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.     

Formulas for the Computation of the WILCOXON Test and other Rank Statistics

A simple program for the computation of the WILCOXON test is introduced, which is suitable for implementation on a programmable desk computer. The allocation of ranks R_i is not necessary (see BERCHTOLD, 1979). In an analogous manner any ranks respectively rank sums also can be calculated, so that any test criterion or statistical estimate called rank statistic, may be completely constituted by computer programs without explicitely ranging of values.     

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%).     

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.     

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%.     

A Generalized Intrinsic Rank Test for k Independent Samples

The original intrinsic rank test is generalized in that the sizes of the k samples may now be arbitrary, and the number of intrinsic rank intervals need not equal the number of samples. Furthermore, the size of these intervals can be made variable, subject only to relatively mild constraints. These generalizations permit the formulation and testing of more specific hypotheses concerning the commonality of the sample distributions. A generalized intrinsic rank function is used to transform the usual ordinal ranks, obtained from the combined samples, into intrinsic ranks. Original sample identity and intrinsic ranks are then cross-tabulated and evaluated as 2-way contingency table.     

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.     

An Algorithm for the Mann-Whitney U-Test

An algorithm and the corresponding FORTRAN-program for the computation of the MANN-WHITNEY U-Test test statistic are introduced which are simpler than the familiar procedure using ranks.     

A Test of Equality of Survival Distributions for Two Sample Censored Grouped Survival Data

A common testing problem for a life table or survival data is to test the equality of two survival distributions when the data is both grouped and censored. Several tests have been proposed in the literature which require various assumptions about the censoring distributions. It is shown that if these conditions are relaxed then the tests may no longer have the stated properties. The maximum likelihood test of equality when no assumptions are made about the censoring marginal distributions is derived. The properties of the test are found and it is compared to the existing tests. The fact that no assumptions are required about the censoring distributions make the test a useful initial testing procedure.     

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 Comparison between the LEHMACHER & WALL Rank Tests and Pyhel's Permutation Test for the Analysis of r Independent Samples of Response Curves

LEHMACHER & WALL'S (1978) example of the application of a rank test for the comparison of two independent samples of response curves is reanalyzed by PYHEL'S (1980) permutation test for the hypothesis of parallelism of response curves. This permutation test is part of a complete evaluation of effects for a split-plot design using the permutation test based procedure by WILLMES & PYHEL (1981). Differences in test decisions are discussed.     

A Multivariate Goodness-of-Fit Test for Stochastically Ordered Distributions

There are a number of nonparametric procedures known for testing goodness-of-fit in the univariate case. Similar procedures can be derived for testing goodness-of-fit in the multivariate case through an application of the theory of statistically equivalent blocks (SEB). The SEB transforms the data into coverages which are distributed as spacings from a uniform distribution on [0,1], under the null hypothesis. In this paper, we present a multivariate nonparametric test of goodness-of-fit based on the SEB when the multivariate distributions under the null hypothesis and the alternative hypothesis are “weakly” ordered. Empirical results are given on the performance of the proposed test in an application to the problem of assessing the reliability of a p-component system.     

鸡胚尿囊膜（HET-CAM）眼刺激替代试验对产品及原料评价的比较

目的比较鸡胚尿囊膜试验（HET-CAM）作为一种眼刺激替代方法对产品或原料的评价。方法采用HET-CAM和兔眼Draize试验方法,对19种原料和23种化妆品及家用洗涤产品眼刺激性进行检测,对体内体外试验的结果进行统计比较。结果比较体内体外试验,原料和产品的kappa系数分别为0.826,0.531;灵敏度分别为100%,81.8%;特异度分别为85.7%,77.8%。结论 HET-CAM可作为Draize试验的替代试验,HET-CAM系统更适合对单纯化学品原料进行眼刺激试验。     

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)     

A sampling study of the size and power of tests for paired survival data

The paired-t, sign, and signed rank tests were compared for samples from a bivariate exponential distribution. Each is a valid α-level test. One test was not uniformly more powerful than the others for all sample sizes, α levels, correlations, and alternative hypotheses considered, but the signed rank test did well consistently. It was always preferable to the sign test and never was appreciably worse than the paired-t test. The relative performance of the tests depends on α as well as the sample size.     

A Note on the Log‐Rank Test in Life Table Analysis with Correlated Observations

Survival data consisting of independent sets of correlated failure times may arise in many situations. For example, we may take repeated observations of the failure time of interest from each patient or observations of the failure time on siblings, or consider the failure times on littermates in toxicological experiments. Because the failure times taken on the same patient or related family members or from the same litter are likely correlated, use of the classical log‐rank test in these situations can be quite misleading with respect to type I error. To avoid this concern, this paper develops two closed‐form asymptotic summary tests, that account for the intraclass correlation between the failure times within patients or units. In fact, one of these two test includes the classical log‐rank test as a special case when the intraclass correlation equals 0. Furthermore, to evaluate the finite‐sample performance of the two tests developed here, this paper applies Monte Carlo simulation and notes that they can actually perform quite well in a variety of situations considered here.     

A Simple Two‐Sample Rank Test for Multivariate Survival Outcomes with Left Truncation and Right Censoring

A distribution‐free two‐sample rank test is proposed for testing for differences between survival distributions in the analysis of biomedical studies in which two groups of subjects are followed over time for a particular outcome, which may recur. This method is motivated by an observational HIV (human immunodeficiency virus) study in which a group of HIV‐seropositive women and a comparable group of HIV‐seronegative women were examined every 6 months for the presence of cervical intraepithelial neoplasia (CIN), the cervical cancer precursor. Women entered the study serially and were subject to random loss to follow‐up. Only women free of CIN at study entry were followed resulting in left‐truncated survival times. If a woman is found to be CIN infected at a later examination, she is treated and then followed until CIN recurs. The two groups of women were compared at both occurrences of CIN on the basis of rank statistics. For the first occurrence of CIN, survival times since the beginning of the study (based on calendar time) are compared. For a recurrence of CIN, survival times since the first development of CIN are compared. The proposed test statistic for an overall difference between the two groups follows a chi‐square distribution with two degrees of freedom. Simulation results demonstrate the usefulness of the proposed test proposed test statistic, which reduces to the Gehan statistic if each person is followed only to the first failure and there is no serial enrollment.     

A FORTRAN Program for the KASTENBAUM-BOWMAN Test

The KASTEITBAUM -BOWMAN test allows the determination of the significance of a mutation frequency. A simple FORTRAN program for the KASTENBAUM -BOWMAN test is given. The subroutines may be incorporated easily into larger programs and allow to calculate the exact P value for the test.