An exact test for trend among binomial proportions based on a modified Baumgartner-Weiß-Schindler statistic

The Cochran-Armitage test is the most frequently used test for trend among binomial proportions. This test can be performed based on the asymptotic normality of its test statistic or based on an exact null distribution. As an alternative, a recently introduced modification of the Baumgartner-Weiß-Schindler statistic, a novel nonparametric statistic, can be used. Simulation results indicate that the exact test based on this modification is preferable to the Cochran-Armitage test. This exact test is less conservative and more powerful than the exact Cochran-Armitage test. The power comparison to the asymptotic Cochran-Armitage test does not show a clear winner, but the difference in power is usually small. The exact test based on the modification is recommended here because, in contrast to the asymptotic Cochran-Armitage test, it guarantees a type I error rate less than or equal to the significance level. Moreover, an exact test is often more appropriate than an asymptotic test because randomization rather than random sampling is the norm, for example in biomedical research. The methods are illustrated with an example data set.     

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.     

A Multisample Rank Test for Location-Scale Parameters

One of the multisample problems is discussed in this article. A new multisample rank tests based on a k-sample Baumgartner statistic are proposed for testing the location-scale parameters. The exact critical values of proposed statistics are calculated. Simulations are used to investigate the power of proposed statistics for various population distributions.     

The multisample Cucconi test

The multisample version of the Cucconi rank test for the two-sample location-scale problem is proposed. Even though little known, the Cucconi test is of interest for several reasons. The test is compared with some Lepage-type tests. It is shown that the multisample Cucconi test is slightly more powerful than the multisample Lepage test. Moreover, its test statistic can be computed analytically whereas several others cannot. A practical application example in experimental nutrition is presented. An R function to perform the multisample Cucconi test is given.     

Comparing two independent incidence rates using conditional and unconditional exact tests

Several unconditional exact tests, which are constructed to control the Type I error rate at the nominal level, for comparing two independent Poisson rates are proposed and compared to the conditional exact test using a binomial distribution. The unconditional exact test using binomial p-value, likelihood ratio, or efficient score as the test statistic improves the power in general, and are therefore recommended. Unconditional exact tests using Wald statistics, whether on the original or square-root scale, may be substantially less powerful than the conditional exact test, and is not recommended. An example is provided from a cardiovascular trial.     

An empirical likelihood ratio based goodness-of-fit test for Inverse Gaussian distributions

The Inverse Gaussian (IG) distribution is commonly introduced to model and examine right skewed data having positive support. When applying the IG model, it is critical to develop efficient goodness-of-fit tests. In this article, we propose a new test statistic for examining the IG goodness-of-fit based on approximating parametric likelihood ratios. The parametric likelihood ratio methodology is well-known to provide powerful likelihood ratio tests. In the nonparametric context, the classical empirical likelihood (EL) ratio method is often applied in order to efficiently approximate properties of parametric likelihoods, using an approach based on substituting empirical distribution functions for their population counterparts. The optimal parametric likelihood ratio approach is however based on density functions. We develop and analyze the EL ratio approach based on densities in order to test the IG model fit. We show that the proposed test is an improvement over the entropy-based goodness-of-fit test for IG presented by Mudholkar and Tian (2002). Theoretical support is obtained by proving consistency of the new test and an asymptotic proposition regarding the null distribution of the proposed test statistic. Monte Carlo simulations confirm the powerful properties of the proposed method. Real data examples demonstrate the applicability of the density-based EL ratio goodness-of-fit test for an IG assumption in practice.     

Book Reviews

The Levene test is a widely used test for detecting differences in dispersion. The modified Levene transformation using sample medians is considered in this article. After Levene's transformation the data are not normally distributed, hence, nonparametric tests may be useful. As the Wilcoxon rank sum test applied to the transformed data cannot control the type I error rate for asymmetric distributions, a permutation test based on reallocations of the original observations rather than the absolute deviations was investigated. Levene's transformation is then only an intermediate step to compute the test statistic. Such a Levene test, however, cannot control the type I error rate when the Wilcoxon statistic is used; with the Fisher–Pitman permutation test it can be extremely conservative. The Fisher–Pitman test based on reallocations of the transformed data seems to be the only acceptable nonparametric test. Simulation results indicate that this test is on average more powerful than applying the t test after Levene's transformation, even when the t test is improved by the deletion of structural zeros.     

Power Comparison of Multivariate Wilcoxon-type Tests Based on the Jurečková–Kalina’s Ranks of Distances

A multivariate two-sample testing problem is one of the most important topics in nonparametric statistics. One of the multivariate two-sample testing problems based on the Jure?ková–Kalina ranks of distance is discussed in this article. Further, a multivariate Wilcoxon-type test is proposed for testing the equality of two continuous distribution functions. Simulations are used to investigate the power of this test for the two-sided alternative with various population distributions. The results show that the proposed test statistic is more suitable than various existing statistics for testing a shift in the locationt and location-scale parameters.     

A new computational approach test for one-way ANOVA under heteroscedasticity

This paper proposes a new test statistic based on the computational approach test (CAT) for one-way analysis of variance (ANOVA) under heteroscedasticity. The proposed test was compared with other popular tests according to type I error and power of tests under different combinations of variances, means, number of groups and sample sizes. As a result, it was observed that the proposed test yields better results than other tests in many cases.     

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.     

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.     

A test of normality using nonparametrlic residuals

In this paper, we develop a test of the normality assumption of the errors using the residuals from a nonparametric kernel regression. Contrary to the existing tests based on the residuals from a parametric regression, our test is thus robust to misspecification of the regression function. The test statistic proposed here is a Bera-Jarque type test of skewness and kurtosis. We show that the test statistic has the usual x²(2) limit distribution under the null hypothesis. In contrast to the results of Rilstone (1992), we provide a set of primitive assumptions that allow weakly dependent observations and data dependent bandwidth parameters. We also establish consistency property of the test. Monte Carlo experiments show that our test has reasonably good size and power performance in small samples and perfornu better than some of the alternative tests in various situations.     

Review of Statistical Filmstrip Programs

We develop a novel nonparametric likelihood ratio test for independence between two random variables using a technique that is free of the common constraints of defining a given set of specific dependence structures. Our methodology revolves around an exact density-based empirical likelihood ratio test statistic that approximates in a distribution-free fashion the corresponding most powerful parametric likelihood ratio test. We demonstrate that the proposed test is very powerful in detecting general structures of dependence between two random variables, including nonlinear and/or random-effect dependence structures. An extensive Monte Carlo study confirms that the proposed test is superior to the classical nonparametric procedures across a variety of settings. The real-world applicability of the proposed test is illustrated using data from a study of biomarkers associated with myocardial infarction. Supplementary materials for this article are available online.     

A test of normality using nonparametrlic residuals

In this paper, we develop a test of the normality assumption of the errors using the residuals from a nonparametric kernel regression. Contrary to the existing tests based on the residuals from a parametric regression, our test is thus robust to misspecification of the regression function. The test statistic proposed here is a Bera-Jarque type test of skewness and kurtosis. We show that the test statistic has the usual x ²(2) limit distribution under the null hypothesis. In contrast to the results of Rilstone (1992), we provide a set of primitive assumptions that allow weakly dependent observations and data dependent bandwidth parameters. We also establish consistency property of the test. Monte Carlo experiments show that our test has reasonably good size and power performance in small samples and perfornu better than some of the alternative tests in various situations.     

Multivariate extension of chi-squared univariate normality test

We propose a multivariate extension of the univariate chi-squared normality test. Using a known result for the distribution of quadratic forms in normal variables, we show that the proposed test statistic has an approximated chi-squared distribution under the null hypothesis of multivariate normality. As in the univariate case, the new test statistic is based on a comparison of observed and expected frequencies for specified events in sample space. In the univariate case, these events are the standard class intervals, but in the multivariate extension we propose these become hyper-ellipsoidal annuli in multivariate sample space. We assess the performance of the new test using Monte Carlo simulation. Keeping the type I error rate fixed, we show that the new test has power that compares favourably with other standard normality tests, though no uniformly most powerful test has been found. We recommend the new test due to its competitive advantages.     

A nonparametric test of symmetry based on the overlapping coefficient

In this paper, we introduce a new nonparametric test of symmetry based on the empirical overlap coefficient using kernel density estimation. Our investigation reveals that the new test is more powerful than the runs test of symmetry proposed by McWilliams [31]. Intensive simulation is conducted to examine the power of the proposed test. Data from a level I Trauma center are used to illustrate the procedures developed in this paper.     

A nonparametric hypothesis test for heteroscedasticity

In this paper, a hypothesis test for heteroscedasticity is proposed in a nonparametric regression model. The test statistic, which uses the residuals from a nonparametric fit of the mean function, is based on an adaptation of the well-known Levene's test. Using the recent theory for analysis of variance when the number of factor levels goes to infinity, the asymptotic distribution of the test statistic is established under the null hypothesis of homocedasticity and under local alternatives. Simulations suggest that the proposed test performs well in several situations, especially when the variance is a nonlinear function of the predictor.     

Nonparametric test for the homogeneity of the overall variability

In this paper, we propose a nonparametric test for homogeneity of overall variabilities for two multi-dimensional populations. Comparisons between the proposed nonparametric procedure and the asymptotic parametric procedure and a permutation test based on standardized generalized variances are made when the underlying populations are multivariate normal. We also study the performance of these test procedures when the underlying populations are non-normal. We observe that the nonparametric procedure and the permutation test based on standardized generalized variances are not as powerful as the asymptotic parametric test under normality. However, they are reliable and powerful tests for comparing overall variability under other multivariate distributions such as the multivariate Cauchy, the multivariate Pareto and the multivariate exponential distributions, even with small sample sizes. A Monte Carlo simulation study is used to evaluate the performance of the proposed procedures. An example from an educational study is used to illustrate the proposed nonparametric test.     

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.     

An empirical likelihood goodness-of-fit test for time series

Summary. Standard goodness-of-fit tests for a parametric regression model against a series of nonparametric alternatives are based on residuals arising from a fitted model. When a parametric regression model is compared with a nonparametric model, goodness-of-fit testing can be naturally approached by evaluating the likelihood of the parametric model within a nonparametric framework. We employ the empirical likelihood for an α -mixing process to formulate a test statistic that measures the goodness of fit of a parametric regression model. The technique is based on a comparison with kernel smoothing estimators. The empirical likelihood formulation of the test has two attractive features. One is its automatic consideration of the variation that is associated with the nonparametric fit due to empirical likelihood's ability to Studentize internally. The other is that the asymptotic distribution of the test statistic is free of unknown parameters, avoiding plug-in estimation. We apply the test to a discretized diffusion model which has recently been considered in financial market analysis.