A Comparison of Algorithms for the Computation of the Mann-Whitney Test

An algorithm is proposed for the computation of the Mann-Whitney Test which only requires the separate ordering of the individual samples and is therefore more suitable than the usual one. It is also more suitable as to computation time than the algorithm described by KUMMER , 1981, if the two sample sizes are of the same magnitude.     

A Modified MANN-WHITNEY Test with Improved Asymptotic Relative Efficiency

A slight modification of the MANN -WHITNEY test is introduced here. When the underlying distribution is normal the ARE of the modified test with respect to t-test is 99.22% (MANN -WHITNEY test: 95.45%). The lower bound of the ARE of the modified test is 92.50% for all distributions with a continuous density function and a finite variance (MANN -WHITNEY test: 86.4%).     

Power and sample size estimation for the Wilcoxon rank sum test with application to comparisons of C statistics from alternative prediction models

Summary .  The Wilcoxon Mann-Whitney (WMW) U test is commonly used in nonparametric two-group comparisons when the normality of the underlying distribution is questionable. There has been some previous work on estimating power based on this procedure ( Lehmann, 1998 , Nonparametrics ). In this article, we present an approach for estimating type II error, which is applicable to any continuous distribution, and also extend the approach to handle grouped continuous data allowing for ties. We apply these results to obtaining standard errors of the area under the receiver operating characteristic curve (AUROC) for risk-prediction rules under H ₁ and for comparing AUROC between competing risk prediction rules applied to the same data set. These results are based on SAS -callable functions to evaluate the bivariate normal integral and are thus easily implemented with standard software.     

A Unifying,Analytic Model for Certain Linear Rank Tests. Part II: Exact Distribution Algorithms

On the basic of the analytic properties of the unifying model developed in the earlier Part I of this presentation, exact distribution algorithms are developed for the Wilcoxon 2-sample sum-of-rank test, the Freund-Ansari scale test, and the Wilcoxon signed-rank test. For the signed-rank test, provisions are made for zero-differences.     

Exact Calculation of Permutational Distributions for Two Independent Samples

A simple Fortran Subroutine is given for the exact calculation of permutational distributions for two independent samples. Important special cases are the Fisher-Pitman randomization test, the Mann-Whitney U-test, the Mantel-Haenszel-test and the exact test for 2×k contingency tables. The algorithm needs polynomial time and is of use even for personal- or microcomputers.     

Nonparametric Location Tests Based on the Empirical Distribution

We present a unified approach to the nonparametric comparison of locations in the one- and two sample case using the empirical distribution functions. This approach reveals how to define a simple test statistic for this comparison in the case of large sample sizes which is equivalent to the known Wilcoxon-Mann-Whitney statistics and is based on the grouping of the data. In the particular case of equal group sizes we find an extremely simple form for this location test.     

A Note on the Mann-Whitney Statistic for Lehmann Alternatives

For the case of the LEHMANN alternatives the paper presents some new facts on the MANN -WHITNEY statistic and, in particular, its variance V(p, m, n), where p = P(x_i<y_i). Explicit formulas for U and V are used to prove, among other things, the following propositions: For any m, n, V is a one-hump function of p, and the hump always lies in the interval (1/2(3 - √5), 1/2(√5 - 1)). If no restrictions are imposed on p the boundaries of this interval are sharp. Given s = m + n, V(1/2, s/2,s/2) is maximal among all values V(p, m, n). The formulas allow, moreover, the improvement of the known bounds for the variance of p? = U/mn.     

水文变异下的黄河流域生态流量

使用t检验和Mann-Whitney U检验对黄河干流7个水文站月均流量进行水文变异分析,探讨了水文变异成因,在此基础上,确定变异前各月月均流量序列最适概率分布函数,将概率密度最大的月平均流量定义为河道内生态流量。经与Tennant法、最小月平均流量发和逐月径流法比较,考虑水文变异的河道内生态流量计算方法是可行、合理的。水文变异后,黄河干流7水文站月均流量普遍减少,月均流量满足河道内生态流量的频率降低。研究结果表明,人类活动是黄河生态系统水环境恶化的重要原因。在流域生态管理中,确保变异后生态流量满足频率与变异前相当。研究对于理解在当前气候变化与人类活动双重影响下,干旱半干旱区流域水资源科学管理具有一定理论与现实意义。     

Evaluation of Noether's Method of Sample Size Determination for the Wilcoxon-Mann-Whitney Test

NOETHER (1987) proposed a method of sample size determination for the Wilcoxon-Mann-Whitney test. To obtain a sample size formula, he restricted himself to alternatives that differ only slightly from the null hypothesis, so that the unknown variance o² of the Mann-Whitney statistic can be approximated by the known variance under the null hypothesis which depends only on n. This fact is frequently forgotten in statistical practice. In this paper, we compare Noether's large sample solution against an alternative approach based on upper bounds of σ² which is valid for any alternatives. This comparison shows that Noether's approximation is sufficiently reliable with small and large deviations from the null hypothesis.     

Partial AUC estimation and regression

Accurate diagnosis of disease is a critical part of health care. New diagnostic and screening tests must be evaluated based on their abilities to discriminate diseased from nondiseased states. The partial area under the receiver operating characteristic (ROC) curve is a measure of diagnostic test accuracy. We present an interpretation of the partial area under the curve (AUC), which gives rise to a nonparametric estimator. This estimator is more robust than existing estimators, which make parametric assumptions. We show that the robustness is gained with only a moderate loss in efficiency. We describe a regression modeling framework for making inference about covariate effects on the partial AUC. Such models can refine knowledge about test accuracy. Model parameters can be estimated using binary regression methods. We use the regression framework to compare two prostate-specific antigen biomarkers and to evaluate the dependence of biomarker accuracy on the time prior to clinical diagnosis of prostate cancer.     

A Distribution-Free Two-Sample Equivalence Test Allowing for Tied Observations

A new testing procedure is derived which enables to assess the equivalence of two arbitrary noncontinuous distribution functions from which unrelated samples are taken as the data to be analyzed. The equivalence region is defined to consist of all pairs (F,G) of distribution functions such that for independent X ∼ F, Y ∼ G the conditional probability of {X > Y} given {X ≠ Y} lies in some short interval around 1/2. The test rejects the null hypothesis of nonequivalence if and only if the standardized distance between the U-statistics estimator of P[X > Y ∣ X ≠ Y] and the center of the equivalence interval (1/2 — ε₁, 1/2 + ε₂) does not exceed a critical upper bound which has to be computed as the α-quantile of a χ²-distribution with one degree of freedom and a random noncentrality parameter proportional to the squared length of that interval. The test is shown to maintain the asymptotic significance level under very weak regularity conditions. Results of an extensive simulation study suggest that its level properties are very satisfactory in small samples as well. The power turns out to be inversely related to the rate P[X = Y] of ties between observations from different samples.     

A New Approach to Equivalence Assessment in Standard Comparative Bioavailability Trials by Means of the Mann-Whitney Statistic

By a suitable transformation of the pairs of observations obtained in the successive periods of the trial, bioequivalence assessment in a standard comparative bioavailability study reduces to testing for equivalence of two continuous distributions from which unrelated samples are available. Let the two distribution functions be given by F(x) = P[X ≤ x], G(y) = P[Y ≤ y] with (X, Y) denoting an independent pair of real-valued random variables. An intuitively appealing way of putting the notion of equivalence of F and G into nonparametric terms can be based on the distance of the functional P[X > Y] from the value it takes if F and G coincide. This leads to the problem of testing the null hypothesis H_o P[X > Y] ≤ 1/2 - ε₁ or P[X > Y] ≥ 1/2 + ε₂ versus H₁ : 1/2 ? ε₁ < P[X > Y] < 1/2 + ∈₂, with sufficiently small ε₁, ε₂ ∈ (0, 1/2). The testing procedure we derive for (₀, H₁) and propose to term Mann-Whitney test for equivalence, consists of carrying out in terms of the U-statistics estimator of P[X > Y] the uniformly most powerful level a test for an interval hypothesis about the mean of a Gaussian distribution with fixed variance. The test is shown to be asymptotically distribution-free with respect to the significance level. In addition, results of an extensive simulation study are presented which suggest that the new test controls the level even with sample sizes as small as 10. For normally distributed data, the loss in power as against the optimal parametric procedure is found to be almost as small as in comparisons between the Mann-Whitney and the t-statistic in the conventional one or two-sided setting, provided the power of the parametric test does not fall short of 80%.     

Is the Exact Test Better than the Asymptotic Test for Testing Marginal Homogeneity in 2 × 2 Tables?

McNemar test is commonly used to test for the marginal homogeneity in 2 × 2 contingency tables. McNemar test is an asymptotic test based either on standard normal distribution or on the chi‐square distribution. When the total sample size is small, an exact version of McNemar test is available based on the binomial probabilities. The example in the paper came from a clinical study to investigate the effect of epidermal growth factor for children who had microvillus inclusion diseases. There were only six observations available. The test results differ between the exact test and the asymptotic test. It is a common belief that with this small sample size the exact test be used. However, we claim that McNemar test performs better than the exact test even when the sample size is small. In order to investigate the performances of McNemar test and the exact test, we identify the parameters that affect the test results and then perform sensitivity analysis. In addition, through Monte Carlo simulation studies we compare the empirical sizes and powers of these tests as well as other asymptotic tests such as Wald test and the likelihood ratio test.     

Adjusting O'Brien's test to control type I error for the generalized nonparametric Behrens-Fisher problem

O'Brien (1984, Biometrics 40, 1079-1087) introduced a simple nonparametric test procedure for testing whether multiple outcomes in one treatment group have consistently larger values than outcomes in the other treatment group. We first explore the theoretical properties of O'Brien's test. We then extend it to the general nonparametric Behrens-Fisher hypothesis problem when no assumption is made regarding the shape of the distributions. We provide conditions when O'Brien's test controls its error probability asymptotically and when it fails. We also provide adjusted tests when the conditions do not hold. Throughout this article, we do not assume that all outcomes are continuous. Simulations are performed to compare the adjusted tests to O'Brien's test. The difference is also illustrated using data from a Parkinson's disease clinical trial.