Multi-sample Location Problem Research Articles

New nonparametric tests based on data depth for multivariate multi-sample location problem

Data depth based nonparametric tests for multivariate multisample location problem are provided which extend the depth based nonparametric test for multivariate two-sample location problem by Shirke and Khorate (2018) [Shirke, D.T. and Khorate, S.D., 2018. Two-sample nonparametric test for testing equality of locations based on data depth. Journal of the Indian Society for Probability and Statistics, 19, pp. 9-23.] to the multi-sample setting. Proposed tests are based on the correlations among the depths of data points. Power comparison of proposed tests with the existing depth based nonparametric tests is provided using a simulation study based on a permutation test. Simulation study reveals that the proposed tests give comparable powers compared to the existing data depth based nonparametric tests. Proposed tests are illustrated with real life data sets.

Advances on Permutation Multivariate Analysis of Variance for big data

Abstract In many applications of the multivariate analyses of variance, the classic parametric solutions for testing hypotheses of equality in population means or multisample and multivariate location problems might not be suitable for various reasons. Multivariate multisample location problems lack a comparative study of the power behaviour of the most important combined permutation tests as the number of variables diverges. In particular, it is useful to know under which conditions each of the different tests is preferable in terms of power, how the power of each test increases when the number of variables under the alternative hypothesis diverges, and the power behaviour of each test as the function of the proportion of true alternative hypotheses. The purpose of this paper is to fill the gap in the literature about combined permutation tests, in particular for big data with a large number of variables. A Monte Carlo simulation study was carried out to investigate the power behaviour of the tests, and the application to a real case study was performed to show the utility of the method.

Nonparametric tests for multivariate multi-sample locations based on data depth

ABSTRACTSeveral nonparametric tests for multivariate multi-sample location problem are proposed in this paper. These tests are based on the notion of data depth, which is used to measure the centrality/outlyingness of a given point with respect to a given distribution or a data cloud. Proposed tests are completely nonparametric and implemented through the idea of permutation tests. Performance of the proposed tests is compared with existing parametric test and nonparametric test based on data depth. An extensive simulation study reveals that proposed tests are superior to the existing tests based on data depth with regard to power. Illustrations with real data are provided.

Smoothed ranks for two or multi-sample location problems

ABSTRACTSmoothed ranks are proposed for two or multi-sample location problems. The regular ranks in Wilcoxon's two sample test are replaced with smoothed ranks, and the shift parameter is estimated. Asymptotic properties of the smoothed rank estimator are shown and a hypothesis test is proposed. Moreover, the smoothed ranks are applied in the Kruskal–Wallis's r-sample test and the power of the test is computed using regular and smoothed ranks. Examples and Monte Carlo simulations show that the smoothed ranks perform similarly to the traditional rank based estimators under contaminated normal or non-normal populations.

Progressive censoring under inverse sampling for nonparametric multi-sample location problem

In this study, we propose a class of distribution-free progressive censoring test procedures for multi-sample location problem using a stage-wise partially sequential sampling technique. For this, we draw a fixed number of sample observations from one of the populations (say, control) and random number of observations from other populations (say, treatments) using suitable stopping rules. At each stage, two types of control groups, termed as fixed control group (FCG) and updated control group (UCG), are considered. Suitable stopping rules are constructed based on quantiles of the FCG and UCG observations separately. At each stage, FCG consists of initial control observations only; while UCG consists of stage-wise updated combined control observations and previous treatment observations. We examined different large sample results of the proposed tests. We also performed numerical studies to compare the performance of the tests based on FCG and UCG procedures. In addition, we compared the performance of the progressive censoring test procedures with the corresponding competitive terminal test procedure numerically.

Applications in business, medical and industrial statistics of bi-aspect nonparametric tests for location problems

Starting from the theory of the Nonparametric Combination of Dependent Permutation Tests (Pesarin, 1992, 2001), Marozzi (2002a, b) proposed two bi-aspect nonparametric tests for the two-sample and the multi-sample location problems. These tests are shown by simulation to be remarkably more powerful than the traditional parametric and permutation competitors (which can be seen as uni-aspect tests) under heavy-tailed and skewed distributions. After a brief presentation of the bi-aspect idea to location testing problems, three actual applications are discussed. The first one is a problem of business statistics and deals with the analysis of time for service calls. The second one is in medical statistics and deals with the analysis of the effect of cigarette smoking on maternal airway function during pregnancy. The third one is in industrial statistics and deals with the analysis of the setting of machines that produce steel ball bearings. The bi-aspect testing allows us to draw deeper and more informative inference than that allowed by traditional competitors.

A bi-aspect nonparametric test for the multi-sample location problem

A bi-aspect nonparametric test for testing hypotheses of location shifts of two populations was proposed in literature. The test is based on the nonparametric combination of dependent tests theory and is obtained by combining the traditional permutation test for the two-sample location problem with a test that takes into account whether a sample observation is or is not greater than the pooled sample median. A natural multi-sample extension of the test is proposed. The extension is shown by simulation to behave very similarly to the bi-aspect test for the two-sample problem. In fact, it is shown that the proposed test is remarkably more powerful than the traditional permutation test for the multi-sample location problem under heavy-tailed distributions like the Cauchy, the half-Cauchy, the 10% and the 30% outlier distributions. When sampling from the double-exponential and the exponential distributions, the proposed test appears to be better on the whole than the traditional permutation test. Under the considered χ 2 distributions, the bi-aspect test is practically as powerful as the traditional permutation test. Whereas under normal, uniform and bimodal distributions it is slightly less powerful. Moreover, the proposed test maintained the type-one error rate close to the nominal significance level and was generally slightly conservative.

Robustness and power of parametric, nonparametric, robustified and adaptive tests—The multi-sample location problem

This paper deals with a survey of different types of tests, parametric, nonparametric, robustified and adaptive ones, and with an application to the two-sided c-sample location problem. Some concepts of robustness are discussed, such as breakdown point, influence function, gross-error sensitivity and especially α- and β-robustness. A robustness study on level α in the case of heteroscedasticity and nonnormal distributions is carried out via Monte Carlo methods and also a power comparison of all the tests considered. It turns out that robustified versions of the F-test and Welch-test where the original observations are replaced by its ranks behave well over a broad class of distributions, symmetric ones with different tail weight and asymmetric ones, but, on the whole, an adaptive test is to prefer.

A general theory of hypothesis testing based on rankings

In nonparametric statistics, a hypothesis testing problem based on the ranks of the data gives rise to two separate permutation sets corresponding to the null and to the alternative hypothesis, respectively. A modification of Critchlow's unified approach to hypothesis testing is proposed. By defining the distance between permutation sets to be the average distance between pairs of permutations, one from each set, various test statistics are derived for the multi-sample location problem and the two-way layout. The asymptotic distributions of the test statistics are computed under both the null and alternative hypotheses. Some comparisons are made on the basis of the asymptotic relative efficiency.

On rank statistics: an approach via metrics on the permutation group

Abstract This paper investigates a general construction that generates many rank test statistics, for most of the standard nonparametric hypothesis testing situations. The construction uses metrics on permutations in a new way: the induced rank test statistics is a distance between two sets of permutations. These sets of permutations have group-theoretic descriptions, and the resulting rank statistic metrizes a quotient space of the permutation group. This construction is applied systematically to the two-sample and multi-sample location problems, the two-way layout problem, the one-sample location problem, the two-sample dispersion problem with equal medians, and the problem of testing for trend. The construction gives rise to many familiar rank test statistics for these six hypothesis testing situations, and also produces some plausible new test statistics. Additionally, the method enables one to ‘extend’ rank statistics to other hypothesis testing situations, in the sense that it creates families of statistics, for the six given settings, that are linked by their use of the same metric. Some connections with standard nonparametric theory are discussed. In the course of implementing the general construction, certain combinatorial and group-theoretic properties of metrics on permutations are studied, that play key roles in the computation of the induced rank statistics.

Partial least squares analysis with cross‐validation for the two‐class problem: A Monte Carlo study

AbstractA method for statistical analysis of two independent samples with respect to difference in location is investigated. The method uses the partial least squares projections to latent structures (PLS) with cross‐validation. The relation to classical methods is discussed and a Monte Carlo study is performed to describe how the distribution of the test‐statistic employed depends on the number of objects, the number of variables, the percentage variance explained by the first PLS‐component and the percentage missing values. Polynomial approximations for the dependency of the 50 per cent and the 5 per cent levels of the test‐statistic on these factors are given. The polynomial for the 50 per cent level is complicated, involving several first‐, second‐ and third‐degree terms, whereas the polynomial for the 5 per cent level is dependent only on the number of objects and the size of the first component. A separate Monte Carlo experiment indicates that a moderate difference in sample size does not affect the distribution of the test‐statistic. The multi‐sample location problem is also studied and the effect of increasing the number of samples on the test‐statistic is shown in simulations.

Robust procedures for multi-sample location problems with unequal group variances

Robust procedures are proposed for testing the equality of several group means without assuming the equality of group variances. These statistics are obtained by modifying Welch's W and Brown-Forsythe's F* using a trimmed mean and a sine-wave M estimator.Approximate distributions of these new statistics are obtained under normality. Their performances are evaluated by Monte Carlo sampling experiments under various long-tailed symmetric distributions

Nonparametric repeated significance tests for one-way anova with adoptation to multiple comparisons

Abstract Based on two-sample rank order statistics, a repeated significance testing procedure for a multi-sample location problem is considered. The asymptotic distribution theory of the proposed tests is given under the null hypothesis as well as under local alternatives. A Bahadur efficiency result of the repeated significance test relative to the terminal test based solely on the target sample size is presented. In the adaptation of the proposed tests to multiple comparisons, an asymptotically equivalent test statistic in terms of the rank estimators of the location parameters is derived from which the Scheffe method of multiple comparisons can be obtained in a convinient way.

M—estimators based repeated significance tests for one—way anova with adaptation to multiple comparisons

A repeated significance testing procedure for a multi—sample location problem based on a general class of M—estimators is considered. The asymptotic distribution theory of the allied test statistics is developed under the null hypothesis as well as under local alternatives. The theory is then adapted to provide a multiple comparison procedure which generalizes the classical Scheffe method to a pseudo—sequential setup. Allied asymptotic relative efficienty results are also considered.

A Class of Rank Order Tests for a General Linear Hypothesis

For a general multivariate linear hypothesis testing problem, a class of permutationally (conditionally) distribution-free tests is proposed and studied. The asymptotic distribution theory of the proposed class of test statistics is studied along with a generalization of the elegant results of Hajek (1968) to the multistatistics and multivariate situations. Asymptotic power and optimality of the proposed tests are established and a characterization of the multivariate multisample location problem [cf. Puri and Sen (1966)] in terms of the proposed linear hypothesis is also considered.