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

Güven, Gamze; Gürer, Özge; Şamkar, Hatice; Şenoğlu, Birdal

doi:10.1080/00949655.2019.1593985

Cited by 4 publications

(4 citation statements)

References 30 publications

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“…However, nonnormal distributions are encountered more frequently in practice. Therefore, Guven et al [5] considered robust version of the Cochran test statistic based on modified maximum likelihood (MML) estimators, denoted as RC M M L , and fiducial based test using RC M M L for testing the equality of means when the underlying distribution is long-tailed symmetric (LTS). MML estimators proposed by Tiku [16,17] are asymptotically equivalent to the maximum likelihood (ML) estimators and more efficient than the LS estimators under nonnormality.…”

Section: G Guvenmentioning

confidence: 99%

“…Remark 1. C LS test statistic given in (2) and RC M M L test statistic given in (5) are asymptotically distributed as chi-square with a − 1 degrees of freedom, see [5,9] for details. However, as mentioned earlier, the null distribution of these test statistics are unknown for small and moderate samples.…”

Section: Rc M M L Test Proposed By Güven Et Al In 2019 Is Given As Fo...mentioning

confidence: 99%

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Testing equality of means in one-way ANOVA using three and four moment approximations

GUVEN

2023

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

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In this study, we focus on two test statistics for testing the equality of treatment means in one-way analysis of variance (ANOVA). The first one is the well known Cochran ($C_{LS}$) test statistic based on least squares (LS) estimators and the second one is robust version of it ($RC_{MML}$) based on modified maximum likelihood (MML) estimators. These two test statistics are asymptotically distributed as chi-square. However, distributions of them are unknown for small samples. Therefore, three-moment chi-square and four moment $F$ approximations to the null distributions of $C_{LS}$ and $RC_{MML}$ are derived inspired by Tiku and Wong [19]. To investigate the small and moderate sample properties of these tests based on the mentioned approximations, an extensive Monte-Carlo simulation study is performed when the underlying distribution is long-tailed symmetric (LTS). Simulation results show that four-moment $F$ approximation provides better approximation than the three-moment chi-square approximation for both $C_{LS}$ and $RC_{MML}$ tests. Therefore, the simulated Type I error rates and powers of the $C_{LS}$ and $RC_{MML}$ test statistics are calculated using four-moment $F$ approximation. According to simulation results, $RC_{MML}$ test is more powerful than the corresponding $C_{LS}$ test.

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Section: G Guvenmentioning

confidence: 99%

Section: Rc M M L Test Proposed By Güven Et Al In 2019 Is Given As Fo...mentioning

confidence: 99%

Testing equality of means in one-way ANOVA using three and four moment approximations

GUVEN

2023

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

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“…However, the performance of the F test can be negatively affected in case of violation of these assumptions [22]. Therefore, various alternative test procedures have been proposed in the literature when the normality and/or homogeneity of variances assumptions are not satisfied; see for example [9,13,14,16,17,19,24,28,34,36,37], etc.…”

Section: Introductionmentioning

confidence: 99%

Testing the equality of treatment means in one-way ANOVA: Short-tailed symmetric error terms with heterogeneous variances

Güven

2022

Hacettepe Journal of Mathematics and Statistics

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We propose two tests based on fiducial and generalized $p$-value approaches for testing the equality of treatment means in one-way analysis of variance (ANOVA). Modified maximum likelihood (MML) estimators are used in the proposed tests. In contrast to least squares (LS) estimators, MML estimators are highly efficient and robust to plausible deviations from an assumed distribution and to mild data anomalies. In this study, error terms are assumed to have short-tailed symmetric (STS) distributions with heterogeneous variances. The performances of the proposed tests are compared with the fiducial based test using bias-corrected LS estimators via an extensive Monte Carlo simulation study. Finally, two real datasets are analyzed for illustrative purposes.

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“…In recent years, there are interest and new applications of one-way ANOVA in many fields. Guven et al [15] proposed a new test for equality of the treatment averages in one-way ANOVA when the assumptions of normality and variance homogeneity are not satisfied where they assumed the error term had a long tailed symmetric distribution with a normal distribution as limiting case. Kim [20] studied one-way ANOVA based on conceptual figures where he showed that the conceptual figures served as a convenient guide to clarify the average difference problems by using treatment and error population variance differences.…”

Section: Introductionmentioning

confidence: 99%

Simultaneous Test for Means: An Unblind Way to the F-test in One-way Analysis of Variance

Elamir

2021

Stat., optim. inf. comput.

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After rejecting the null hypothesis in the analysis of variance, the next step is to make the pairwise comparisons to find out differences in means. The purpose of this paper is threefold. The foremost aim is to suggest expression for calculating decision limit that enables us to collect the test and pairwise comparisons in one step. This expression is proposed as the ratio of between square for each treatment and within sum of squares for all treatments. The second aim is to obtain the sampling distribution of the proposed ratio under the null hypothesis. This sampling distribution is derived exactly as the beta distribution of the second type. The third aim is to use beta distribution of second type and adjusted p-values to create adjusted points and decision limit. Therefore, reject the null hypothesis of equal means if any adjusted point falls outside the decision limit. Simulation study is conducted to compute type I error. The results show that the proposed method controls the type I error near the nominal values using Benjamini-Hochberg’adjusted p-values. Two applications are given to show the benefits of the proposed method.

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

Cited by 4 publications

References 30 publications

Testing equality of means in one-way ANOVA using three and four moment approximations

Testing equality of means in one-way ANOVA using three and four moment approximations

Testing the equality of treatment means in one-way ANOVA: Short-tailed symmetric error terms with heterogeneous variances

Simultaneous Test for Means: An Unblind Way to the F-test in One-way Analysis of Variance

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