Comparison of unweighted and weighted rank based tests for an ordered alternative in randomized complete block designs

In randomized complete block designs, a monotonic relationship among treatment groups may already be established from prior information, e.g., a study with different dose levels of a drug. The test statistic developed by Page and another from Jonckheere and Terpstra are two unweighted rank based tests used to detect ordered alternatives when the assumptions in the traditional two-way analysis of variance are not satisfied. We consider a new weighted rank based test by utilizing a weight for each subject based on the sample variance in computing the new test statistic. The new weighted rank based test is compared with the two commonly used unweighted tests with regard to power under various conditions. The weighted test is generally more powerful than the two unweighted tests when the number of treatment groups is small to moderate.     

Linear signed rank tests for hultivariate location

Previously proposed linear signed rank tests for multivariate location are not invariant under linear transformations of the observations, The asymptotic relative efficiencies of the tests 2 with respect to Hotelling's T²test depend on the direction of shift and the covariance matrix of the alternative distributions. For distributions with highly correlated components, the efficiencies of some of these tests can be arbitrarily low; they approach zero for certain multivariate normal alternatives, This article proposes a transformation of the data to be performed prior to standard linear signed rank tests, The resulting procedures have attractive power and efficiency properties compared to the original tests, In particular, for elliptically symmetric contiguous alternafives, the efficiencies of the new tests equal those of corresponding univariate linear signed rank tests with respect to the t test.     

THE POWER AND SIZE OF NONPARAMETRIC TESTS FOR COMMON DISTRIBUTIONAL CHARACTERISTICS

This paper considers the power and size properties of some well known nonparametric linear rank tests for location and scale as well as the Kolmogorov-Smirnov omnibus test and proposed alternatives to it. Independence between some classes of linear rank tests is established facilitating their joint application. Monte Carlo study confirms the asymptotic power properties of the linear rank tests but raises concerns about their application in more general and practically relevant circumstances. It also indicates that the new omnibus tests constitute viable alternatives with superior properties to the Kolmogorov-Smirnov test in certain circumstances.     

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 note on the nonparametric test based on theL 1-version of the Cramér-von Mises statistic

We consider the test based on theL ₁-version of the Cramér-von Mises statistic for the nonparametric two-sample problem. Some quantiles of the exact distribution under H₀ of the test statistic are computed for small sample sizes. We compare the test in terms of power against general alternatives to other two-sample tests, namely the Wilcoxon rank sum test, the Smirnov test and the Cramér-von Mises test in the case of unbalanced small sample sizes. The computation of the power is rather complicated when the sample sizes are unequal. Using Monte Carlo power estimates it turns out that the Smirnov test is more sensitive to non stochastically ordered alternatives than the new test. And under location-contamination alternatives the power estimates of the new test and of the competing tests are equal.     

A class of multivariate distribution-free tests of independence based on graphs

A class of distribution-free tests is proposed for the independence of two subsets of response coordinates. The tests are based on the pairwise distances across subjects within each subset of the response. A complete graph is induced by each subset of response coordinates, with the sample points as nodes and the pairwise distances as the edge weights. The proposed test statistic depends only on the rank order of edges in these complete graphs. The response vector may be of any dimensions. In particular, the number of samples may be smaller than the dimensions of the response. The test statistic is shown to have a normal limiting distribution with known expectation and variance under the null hypothesis of independence. The exact distribution free null distribution of the test statistic is given for a sample of size 14, and its Monte-Carlo approximation is considered for larger sample sizes. We demonstrate in simulations that this new class of tests has good power properties for very general alternatives.     

The power of the exponential scores test for an ordinary renewal process against trend alternatives

The power properties of a statistic based on the use of exponential scores which may be used for testing whether a series of events occurring in time form an ordinary renewal process against trend alternatives are examined. Small sample power comparisons under a Lehmann trend alternative are made with an alternative nonparametric test based on a rank trend statistic and with the parametric test when the intervals are exponentially distributed. Finally, some asymptotic efficiency results are developed for limiting trend alternatives.     

A nonparametric test for interaction in two‐way layouts

The authors present a new nonparametric approach to test for interaction in two‐way layouts. Based on the concept of composite linear rank statistics, they combine the correlated row and column ranking information to construct the test statistic. They determine the limiting distributions of the proposed test statistic under the null hypothesis and Pitman alternatives. They also propose consistent estimators for the limiting covariance matrices associated with the test. They illustrate the application of their test in practical settings using a microarray data set.     

The two-sample problem with multivariate censored data: a new rank test family

A new rank test family is proposed to test the equality of two multivariate failure times distributions with censored observations. The tests are very simple: they are based on a transformation of the multivariate rank vectors to a univariate rank score and the resulting statistics belong to the familiar class of the weighted logrank test statistics. The new procedure is also applicable to multivariate observations in general, such as repeated measures, some of which may be missing. To investigate the performance of the proposed tests, a simulation study was conducted with bivariate exponential models for various censoring rates. The size and power of these tests against Lehmann alternatives were compared to the size and power of two other tests (Wei and Lachin, 1984 and Wei and Knuiman, 1987). In all simulations the new procedures provide a relatively good power and an accurate control over the size of the test. A real example from the National Cooperative Gallstone Study is given     

Independence test in high-dimension using distance correlation and power enhancement technique

Abstract

This paper is concerned with independence test in high-dimension. A new test statistic is proposed with two terms: one is based on the modified distance correlation statistic, the other is constructed to enhance the power under sparse alternatives. Asymptotic properties of the test statistic are discussed under some regular conditions. The finite-sample simulations exhibit its superiority over some existing procedures. Finally, a real data example illustrates the proposed test.     

Linear rank tests under general alternatives,with application to summary statistics computed from repeated measures data

Linear rank tests are used extensively for comparing two or more groups of continuous outcomes. Tests in this class retain proper test size with minimal assumptions and can have high efficiency towards an alternative of interest. In recent years, these tests have been increasingly used in settings where an individual's observation is itself a scalar summary of several outcome measures. Here, simple distributional structures on the outcome variables can lead to complex differences between the distributions of summary statistics of the comparison groups. The local asymptotic power of linear rank tests when the groups are assumed to differ by a location or scale alternative has been studied in detail. However, not much is known about their behavior for other types of alternatives. To address this, we derive the asymptotic distribution of linear rank tests under a general contiguous alternative and then investigate the implications for location–scale families and more general settings, including an example drawn from an AIDS clinical trial where the continuous outcome is a summary statistic computed from repeated measures of a biological marker.     

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     

A rank test for bivariate location and scale problem for elliptically symmetric populations ∗

A nonparametric rank test for the two-sample location and scale bivariate problem is proposed. The asymptotic distribution of the statistic of the test is derived under the null hypothesis and under a class of contiguous alternatives. The asymptotic relative efficiency is given and a simulation study gives the performance of the test and some competitors.     

A Correlation Test for Normality Based on the Lévy Characterization

A powerful test of fit for normal distributions is proposed. Based on the Lévy characterization, the test statistic is the sample correlation coefficient of normal quantiles and sums of pairs of observations from a random sample. Since the test statistic is location-scale invariant, critical values can be obtained by simulation without estimating any parameters. It is proved that this test is consistent. A power comparison study including some directed tests shows that the proposed test is competitive, it is more powerful than the well-known Jarque–Bera test, and it is comparable to Shapiro–Wilk test against a number of alternatives.     

Small sample power of multivariate nonparametric tests for monotone trend

Bhattacharyya and Kioiz (1966) propose two multivariate nonparametric tests for monotone trend, one involving coordinate-wise Mann statistics and the other, coordinate-wise Spearman statistics. Dietz and Killeen (1981) propose a different test statistic based on coordinate-wise Mann statistics. The Pitman asymptotic relative efficiency of all three tests with respect to a normal theory competitor equals the cube root of the efficiency of a multivariate signed rank test with respect to Hotelling's T2. In this article, the small sample power of the nonparametric tests, the normal theory test, and a Bonferroni approach involving coordinate-wise univariate Mann or Spearman tests is examined in a simulation study. The Mann statistic of Dietz and Killeen and the Spearman statistic of Bhattacharyya and Klotz are found to perform well under both null and alternative hypotheses     

Linear Signed Rank Test for Model Selection

In this article, we consider a linear signed rank test for non-nested distributions in the context of the model selection. Introducing a new test, we show that, it is asymptotically more efficient than the Vuong test and the test statistic based on B statistic introduced by Clarke. However, here, we let the magnitude of the data give a better performance to the test statistic. We have shown that this test is an unbiased one. The results of simulations show that the rank test has the greater statistical power than the Vuong test where the underline distributions is symmetric.     

Nonparametric analysis of treatment effects with missing observations

This paper develops a test for comparing treatment effects when observations are missing at random for repeated measures data on independent subjects. It is assumed that missingness at any occasion follows a Bernoulli distribution. It is shown that the distribution of the vector of linear rank statistics depends on the unknown parameters of the probability law that governs missingness, which is absent in the existing conditional methods employing rank statistics. This dependence is through the variance–covariance matrix of the vector of linear ranks. The test statistic is a quadratic form in the linear rank statistics when the variance–covariance matrix is estimated. The limiting distribution of the test statistic is derived under the null hypothesis. Several methods of estimating the unknown components of the variance–covariance matrix are considered. The estimate that produces stable empirical Type I error rate while maintaining the highest power among the competing tests is recommended for implementation in practice. Simulation studies are also presented to show the advantage of the proposed test over other rank-based tests that do not account for the randomness in the missing data pattern. Our method is shown to have the highest power while also maintaining near-nominal Type I error rates. Our results clearly illustrate that even for an ignorable missingness mechanism, the randomness in the pattern of missingness cannot be ignored. A real data example is presented to highlight the effectiveness of the proposed method.     

A Comparative Study of Some Modified Chi-Squared Tests

Some recent results in the theory and applications of modified chi-squared goodness-of-fit tests are briefly discussed. It seems that for the first time power of modified chi-squared type tests for the logistic and three-parameter Weibull distributions based on moment type estimators is studied. Power of different modified tests against some alternatives for equiprobable fixed or random grouping intervals, and for Neyman–Pearson classes is investigated. It is shown that power of test statistic essentially depends on the quantity of Fisher's sample information this statistic uses. Some recommendations on implementing modified chi-squared type tests are given.     

On the Power of Bootstrapped Specification Tests

Abstract

Decisions based on econometric model estimates may not have the expected effect if the model is misspecified. Thus, specification tests should precede any analysis. Bierens' specification test is consistent and has optimality properties against some local alternatives. A shortcoming is that the test statistic is not distribution free, even asymptotically. This makes the test unfeasible. There have been many suggestions to circumvent this problem, including the use of upper bounds for the critical values. However, these suggestions lead to tests that lose power and optimality against local alternatives. In this paper we show that bootstrap methods allow us to recover power and optimality of Bierens' original test. Bootstrap also provides reliable p-values, which have a central role in Fisher's theory of hypothesis testing. The paper also includes a discussion of the properties of the bootstrap Nonlinear Least Squares Estimator under local alternatives.     

Linear hypothesis testing for weighted functional data with applications

In socioeconomic areas, functional observations may be collected with weights, called weighted functional data. In this paper, we deal with a general linear hypothesis testing (GLHT) problem in the framework of functional analysis of variance with weighted functional data. With weights taken into account, we obtain unbiased and consistent estimators of the group mean and covariance functions. For the GLHT problem, we obtain a pointwise F-test statistic and build two global tests, respectively, via integrating the pointwise F-test statistic or taking its supremum over an interval of interest. The asymptotic distributions of test statistics under the null and some local alternatives are derived. Methods for approximating their null distributions are discussed. An application of the proposed methods to density function data is also presented. Intensive simulation studies and two real data examples show that the proposed tests outperform the existing competitors substantially in terms of size control and power.