An extension of the median test to analysis of variance

In a k-way analysis of variance model, the major concern is testing for main effects and for the presence of interaction between the factors. When the assumptions of normality and equal variances are satisfied, the appropriate test to use is the usual F-test for ANOVA. However, when the normality assumption is not satisfied then a robust or nonparametric test is needed to conduct the analysis. In this paper a nonparametric method based on cell counts is proposed. Each cell is divided into L subcells based on predetermined outpoints and the resulting frequencies are laid out in a contingency table. Then the Pearson x² and tne likelihood ratio tests are performed. A comparison with the classical ANOVA F-test indicates that the proposed method is preferable when the data comes from a thick-tailed highly skewed distribution.     

Improved tests for homogeneity of variances

Equality of variances is one of the key assumptions of analysis of variances (ANOVA). There are several testing procedures available to validate this assumption, but it is rare to find a test procedure which controls the type I error rate while providing high statistical power. In this article, we introduce a bootstrap test based on the ratio of mean absolute deviances (RMD). We also propose a two-stage testing procedure where we first quantify the skewness of the distributions and then choose an appropriate test for homogeneity of variances. The performances of these test procedures are studied via a simulation study.     

Equivalent inferences using transformations

Justice (1977) has presented a Leviiison-typc- solution for two dimensional Wiener filtering problem. Since the solution is based on the Szego polynomials, it is necessary to calculate the bivariate Szego polynomials to obtain it In this paper, an alternative solution of the problem is proposed, which is based on the block Cholesky decomposition of the inverse of a Hermi-tian block Toeplitz matrix. Since the block Choiesky decomposition can be accomplished through the Whittle algorithm, the new solution is easy to implement into a computer program.     

Confidence intervals for the between group variance in the unbalanced one-way random effects model of anaylsis of variance

A confidence interval for the between group variance is proposed which is deduced from Wald'sexact confidence interval for the rtio of the two variance components in the one-way random effects model and the exact confidence interval for the error variance resp.an unbiased estimator of the error variance. In a simulation study the confidence coeffecients for these two intervals are compared with the confidence coefficients of two other commonly used confidence intervals. There the confidence interval derived here yields confidence coefficiends which are always greater than the prescriped level.     

Decision and choice in transforming data

Statistics are developed for predicting the effect of data transformations on the F statistic when the assumptions of homoscedasticity and normality underlying the AN OVA are not necessarily satisfied. These statistics are useful for determining whether and how to transform, They are developed by partitioning the change in the observed value of the jF-statistic under the transformation, into two expressions, one of which depends on the "truth" of HQ while the other does not. Using this partition, desirable properties are derived for transformations. Criteria are developed defining transformations which tend to preserve the type 1 error while increasing power when needed. Using these criteria, the notion of model robustness is introduced. It is shown that the Box-Cox methodology for selecting a power transform may, under certain conditions, produce a transformation which does not permit inferences to be made about the parent population from the transformed population. An alternative approach suggested here does permit such inferences.     

Tests for homogeneity of variance

Six procedures which convert tests of homogeneity of variance into tests for mean equality for independent groups are compared. The tests are the analysis of variance (ANOVA) and Welch F statistics. The Welch statistics are included since it was anticipated that ANOVA would not provide a robust test when samples of unequal sizes are obtained from non-normal populations. However, the Welch tests are not found to be uniformly preferrable. In addition, a prior recommendation for Miller's jackknife procedure is not supported for the unequal sample size case. The data indicates that the current tests for variance heterogeneity are either sensitive to non-normality or, if robust, lacking in power. Therefore, these tests cannot be recommended for the purpose of testing the validity of the ANOVA homogeneity assumption.     

Some partially variance balanced structurally incomplete row-column designs

Earlier results by the authors are used to provide the intrablock analysis for row-column designs that have observations at nodes of the row-column lattice, the design being structurally incomplete when some nodes are empty. Construction, properties, and intrablock analyses of some special b× b row-column designs with b empty nodes taken along the principal diagonal of the lattice are developed. The designs discussed have m > 1 associate classes and are said to be partially variance balanced. The special designs fall in two classes and are shown to be nearly optimal in a specified class of designs. A small catalog of designs constructed is provided and they should be useful when empty nodes do not represent wasted experimental units, perhaps because the row and column assignments of treatments are sequenced.     

Estimation of variance in bivariate normal distribution after the preliminary test of homogeneity

A preliminary test estimator of variance in the bivariate normal distribution is proposed after the Pitman–Morgan test of homogeneity of two variances. The bias and mean square error of the estimator are derived. The relative efficiency (RE) of the preliminary test estimator is studied. Computations and 3D graphs of RE for different parameters are analyzed. In order to get the maximum RE, recommendations of the significance level for the preliminary test are given for various sample sizes by using the max–min criterion.     

C472. The expected value of a conditional variance: An upper bound

The article derives Bartlett corrections for improving the chi-square approximation to the likelihood ratio statistics in a class of symmetric nonlinear regression models. This is a wide class of models which encompasses the t model and several other symmetric distributions with longer-than normal tails. In this paper we present, in matrix notation, Bartlett corrections to likelihood ratio statistics in nonlinear regression models with errors that follow a symmetric distribution. We generalize the results obtained by Ferrari, S. L. P. and Arellano-Valle, R. B. (1996). Modified likelihood ratio and score tests in linear regression models using the t distribution. Braz. J. Prob. Statist., 10, 15–33, who considered a t distribution for the errors, and by Ferrari, S. L. P. and Uribe-Opazo, M. A. (2001). Corrected likelihood ratio tests in a class of symmetric linear regression models. Braz. J. Prob. Statist., 15, 49–67, who considered a symmetric linear regression model. The formulae derived are simple enough to be used analytically to obtain several Bartlett corrections in a variety of important models. We also present simulation results comparing the sizes and powers of the usual likelihood ratio tests and their Bartlett corrected versions.     

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.     

On some test statistics for testing homogeneity of variances: a comparative study

In this paper, we have reviewed 25 test procedures that are widely reported in the literature for testing the hypothesis of homogeneity of variances under various experimental conditions. Since a theoretical comparison was not possible, a simulation study has been conducted to compare the performance of the test statistics in terms of robustness and empirical power. Monte Carlo simulation was performed for various symmetric and skewed distributions, number of groups, sample size per group, degree of group size inequalities, and degree of variance heterogeneity. Using simulation results and based on the robustness and power of the tests, some promising test statistics are recommended for practitioners.     

On some tests for independence in nonnormal situations: neyman'S C(A)test

A study is made of Neyman's C(a) test for testing independence in nonnormal situations. It is shown that it performs very well both in terms of the level of significance and the powereven for smallvalues of the samplesize. Also, in the case of the bivariate Polsson distribution, itis shown that Fisher's z and Student's t transforms of the sample correlation coefficient are good competitors for Neyman's procedure.

     

A circular-cone test for testing homogeneity against a simple tree order

Several methods exist for the problem of testing the equality of several treatments against the one-sided alternative that the treatments are better than the control. These methods include Dunnett's test, Bartholomew's likelihood-ratio test, the Abelson-Tukey-Schaafsma-Smid optimal-contrast test, and the multiple-contrast test of Mukerjee, Robertson, and Wright. A new test is proposed based on an approximation of the likelihood-ratio test of Bartholomew. This test involves using a circular cone in place of the alternative-hypothesis cone. The circular-cone test has excellent power characteristics similar to those of Bartholomew's test. Moreover, it has the advantages of being simpler to compute and may be used with unequal sample sizes.     

A fully nonparametric diagnostic test for homogeneity of variances

The authors propose a new nonparametric diagnostic test for checking the constancy of the conditional variance function σ²(x) in the regression model Y_i = m(x_i) + σ(x_i)?_i, i = 1,…, m. Their test, which does not assume a known parametric form for the conditional mean function m(x), is inspired by a recent asymptotic theory in the analysis of variance when the number of factor levels is large. The authors demonstrate through simulations the good finite‐sample properties of the test and illustrate its use in a study on the effect of drug utilization on health care costs.     

Interval estimates for variance components

A new approach to inference on variance components is propounded. This approach not only gives a new justification for Fisher's fiducial, distribution for the “between classes” component, but also leads to a new distribution for the “within classes” component. This latter distribution is studied, and has some intuitively very reasonable properties. Numerical results are given.     

A new random permutation test in ANOVA models

This paper developed an exact method of random permutations when testing both interaction and main effects in the two-way ANOVA model. The method of this paper can be regarded as a much improved model when compared with those of the previous studies such as Still and White (1981) and ter Braak (1992). We further conducted a simulation experiment in order to check the statistical performance of the proposed method. The proposed method works relatively well for small sample sizes compare with the existing methods. This work was supported by Korea Science and Engineering Foundation Grant (R14-2003-002-0100)     

Some competitors of the mood test for the two-sample scale problem

Let X_l,…,X_n (Y_l,…,Y_m) be a random sample from an absolutely continuous distribution with distribution function F(G).A class of distribution-free tests based on U-statistics is proposed for testing the equality of F and G against the alternative that X's are more dispersed then Y's. Let 2 ? C ? n and 2 ? d ? m be two fixed integers. Let ?_c,d(X_il,…,X_ic ; Y_jl,…,X_jd)=1(-1)when max as well as min of {X_il,…,X_ic ; Y_jl,…,Y_jd } are some X_i's (Y_j's)and zero oterwise. Let S_c,d be the U-statistic corresponding to ?_c,d.In case of equal sample sizes, S22 is equivalent to Mood's Statistic.Large values of S_c,d are significant and these tests are quite efficient