A DISTRIBUTION-FREE TEST FOR TWO-WAY LAYOUT1

Many nonparametric tests have been proposed for the hypothesis of no row (treatment) effect in a one-way layout design. Examples of such tests are Kruskal-Wallis H-test, Bhapkar's (1961) V-test and Deshpande's (1965) L-test. However not many tests are available for testing the same hypothesis in a two-way layout design without interaction. Perhaps the only “established” test is the one due to Friedman (1937). However, it applies to the case of one observation per cell only. In this paper, a new distribution-free test is proposed for the hypothesis of row effect in a two-way layout design. It applies to the case of several observations per cell, not necessarily equal. The asymptotic efficiency of the proposed test relative to other tests is studied.     

An adaptive test for the one-way layout

An adaptive test is proposed for the one-way layout. This test procedure uses the order statistics of the combined data to obtain estimates of percentiles, which are used to select an appropriate set of rank scores for the one-way test statistic. This test is designed to have reasonably high power over a range of distributions. The adaptive procedure proposed for a one-way layout is a generalization of an existing two-sample adaptive test procedure. In this Monte Carlo study, the power and significance level of the F-test, the Kruskal-Wallis test, the normal scores test, and the adaptive test were evaluated for the one-way layout. All tests maintained their significance level for data sets having at least 24 observations. The simulation results show that the adaptive test is more powerful than the other tests for skewed distributions if the total number of observations equals or exceeds 24. For data sets having at least 60 observations the adaptive test is also more powerful than the F-test for some symmetric distributions.     

Nonparametric rank-based test procedures for non-additive models in the two-way layout I. no replications

One of the major unresolved problems in the area of nonparametric statistics is the need for satisfactory rank-based test procedures for non-additive models in the two-way layout, especially when there is only one observation on each combination of the levels of the experimental factors. In this paper we consider an arbitrary non-additive model for the two-way layout with n levels of each factor. We utilize both alignment and ranking of the data together with basic properties of Latin squares to develop rank tests for interaction (non-additivity). Our technique involves first aligning within one of the main effects, ranking within the other main effects (columns and rows) and then adding the resulting ranks within “interaction bands” corresponding to orthogonal partitions of the interaction for the model, as denoted by the letters of an n × n Latin square. A Friedman-type statistic is then computed on the resulting sums. This is repeated for each of (n?1) mutually orthogonal Latin squares (thus accounting for all the interaction degrees of freedom). The resulting (n?1) Friedman-type statistics are finally combined to obtain an overall test statistic. The necessary null distribution tables for applying the proposed test for non-additivity are presented and we discuss the results of a Monte Carlo simulation study of the relative powers of this new procedure and other (parametric and nonparametric) procedures designed to detect interaction in a two-way layout with one observation per cell.     

Adaptive Neyman's smooth tests of homogeneity of two samples of survival data

The problem of testing whether two samples of possibly right-censored survival data come from the same distribution is considered. The aim is to develop a test which is capable of detection of a wide spectrum of alternatives. A new class of tests based on Neyman's embedding idea is proposed. The null hypothesis is tested against a model where the hazard ratio of the two survival distributions is expressed by several smooth functions. A data-driven approach to the selection of these functions is studied. Asymptotic properties of the proposed procedures are investigated under fixed and local alternatives. Small-sample performance is explored via simulations which show that the power of the proposed tests appears to be more robust than the power of some versatile tests previously proposed in the literature (such as combinations of weighted logrank tests, or Kolmogorov–Smirnov tests).     

Pearson-type goodness-of-fit tests for regression

A procedure for testing the goodness of fit of linear regression models is introduced. For a given partition of the real line into cells, the proposed test is a quadratic form based on the vector of observed minus expected frequencies of the residuals obtained by maximum-likelihood estimation of the regression parameters. The quadratic form is of the same computational difficulty as the traditional Pearson-type tests with uncensored data. A statistic based on only one cell is particularly easy to apply and is used for testing the normality assumption in a real data set from astronomy. A simulation study examines the finite-sample properties of the proposed tests.     

A Generalization of Shapiro–Wilk's Test for Multivariate Normality

A goodness-of-fit test for multivariate normality is proposed which is based on Shapiro–Wilk's statistic for univariate normality and on an empirical standardization of the observations. The critical values can be approximated by using a transformation of the univariate standard normal distribution. A Monte Carlo study reveals that this test has a better power performance than some of the best known tests for multinormality against a wide range of alternatives.     

Tests for compound periodicities and estimating a non linear function

We propose two tests for testing compound periodicities which are the uniformly most powerful invariant decision procedures against simple periodicities. The second test can provide an excellent estimation of a compound periodic non linear function from observed data. These tests were compared with the tests proposed by Fisher and Siegel by Monte Carlo studies and we found that all the tests showed high power and high probability of a correct decision when all the amplitudes of underlying periods were the same. However, if there are at least several different periods with unequal amplitudes, then the second test proposed always showed high power and high probability of a correct decision, whereas the tests proposed by Fisher and Siegel gave 0 for the power and 0 for the probability of a correct decision, whatever the standard deviation of pseudo normal random numbers. Overall, the second test proposed is the best of all in view of the probability of a correct decision and power.     

Rank-based test procedures for interaction in the two-way layout with one observation per cell

New aligned-rank test procedures for the composite null hypothesis of no interaction effects (without placing restrictions on the two main effects) against appropriate composite general alternatives are developed for the standard two-way layout with a single observation per cell. Relative power performances of the two new aligned-rank procedures and existing tests due to Tukey (1949) and to de Kroon & van der Laan (1981) are examined via Monte Carlo simulation. Extensive power studies conducted on the 5 × 6 and 5 × 9 two-way layouts with one observation per cell show superior performance of the new procedures for a variety of interaction effects. Simulated critical values for the new procedures are provided in settings where the number of levels for each of the factors is between 3 and 9, inclusive.     

Modification of Tukey's additivity test

In this paper we discuss testing for an interaction in the two-way ANOVA with just one observation per cell. The known results are reviewed and a simulation study is performed to evaluate type I and type II risks of the tests. It is shown that the Tukey and Mandel additivity tests have very low power in case of more general interaction scheme. A modification of Tukey's test is developed to resolve this issue. All tests mentioned in the paper have been implemented in R package Additivity Tests.     

Nonparametric tests of hypotheses for umbrella alternatives

The author proposes a general method for constructing nonparametric tests of hypotheses for umbrella alternatives. Such alternatives are relevant when the treatment effect changes in direction after reaching a peak. The author's class of tests is based on the ranks of the observations. His general approach consists of defining two sets of rankings: the first is induced by the alternative and the other by the data itself. His test statistic measures the distance between the two sets. The author determines the asymptotic distribution for some special cases of distances under both the null and the alternative hypothesis when the location of the peak is known or unknown. He shows the good power of his tests through a limited simulation study     

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     

A new test for two-sample location problem based on empirical distribution function

We propose a new test for testing the equality of location parameter of two populations based on empirical distribution function (ECDF). The test statistics is obtained as a power divergence between two ECDFs. The test is shown to be distribution free, and its null distribution is obtained. We conducted empirical power comparison of the proposed test with several other available tests in the literature. We found that the proposed test performs better than its competitors considered here under several population structures. We also used two real datasets to illustrate the procedure.     

A NEW TEST FOR EQUALITY OF VARIANCES FOR k NORMAL POPULATIONS

ABSTRACT

A simple test based on Gini's mean difference is proposed to test the hypothesis of equality of population variances. Using 2000 replicated samples and empirical distributions, we show that the test compares favourably with Bartlett's and Levene's test for the normal population. Also, it is more powerful than Bartlett's and Levene's tests for some alternative hypotheses for some non-normal distributions and more robust than the other two tests for large sample sizes under some alternative hypotheses. We also give an approximate distribution to the test statistic to enable one to calculate the nominal levels and P-values.     

An extensive power evaluation of a novel two-sample density-based empirical likelihood ratio test for paired data with an application to a treatment study of attention-deficit/hyperactivity disorder and severe mood dysregulation

In many case-control studies, it is common to utilize paired data when treatments are being evaluated. In this article, we propose and examine an efficient distribution-free test to compare two independent samples, where each is based on paired observations. We extend and modify the density-based empirical likelihood ratio test presented by Gurevich and Vexler [7] to formulate an appropriate parametric likelihood ratio test statistic corresponding to the hypothesis of our interest and then to approximate the test statistic nonparametrically. We conduct an extensive Monte Carlo study to evaluate the proposed test. The results of the performed simulation study demonstrate the robustness of the proposed test with respect to values of test parameters. Furthermore, an extensive power analysis via Monte Carlo simulations confirms that the proposed method outperforms the classical and general procedures in most cases related to a wide class of alternatives. An application to a real paired data study illustrates that the proposed test can be efficiently implemented in practice.     

Exact Statistical Tests for Heterogeneity of Frequencies Based on Extreme Values

Sophisticated statistical analyses of incidence frequencies are often required for various epidemiologic and biomedical applications. Among the most commonly applied methods is the Pearson's χ² test, which is structured to detect non specific anomalous patterns of frequencies and is useful for testing the significance for incidence heterogeneity. However, the Pearson's χ² test is not efficient for assessing the significance of frequency in a particular cell (or class) to be attributed to chance alone. We recently developed statistical tests for detecting temporal anomalies of disease cases based on maximum and minimum frequencies; these tests are actually designed to test of significance for a particular high or low frequency. The purpose of this article is to demonstrate merits of these tests in epidemiologic and biomedical studies. We show that our proposed methods are more sensitive and powerful for testing extreme cell counts than is the Pearson's χ² test. This feature could provide important and valuable information in epidemiologic or biomeidcal studies. We elucidated and illustrated the differences in sensitivity among our tests and the Pearson's χ² test by analyzing a data set of Langerhans cell histiocytosis cases and its hypothetical sets. We also computed and compared the statistical power of these methods using various sets of cell numbers and alternative frequencies. The investigation of statistical sensitivity and power presented in this work will provide investigators with useful guidelines for selecting the appropriate tests for their studies.     

Generalized orthogonal contrast tests for homogeneity of ordered means

Several procedures have been proposed for testing equality of ordered means. The best-known of these is the likelihood-ratio test introduced by Bartholomew, which possesses generally superior power characteristics to those of its competitors. Difficulties in implementing this test have led to the development of alternative approaches, such as tests based on single and multiple contrasts. Some recent approaches have utilized approximations to the polyhedral cone defining the restricted parameter space, including those of Akkerboom (circular cone) and Mudholkar & McDermott (orthant). This article proposes a class of tests based on an improved orthant approximation to the polyhedral cone. These tests may be viewed as generalizations of the orthogonal contrast test proposed by Mukerjee, Robertson & Wright. Studies of the power functions of several competing tests indicate that the generalized orthogonal contrast tests are effective alternatives to the likelihood-ratio test, especially when the latter is difficult to implement.     

A maxmin linear test of normal means and its application to lachin’s data

For stochastic ordering tests for normal distributions there exist two well known types of tests. One of them is based on the maximum likelihood ratio principle, the other is the most stringent somewhere most powerful test of Schaafsma and Smid(for a comprehensive treatment see Robertson, Wright and Dykstra(1988), for the latter test also Shi and Kudo(1987)). All these tests are in general numerically tedious. Wei, Lachin(1984)and particularly Lachin(1992)formulate a simple and easily computable test. However, it is not known so far for which sort of ordered alternatives his test is optimal

In this paper it is shown that his procedure is a maxmin test for reasonable subalternatives, provided the covariance matrix has nonnegative row sums. If this property is violated then his procedure can be altered in such a manner that the resul ting test again is a maxmin test. An example is glven where the modified procedure even in the least favourable case leads to a nontrifling increase in power. The fact that Lachins test resp. the modified version are maxmin tests on appropriate subalternatives amounts to the property that they are maxmin tests on subhypotheses which are relevant in practical applications.     

A moment test to identify uniformity

An omnibus test of uniformity based upon the ratios of sample moments and population moments is introduced. Results of a monte carlo power study show that for two types of alternatives considered, the proposed test has good power in comparison with Neyman's test N ₂Greenwood's test, Kolmogorov-Smirnov test, and Chi-squared test.     

Accent on Teaching Materials

A linear combination test for combining several tests of the correlation coefficient in the bivariate normal distribution is proposed. The linear combination test is compared with the well-known Fisher method of combining tests. It is shown by a Monte Carlo study that the linear combination test has a larger power.     

ON TESTS FOR INTERACTION IN A NONREPLICATED TWO-WAY LAYOUT

The paper proposes tests for interaction in a two-way table with one observation per cell. The power of these tests is independent of the additive main effects in the linear model. This is an advantage compared to a test suggested earlier, which has low power if main effects are very variable.