Marco Marozzi

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

Permutation methods are prized for their lack of assumptions concerning distributions of variables. A bi-aspect permutation test based on the Nonparametric Combination of Dependent Tests theory is developed for testing hypotheses of location shifts of two independent populations. The test is obtained by combining the traditional permutation test with a test that takes into account whether a sample observation is less than or equal to, or greater than the pooled sample median. The procedure to compute the proposed test is presented. The type-one error rate and power of the test are investigated for many distributions and sample-size settings via Monte Carlo simulations. These simulations show that the proposed test is remarkably more powerful than the traditional permutation test under heavy-tailed distributions like the Cauchy, the half-Cauchy, a 10% and a 30% outlier distribution. 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 normal, uniform, a chi-squared and a bimodal distribution, the bi-aspect test is practically as powerful as the traditional permutation test. Moreover, in these simulations the proposed test maintained its type-one error rate close to the nominal significance level.

Some notes on the location–scale Cucconi test

The best known and most used rank test for the location–scale problem is due to Lepage [Y. Lepage, A combination of Wilcoxons and Ansari–Bradleys statistics, Biometrika 58 (1971), pp. 213–217.], but this paper is focused on the location–scale rank test of Cucconi [O. Cucconi, Un nuovo test non parametrico per il confronto tra due gruppi campionari, Giorn. Econom. XXVII (1968), pp. 225–248.], proposed earlier but not nearly as well-known. The test is of interest because, contrary to the other location–scale tests, it is not a quadratic form combining a test for location and a test for scale differences, and it is based on squared ranks and squared contrary-ranks. Moreover, it is easier to compute the test of Cucconi than those of Lepage, Manly–Francis, Büning–Thadewald, Neuhäuser, Büning and Murakami. Exact critical values for the test have been computed for the very first time. The power of the Cucconi test has been studied for the very first time and compared with that of the Lepage and other tests that include several Podgor–Gastwirth efficiency robust tests. Simulations show that the test of Cucconi maintains a size very close to α and is more powerful than the Lepage test, and therefore should be taken into account as a better alternative when it is not possible to develop an efficiency robust procedure for the problem at hand. The simulation study considers also the case of different shapes for the parent distributions, and the case of tied observations which is generally not considered in power studies. The presence of ties does not lower the performance of the Cucconi test, the contrary happens for the Lepage test. The tests are applied to real and fictitious biomedical data.

Multivariate tri-aspect non-parametric testing

Permutation tests are prized for their lack of assumptions concerning distribution of underlying populations. The (usual) permutation test for the two-sample location problem based on comparison of sample means is generally effective with regular, roughly symmetric, unimodal, and light-tailed distributions, whereas it might not be so with highly asymmetric and/or heavy-tailed distributions. Another drawback is that it is not consistent for distributions for which first and second moments do not exist. Marozzi [Marozzi, M., 2004, A bi-aspect nonparametric test for the two-sample location problem. Computational Statistics and Data Analysis, 44, 639–648.] proposed a bi-aspect non-parametric test for comparing two populations obtained by non-parametric combining the usual permutation test (which addresses the numerical aspect X i ) and a test based on comparison of frequencies over the pooled median (which addresses the categorical aspect related to the comparison of sample units with the pooled sample median). Unlike the usual permutation test, the bi-aspect test is consistent for every distribution and is very powerful with highly-skewed and/or heavy-tailed distributions. In the paper, the bi-aspect testing idea is extended by also considering the aspect based on ranks, with the role of third aspect. A simulation study with many sample size and distribution settings shows that the tri-aspect test is more powerful than the bi-aspect one. Moreover, the multivariate problem is addressed and formal proofs of exactness, unbiasedness, and consistency are given.

A distribution-free phase-II CUSUM procedure for monitoring service quality

A new single distribution-free Phase-II CUSUM procedure based on the Cucconi statistic for simultaneously monitoring shifts in the unknown location and scale parameters of a process is proposed. The procedure does not require the assumption of normal data. Control limits are tabulated for practical implementation of the procedure. The in-control and out-of-control performance of the procedure are comprehensively examined in terms of mean, median, variability and some percentiles of the corresponding run length distribution. A thorough comparison with the Shewhart-type procedures based on the Cucconi and Lepage statistics as well as with the CUSUM procedure based on the Lepage statistic is presented. The proposed procedure is illustrated by analysing the service quality of the Vancouver City Call Centre.

Nonparametric Simultaneous Tests for Location and Scale Testing: A Comparison of Several Methods

The two-sample location-scale problem arises in many situations like climate dynamics, bioinformatics, medicine, and finance. To address this problem, the nonparametric approach is considered because in practice, the normal assumption is often not fulfilled or the observations are too few to rely on the central limit theorem, and moreover outliers, heavy tails and skewness may be possible. In these situations, a nonparametric test is generally more robust and powerful than a parametric test. Various nonparametric tests have been proposed for the two-sample location-scale problem. In particular, we consider tests due to Lepage, Cucconi, Podgor-Gastwirth, Neuhäuser, Zhang, and Murakami. So far all these tests have not been compared. Moreover, for the Neuhäuser test and the Murakami test, the power has not been studied in detail. It is the aim of the article to review and compare these tests for the jointly detection of location and scale changes by means of a very detailed simulation study. It is shown that both the Podgor–Gastwirth test and the computationally simpler Cucconi test are preferable. Two actual examples within the medical context are discussed.

Multivariate tests based on interpoint distances with application to magnetic resonance imaging

The multivariate location problem is addressed. The most familiar method to address the problem is the Hotelling test. When the hypothesis of normal distributions holds, the Hotelling test is optimal. Unfortunately, in practice the distributions underlying the samples are generally unknown and without assuming normality the finite sample unbiasedness of the Hotelling test is not guaranteed. Moreover, high-dimensional data are increasingly encountered when analyzing medical and biological problems, and in these situations the Hotelling test performs poorly or cannot be computed. A test that is unbiased for non-normal data, for small sample sizes as well as for two-sided alternatives and that can be computed for high-dimensional data has been recently proposed and is based on the ranks of the interpoint Euclidean distances between observations. Five modifications of this test are proposed and compared to the original test and the Hotelling test. Unbiasedness and consistency of the tests are proven and the problem of power computation is addressed. It is shown that two of the modified interpoint distance-based tests are always more powerful than the original test. Particularly, the modified test based on the Tippett criterium is suggested when the assumption of normality is not tenable and/or in case of high-dimensional data with complex dependence structure which are typical in molecular biology and medical imaging. A practical application to a case-control study where functional magnetic resonance imaging is used is discussed.

Distribution-free Lepage Type Circular-grid Charts for Joint Monitoring of Location and Scale Parameters of a Process

In the last 5 years, research works on distribution-free (nonparametric) process monitoring have registered a phenomenal growth. A Google Scholar database search on early September 2015 reveals 246 articles on distribution-free control charts during 2000–2009 and 466 articles in the following years. These figures are about 1400 and 2860 respectively if the word ‘nonparametric’ is used in place of ‘distribution-free’. Distribution-free charts do not require any prior knowledge about the process parameters. Consequently, they are very effective in monitoring various non-normal and complex processes. Traditional process monitoring schemes use two separate charts, one for monitoring process location and the other for process scale. Recently, various schemes have been introduced to monitor the process location and process scale simultaneously using a single chart. Performance advantages of such charts have been clearly established. In this paper, we introduce a new graphical device, namely, circular-grid charts, for simultaneous monitoring of process location and process scale based on Lepage-type statistics. We also discuss general form of Lepage statistics and show that a new modified Lepage statistic is often better than the traditional of Lepage statistic. We offer a new and attractive post-signal follow-up analysis. A detailed numerical study based on Monte-Carlo simulations is performed, and some illustrations are provided. A clear guideline for practitioners is offered to facilitate the best selection of charts among various alternatives for simultaneous monitoring of location-scale. The practical application of the charts is illustrated. Copyright

Nonparametric Hypothesis Testing: Rank and Permutation Methods with Applications in R

Description: A novel presentation of rank and permutation tests, with accessible guidance to applications in R Nonparametric testing problems are frequently encountered in many scientific disciplines, such as engineering, medicine and the social sciences. This book summarizes traditional rank techniques and more recent developments in permutation testing as robust tools for dealing with complex data with low sample size. Key Features: Examines the most widely used methodologies of nonparametric testing. Includes extensive software codes in R featuring worked examples, and uses real case studies from both experimental and observational studies. Presents and discusses solutions to the most important and frequently encountered real problems in different fields. Features a supporting website containing all of the data sets examined in the book along with ready to use R software codes. Nonparametric Hypothesis Testing combines an up to date overview with useful practical guidance to applications in R, and will be a valuable resource for practitioners and researchers working in a wide range of scientific fields including engineering, biostatistics, psychology and medicine.

Multivariate multidistance tests for high‐dimensional low sample size case‐control studies

A class of multivariate tests for case-control studies with high-dimensional low sample size data and with complex dependence structure, which are common in medical imaging and molecular biology, is proposed. The tests can be applied when the number of variables is much larger than the number of subjects and when the underlying population distributions are heavy-tailed or skewed. As a motivating application, we consider a case-control study where phase-contrast cinematic cardiovascular magnetic resonance imaging has been used to compare many cardiovascular characteristics of young healthy smokers and young healthy non-smokers. The tests are based on the combination of tests on interpoint distances. It is theoretically proved that the tests are exact, unbiased and consistent. It is shown that the tests are very powerful under normal, heavy-tailed and skewed distributions. The tests can also be applied to case-control studies with high-dimensional low sample size data from other medical imaging techniques (like computed tomography or X-ray radiography), chemometrics and microarray data (proteomics and transcriptomics).

Multivariate bi-aspect testing for the two-sample location problem

Abstract The multivariate extension of the bi-aspect nonparametric testing procedure for the two-sample location problem presented in Marozzi (2004) is discussed. Two solutions are presented: the former is focused on each variable, the latter is focused on each of the two aspects involved in the bi-aspect (the categorical and the numerical one). Formal proofs of exactness, unbiasedness, and consistency of the multivariate tests are given. Such properties hold even when population first and second moments do not exist.