometrics 54, 706-711), our empirical studies show that the existing asymptotic procedures may not attain a prespecified confidence level even in moderate sample-size designs (e.g., n = 50). Our exact unconditional procedures on the other hand do not suffer from this problem. Hence, the asymptotic procedures should be applied with caution. We propose two approximate unconditional confidence interval construction methods that outperform the existing asymptotic ones in terms of coverage probability and expected interval width. Also, we empirically demonstrate that the approximate unconditional tests are more powerful than their associated exact unconditional tests. A real data set from a two-step tuberculosis testing study is used to illustrate the methodologies.     

Strain-dependent differences in responses to exercise training in inbred and hybrid mice

The aim of this study was to characterize the response to exercise training in several mouse strains and estimate the genetic contribution to phenotypic variation in the responses to exercise training. Male mice from three inbred strains [C57Bl/6J (BL6), FVB/NJ (FVB), and Balb/cJ (Balb/c)] and three hybrid F(1) strains [CB6F1/J (CB6 = female Balb/c x male BL6), B6F F(1) (female BL6 x male FVB), and FB6 F(1) (female FVB x male BL6)] completed an exercise performance test before and after a 4-wk treadmill running program. Distance was used as the primary estimate of endurance exercise performance. FVB mice showed the greatest response to training, with five- to sevenfold greater increases in distance run compared with BL6 and Balb/c strains. Specifically, BL6, FVB, and Balb/c strains increased distance by 33, 172, and 23%, respectively. A similar pattern of changes across strains was observed for run time (17, 87, and 11%) and work (99, 287, and 57%). As a group, F(1) hybrid mice derived from BL6 and FVB strains showed an intermediate response to training (61%). However, further analysis indicated that training responses in FB6 F(1) mice (80%) were approximately 2.5-fold greater than responses in B6F F(1) mice (33%, P = 0.08). A similar pattern of changes between FB6 and B6F F(1) mice was observed for run time (44.5 and 17%) and work (141 and 59%). These data demonstrate that there are large strain-dependent differences in training responses among inbred mouse strains, suggesting that genetic background contributes significantly to adaptation to exercise. Furthermore, the contrasting responses in B6F and FB6 F(1) strains show that a maternal component strongly influences strain-dependent differences in training responses.     

Statistical analysis of occupancy rates for overdispersed populations by redistribution procedures: Application to the european corn borer EGG masses distribution

Standard statistical analyses of distributions of individuals from contingency tables are generally invalid if the individuals are not distributed independently of each other. In this paper, we discuss a method of testing hypotheses about classification category occupancy rates for overdispersed population or for population whose individuals are distributed by groups rather than lonely. These methods are based on population redistribution simulations and provide valid, exact and powerful tests in situations for which classical methods are not appropriate. Illustrations are given from the European Corn Borer eggs data.