A fiducial-based approach to the one-way ANOVA in the presence of nonnormality and heterogeneous error variances

In this study, we propose a new test for testing the equality of the treatment means in one-way ANOVA when the usual normality and the homogeneity of variances assumptions are not met. In developing the proposed test, we benefit from the Fisher's fiducial inference [1–3]. Distribution of the error terms is assumed to be long-tailed symmetric (LTS) which includes the normal distribution as a limiting case. Modified maximum likelihood (MML) estimators are used in the test statistics rather than the traditional least squares (LS) estimators, since LS estimators have very low efficiencies under nonnormal distributions, see Tiku [4] for the details of MML methodology. An extensive Monte Carlo simulation study is done to compare the efficiency of the proposed test with the corresponding test based on normal theory, see Li et al. [5]. Finally, we give a real life example to show the applicability of the proposed methodology.     

Dimension test approach of heteroscedasticity in the linear model

Heteroscedasticity testing has a long history and is still an important matter in the linear model. There exist many types of tests, but they are limited in use to their own specific cases and sensitive to normality. Here, we propose a dimension test approach to heteroscedasticity. The proposed test overcomes the shortcomings of the existing methods, so that it is robust to normality and is unified in sense that it is applicable in the linear model with multi-dimensional response. Numerical studies confirm that the proposed test is favorable over the existing tests with moderate sample sizes, and real data analysis is presented.     

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.     

Uniformly implementable small sample integrated likelihood ratio test for one-way and two-way ANOVA under heteroscedasticity and normality

AStA Advances in Statistical Analysis - ANOVA under normally distributed response and heteroscedastic variances is commonly encountered in biological, behavioral, educational and agricultural...     

Robust estimation and testing in one-way ANOVA for Type II censored samples: skew normal error terms

Censoring can be occurred in many statistical analyses in the framework of experimental design. In this study, we estimate the model parameters in one-way ANOVA under Type II censoring. We assume that the distribution of the error terms is Azzalini's skew normal. We use Tiku's modified maximum likelihood (MML) methodology which is a modified version of the well-known maximum likelihood (ML) in the estimation procedure. Unlike ML methodology, MML methodology is non-iterative and gives explicit estimators of the model parameters. We also propose new test statistics based on the proposed estimators. The performances of the proposed estimators and the test statistics based on them are compared with the corresponding normal theory results via Monte Carlo simulation study. A real life data is analysed to show the implementation of the methodology presented in this paper at the end of the study.     

Testing for multivariate heteroscedasticity

In this article, we propose a testing technique for multivariate heteroscedasticity, which is expressed as a test of linear restrictions in a multivariate regression model. Four test statistics with known asymptotical null distributions are suggested, namely the Wald, Lagrange multiplier (LM), likelihood ratio (LR) and the multivariate Rao F-test. The critical values for the statistics are determined by their asymptotic null distributions, but bootstrapped critical values are also used. The size, power and robustness of the tests are examined in a Monte Carlo experiment. Our main finding is that all the tests limit their nominal sizes asymptotically, but some of them have superior small sample properties. These are the F, LM and bootstrapped versions of Wald and LR tests.     

Testing the equality of several log-normal means based on a computational approach

In this study, we propose a new test based on a computational approach to test the equality of several log-normal means. We compare this test with some existing methods in terms of the type-I error rate and power using Monte Carlo simulations for varying values of number of groups and sample sizes. The simulation results indicate that the proposed test could be suggested as a good alternative for testing the equality of several log-normal means.     

A note on fiducial generalized pivots for in one-way heteroscedastic ANOVA with random effects

The paper deals with generalized confidence intervals for the between-group variance in one-way heteroscedastic (unbalanced) ANOVA with random effects. The approach used mimics the standard one applied in mixed linear models with two variance components, where interval estimators are based on a minimal sufficient statistic derived after an initial reduction by the principle of invariance. A minimal sufficient statistic under heteroscedasticity is found to resemble its homoscedastic counterpart and further analogies between heteroscedastic and homoscedastic cases lead us to two classes of fiducial generalized pivots for the between-group variance. The procedures suggested formerly by Wimmer and Witkovský [Between group variance component interval estimation for the unbalanced heteroscedastic one-way random effects model, J. Stat. Comput. Simul. 73 (2003), pp. 333–346] and Li [Comparison of confidence intervals on between group variance in unbalanced heteroscedastic one-way random models, Comm. Statist. Simulation Comput. 36 (2007), pp. 381–390] are found to belong to these two classes. We comment briefly on some of their properties that were not mentioned in the original papers. In addition, properties of another particular generalized pivot are considered.     

Bootstrap test for a structural break under possible heteroscedasticity

In this article, we consider the Wald test statistic for testing equality between the sets of regression coefficients in two linear regression models when the disturbance variances may possibly be unequal. This test can be also used as a test for a structural break. However, it is well known that the test based on the Wald test statistic suffers from severe size distortion in small sample when the disturbance variances of the two regression models are unequal. Our simulation results show that substantial improvements are made when the bootstrap methods are applied.     

A note on Cochran test for homogeneity in one-way ANOVA and meta-analysis

In this paper we provide a formal yet simple and straightforward proof of the asymptotic χ² distribution for Cochran test statistic. Then, we show that the general form of this type of test statistics is invariant for the choice of weights. This fact is important since in practice many such test statistics are constructed with more complicated forms which usually require calculating generalized inverse matrices. Based on our results, we can simplify the construction of the test statistics. More importantly, properties such as anti-conservativeness of this type of test statistics can be drawn from Cochran test statistic. Furthermore, one can improve the performance of the tests by using some modified statistics with correction for small sample size situations.     

A nonparametric test of conditional autoregressive heteroscedasticity for threshold autoregressive models

Threshold autoregressive models are widely used in time‐series applications. When building or using such a model, it is important to know whether conditional heteroscedasticity exists. The authors propose a nonparametric test of this hypothesis. They develop the large‐sample theory of a test of nonlinear conditional heteroscedasticity adapted to nonlinear autoregressive models and study its finite‐sample properties through simulations. They also provide percentage points for carrying out this test, which is found to have very good power overall.     

Computational approach test for inference about several correlation coefficients: Equality and common

In this article, we consider inference about the correlation coefficients of several bivariate normal distributions. We first propose computational approach tests for testing the equality of the correlation coefficients. In fact, these approaches are parametric bootstrap tests, and simulation studies show that they perform very satisfactory, and the actual sizes of these tests are better than other existing approaches. We also present a computational approach test and a parametric bootstrap confidence interval for inference about the parameter of common correlation coefficient. At the end, all the approaches are illustrated using two real examples.     

A Bayes Factor for Replications of ANOVA Results

ABSTRACT

With an increasing number of replication studies performed in psychological science, the question of how to evaluate the outcome of a replication attempt deserves careful consideration. Bayesian approaches allow to incorporate uncertainty and prior information into the analysis of the replication attempt by their design. The Replication Bayes factor, introduced by Verhagen and Wagenmakers (2014 Verhagen, J., and Wagenmakers, E.-J. (2014), “Bayesian Tests to Quantify the Result of a Replication Attempt,” Journal of Experimental Psychology: General, 143, 1457–1475. DOI: 10.1037/a0036731.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), provides quantitative, relative evidence in favor or against a successful replication. In previous work by Verhagen and Wagenmakers (2014 Verhagen, J., and Wagenmakers, E.-J. (2014), “Bayesian Tests to Quantify the Result of a Replication Attempt,” Journal of Experimental Psychology: General, 143, 1457–1475. DOI: 10.1037/a0036731.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), it was limited to the case of t-tests. In this article, the Replication Bayes factor is extended to F-tests in multigroup, fixed-effect ANOVA designs. Simulations and examples are presented to facilitate the understanding and to demonstrate the usefulness of this approach. Finally, the Replication Bayes factor is compared to other Bayesian and frequentist approaches and discussed in the context of replication attempts. R code to calculate Replication Bayes factors and to reproduce the examples in the article is available at https://osf.io/jv39h/.     

An approximate degrees of freedom test for heteroscedastic two-way ANOVA

Heteroscedastic two-way ANOVA are frequently encountered in real data analysis. In the literature, classical F-tests are often blindly employed although they are often biased even for moderate heteroscedasticity. To overcome this problem, several approximate tests have been proposed in the literature. These tests, however, are either too complicated to implement or do not work well in terms of size controlling. In this paper, we propose a simple and accurate approximate degrees of freedom (ADF) test. The ADF test is shown to be invariant under affine-transformations, different choices of contrast matrix for the same null hypothesis, or different labeling schemes of cell means. Moreover, it can be conducted easily using the usual F-distribution with one unknown degree of freedom estimated from the data. Simulations demonstrate that the ADF test works well in various cell sizes and parameter configurations but the classical F-tests work badly when the cell variance homogeneity assumption is violated. A real data example illustrates the methodologies.     

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.     

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)     

A new generalized confidence interval for the among-group variance in the heteroscedastic one-way random effects model

Based on the generalized inference idea, a new kind of generalized confidence intervals is derived for the among-group variance component in the heteroscedastic one-way random effects model. We construct structure equations of all variance components in the model based on their minimal sufficient statistics; meanwhile, the fiducial generalized pivotal quantity (FGPQ) can be obtained through solving an implicit equation of the parameter of interest. Then, the confidence interval is derived naturally from the FGPQ. Simulation results demonstrate that the new procedure performs very well in terms of both empirical coverage probability and average interval length.     

Empirical likelihood based diagnostics for heteroscedasticity in semiparametric varying-coefficient partially linear errors-in-variables models

In this paper, we propose an empirical likelihood based diagnostic technique for heteroscedasticity in the semiparametric varying-coefficient partially linear errors-in-variables models. Under mild conditions, a nonparametric version of Wilk’s theorem is derived. Simulation results reveal that our test performs well in both size and power.     

ANOVA models for Brownian motion

We investigate longitudinal models having Brownian-motion covariance structure. We show that any such model can be viewed as arising from a related “timeless” classical linear model where sample sizes correspond to longitudinal observation times. This relationship is of practical impact when there are closed-form ANOVA tables for the related classical model. Such tables can be directly transformed into the analogous tables for the original longitudinal model. We in particular provide complete results for one-way fixed and random effects ANOVA on the drift parameter in Brownian motion, and illustrate its use in estimating heterogeneity in tumor growth rates.