Robust nonparametric tests for the two-sample location problem

We construct and investigate robust nonparametric tests for the two-sample location problem. A test based on a suitable scaling of the median of the set of differences between the two samples, which is the Hodges-Lehmann shift estimator corresponding to the Wilcoxon two-sample rank test, leads to higher robustness against outliers than the Wilcoxon test itself, while preserving its efficiency under a broad range of distributions. The good performance of the constructed test is investigated under different distributions and outlier configurations and compared to alternatives like the two-sample t-, the Wilcoxon and the median test, as well as to tests based on the difference of the sample medians or the one-sample Hodges-Lehmann estimators.     

A CLASS OF NONPARAMETRIC TESTS FOR THE ONE-SAMPLE LOCATION PROBLEM

A class of “optimal”U-statistics type nonparametric test statistics is proposed for the one-sample location problem by considering a kernel depending on a constant a and all possible (distinct) subsamples of size two from a sample of n independent and identically distributed observations. The “optimal” choice of a is determined by the underlying distribution. The proposed class includes the Sign and the modified Wilcoxon signed-rank statistics as special cases. It is shown that any “optimal” member of the class performs better in terms of Pitman efficiency relative to the Sign and Wilcoxon-signed rank statistics. The effect of deviation of chosen a from the “optimal” a on Pitman efficiency is also examined. A Hodges-Lehmann type point estimator of the location parameter corresponding to the proposed “optimal” test-statistics is also defined and studied in this paper.     

Data Driven Rank Test for Two-Sample Problem

Traditional linear rank tests are known to possess low power for large spectrum of alternatives. In this paper we introduce a new rank test possessing a considerably larger range of sensitivity than linear rank tests. The new test statistic is a sum of squares of some linear rank statistics while the number of summands is chosen via a data-based selection rule. Simulations show that the new test possesses high and stable power in situations when linear rank tests completely break down, while simultaneously it has almost the same power under alternatives which can be detected by standard linear rank tests. Our approach is illustrated by some practical examples. Theoretical support is given by deriving asymptotic null distribution of the test statistic and proving consistency of the new test under essentially any alternative.     

Generalized Mann–Whitney Type Tests for Microarray Experiments

New statistical procedures are introduced to analyse typical microRNA expression data sets. For each separate microRNA expression, the null hypothesis to be tested is that there is no difference between the distributions of the expression in different groups. The test statistics are then constructed having certain type of alternatives in mind. To avoid strong (parametric) distributional assumptions, the alternatives are formulated using probabilities of different orders of pairs or triples of observations coming from different groups, and the test statistics are then constructed using corresponding several‐sample U‐statistics, natural estimates of these probabilities. Classical several‐sample rank test statistics, such as the Kruskal–Wallis and Jonckheere–Terpstra tests, are special cases in our approach. Also, as the number of variables (microRNAs) is huge, we confront a serious simultaneous testing problem. Different approaches to control the family‐wise error rate or the false discovery rate are shortly discussed, and it is shown how the Chen–Stein theorem can be used to show that family‐wise error rate can be controlled for cluster‐dependent microRNAs under weak assumptions. The theory is illustrated with an analysis of real data, a microRNA expression data set on Finnish (aggressive and non‐aggressive) prostate cancer patients and their controls.     

Non parametric Bayesian analysis of the two-sample problem with censoring

Testing for differences between two groups is a fundamental problem in statistics, and due to developments in Bayesian non parametrics and semiparametrics there has been renewed interest in approaches to this problem. Here we describe a new approach to developing such tests and introduce a class of such tests that take advantage of developments in Bayesian non parametric computing. This class of tests uses the connection between the Dirichlet process (DP) prior and the Wilcoxon rank sum test but extends this idea to the DP mixture prior. Here tests are developed that have appropriate frequentist sampling procedures for large samples but have the potential to outperform the usual frequentist tests. Extensions to interval and right censoring are considered and an application to a high-dimensional data set obtained from an RNA-Seq investigation demonstrates the practical utility of the method.     

An extended class of permutation techniques for matched pairs

A class of matched-pairs permutation techniques based on distances between each pair of observed signed values is considered. Although many commonly-used inference techniques for matched pairs are members of this class, some of the more appealing inference techniques among this class have received very little attention. Two new simple rank tests of this class jointly possess both intuitive properties and location-alternative power characteristics which appear more appealing than the corresponding characteristics of either the sign test or the Wllcoxon signed-ranks test. In particular, power comparisons based on slmula-tions indicate that these new rank tests are jointly as good or even vastly superior to the sign test or the Wilcoxon signed-ranks test for location alternatives involving five symmetric distributions. The five distributions selected for these com-parisons include the Laplace, logistic, normal, uniform and a U-shaped distribution     

SEMI-PARAMETRIC ANALYSIS OF COVARIANCE UNDER DEPENDENCE CONDITIONS WITHIN EACH GROUP

Consider the problem of covariance analysis based on regression models whose regression function is the sum of a linear and a non-parametric component. We propose a parametric and a non-parametric statistical test to compare the effects of the linear and non-parametric components, respectively, on the response variable in   L ≥ 2  groups. Serially correlated errors within each group are allowed. The first (second) test compares the differences between the estimates of the parametric (non-parametric) components of each group by means of a Mahalanobis  ( L ²)  distance. The asymptotic distribution of each statistic under the null hypothesis is obtained. A modest simulation study and an application to a real data set illustrate our methodology.     

Smoothed alternatives of the two-sample median and Wilcoxon's rank sum tests

We discuss smoothed rank statistics for testing the location shift parameter of the two-sample problem. They are based on discrete test statistics – the median and Wilcoxon's rank sum tests. For the one-sample problem, Maesono et al. [Smoothed nonparametric tests and their properties. arXiv preprint. 2016; ArXiv:1610.02145] reported that some nonparametric discrete tests have a problem with their p-values because of their discreteness. The p-values of Wilcoxon's test are frequently smaller than those of the median test in the tail area. This leads to an arbitrary choice of the median and Wilcoxon's rank sum tests. To overcome this problem, we propose smoothed versions of those tests. The smoothed tests inherit the good properties of the original tests and are asymptotically equivalent to them. We study the significance probabilities and local asymptotic powers of the proposed tests.     

The chernoff efficiency of the wilcoxon rank test

Exact limiting Chernoff efficiencies of the Wilcoxon rank test are derived using Hoadley's results. Efficiency curves are derived for the two-sample Wilcoxon rank test relative to the two-sample t test for normal shift alternatives when the null hypothesis is that of common normality. The comparisons with Bahadur efficiency and small sample Hodges-Lehmann efficiency-are also made.     

Amplification of Sensitivity Analysis in Matched Observational Studies

A sensitivity analysis displays the increase in uncertainty that attends an inference when a key assumption is relaxed. In matched observational studies of treatment effects, a key assumption in some analyses is that subjects matched for observed covariates are comparable, and this assumption is relaxed by positing a relevant covariate that was not observed and not controlled by matching. What properties would such an unobserved covariate need to have to materially alter the inference about treatment effects? For ease of calculation and reporting, it is convenient that the sensitivity analysis be of low dimension, perhaps indexed by a scalar sensitivity parameter, but for interpretation in specific contexts, a higher dimensional analysis may be of greater relevance. An amplification of a sensitivity analysis is defined as a map from each point in a low dimensional sensitivity analysis to a set of points, perhaps a 'curve,' in a higher dimensional sensitivity analysis such that the possible inferences are the same for all points in the set. Possessing an amplification, an investigator may calculate and report the low dimensional analysis, yet have available the interpretations of the higher dimensional analysis.     

Weighted Mean Survival Test Statistics: a Class of Distance Tests for Censored Survival Data

A class of test statistics is introduced which is sensitive against the alternative of stochastic ordering in the two-sample censored data problem. The test statistics for evaluating a cumulative weighted difference in survival distributions are developed while taking into account the imbalances in base-line covariates between two groups. This procedure can be used to test the null hypothesis of no treatment effect, especially when base-line hazards cross and prognostic covariates need to be adjusted. The statistics are semiparametric, not rank based, and can be written as integrated weighted differences in estimated survival functions, where these survival estimates are adjusted for covariate imbalances. The asymptotic distribution theory of the tests is developed, yielding test procedures that are shown to be consistent under a fixed alternative. The choice of weight function is discussed and relies on stability and interpretability considerations. An example taken from a clinical trial for acquired immune deficiency syndrome is presented.     

Exact tests based on the Baumgartner-Weiß-Schindler statistic—A survey

It is the purpose of this paper to review recently-proposed exact tests based on the Baumgartner-Weiß-Schindler statistic and its modification. Except for the generalized Behrens-Fisher problem, these tests are broadly applicable, and they can be used to compare two groups irrespective of whether or not ties occur. In addition, a nonparametric trend test and a trend test for binomial proportions are possible. These exact tests are preferable to commonly-applied tests, such as the Wilcoxon rank sum test, in terms of both type I error rate and power.     

WILCOXON-TYPE RANK-SUM PRECEDENCE TESTS

This paper introduces Wilcoxon‐type rank‐sum precedence tests for testing the hypothesis that two life‐time distribution functions are equal. They extend the precedence life‐test first proposed by Nelson in 1963. The paper proposes three Wilcoxon‐type rank‐sum precedence test statistics—the minimal, maximal and expected rank‐sum statistics—and derives their null distributions. Critical values are presented for some combinations of sample sizes, and the exact power function is derived under the Lehmann alternative. The paper examines the power properties of the Wilcoxon‐type rank‐sum precedence tests under a location‐shift alternative through Monte Carlo simulations, and it compares the power of the precedence test, the maximal precedence test and Wilcoxon rank‐sum test (based on complete samples). Two examples are presented for illustration.     

The analysis of multicentre clinical trials when there is heterogeneity between centres

Two-treatment multicentre clinical trials are very common in practice. In cases where a non-parametric analysis is appropriate, a rank-sum test for grouped data called the van Elteren test can be applied. As an alternative approach, one may apply a combination test such as Fisher's combination test or the inverse normal combination test (also called Liptak's method) in order to combine centre-specific P-values. If there are no ties and no differences between centres with regard to the groups’ sample sizes, the inverse normal combination test using centre-specific Wilcoxon rank-sum tests is equivalent to the van Elteren test. In this paper, the van Elteren test is compared with Fisher's combination test based on Wilcoxon rank-sum tests. Data from two multicentre trials as well as simulated data indicate that Fisher's combination of P-values is more powerful than the van Elteren test in realistic scenarios, i.e. when there are large differences between the centres’ P-values, some quantitative interaction between treatment and centre, and/or heterogeneity in variability. The combination approach opens the possibility of using statistics other than the rank sum, and it is also a suitable method for more complicated designs, e.g. when covariates such as age or gender are included in the analysis.     

On a class of nonparametric tests for the treatment vs control problem

The classical problem of testing treatment versus control is revisited by considering a class of test statistics based on a kernel that depends on a constant ‘a’. The proposed class includes the celebrated Wilcoxon-Mann-Whitnet statistics as a special case when ‘a’=1. It is shown that, with optimal choice of ‘a’ depending on the underlying distribution, the optimal member performs better (in terms of Pitman efficiency) than the Wilcoxon-Mann-Whitney and the Median tests for a wide range of underlying distributions. An extended Hodges-Lehmann type point estimator of the shift prameter corresponding to the proposed ‘optimal’ test statistic is also derived.     

Marginal homogeneity,symmetry, and independence

A contingency table of the mc form provides a convenient summary of data when c individuals in a matched set9 each belonging to a different one of c classifications, are identified as belonging to one of m categories, A study in which matched sets (c=3) of 1 case, 1 hospital control, and 1 neighborhood control are classified into one of m=4 occupational categories would be an example, Independence in the cxm tables for each of the matched sets implies symmetry in the summary mc table with consequent marginal homogeneity. Adaptation of the Mantel-Haenszel procedure for testing independence to the case of many cxm tables so as to yield a chi square with (cl)(ml) degrees of freedom (DF) provides a test of marginal homogeneity in the summary mc table. This can be viewed as a test of symmetry directed against alternatives which would make for marginal inhomogeneity and can differ     

Linear rank tests for censored survival data

We describe a class of rank test procedures for the two-sample problem with right censored survival data. The class of tests is directly generalized from the linear rank tests by assigning each observation a rank according to its corresponding Wilcoxon scores. It allows a flexible choice of score functions, in particular, those powerful against scale differences between the two survival distributions. Monte Carlo simulations have shown that some members of this class have great power in detecting crossing-curve alternatives (alternatives where underlying survival curves cross over). The class also contains tests essentially equivalent to the Gehan-Wilcoxon and the logrank tests.     

A weighted test of internal symmetry

This study examines extensions of McNemar's Test with multinomial responses, and proposes a linear weighting scheme, based on the distance of the response change, that is applied to one of these extensions (Bowker's test). This weighted version of Bowker's test is then appropriate for ordinal response variables. A Monte Carlo simulation was conducted to examine the Type I error rate of the weighted Bowker's test for a cross-classification table based on a five-category ordinal response scale. The weighted Bowker's test was also applied to a data set involving change in student attitudes towards mathematics. The results of the weighted Bowker's test were compared with the results of Bowker's test applied to the same set of data.     

The matched pair sign test using bivariate ranked set sampling for different ranking based schemes

Bivariate rank set sample (BVRSS) matched pair sign test is introduced and investigated for different ranking based schemes. We show that this test is asymptotically more efficient and more powerful than its counterpart sign test based on a bivariate simple random sample (BVSRS) for different ranking schemes. The asymptotic null distribution and the efficiency of the test are derived. Pitman’s asymptotic relative efficiency is used to compare the asymptotic performance of the matched pair sign test using BVRSS versus using BVSRS in all ranking cases. For small sample sizes, the bootstrap method is used to estimate P-values. Numerical comparisons are used to gain insight about the efficiency of the BVRSS sign test compared to the BVSRS sign test. Our numerical and theoretical results indicate that using any ranking scheme of BVRSS for the matched pair sign test is more efficient than using BVSRS.     

Letters to the Editor

Although several authors have indicated that the median test has low power in small samples, it continues to be presented in many statistical textbooks, included in a number of popular statistical software packages, and used in a variety of application areas. We present results of a power simulation study that shows that the median test has noticeably lower power, even for the double exponential distribution for which it is asymptotically most powerful, than other readily available rank tests. We suggest that the median test be “retired” from routine use and recommend alternative rank tests that have superior power over a relatively large family of symmetric distributions.