A Program for the WILCOXON Signed Ranks Test

A BASIC-program for the computation of the Wilcoxon test statistic is introduced. It is more simple than the usual ones, because it is based on a more simple kind of computation.     

A novel index of functional connectivity: phase lag based on Wilcoxon signed rank test

Phase synchronization has been an effective measurement of functional connectivity, detecting similar dynamics over time among distinct brain regions. However, traditional phase synchronization-based functional connectivity indices have been proved to have some drawbacks. For example, the phase locking value (PLV) index is sensitive to volume conduction, while the phase lag index (PLI) and the weighted phase lag index (wPLI) are easily affected by noise perturbations. In addition, thresholds need to be applied to these indices to obtain the binary adjacency matrix that determines the connections. However, the selection of the thresholds is generally arbitrary. To address these issues, in this paper we propose a novel index of functional connectivity, named the phase lag based on the Wilcoxon signed-rank test (PLWT). Specifically, it characterizes the functional connectivity based on the phase lag with a weighting procedure to reduce the influence of volume conduction and noise. Besides, it automatically identifies the important connections without relying on thresholds, by taking advantage of the framework of the Wilcoxon signed-rank test. The performance of the proposed PLWT index is evaluated on simulated electroencephalograph (EEG) datasets, as well as on two resting-state EEG datasets. The experimental results on the simulated EEG data show that the PLWT index is robust to volume conduction and noise. Furthermore, the brain functional networks derived by PLWT on the real EEG data exhibit a reasonable scale-free characteristic and high test–retest (TRT) reliability of graph measures. We believe that the proposed PLWT index provides a useful and reliable tool to identify the underlying neural interactions, while effectively diminishing the influence of volume conduction and noise.     

Exact Calculation of Permutational Distributions for Two Dependent Samples I

A simple Fortran subroutine is given for the calculation of permutational distributions. Important special cases are Fisher's randomization test, the Wilcoxon signed ranks test and the sign test. The algorithm works in polynomial time. Thus it can be used even for micro-computers within justifiable time limits.     

An omnibus consistent adaptive percentile modified Wilcoxon rank sum test with applications in gene expression studies

Summary We present an adaptive percentile modified Wilcoxon rank sum test for the two‐sample problem. The test is basically a Wilcoxon rank sum test applied on a fraction of the sample observations, and the fraction is adaptively determined by the sample observations. Most of the theory is developed under a location‐shift model, but we demonstrate that the test is also meaningful for testing against more general alternatives. The test may be particularly useful for the analysis of massive datasets in which quasi‐automatic hypothesis testing is required. We investigate the power characteristics of the new test in a simulation study, and we apply the test to a microarray experiment on colorectal cancer. These empirical studies demonstrate that the new test has good overall power and that it succeeds better in finding differentially expressed genes as compared to other popular tests. We conclude that the new nonparametric test is widely applicable and that its power is comparable to the power of the Baumgartner‐Weiß‐Schindler test.     

EDISON-WMW: Exact Dynamic Programing Solution of the Wilcoxon–Mann–Whitney Test

In many research disciplines, hypothesis tests are applied to evaluate whether findings are statistically significant or could be explained by chance. The Wilcoxon–Mann–Whitney(WMW) test is among the most popular hypothesis tests in medicine and life science to analyze if two groups of samples are equally distributed. This nonparametric statistical homogeneity test is commonly applied in molecular diagnosis. Generally, the solution of the WMW test takes a high combinatorial effort for large sample cohorts containing a significant number of ties. Hence, P value is frequently approximated by a normal distribution. We developed EDISON-WMW, a new approach to calculate the exact permutation of the two-tailed unpaired WMW test without any corrections required and allowing for ties. The method relies on dynamic programing to solve the combinatorial problem of the WMW test efficiently. Beyond a straightforward implementation of the algorithm, we presented different optimization strategies and developed a parallel solution. Using our program,the exact P value for large cohorts containing more than 1000 samples with ties can be calculated within minutes. We demonstrate the performance of this novel approach on randomly-generated data, benchmark it against 13 other commonly-applied approaches and moreover evaluate molecular biomarkers for lung carcinoma and chronic obstructive pulmonary disease(COPD). We foundthat approximated P values were generally higher than the exact solution provided by EDISONWMW. Importantly, the algorithm can also be applied to high-throughput omics datasets, where hundreds or thousands of features are included. To provide easy access to the multi-threaded version of EDISON-WMW, a web-based solution of our algorithm is freely available at http://www.ccb.uni-saarland.de/software/wtest/.     

Truncation of a Two Sample Sequential Wilcoxon Test

To reduce the number of patients needed in a clinical trial, a sequential plan is sometimes used. This paper deals with the truncation of a sequential two sample Wilcoxon test which was originally an open plan (SKOVLUND and WALLØE , 1988). The truncated boundaries have been developed by stochastic simulation. These boundaries form a triangular continuation region. The simulated constants of the boundaries are presented in a table which may be used directly. Boundaries for truncation at 50, 100, 150 and 200 patients are developed. The chosen significance level may be 0.01, 0.025 or 0.05, and the power 0.95. To extend the applicability to other treatment differences than those tabulated, smooth curves are fitted to the simulated points. These make the test applicable to any treatment difference within the simulated range.     

ArraySolver: an algorithm for colour-coded graphical display and Wilcoxon signed-rank statistics for comparing microarray gene expression data

The massive surge in the production of microarray data poses a great challenge for proper analysis and interpretation. In recent years numerous computational tools have been developed to extract meaningful interpretation of microarray gene expression data. However, a convenient tool for two-groups comparison of microarray data is still lacking and users have to rely on commercial statistical packages that might be costly and require special skills, in addition to extra time and effort for transferring data from one platform to other. Various statistical methods, including the t-test, analysis of variance, Pearson test and Mann-Whitney U test, have been reported for comparing microarray data, whereas the utilization of the Wilcoxon signed-rank test, which is an appropriate test for two-groups comparison of gene expression data, has largely been neglected in microarray studies. The aim of this investigation was to build an integrated tool, ArraySolver, for colour-coded graphical display and comparison of gene expression data using the Wilcoxon signed-rank test. The results of software validation showed similar outputs with ArraySolver and SPSS for large datasets. Whereas the former program appeared to be more accurate for 25 or fewer pairs (n 相似文献   

A New Testing Approach for Comparing the Overall Homogeneity of Survival Curves

The assessment of overall homogeneity of time‐to‐event curves is a key element in survival analysis. The currently commonly used methods, e.g., log‐rank and Wilcoxon tests, may have a significant loss of statistical testing power under certain circumstances. In this paper a new statistical testing approach is developed to compare the overall homogeneity of survival curves. The proposed new method has greater power than the commonly used tests to detect overall differences between crossing survival curves. The small‐sample performance of the new test is investigated under a variety of situations by means of Monte Carlo simulations. Furthermore, the applicability of the proposed testing approach is illustrated by a real data example from a kidney dialysis trial. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

ArrayVigil: a methodology for statistical comparison of gene signatures using segregated-one-tailed (SOT) Wilcoxon's signed-rank test

Due to versatile diagnostic and prognostic fidelity molecular signatures or fingerprints are anticipated as the most powerful tools for cancer management in the near future. Notwithstanding the experimental advancements in microarray technology, methods for analyzing either whole arrays or gene signatures have not been firmly established. Recently, an algorithm, ArraySolver has been reported by Khan for two-group comparison of microarray gene expression data using two-tailed Wilcoxon signed-rank test. Most of the molecular signatures are composed of two sets of genes (hybrid signatures) wherein up-regulation of one set and down-regulation of the other set collectively define the purpose of a gene signature. Since the direction of a selected gene's expression (positive or negative) with respect to a particular disease condition is known, application of one-tailed statistics could be a more relevant choice. A novel method, ArrayVigil, is described for comparing hybrid signatures using segregated-one-tailed (SOT) Wilcoxon signed-rank test and the results compared with integrated-two-tailed (ITT) procedures (SPSS and ArraySolver). ArrayVigil resulted in lower P values than those obtained from ITT statistics while comparing real data from four signatures.     

Signed-Rank Tests in the Presence of Ties (with extended tables)

We begin with a review of the areas of application of the signed-rank tests (SRTs) and we conclude that the results are exact only if no ties of non-null differences exist. In order to apply the SRTs according to WILCOXON and according to PRATT also in the presence of ties, by assigning midranks, we derive their null distributions. As special cases the null distributions for the problem without ties are obtained. In order to save the practising statistician the time-consuming calculations of the distribution functions, we compute tables of critical values (for reasons of volume they will be published as part of the reprints only). For N₀ = 0 (1) 5 null differences and M = = 1(1) 10 non-null differences the critical values of all distributions with all possible tie vectors are calculated. Instructions are provided and an example serves to illustrate the use of the table. The extension of the tables are obtained by means of counting formulas given in the text. Approximations are provided in order to make the application of tests possible for larger samples as well. It is shown that the approximation of the null distribution in the presence of ties by the null distributions under the assumption of no ties in some cases overstates and sometimes understates the exact rejection probability. For N₀ = 0 (1) 10 and M = 1 (1) 10 all distributions with all possible tie vectors for the SRTs with WILCOXON and PRATT ranking are examined with respect to the lattice type of the test statistic. The result is given in table 6. It is evident that the portion of PRATT -distributions with lattice character decreases as the number of null differences increases. Continuity corrections are obtained for the asymptotic normal distribution which take into account the lattice character of the distribution of the test statistic.     

Exact log-rank tests for unequal follow-up

The asymptotic log-rank and generalized Wilcoxon tests are the standard procedures for comparing samples of possibly censored survival times. For comparison of samples of very different sizes, an exact test is available that is based on a complete permutation of log-rank or Wilcoxon scores. While the asymptotic tests do not keep their nominal sizes if sample sizes differ substantially, the exact complete permutation test requires equal follow-up of the samples. Therefore, we have developed and present two new exact tests also suitable for unequal follow-up. The first of these is an exact analogue of the asymptotic log-rank test and conditions on observed risk sets, whereas the second approach permutes survival times while conditioning on the realized follow-up in each group. In an empirical study, we compare the new procedures with the asymptotic log-rank test, the exact complete permutation test, and an earlier proposed approach that equalizes the follow-up distributions using artificial censoring. Results confirm highly satisfactory performance of the exact procedure conditioning on realized follow-up, particularly in case of unequal follow-up. The advantage of this test over other options of analysis is finally exemplified in the analysis of a breast cancer study.     

Robustness of Statistical Tests in the Two-Sample Location Problem

A simulation study was performed to compare three statistical tests with respect to their performances in the two-sample location problem for contaminated normal distributions. The three tests were: the t-test, the rank-transformed t-test, and the Wilcoxon rank-sum test. The results showed the t-test to be inferior to the other two tests.     

Power Estimation for Two-Sample Tests Using Importance and Antithetic Resampling

Collings and Hamilton (1988), described a uniform bootstrap method that is applied on observed or pilot data in order to approximate the power of the two-sample Wilcoxon test for location shift alternatives. In this paper we demonstrate how importance and antithetic resampling can be used to substantially reduce the amount of computation needed to approximate the power of the two-sample tests for location shift and scale alternatives. Importance and antithetic bootstrap resampling methods are applied to simulated data of different sample sizes from a variety of distributions as well as to data from the Iowa 65+ Rural Health Study. Also, a suggestion is given for using a combination of importance and antithetic resampling for approximating the power of two-sample tests.     

Power and sample size estimation for the clustered wilcoxon test

The Wilcoxon rank sum test is widely used for two-group comparisons of nonnormal data. An assumption of this test is independence of sampling units both within and between groups, which will be violated in the clustered data setting such as in ophthalmological clinical trials, where the unit of randomization is the subject, but the unit of analysis is the individual eye. For this purpose, we have proposed the clustered Wilcoxon test to account for clustering among multiple subunits within the same cluster (Rosner, Glynn, and Lee, 2003, Biometrics 59, 1089-1098; 2006, Biometrics 62, 1251-1259). However, power estimation is needed to plan studies that use this analytic approach. We have recently published methods for estimating power and sample size for the ordinary Wilcoxon rank sum test (Rosner and Glynn, 2009, Biometrics 65, 188-197). In this article we present extensions of this approach to estimate power for the clustered Wilcoxon test. Simulation studies show a good agreement between estimated and empirical power. These methods are illustrated with examples from randomized trials in ophthalmology. Enhanced power is achieved with use of the subunit as the unit of analysis instead of the cluster using the ordinary Wilcoxon rank sum test.     

基于转录调控模体的人不同组织基因差异性的统计分析

转录调控是基因表达调控的主要过程,而转录调控模体使用的差异性可能是导致基因组织特异性的因素之一.本文提出一种不同组织基因调控差异性的统计分析方法,首先结合泊松分布和主成分分析提取基因启动子中过表达模体作为潜在的转录因子结合位点.基于这些位点通过Wilcoxon秩和检验获得不同组织基因结构的差异性.再用超几何分布确定出现次数显著的模体作为组织基因的特有模体,并分析特有模体的碱基特征及在启动子序列中的位置分布.将特有模体与TRANSFAC数据库进行对照,得到潜在的调控组织特异性基因的转录因子结合位点.以人管家基因及30个组织特异性基因为分析对象,得到不同组织调控模体使用的差异性信息.     

Generalized Wilcoxon Rank Tests and Sensitivity Curves

In this paper we use the sensitivity curves of TUKEY (1977) and the change of decision point (cdp) (a modified version of the breakdown point of YLVISAKER, 1977), supplemented by simulation studies to acquire knowledge about sensitivity in generalized Wilcoxon rank test statistics. Sensitivity depends on balanced or unbalanced sample size cases, censoring, combinations of failure distributions and sources of errors in the data. It is important to consider the quality of the data, and the results show that cdp and some properties of the sensitivity curves may serve as a hint when selecting a test statistic and when making a decision for a given test statistic.     

The Determination of Sample Sizes in the Comparison of Two Multinomial Proportions from Ordered Categories

We consider sample size determination for ordered categorical data when the alternative assumption is the proportional odds model. In this paper the sample size formula proposed by Whitehead (Statistics in Medicine, 12 , 2257–2271, 1993) is compared with the methods based on exact and asymptotic linear rank tests with Wilcoxon and trend scores. We show that Whitehead's formula, which is based on a normal approximation, works well when the sample size is moderate to large but recommend the exact method with Wilcoxon scores for small sample sizes. The consequences of misspecification in models are also investigated.     

A Nonparametric Test for Random Dropouts

The problem of dropout is a common one in longitudinal studies. One usually assumes for the analysis that dropout is at random. There are some tests to investigate this assumption. But these tests depend on normally distributed data or lack power, cf. Listing and Schlittgen (1998). We here propose an overall test which combines several Wilcoxon rank sum tests. The alternative hypothesis states that there is a tendency for larger (smaller) values of the target variable the last time the probands show up. The test is applicable with many ties also. It proves to perform well, compared to the test developed for normally distributed data, as well as to a test for completely missing at random which is proposed by Little (1988). An application to real data is given too.     

A new location-scale test based on a combination of the ideas of Levene and Lepage

Lepage's test combines the Wilcoxon rank-sum and the Ansari-Bradley statistics. We propose to replace the latter statistic by a Wilcoxon rank-sum calculated after Levene's transformation. We use the medians for this transformation, i.e. absolute deviations from sample medians are calculated. The new location-scale test can be carried out as a permutation test based on permutations of the original observations, the Levene transformation has to be applied for each permutation in an intermediate step to calculate the test statistic. Simulations indicate that the new test can be more powerful than an O'Brien-type test and Lepage's test, the latter is the standard nonparametric location-scale test. The new test is illustrated using real data about colony sizes of yellow-eyed penguins and an SAS program to perform the test is freely available.     

A signed-rank test for clustered data

Summary .   We consider the problem of comparing two outcome measures when the pairs are clustered. Using the general principle of within-cluster resampling, we obtain a novel signed-rank test for clustered paired data. We show by a simple informative cluster size simulation model that only our test maintains the correct size under a null hypothesis of marginal symmetry compared to four other existing signed rank tests; further, our test has adequate power when cluster size is noninformative. In general, cluster size is informative if the distribution of pair-wise differences within a cluster depends on the cluster size. An application of our method to testing radiation toxicity trend is presented.