A Robust Genome‐Wide Scan Statistic of the Wellcome Trust Case–Control Consortium

Summary In genome‐wide association (GWA) studies, test statistics that are efficient and robust across various genetic models are preferable, particularly for studying multiple diseases in the Wellcome Trust Case–Control Consortium ( WTCCC, 2007 , Nature 447 , 661–678). A new test statistic, the minimum of the p‐values of the trend test and Pearson's test, was considered by the WTCCC. It is referred to here as MIN2. Because the minimum of two p‐values is no longer a valid p‐value itself, the WTCCC only used it to rank single nucleotide polymorphisms (SNPs) but did not report the p‐values of the associated SNPs when MIN2 was used for ranking. Given its importance in practice, we derive the asymptotic null distribution of MIN2, study some of its analytical properties related to GWA studies, and compare it with existing methods (the trend test, Pearson's test, MAX3, and the constrained likelihood ratio test [CLRT]) by simulations across a wide range of possible genetic models: the recessive (REC), additive (ADD), multiplicative (MUL), dominant (DOM), and overdominant models. The results show that MAX3 and CLRT have greater efficiency robustness than other tests when the REC, ADD/MUL, and DOM models are possible, whereas Pearson's test and MIN2 have greater efficiency robustness if the possible genetic models also include the overdominant model. We conclude that robust tests (MAX3, MIN2, CLRT, and Pearson's test) are preferable to a single trend test for initial GWA studies. The four robust tests are applied to more than 100 SNPs associated with 11 common diseases identified by the two WTCCC GWA studies.     

Tests for Homogeneity of Variances Using Robust Weighted Likelihood Estimates

The classical normal-theory tests for testing the null hypothesis of common variance and the classical estimates of scale have long been known to be quite nonrobust to even mild deviations from normality assumptions for moderate sample sizes. Levene (1960) suggested a one-way ANOVA type statistic as a robust test. Brown and Forsythe (1974) considered a modified version of Levene's test by replacing the sample means with sample medians as estimates of population locations, and their test is computationally the simplest among the three tests recommended by Conover , Johnson , and Johnson (1981) in terms of robustness and power. In this paper a new robust and powerful test for homogeneity of variances is proposed based on a modification of Levene's test using the weighted likelihood estimates (Markatou , Basu , and Lindsay , 1996) of the population means. For two and three populations the proposed test using the Hellinger distance based weighted likelihood estimates is observed to achieve better empirical level and power than Brown-Forsythe's test in symmetric distributions having a thicker tail than the normal, and higher empirical power in skew distributions under the use of F distribution critical values.     

A Note Concerning “Some Nonparametric Tests of Predicted Order”

SARRIS and WILKENING (1977) have recently proposed some non-parametric trend tests, which they view as extensions of MOSTELLER 'S test of predicted order. The present paper notes some errors in SARRIS and WILKENING 's determination of significance levels for these tests, and describes how these errors may be corrected.     

TESTING FOR PHYLOGENETIC SIGNAL IN BIOLOGICAL TRAITS: THE UBIQUITY OF CROSS‐PRODUCT STATISTICS

To evaluate rates of evolution, to establish tests of correlation between two traits, or to investigate to what degree the phylogeny of a species assemblage is predictive of a trait value so‐called tests for phylogenetic signal are used. Being based on different approaches, these tests are generally thought to possess quite different statistical performances. In this article, we show that the Blomberg et al. K and K*, the Abouheif index, the Moran's I, and the Mantel correlation are all based on a cross‐product statistic, and are thus all related to each other when they are associated to a permutation test of phylogenetic signal. What changes is only the way phylogenetic and trait similarities (or dissimilarities) among the tips of a phylogeny are computed. The definitions of the phylogenetic and trait‐based (dis)similarities among tips thus determines the performance of the tests. We shortly discuss the biological and statistical consequences (in terms of power and type I error of the tests) of the observed relatedness among the statistics that allow tests for phylogenetic signal. Blomberg et al. K* statistic appears as one on the most efficient approaches to test for phylogenetic signal. When branch lengths are not available or not accurate, Abouheif's C_mean statistic is a powerful alternative to K*.     

An Adaptive Location‐Scale Test

Increasing locations are often accompanied by an increase in variability. In this case apparent heteroscedasticity can indicate that there are treatment effects and it is appropriate to consider an alternative involving differences in location as well as in scale. As a location‐scale test the sum of a location and a scale test statistic can be used. However, the power can be raised through weighting the sum. In order to select values for this weighting an adaptive design with an interim analysis is proposed: The data of the first stage are used to calculate the weights and with the second stage's data a weighted location‐scale test is carried out. The p‐values of the two stages are combined through Fisher's combination test. With a Lepage‐type location‐scale test it is illustrated that the resultant adaptive test can be more powerful than the ‘optimum’ test with no interim analysis. The principle to calculate weights, which cannot be reasonably chosen a priori, with the data of the first stage may be useful for other tests which utilize weighted statistics, too. Furthermore, the proposed test is illustrated with an example from experimental ecology.     

A Robust Test for Multi-Sample Location Problems with Unequal Group Variances and Nonnormality

A robust test (to be referred to as M^* test) is proposed for testing equality of several group means without assuming normality and equality of variances. This test statistic is obtained by combining Tiku's MML robust procedure with the James statistic. Monte Carlo simulation studies indicate that the M^* test is more powerful than the Welch test, the James test, and the tests based on Huber's M-estimators over a wide range of nonnormal universes. It is also more powerful than the Brown and Forsythe test under most of nonnormal distributions and has substantially the same power as the Brown and Forsythe test under normal distribution. Comparing with Tan-Tabatabai test, M^* is almost as powerful as Tan-Tabatabai test.     

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.     

A Comparison of Robust Estimators in the Presence of Arbitrarily Right Censored Data

Robust estimation of a location parameter is considered when the data from an unknown symmetric population are subject to arbitrary right-censorship. Comparisons are made between various M-estimators, several L-estimators (trimmed means), and the Kaplan-Meier median. Ten sampling distributions, two uniform censoring distributions, and three sample sizes are examined. A Cauchy censoring distribution is also considered when the sample size is equal to twenty for each of the ten sampling distributions. Performance is based on the estimated mean square error.     

塔克尔莫乎尔沙漠银沙槐群落主要植物的种间关系

根据野外调查的样地数据资料,运用2×2列联表的Fisher精确检验、Pearson相关系数和Spearman秩相关系数检验,并结合DCA排序,分析塔克尔莫乎尔沙漠银沙槐群落主要植物种群之间的种间关系。结果表明:Fisher精确检验有3个种对呈显著正关联,3个种对呈显著负关联;Pearson相关系数检验有1个种对呈显著正相关,12个种对呈显著负相关;Spearman秩相关系数检验,有14个种对存在显著相关,其中负相关13对,占所有种对数的14.29%,正相关1对,占所有种对数的1.1%。银沙槐群落种间关系相对松散,反映出植物群落随着环境变化其物种组成也发生相应变化,正向着相对稳定独立的无关联方向发展。     

Inverse Probability of Censoring Weighted Estimates of Kendall's τ for Gap Time Analyses

Summary In life history studies, interest often lies in the analysis of the interevent, or gap times and the association between event times. Gap time analyses are challenging however, even when the length of follow‐up is determined independently of the event process, because associations between gap times induce dependent censoring for second and subsequent gap times. This article discusses nonparametric estimation of the association between consecutive gap times based on Kendall's τ in the presence of this type of dependent censoring. A nonparametric estimator that uses inverse probability of censoring weights is provided. Estimates of conditional gap time distributions can be obtained following specification of a particular copula function. Simulation studies show the estimator performs well and compares favorably with an alternative estimator. Generalizations to a piecewise constant Clayton copula are given. Several simulation studies and illustrations with real data sets are also provided.     

The Statistical Significance of Mutation Frequencies

TRAUT has proposed a method for determining the statistical significance between a mutation frequency observed after treatment with a potential mutagen and a control frequency. The method is developed further by us for an easier practical use. A table for the minimal sample size in the region of interest for the sex-linked recessive lethal test in Drosophila melanogaster has been calculated. TRACT'S test is compared with FISHER'S exact test and the KASTENBAUM -BOWMAN test. TRAUT'S test turns out to be more sensitive than the other tests. This observation strongly supports TRAUT'S conclusion that his test should only be used if very accurate determinations of the spontaneous frequencies are available.     

Species co‐occurrence analysis: pairwise versus matrix‐level approaches

Veech (2013, Global Ecology and Biogeography, 22 , 252–260) introduced a formula to calculate the probability of two species co‐occurring in various sites under the assumption of statistical independence between the two distributional patterns. He presented his model as a new procedure, a ‘pairwise approach’, different from analyses of whole presence–absence matrices to examine patterns of co‐occurrence. Here I show that: (1) Veech's method is identical to Fisher's exact test, a standard procedure for measuring the statistical association between two discrete variables; (2) in a broad sense, the pairwise approach is very similar to early analyses of spatial association, such as the one advanced by Forbes in 1907; (3) implicit in Veech's formula is a sampling scheme that is indistinguishable from well‐known matrix‐level null models that randomize the distribution of species among equiprobable sites; (4) pairwise co‐occurrence patterns can be analysed using any matrix‐level null model, so pairwise comparisons are not limited to using Veech's formula. The methodological distinction that Veech proposed between pairwise and matrix‐level approaches does not in fact exist, although the conceptual distinction between the two approaches is still a debated topic.     

chifish: a computer program testing for genetic heterogeneity at multiple loci using chi‐square and Fisher's exact test

chifish is a 32‐bit Windows/DOS program evaluating divergence at multiple gene loci. It tests the hypothesis of no difference at any locus both by means of Pearson's traditional chi‐square and by using Fisher's method of combining P values obtained by Fisher's exact test. Input data are read from a file formatted for genepop . Commonly used population genetics software do not perform chi‐square tests, and the simultaneous application of both techniques aids in situations where poor power of the ‘exact approach’ may prevent detection of true differentiation (e.g. few populations and few alleles per locus).     

A remark on a paper by misra

MISRA (1978) sets confidence intervals for a double linear compound of multivariate normal regression coefficients by using ROY'S maximum root test criterion. The exact test statistic to be used is STUDENT'S t. The t statistic gives narrower confidence bounds than those given by ROY's maximum root statistic. A result given by MORRISON (1975, p. 18, equation 10) for profile analysis is also obtained by using the STUDENT'S t test.     

Of rowing boats,ocean liners and tests of the anova homogeneity of variance assumption

Abstract One of the assumptions of analysis of variance (anova ) is that the variances of the groups being compared are approximately equal. This assumption is routinely checked before doing an analysis, although some workers consider anova robust and do not bother and others avoid parametric procedures entirely. Two of the more commonly used heterogeneity tests are Bartlett's and Cochran's, although, as for most of these tests, they may well be more sensitive to violations of the anova assumptions than is anova itself. Simulations were used to examine how well these two tests protected anova against the problems created by variance heterogeneity. Although Cochran's test performed a little better than Bartlett's, both tests performed poorly, frequently disallowing perfectly valid analyses. Recommendations are made about how to proceed, given these results.     

The generalized Fisher's combination and accurate p-value calculation under dependence

Combining dependent tests of significance has broad applications but the related p-value calculation is challenging. For Fisher's combination test, current p-value calculation methods (eg, Brown's approximation) tend to inflate the type I error rate when the desired significance level is substantially less than 0.05. The problem could lead to significant false discoveries in big data analyses. This paper provides two main contributions. First, it presents a general family of Fisher type statistics, referred to as the GFisher, which covers many classic statistics, such as Fisher's combination, Good's statistic, Lancaster's statistic, weighted Z-score combination, and so forth. The GFisher allows a flexible weighting scheme, as well as an omnibus procedure that automatically adapts proper weights and the statistic-defining parameters to a given data. Second, the paper presents several new p-value calculation methods based on two novel ideas: moment-ratio matching and joint-distribution surrogating. Systematic simulations show that the new calculation methods are more accurate under multivariate Gaussian, and more robust under the generalized linear model and the multivariate t-distribution. The applications of the GFisher and the new p-value calculation methods are demonstrated by a gene-based single nucleotide polymorphism (SNP)-set association study. Relevant computation has been implemented to an R package GFisher available on the Comprehensive R Archive Network.     

An Approximate Solution to the Behrens-Fisher Problem

An approximate and practical solution is proposed for the Behrens-Fisher problem. This solution is compared to the solutions considered by Mehta and Srinivasan (1970) and Welch's (1937) approximate t-test in terms of the stability of the size and magnitude of the power. It is shown that the stability of the size of the new test is better than that of Welch's t when at least one of the sample sizes is small. When the sample sizes are moderately large or large the sizes and powers of all the recommended tests are almost the same.     

Tables and Applications of the Bonferroni t-Statistics: A Revision of Dunn's Simultaneous t-Tests

Tables for Bonferroni-adjusted significance levels of Student's t are provided for r = 3(1)30(5)50 etc. and alphas of 0.05, 0.01 and 0.001. The tables are suggested for various applications, for example in replacing analysis of variance of k samples by r simultaneous t tests. Use of the tables is shown by numerical examples. The adjustment of alpha for improving the efficiency of testing is made by WILKINSON for orthogonal comparisons and by HOLM for nonorthogonal comparisons.     

A Comparison of the Three Conditional Exact Tests in Two‐way Contingency Tables Using the Unconditional Exact Power

The conditional exact tests of homogeneity of two binomial proportions are often used in small samples, because the exact tests guarantee to keep the size under the nominal level. The Fisher's exact test, the exact chi‐squared test and the exact likelihood ratio test are popular and can be implemented in software StatXact. In this paper we investigate which test is the best in small samples in terms of the unconditional exact power. In equal sample cases it is proved that the three tests produce the same unconditional exact power. A symmetry of the unconditional exact power is also found. In unequal sample cases the unconditional exact powers of the three tests are computed and compared. In most cases the Fisher's exact test turns out to be best, but we characterize some cases in which the exact likelihood ratio test has the highest unconditional exact power. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Randomization Modelling of the Crossover Experiment for Clinical Trials

The two-period cross-over experiment for clinical trials has been examined by several writers following a Gaussian linear model approach. Some authors have expressed interest in the “derivation of the finite permutation model” and have pointed out that the randomization approach to modeling the two-period cross-over design “would highlight the importance of randomizing the subjects to the two groups as a basis for inference”. However, in the literature, there is no development of the randomization approach to this important design. In this paper, after a statement of the experimental design and formulation of the observation random variables of the finite population, two additive randomization models—one with residual effects, the other without—which are the analogues of Grizzle's Gaussian models, are derived. Statistical inference is developed for these randomization models and the results are compared with those of the corresponding Gaussian models. Also, exact inference based upon Fischer's approach is presented.