Rank-based Tests Research Articles

Robust Testing of Paired Outcomes Incorporating Covariate Effects in Clustered Data with Informative Cluster Size

Paired outcomes are common in correlated clustered data where the main aim is to compare the distributions of the outcomes in a pair. In such clustered paired data, informative cluster sizes can occur when the number of pairs in a cluster (i.e., a cluster size) is correlated to the paired outcomes or the paired differences. There have been some attempts to develop robust rank-based tests for comparing paired outcomes in such complex clustered data. Most of these existing rank tests developed for paired outcomes in clustered data compare the marginal distributions in a pair and ignore any covariate effect on the outcomes. However, when potentially important covariate data is available in observational studies, ignoring these covariate effects on the outcomes can result in a flawed inference. In this article, using rank based weighted estimating equations, we propose a robust procedure for covariate effect adjusted comparison of paired outcomes in a clustered data that can also address the issue of informative cluster size. Through simulated scenarios and real-life neuroimaging data, we demonstrate the importance of considering covariate effects during paired testing and robust performances of our proposed method in covariate adjusted paired comparisons in complex clustered data settings.

Glycosylated haemoglobin as an indicator of diabetes control in pregnancy: A 10-year review of the relationship between HbA1c trends and delivery outcome in type I and type II diabetes

BackgroundPregestational diabetes mellitus (PGDM) confers an increased risk of adverse maternal and neonatal outcomes [1,2]. Glycaemic control in the medium and long term is commonly evaluated by examining glycosylated haemoglobin (HbA1c) levels. However, the value of HbA1c in pregnancy may be diminished by increased level of red cell turnover characteristic of pregnancy [3,4]. We sought to examine the impact of HbA1c in the first trimester and pre-delivery, and the within-patient change throughout gestation on mode of delivery and birthweight in pregnancies complicated by a pre-pregnancy diagnosis of type I or type II diabetes. MethodsA 10-year consecutive cohort of pregnancies complicated by PGDM, from Jan 2010 until Dec 2019, was examined for HbA1c data in the first trimester and within 6 weeks of delivery. Perinatal outcome data, including gestational age at delivery, mode of delivery and birthweight centile, were obtained from hospital records. The Spearman Rank correlation was used to correlate HcA1c levels in the first trimester with birthweight centiles. Non-parametric summaries and rank-based tests, Signed-rank test and Kruskal-Wallis test, were used to compare Hba1c levels. ResultsDuring the 10-year study period, a consecutive cohort of 396 pregnancies that attained a viable gestational age (>24 weeks’ gestation) and complicated by pregestational diabetes was identified; representing 81 % of the population of pregestational diabetic pregnancies managed by this service during the study period. The median [IQR] HbA1c levels (mmol/mol) in the first trimester, pre-delivery and the differential across gestation were 51 [19] mmol/mol, 43 [11] mmol/mol and −8 [13] mmol/mol, respectively. A statistically significant reduction in HbA1c levels throughout gestation was observed (p < 0.001). The median [IQR] birthweight centile was 69 [50 – 96]. The distributions in HbA1c levels and birthweight centiles were heavily skewed. No correlation was identified between HbA1c levels and mode of delivery. ConclusionNeither baseline HbA1c levels, pre-delivery values, nor trends across gestation appear to impact birthweight centile or mode of delivery in PGDM. While optimising glycaemic control can affect the long term health of the mother, these indices cannot be relied upon to reflect the impact of glycaemic control on fetal growth aberrations that influence mode of delivery.

Rank-based change-point analysis for long-range dependent time series

We consider change-point tests based on rank statistics to test for structural changes in long-range dependent observations. Under the hypothesis of stationary time series and under the assumption of a change with decreasing change-point height, the asymptotic distributions of corresponding test statistics are derived. For this, a uniform reduction principle for the sequential empirical process in a two-parameter Skorohod space equipped with a weighted supremum norm is proved. Moreover, we compare the efficiency of rank tests resulting from the consideration of different score functions. Under Gaussianity, the asymptotic relative efficiency of rank-based tests with respect to the CuSum test is 1, irrespective of the score function. Regarding the practical implementation of rank-based change-point tests, we suggest to combine self-normalized rank statistics with subsampling. The theoretical results are accompanied by simulation studies that, in particular, allow for a comparison of rank tests resulting from different score functions. With respect to the finite sample performance of rank-based change-point tests, the Van der Waerden rank test proves to be favorable in a broad range of situations. Finally, we analyze data sets from economy, hydrology, and network traffic monitoring in view of structural changes and compare our results to previous analysis of the data.

Towards lithology mapping in semi-arid areas using time-series Landsat-8 data

Previous remote sensing studies have used optical images to map lithology, commonly based on a single-date product to extract spectral diagnostic features or other geology-related features to aid interpretation. Yet, the performance of time-series or multi-date optical images in lithology mapping has rarely been discussed. In this study, we employed time-series (TS) Landsat-8 products to map the lithology in a semi-arid area of Xinjiang, China. First, we extracted TS reflectance from Landsat-8 multispectral data, and then extracted TS surface moisture, greenness, and brightness via tasseled cap transformation (TCT), a method for transforming the spectral bands of optical images into components (brightness, greenness, and moisture) that can be physically interpreted. Moreover, the land surface temperature (LST) was directly extracted from Landsat-8 Collection 2 products. In total, five surface parameters (TS reflectance, TS moisture, TS greenness, TS brightness, and TS LST), were collected. Then, the rank-based non-parametric Kruskal Wallis rank-sum statistical test was applied, which demonstrated a significant difference between the TS surface parameters among different rock units. Furthermore, different combinations of the TS surface parameters were stacked separately and served as different input features for the random forest classifier. Finally, the performance of TS surface parameters in mapping lithology was evaluated. The classification results showed that (1) TS reflectance outperformed single-date reflectance for mapping rock units; (2) the combination of TS brightness, greenness, and moisture gave a comparable classification result, relative to TS reflectance; (3) TS LST performed best when only one single surface parameter was used to map lithology, and (4) the combined use of LST and all three TCT components achieved the highest accuracy (85.26%) with a kappa coefficient of 0.77. Although surface parameters derived from rock units changed inconspicuously or irregularly over time, these variations captured by TS Landsat-8 data enabled the improvement of the classification accuracy. Overall, our study illustrates the great potential and benefits of using TS Lansat-8 data to map rock units in arid domains.

Spatial and temporal parasite dynamics: microhabitat preferences and infection progression of two co-infecting gyrodactylids

BackgroundMathematical modelling of host-parasite systems has seen tremendous developments and broad applications in theoretical and applied ecology. The current study focuses on the infection dynamics of a gyrodactylid-fish system. Previous experimental studies have explored the infrapopulation dynamics of co-infecting ectoparasites, Gyrodactylus turnbulli and G. bullatarudis, on their fish host, Poecilia reticulata, but questions remain about parasite microhabitat preferences, host survival and parasite virulence over time. Here, we use more advanced statistics and a sophisticated mathematical model to investigate these questions based on empirical data to add to our understanding of this gyrodactylid-fish system.MethodsA rank-based multivariate Kruskal-Wallis test coupled with its post-hoc tests and graphical summaries were used to investigate the spatial and temporal parasite distribution of different gyrodactylid strains across different host populations. By adapting a multi-state Markov model that extends the standard survival models, we improved previous estimates of survival probabilities. Finally, we quantified parasite virulence of three different strains as a function of host mortality and recovery across different fish stocks and sexes.ResultsWe confirmed that the captive-bred G. turnbulli and wild G. bullatarudis strains preferred the caudal and rostral regions respectively across different fish stocks; however, the wild G. turnbulli strain changed microhabitat preference over time, indicating microhabitat preference of gyrodactylids is host and time dependent. The average time of host infection before recovery or death was between 6 and 14 days. For this gyrodactylid-fish system, a longer period of host infection led to a higher chance of host recovery. Parasite-related mortalities are host, sex and time dependent, whereas fish size is confirmed to be the key determinant of host recovery.ConclusionFrom existing empirical data, we provided new insights into the gyrodactylid-fish system. This study could inform the modelling of other host-parasite interactions where the entire infection history of the host is of interest by adapting multi-state Markov models. Such models are under-utilised in parasitological studies and could be expanded to estimate relevant epidemiological traits concerning parasite virulence and host survival.Graphical

Characteristics associated with composite surgical failure over 5 years of women in a randomized trial of sacrospinous hysteropexy with graft vs vaginal hysterectomy with uterosacral ligament suspension

Among women with symptomatic uterovaginal prolapse undergoing vaginal surgery in the Vaginal hysterectomy with Native Tissue Vault Suspension vs Sacrospinous Hysteropexy with Graft Suspension (Study for Uterine Prolapse Procedures Randomized Trial) trial, sacrospinous hysteropexy with graft (hysteropexy) resulted in a lower composite surgical failure rate than vaginal hysterectomy with uterosacral suspension over 5 years. This study aimed to identify factors associated with the rate of surgical failure over 5 years among women undergoing sacrospinous hysteropexy with graft vs vaginal hysterectomy with uterosacral suspension for uterovaginal prolapse. This planned secondary analysis of a comparative effectiveness trial of 2 transvaginal apical suspensions (NCT01802281) defined surgical failure as either retreatment of prolapse, recurrence of prolapse beyond the hymen, or bothersome prolapse symptoms. Baseline clinical and sociodemographic factors for eligible participants receiving the randomized surgery (N=173) were compared across categories of failure (≤1 year, >1 year, and no failure) with rank-based tests. Factors with adequate prevalence and clinical relevance were assessed for minimally adjusted bivariate associations using piecewise exponential survival models adjusting for randomized apical repair and clinical site. The multivariable model included factors with bivariate P<.2, additional clinically important variables, apical repair, and clinical site. Backward selection determined final retained risk factors (P<.1) with statistical significance evaluated by Bonferroni correction (P<.005). Final factors were assessed for interaction with type of apical repair at P<.1. Association is presented by adjusted hazard ratios and further illustrated by categorization of risk factors. In the final multivariable model, body mass index (increase of 5 kg/m2: adjusted hazard ratio, 1.7; 95% confidence interval, 1.3-2.2; P<.001) and duration of prolapse symptoms (increase of 1 year: adjusted hazard ratio, 1.1; 95% confidence interval, 1.0-1.1; P<.005) were associated with composite surgical failure, where rates of failure were 2.9 and 1.8 times higher in women with obesity and women who are overweight than women who have normal weight and women who are underweight (95% confidence intervals, 1.5-5.8 and 0.9-3.5) and 3.0 times higher in women experiencing >5 years prolapse symptoms than women experiencing ≤5 years prolapse symptoms (95% confidence interval, 1.8-5.0). Sacrospinous hysteropexy with graft had a lower rate of failure than hysterectomy with uterosacral suspension (adjusted hazard ratio, 0.6; 95% confidence interval, 0.4-1.0; P=.05). The interaction between symptom duration and apical repair (P=.07) indicated that failure was less likely after hysteropexy than hysterectomy for those with ≤5 years symptom duration (adjusted hazard ratio, 0.5; 95% confidence interval, 0.2-0.9), but not for those with >5 years symptom duration (adjusted hazard ratio, 1.0; 95% confidence interval 0.5-2.1). Obesity and duration of prolapse symptoms have been determined as risk factors associated with surgical failure over 5 years from transvaginal prolapse repair, regardless of approach. Providers and patients should consider these modifiable risk factors when discussing treatment plans for bothersome prolapse.

On universally consistent and fully distribution-free rank tests of vector independence

Rank correlations have found many innovative applications in the last decade. In particular, suitable rank correlations have been used for consistent tests of independence between pairs of random variables. Using ranks is especially appealing for continuous data as tests become distribution-free. However, the traditional concept of ranks relies on ordering data and is, thus, tied to univariate observations. As a result, it has long remained unclear how one may construct distribution-free yet consistent tests of independence between random vectors. This is the problem addressed in this paper, in which we lay out a general framework for designing dependence measures that give tests of multivariate independence that are not only consistent and distribution-free but which we also prove to be statistically efficient. Our framework leverages the recently introduced concept of center-outward ranks and signs, a multivariate generalization of traditional ranks, and adopts a common standard form for dependence measures that encompasses many popular examples. In a unified study, we derive a general asymptotic representation of center-outward rank-based test statistics under independence, extending to the multivariate setting the classical Hájek asymptotic representation results. This representation permits direct calculation of limiting null distributions and facilitates a local power analysis that provides strong support for the center-outward approach by establishing, for the first time, the nontrivial power of center-outward rank-based tests over root-n neighborhoods within the class of quadratic mean differentiable alternatives.

Using space filling curves to compare two multivariate distributions with distribution-free tests

This paper proposes new tests to compare two multivariate probability distributions. Since basic ranks do not canonically exist in Rd, it is impossible to have a natural multivariate generalisation of rank-based tests such as the two-sample Kolmogorov–Smirnov test. We thus rely on recent measure transportation theory to transform this d−dimensional problem into one-dimensional classical tests by using space filling curves. To foster lower computation time, we develop distribution-free tests so as to avoid computing critical values for any particular problem. We demonstrate their theoretical validity and compare them to each other and to the existing distribution-free techniques via extensive simulations. We show that they are computationally efficient and most of the time outperform existing techniques when measuring the corresponding power functions. Finally, we apply the proposed tests to a dataset about stars’ luminosity and temperature.

Nitrogen Cycling Microbial Diversity and Operational Taxonomic Unit Clustering: When to Prioritize Accuracy Over Speed.

BackgroundAssessments of the soil microbiome provide valuable insight to ecosystem function due to the integral role microorganisms play in biogeochemical cycling of carbon and nutrients. For example, treatment effects on nitrogen cycling functional groups are often presented alongside one another to demonstrate how agricultural management practices affect various nitrogen cycling processes. However, the functional groups commonly evaluated in nitrogen cycling microbiome studies range from phylogenetically narrow (e.g., N-fixation, nitrification) to broad [e.g., denitrification, dissimilatory nitrate reduction to ammonium (DNRA)]. The bioinformatics methods used in such studies were developed for 16S rRNA gene sequence data, and how these tools perform across functional genes of different phylogenetic diversity has not been established. For example, an OTU clustering method that can accurately characterize sequences harboring comparatively little diversity may not accurately resolve the diversity within a gene comprised of a large number of clades. This study uses two nitrogen cycling genes, nifH, a gene which segregates into only three distinct clades, and nrfA, a gene which is comprised of at least eighteen clades, to investigate differences which may arise when using heuristic OTU clustering (abundance-based greedy clustering, AGC) vs. true hierarchical OTU clustering (Matthews Correlation Coefficient optimizing algorithm, Opti-MCC). Detection of treatment differences for each gene were evaluated to demonstrate how conclusions drawn from a given dataset may differ depending on clustering method used.ResultsThe heuristic and hierarchical methods performed comparably for the more conserved gene, nifH. The hierarchical method outperformed the heuristic method for the more diverse gene, nrfA; this included both the ability to detect treatment differences using PERMANOVA, as well as higher resolution in taxonomic classification. The difference in performance between the two methods may be traced to the AGC method’s preferential assignment of sequences to the most abundant OTUs: when analysis was limited to only the largest 100 OTUs, results from the AGC-assembled OTU table more closely resembled those of the Opti-MCC OTU table. Additionally, both AGC and Opti-MCC OTU tables detected comparable treatment differences using the rank-based ANOSIM test. This demonstrates that treatment differences were preserved using both clustering methods but were structured differently within the OTU tables produced using each method.ConclusionFor questions which can be answered using tests agnostic to clustering method (e.g., ANOSIM), or for genes of relatively low phylogenetic diversity (e.g., nifH), most upstream processing methods should lead to similar conclusions from downstream analyses. For studies involving more diverse genes, however, care should be exercised to choose methods that ensure accurate clustering for all genes. This will mitigate the risk of introducing Type II errors by allowing for detection of comparable treatment differences for all genes assessed, rather than disproportionately detecting treatment differences in only low-diversity genes.

A rank-based high-dimensional test for equality of mean vectors

The Wilcoxon signed-rank test and the Wilcoxon-Mann-Whitney test are two commonly used rank-based methods for one- and two-sample tests when the one-dimensional data are not normally distributed. The new rank-based nonparametric tests for equality of mean vectors are proposed in the high-dimensional settings. To overcome the technical challenges in data sorting, the new statistics are constructed by taking the sum of the Wilcoxon signed-rank or Wilcoxon-Mann-Whitney test statistics from each dimension of the data. The asymptotic properties of the proposed test statistics are investigated under the null and local alternative hypotheses. Simulation studies show that the new tests perform as well as the state-of-the-art methods when the high-dimensional data are normally distributed, but they turn out to be more powerful when the normality assumption is violated. Finally, the new testing methods are also applied to a human peripheral blood mononuclear cells gene expression data set for demonstrating their usefulness in practice.

A generalized Wilcoxon–Mann–Whitney type test for multivariate data through pairwise distance

The Wilcoxon–Mann–Whitney test is designed to test homogeneity of two random samples in the univariate case. It is very powerful to detect location shifts yet may lose power completely when there exist scale differences. We generalize the classic Wilcoxon–Mann–Whitney test through using pairwise distances of all observations. The generalized test can be readily used even when the random observations are multivariate. It is also very powerful in the presence of scale differences. The generalized test is in spirit to compare difference between the distribution functions of two random samples. It is mn/(m+n)-consistent under the strong null and local alternatives, and root-mn/(m+n)-consistent under fixed alternatives, where m,n stand for the respective sizes of the two random samples. The power of the generalized test is asymptotically independent of m/n, the size ratio of the two random samples. This indicates that the generalized test has nontrivial power as long as the sample sizes are not extremely unbalanced. We demonstrate the theoretical properties of our improved rank-based two-sample test through comprehensive numerical studies.

Antibacterial and Antiadhesion Effects of Psidium guajava Fractions on a Multispecies Biofilm Associated with Periodontitis

Objective: To assess the antibacterial activity of Psidium guajava fractions and their effects on adhesion of a multispecies biofilm consisting of Streptococcus gordonii, Fusobacterium nucleatum, and Porphyromonas gingivalis in vitro. Material and Methods: Guava leaves were obtained from the mountains of northern Peru, where they grow wild and free of pesticides. The antimicrobial activity of 25 mg/mL petroleum ether, 25 mg/mL dichloromethane and 25 mg/mL methanol fractions of P. guajava was evaluated by measuring inhibition halos, as well as the effect on the adhesion of multispecies biofilms at 4, 7 and 10 days of growth by measuring the optical density. In addition, antimicrobial susceptibility was compared using the Kruskal-Wallis test and its multiple comparison tests, and differences in mean biofilm adhesion between each fraction were assessed by repeated measures analysis and the Tukey multiple comparison test. Results: The rank-based Kruskal-Wallis test highlighted differences in the effects of the fractions on the zone of inhibition for each oral bacterium, including S. gordonii (p=0.000), F. nucleatum (p=0.000), and P. gingivalis (p=0.000), the Tukey test showed that the group treated with 0.12% chlorhexidine exhibited the least amount of adhesion, followed by the group treated with the 1.56 mg/mL methanol fraction. Conclusion: The methanol fraction of P. guajava had an antibacterial effect on S. gordonii and P. gingivalis, and the 1.56 mg/mL methanol fraction decreased biofilm adhesion.

Explore the Approaches to Corporate Social Responsibility Implemented by E-Commerce Platforms in China During the Early Stage of COVID-19: A Mixed-Methods Content Analysis.

The aim of this study was to explore (a) the approaches to corporate social responsibility (CSR) implemented by e-commerce platforms in China during the early stage of coronavirus disease 2019 (ESCOVID-19) and (b) the factors associated with the platforms' choice of these approaches. We collected the CSR data from the Internet during ESCOVID-19. Conventional content analysis was used to develop the targeted approaches. Finally, based on the frequency analysis of each approach, rank-based nonparametric testing was conducted to answer objective (b). Three main approaches (ie, donative CSR actions, responsive CSR actions, and strategic CSR actions) and 8 subapproaches were implemented. The strategic approach was the most frequently used approach. The platforms with higher market size implemented more donative and strategic CSR actions than did the platforms with lower market size. Donative CSR actions were significantly implemented in the earlier period, while strategic CSR actions were significantly implemented in the later period. Our research highlights the importance of e-commerce platforms to help the public survive and identifies that market size and time were associated with the platforms' CSR choice. The design of prevention and control policies should incorporate and support e-commerce platforms and evaluate the factors when confronting future public health crises.

Rank-based test for slope homogeneity in high-dimensional panel data models

Rank-based test for slope homogeneity in high-dimensional panel data models

Pseudo-Ranks: The Better Way of Ranking?

Rank-based methods are frequently used in the life sciences, and in the empirical sciences in general. Among the best-known examples of nonparametric rank-based tests are the Wilcoxon-Mann-Whitney test and the Kruskal–Wallis test. However, recently, potential pitfalls and paradoxical results pertaining to the use of traditional rank-based procedures for more than two samples have been highlighted, and the so-called pseudo-ranks have been proposed as a remedy for this type of problems. The aim of the present article is twofold: First, we show that pseudo-ranks might also behave counterintuitively when splitting up groups. Second, since the use of pseudo-ranks leads to a slightly different interpretation of the results, we provide some guidance regarding the decision for one or the other approach, in particular with respect to interpretability and generalizability of the findings. It turns out that the choice of the reference distribution, to which the individual groups are compared, is crucial. The practically relevant implications of these aspects are illustrated by a discussion of a dataset from epilepsy research. Summing up, one should decide based on thorough case-by-case considerations whether ranks or pseudo-ranks are appropriate.

A New Change-Point Rank Tests

A new rank-based test statistics are proposed for the problem of a possible change in the distribution of independent observations. We extend the two-sample test statistic of Damico (2004) to the change point setup. The finite sample critical values of the proposed tests is estimated. We also conduct a Monte Carlo simulation to compare the powers of the new tests with their competitors. Using the Nile data of Cobb (1978), we demonstrate the applicability of the new tests.

The Effects of Multitasking and Tools Carried by Travelers Onboard on the Perceived Trip Time

Travelers conduct onboard activities while using the tools they bring with them onboard to convert part of their travel time to a productive time. Productive travel time contributes to the reduction in the disutility of travel time. This paper discusses the influence of travelers’ onboard activities and the tools carried by travelers on the perceived trip time. 10 onboard activities and 12 tools carried by travelers are introduced and studied in this work. A questionnaire focusing on the main trip of each respondent in urban areas is conducted, where a sample size of 525 participants is collected. Statistical methods such as central tendency, chi-square, exploratory factor analysis (EFA), rank-based nonparametric test, and multivariate analysis of variance (MANOVA) are applied. The main findings are the following: almost all of the onboard activities and the tools carried by travelers impact the trip time positively (i.e., the perception is enhanced). For each transport mode, the most frequent onboard activities that impact the trip time positively is obtained, and the connection between each onboard activity and each tool carried by travelers is found (i.e., moderate to strong association). EFA uncovers the underlying relationship between those onboard activities and those tools carried by travelers that influence travelers’ perception. In this case, instead of the full list, fewer onboard activities and tools carried by travelers are produced to simplify the finding of their impacts on the perceived trip time. The participation in onboard activity is ranked across certain groups, such as the tendency of women to be engaged in onboard activities is higher than men’s tendency. Regarding the positive impact on trip time, a statistical difference is demonstrated between groups, where the use of the tools carried by travelers is varied across the transport mode, trip purpose, and trip time, gender, age, education, and job variable. Besides, the involvement in onboard activities is statistically dependent across the transport mode, gender, income, and car ownership variable. The output of this study helps decision-makers and mobility planners in understanding the behavior of travelers onboard in more detail, such as the availability of onboard tools affecting the choice of transport mode.

Particle swarm optimization algorithm using complex-order derivative concept: A comprehensive study

This paper presents a comprehensive study of the Particle Swarm Optimization (PSO) algorithm, called complex-order PSO (CPSO). In the core of new set of algorithms, we employ the complex-order derivative and the conjugate order differential concepts in the position and velocity adaption mechanisms. To determine the influence of the control parameters on the quality of the results, a sensitivity analysis is conducted. A number of value- and rank-based tests assesses the algorithms’ performance. For a suite of benchmark functions, the standard deviation and the mean best of the results are reported. Additionally, the Friedman test specifies the average ranking from the obtained results. The effect of the complex-order operation and the population size are analyzed using the Taguchi test. An application example illustrates the performance of the CPSO.

Testing for equality of distributions using the concept of (niche) overlap

In this paper, we propose a new non-parametric test for equality of distributions. The test is based on the recently introduced measure of (niche) overlap and its rank-based estimator. As the estimator makes only one basic assumption on the underlying distribution, namely continuity, the test is universal applicable in contrast to many tests that are restricted to only specific scenarios. By construction, the new test is capable of detecting differences in location and scale. It thus complements the large class of rank-based tests that are constructed based on the non-parametric relative effect. In simulations this new test procedure obtained higher power and lower type I error compared to two common tests in several settings. The new procedure shows overall good performance. Together with its simplicity, this test can be used broadly.

A rank-based test for monotone trends in k-sample multivariate problems

In many practical applications, a multivariate outcome is measured on two or more groups of subjects. When these variables are analyzed separately, the information is not fully exhausted, as possible dependencies among the endpoints are not considered. Within parametric frameworks, a number of literature exists to test such multivariate outcomes. Also, a few methods have been proposed using a fully non parametric approach. It is of particular interest to test the hypothesis of no treatment effect against conjectured patterns that are associated with treatment efficacy. We proposed a rank-based test for detecting pre-specified alternative patterns across all endpoints. The test statistic we have derived is a multivariate generalization of a recent non parametric univariate test for alternative patterns. Since we do not require the continuity of the distribution functions, the newly proposed test is applied to data with ties, in particular, to multivariate-ordered categorical data. Moreover, the test is invariant under monotone transformations of the responses and the weights describing the alternative pattern. Finite sample performance of the proposed test is assessed through a simulation study and comparisons are made with the most popular test for one-sided multivariate problems. The application of the test statistic is demonstrated with an electroencephalogram dataset.