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The objective of this study was to assess aspects of the social behavior of a mixed-breed herd of beef cows as a potential source for stress and economic losses. Angus (AN; N = 10), Brahman (BR; N = 10), and Senepol (SE; N = 10) cows were assigned to two groups (N = 15 each containing equal breed numbers) on separate pastures. Agonistic interactions (win/loss) during feeding were recorded daily for 45 days. Dominance values were estimated as the proportion of individuals dominated to total herdmates. From this, individuals were placed into social categories based upon linear ranking as follows: dominants (D), intermediate (I), and subordinates (S). Breed influenced (P

This article presents a method for determining which of two tests may offer an advantage in power when their asymptotic relative efficiencies (ARE) are the same. The method maintains some of the desirable properties of ARE, such as ease of calculation and singular result based upon asymptotic properties of the tests. However, using higher-order Taylor expansions of the test statistics’ means, the procedure offers additional insight into the finite sample size behavior of the tests. Two examples demonstrate the method.

In this article, we consider the problem of constructing confidence intervals for a population median when the underlying population is discrete. We describe seven methods of assigning confidence levels to order statistic based confidence intervals, all of which are easy to implement. A simulation study shows that, with discrete populations, it is possible to obtain consistently more accurate confidence levels and shorter intervals compared to the ones reported by the classical method which is implemented in commercial software. More precisely, the best results are obtained by inverting a two-tailed sign test that properly takes into account tied observations. Some real data examples illustrate the use of these confidence intervals.

The problem of calculating power and sample size for the Wilcoxon signed-rank test is discussed. The exact variance large-sample method is examined and explicit formulas are derived for observations from uniform, normal and Laplace distributions. Numerical results are presented to evaluate the exact variance procedure and compare its performance with two simplified approximations that have been suggested in the statistical literature. From the simulation results, it is evident that the exact variance approach is more accurate than the two approximate methods. To facilitate practical use, tabulated values of the estimated sample sizes are provided.

In certain designed experiments a sum of independent Wilcoxon signed rank statistics can be used to establish preference between two competing resources. The statistic incorporates not only which resource was selected but also the order of selection. The test can be implemented using modern statistical packages.

This article considers power and sample size calculations for the Wilcoxon–Mann–Whitney test. An exact variance large-sample method is introduced and explicit formulas are derived for the uniform, ...

A class of affine-invariant test statistics, including a sign test and a related family of signed-rank tests, is proposed for randomized complete block designs with one observation per treatment. This class is obtained by using the transformation–retransformation approach of Chakraborty, Chaudhuri and Oja along with a directional transformation due to Tyler. Under the minimal assumption of directional symmetry of the underlying distribution, the null asymptotic distribution of the sign test statistic is shown to be chi-square with p-1 degrees of freedom. The same null distribution is also proved for the family of signed-rank statistics under the assumption of symmetry of the underlying distribution. The Pitman asymptotic relative efficiencies of the tests, relative to Hotelling–Hsu's T2 are established. Several score functions are discussed including a simple linear score function and the optimal normal score function. The test based on the linear score function is compared to the other members of this family and other statistics in the literature through efficiency calculations and Monte Carlo simulations. This statistic has an excellent performance over a wide range of distributions and for small as well as large dimensions.

A class of affine-invariant signed rank test statistics is considered for the one-sample multivariate location problem. This class is obtained by modifying Randles’ sign test which uses the transformation–retransformation approach of Chakraborty, Chaudhuri and Oja along with a directional transformation due to Tyler. It contains test statistics that have been previously studied for this problem. For the class of symmetric distributions, the asymptotic null distribution of the proposed class of statistics is shown to be chi-square. One particular member of this class of tests is recommended. This specific statistic is obtained using a linear function of the ranks of Euclidean norms of the transformed vectors. Comparisons between this new test statistic and several leading competitors are made through efficiency calculations and Monte Carlo studies. This new statistic has strong efficiencies over a wide spectrum of distributions, ranging from very light-tailed distributions to very heavy-tailed ones. It per...

SUMMARY A robust affine equivariant estimator of location for multivariate data is proposed which becomes the univariate median for data of dimension one. The estimator is robust in the sense that it has a bounded influence function, a positive breakdown value and has high efficiency compared to the sample mean for heavy-tailed distributions. Perhaps its greatest strength is that, unlike other affine equivariant multivariate medians, it is easily computed for data in any practical dimension.

A class of affine-invariant signed rank test statistics is considered for the one-sample multivariate location problem. This class is obtained by modifying Randles’ sign test which uses the transfo...

A multivariate nonparametric statistic is proposed for testing independence among many vectors. The statistic is an extension of the interdirection quadrant statistic introduced by Gieser and Randles for the case of two vectors. The proposed statistic is affine-invariant under a class of nonsingular linear transformations and has an asymptotic chi-square distribution under the null hypothesis of independence when each vector has an elliptically symmetric distribution. It is shown that the proposed test performs better than other tests in the literature when the underlying distributions are heavy-tailed

A simpler multivariate sign test is proposed that uses the transformation-retransformation approach of Chakraborty, Chaudhuri, and Oja together with a directional transformation due to Tyler. This produces a multivariate sign test that is practical to apply to data of any dimension, makes minimal assumptions about the underlying distribution, and has a small-sample distribution-free property over a broad class of population models. It is shown to perform very well in comparison to Hotelling's T 2 and other multivariate sign tests for heavy-tailed and skewed distributions.

Nonparametric distribution-free tests for interactions and main effects are described for certain factorial repeated measures designs. These tests are created by applying nonparametric multivariate one-sample tests to contrasts in the repeated measures. The performance of the testsare compared to ANOVA and to Hotelling's T2 computed on the contrasts.

A new statistic, [Qcirc] n , based on interdirections is proposed for testing whether two vector-valued quantities are dependent. The statistic, which has an intuitive invariance property, reduces to the quadrant statistic when the two quantities are each univariate. Under the null hypothesis of independence, [Qcirc] n has a limiting chi-squared distribution when each vector is elliptically symmetric. The new statistic is compared to the classical normal theory competitor—Wilks' likelihood ratio criterion—and a componentwise quadrant statistic. Using a novel model of dependence between the vectors, Pitman asymptotic relative efficiencies (ARE's) are computed. The Pitman ARE's indicate that [Qcirc] n compares favorably to Wilks' likelihood ratio criterion when the vectors have heavy-tailed elliptically symmetric distributions and is uniformly better than the componentwise quadrant statistic when the vectors are spherically symmetric. A simulation study demonstrates that [Qcirc] n performs better t...

Interdirection tests are proposed for a simple repeated-measures design. The test statistics proposed are applications of the one-sample interdirection sign test and interdirection signed-rank test to a repeated-measurement setting. The interdirection sign test has a small-sample distribution-free property and includes the two-sided univariate sign test and Blumen's bivariate sign test as special cases. The interdirection signed-rank test includes the two-sided univariate Wilcoxon signed-rank test as a special case. The proposed statistics are shown to have a limiting X p−1 2 null distribution when the underlying distribution is elliptically symmetric. In addition, the asymptotic distributions of the proposed statistics under certain contiguous alternatives are obtained for elliptically symmetric distributions with a particular density function form. Pitman asymptotic relative efficiencies and Monte Carlo studies show the proposed interdirection tests to be robust as compared to several competito...

An affine-invariant signed sum test based on interdirections is proposed for the one-sample multivariate location problem. The test proposed, including the two-sided univariate Wilcoxon signed-rank test as a special case, is somewhat like applying the interdirection sign test to the sums of pairs of observed vectors. The proposed statistic is shown to have a limiting x 2 p null distribution when the underlying distribution is elliptically symmetric. In addition, the asymptotic distribution of the statistic under certain contiguous alternatives is obtained for elliptically symmetric distributions with a particular density function form. Comparisons made between the proposed test and Hotelling's T 2 via Pitman asymptotic relative efficiencies show the signed sum test performs better than Hotelling's T 2 when the underlying distribution is heavy-tailed.

An affine-invariant signed-rank test is proposed for the one-sample multivariate location problem. The test suggested is a modification of Randles's multivariate sign test based on interdirections, which extends Blumen's bivariate procedure to the multidimensional setting. Comparisons are made between the proposed statistic and several competitors via Pitman asymptotic relative efficiencies and Monte Carlo results. The signed-rank statistic appears to be robust. It performs better than its competitors when the distribution is light-tailed, and virtually as well as Hotelling's T 2 under multivariate normality. For heavy-tailed distributions the signed-rank statistic performs better than Hotelling's T 2 but not as well as Randles's statistic.

A multivariate affinc-invariant family of rank tests is proposed for the two sample location problem. The class of statistics introduced is built upon Randles' multivariate one-sample sign statistic based on interdirections and the multivariate one-sample signed-rank statistic of Peters and Randles. Asymptotic relative efficiencies are obtained which indicate that selected members of the class perform very well for a broad class of distributions. Further comparisons are made among several statistics using Monte Carlo results.

Robert Vincent Hogg was born on November 8, 1924 in Hannibal, Missouri. He earned a Ph.D. in statistics at the University of Iowa in 1950, where his advisor was Allen Craig. Following graduation, he joined the mathematics faculty at the University of Iowa. He was the founding Chair when the Department of Statistics was created at Iowa in 1965 and he served in that capacity for 19 years. At Iowa he also served as Chair of the Quality Management and Productivity Program and the Hanson Chair of Manufacturing Productivity. He became Professor Emeritus in 2001 after 51 years on the Iowa faculty. He is a Fellow of the Institute of Mathematical Statistics and the American Statistical Association plus an Elected Member of the International Statistical Institute. He was President of the American Statistical Association (1988) and chaired two of its winter conferences (1992, 1994). He received the ASA's Founder's Award (1991) and the Gottfried Noether Award (2001) for contributions to nonparametric statistics. His publications through 1996 are described in Communications in Statistics--Theory and Methods (1996), 2467--2481. This interview was conducted on April 14, 2004 at the Department of Statistics, University of Florida, Gainesville, Florida, and revised in the summer of 2006., Published at http://dx.doi.org/10.1214/088342306000000637 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

New test statistics are proposed for testing whether two random vectors are independent. Gieser and Randles, as well as Taskinen, Kankainen, and Oja have introduced and discussed multivariate extensions of the quadrant test of Blomqvist. This article serves as a sequel to this work and presents new multivariate extensions of Kendall's tau and Spearman's rho statistics. Two different approaches are discussed. First, interdirection proportions are used to estimate the cosines of angles between centered observation vectors and between differences of observation vectors. Second, covariances between affine-equivariant multivariate signs and ranks are used. The test statistics arising from these two approaches appear to be asymptotically equivalent if each vector is elliptically symmetric. The spatial sign versions are easy to compute for data in common dimensions, and they provide practical, robust alternatives to normal-theory methods. Asymptotic theory is developed to approximate the finite-sample null distributions as well, as to calculate limiting Pitman efficiencies. Small-sample null permutation distributions are also described. A simple simulation study is used to compare the proposed tests with the classical Wilks test. Finally, the theory is illustrated by an example. peerReviewed

(2003). Applied Nonparametric Statistical Methods. Technometrics: Vol. 45, No. 4, pp. 369-369.

The limiting distribution of the chi-square goodness-of-fit statistic Tn under alternatives is noncentral chi-square if the alternative probabilities approach the null probabilities at an appropriate rate as n → ∞. It is shown that for fixed alternative probabilities, the limiting distribution of (Tn − μ n )/σ n is standard normal. Both of these asymptotic results can be used to approximate the power of the goodness-of-fit test. Numerical comparisons between these two approximations indicate that for large values of the true power, the normal approximation is best, but for moderate values of power, the chi-square approximation is best.

Distribution-free tests are investigated for the one-sample multivariate location problem. Counts, called interdirections, which measure the angular distance between two observation vectors relative to the positions of the other observations, are introduced. These counts are invariant under nonsingular linear transformations and have a small-sample distribution-free property over a broad class of population models, called distributions with elliptical directions, which includes all elliptically symmetric populations and many skewed populations. A sign test based on interdirections is described, including, as special cases, the two-sided univariate sign test and Blumen's bivariate sign test. The statistic is shown to have a limiting χ2 p null distribution and, because it is based on interdirections, it is also seen to be invariant and to have a small-sample distribution-free property. Pitman asymptotic relative efficiencies and a Monte Carlo study show the test to perform well compared with Hotell...

A robust adaptive estimation procedure for location estimation problems is developed. Classification in this procedure is done on the basis of skewness and tailweight, using two statistics that are ratios of linear functions of sample order statistics. The associated estimators are of the general type known as M-estimators, Following the development of the adaptive location estimation procedure, an application to the k population selection problem is given. Monte Carlo results show the superiority of the adaptive procedure to the sample means procedure, the rank sum procedure, and the previously developed adaptive procedure of Randles, Ramberg, and Hogg.

An asymptotically distribution-free procedure is proposed for testing whether a univariate population is symmetric about some unknown value versus a broad class of asymmetric distribution alternatives. The consistency class of the test is discussed and two competing tests are described, one based on the sample skewness, and the other on Gupta's nonparametric procedure. A Monte Carlo study shows that the proposed test is superior to either competitor since it maintains the designated α levels fairly accurately even for sample sizes as small as 20, while displaying good power for detecting asymmetric distributions.

Asymptotic properties are developed for two classes of cross-validation assessment statistics. The two correspond to a large number of small partition sets and a small number of large partitions sets. Conditions implying weak convergence, strong convergence, and asymptotic normality are presented for each class. A discriminant analysis misclassification estimator and a regression analysis estimator of absolute error are used to illustrate the general theory.

A consistent estimator for the variance of Kendall's tau is proposed which allows for testing the hypothesis of no correlation in a bivariate distribution. The null distribution of the test statistic is tabulated under independence, and the properties of the test are discussed.

The null distribution of a test statistic computed on residuals can be affected by the dependence among the residuals. It might be expected, however, that the dependence decreases as the sample size increases, so that the limiting null distribution is the same as if the test were based on independent observations. The author investigates whether this large sample equivalence holds for tests of correlation, scale difference, regression, and location difference. Particular emphasis is placed on showing when rank tests based on residuals have a strongly asymptotically distribution-free property. U statistic theorems are used to investigate these issues.

A conditional test is proposed that uses the null hypothesis of bivariate symmetry to detect unequal marginal scales in a bivariate population. The asymptotic normality of the test statistic and the consistency of the test are established. Asymptotic relative efficiency comparisons are made with the standard parametric test in both the bivariate normal case and in a trivariate reduction model case and with two of Sen's (1967) tests in the bivariate normal case. The good performance of the test is also illustrated via a Monte Carlo study.

This paper reviews discrimination procedures which provide distribution-free control over the individual misclassification probabilities. Particular emphasis is placed on the two-population rank method developed by Broffitt, Randles and Hogg, which utilizes the general formulation of Quesenberry and Gessaman. It is shown that the rank method extends from two to three or more populations in a natural and flexible fashion. A Monte Carlo study compares two suggested extensions with others proposed by Broffitt.

Two new methods of constructing robust linear and quadratic discriminant functions are introduced. The first is a generalization of Fisher's procedure for finding a linear discriminant function. It places less weight on those observations that are far from the overlapping regions of the two populations. The second new method substitutes M-estimates of the means and the covariance matrices into the usual expressions for the linear and quadratic discriminant functions. Monte Carlo results indicate lower misclassification probabilities for these schemes compared to Fisher's linear discriminant function in cases of heavy-tailed or contaminated distributions.

A model-free rank procedure is proposed for the two-population discrimination problem that enables the practitioner to better control the balance between the two probabilities of misclassification. The method is applied to the discriminant functions resulting from normal assumptions and also to an adaptive one which is a weighted average of the linear and quadratic discriminant functions, where the weights are determined from the data. A Monte Carlo study shows that the rank method can greatly improve the balance between the two misclassification probabilities while keeping their average comparatively small.

An adaptive distribution-free test is proposed for the two-sample location problem. First, the data are used to assess the tailweight and skewness of the underlying distributions. This leads to the selection and then application, with the same data, of one of several common rank tests for shift, such as the Mann-Whitney-Wilcoxon test. The preliminary selection is made in a way that insures the testing procedure is distribution-free. A Monte Carlo study shows that the adaptive test has excellent power over a wide class of distributions and is preferable to certain prominent nonadaptive tests.

An adaptive procedure for selecting the population with the largest (smallest) location parameter is given. A Monte Carlo sampling study is presented indicating that this procedure performs as well as the means procedure when the underlying distribution is medium tailed like the normal. It is shown to be superior to both the means procedure and the rank sum procedure when the underlying distribution is either very light tailed (e.g., the uniform distribution) or very heavy tailed (e.g., the Cauchy distribution).

Some adaptive distribution-free tests about location parameters of symmetric distributions are processed for the one-and two sample cases. The preliminary inferences in these adaptive schemes is to classify the underlying distributions as having light, medium, or heave rails with a statistic that is assentialy the range of the sample divided by the mean deviation from the sample median. If this inference suggests medium tails, a wilcoxon-type test is used; for heavy tails, a median-type test is used. A modified Wilcoxon test (alirnating middle sample itams) is processed for ligntatailed distributions and portions of its null distribution are trasulated. The favorable performance of these adaotive rets is demanstrated by a monta carla study.

Some interesting examples of uncorrelated simple linear rank statistics are listed. Consideration is then given to conditions under which general rank statistics are uncorrelated or have joint distributions which enjoy certain symmetric properties. Two simple linear rank statistics that are uncorrelated are found, under suitable restrictions, to be asymptotically independent. This is followed by some illustrations in which two rank statistics are exactly independent. The importance of these results about independence in actual testing is then noted.