Research Article
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Year 2021, Volume: 70 Issue: 2, 1036 - 1054, 31.12.2021
https://doi.org/10.31801/cfsuasmas.800452

Abstract

References

  • Bradley, J. V., Robustness?, British Journal of Mathematical and Statistical Psychology, 31 (1978), 144-152. https://doi.org/10.1111/j.2044-8317.1978.tb00581.x.
  • Brunner, E., Dette, H., Munk, A., Box-type approximations in nonparametric factorial designs, Journal of the American Statistical Association, 92 (1997), 1494-150. https://doi.org/10.1080/01621459.1997.10473671.
  • Cavus, M., Yazici, B., Sezer, A., Modified tests for comparison of group means under heteroskedasticity and non-normality caused by outlier(s), Hacettepe Journal of Mathematics and Statistics, 46 (2017), 493-510. https://doi.org/10.15672/HJMS.2017.417.
  • Cavus, M., Yazici, B., Sezer, A., Analyzing regional export data by the modified generalized F-test, International Journal of Economic and Administrative Studies, (2018), 541-552. https://doi.org/10.18092/ulikidince.348070.
  • Cavus, M., Yazici, B., Testing the equality of normal distributed and independent groups’ means under unequal variances by doex package, The R Journal, 12 (2020), 134-154. https://doi.org/10.32614/RJ-2021-008.
  • Cavus, M., Yazici, B., Sezer, A., Penalized power approach to compare the power of the tests when Type I error probabilities are different, Communications in Statistics - Simulation and Computation, 50 (7) (2021), 1912-1926. https://doi.org/10.1080/03610918.2019.1588310.
  • Cribbie, R. A., Fiksenbaum, L., Keselman, H. J., Wilcox, R. R., Effect of non-normality on test statistics for one-way independent group designs, British Journal of Mathematical and Statistical Psychology, 65 (2012), 56-73. https://doi.org/10.1111/j.2044-8317.2011.02014.x.
  • Cribbie, R. A., Wilcox, R. R., Bewell, C., and Keselman, H. J., Tests for treatment group equality when data are nonnormal and heteroscedastic, Journal of Modern Applied Statistical Methods, 6 (2007), 117-132. https://doi.org/10.22237/jmasm/1177992660.
  • Fagerland, M. W., Sandvik, L., The Wilcoxon-Mann-Whitney test under scrutiny, Statistics in Medicine, 28 (2009), 1487-1497. https://doi.org/10.1002/sim.3561.
  • Huber, P. J., Robust estimation of a location parameter, Annals of Mathematical Statistics, 35 (1964), 73-101. https://doi.org/10.1214/aoms/1177703732.
  • Huber, P. J. Robust Statistics. New York: John Wiley and Sons, 2013. ISBN: 978-0-470-12990-6.
  • Karagoz, D., Saracbasi, T., Robust Brown-Forsythe and robust modified Brown-Forsythe ANOVA tests under heteroscedasticity for contaminatedWeibull distribution, Revista Colombiana de Estadistica, 39 (2016), 17-32. https://doi.org/10.15446/rce.v39n1.55135.
  • Keselman, H.J., Wilcox, R. R., Othman, A. R., Fradette, K., Trimming, transforming statistics and bootstrapping: circumventing the biasing effects of heteroscedasticity and non-normality, Journal of Modern Applied Statistical Methods, 1 (2002), 288-309. https://doi.org/10.22237/jmasm/1036109820.
  • Kruskal, W. H., Wallis, W. A., Use of ranks in one-criterion variance analysis, Journal of the American Statistical Association, 47 (1952), 583-621. https://doi.org/10.2307/2280779.
  • Kulinskaya, E., Dollinger, M. B., Robust weighted one-way ANOVA: improved approximation and efficiency, Journal of Statistical Planning and Inference, 137 (2007), 462-472. https://doi.org/10.1016/j.jspi.2006.01.008.
  • Luh, W. M., Guo, J. H., A powerful transformation trimmed mean method for one-way fixed effects ANOVA model under non-normality and inequality of variances, British Journal of Mathematical and Statistical Psychology, 52 (1999), 303-320. https://doi.org/10.1348/000711099159125.
  • Luh, W. M., Guo, J. H., Heteroscedastic test statistics for one-way analysis of variance: the trimmed means and Hall’s transformation conjunction. The Journal of Experimental Education, 74 (2005), 75-100. https://doi.org/10.3200/JEXE.74.1.75-100.
  • Ochuko, T. K., Abdullah, S., Zain, Z. B., Yahaya, S. S. S., The modification and evaluation of the Alexander-Govern tests in terms of power, Modern Applied Science, 9 (2015), 1-21. https://doi.org/10.5539/mas.v9n13p1.
  • Oshima, T., Algina, J., Type I error rates for James’s second-order test and Wilcox’s HM test under heteroscedasticity and non-normality, British Journal of Mathematical and Statistical Psychology, 45 (1992), 255-263. https://doi.org/10.1111/j.2044-8317.1992.tb00991.x.
  • Ozdemir, A. F., Wilcox, R. R. Yildiztepe, E., Comparing j independent groups with a method based on trimmed mean, Communications in Statistics-Simulation and Computation, 47 (2018), 852-863. https://doi.org/10.1080/03610918.2017.1295152.
  • Weerahandi, S., ANOVA under unequal error variances. Biometric, 51 (1995), 589-599. https://doi.org/10.2307/2532947.
  • Yazici, B., Cavus, M., A comparative study of computation approaches of the generalized F-test. Journal of Applied Statistics, 48 (2021), 2906-2919. https://doi.org/10.1080/02664763.2021.1939660.
  • Yusof, Z., Abdullah, S., Yahaya, S. S. S., Comparing the performance of modified Ft statistic with ANOVA and Kruskal Wallis test. Applied Mathematics and Information Sciences, 7 (2013), 403-408. https://doi.org/10.12785/amis/072L04.
  • Zhu, D., Zinde-Walsh, V., Properties of estimation of asymmetric exponential power distribution. Journal of Econometrics, 148 (2009), 89-99. https://doi.org/10.1016/j.jeconom.2008.09.038.

A revised generalized F-test for testing the equality of group means under non-normality caused by skewness

Year 2021, Volume: 70 Issue: 2, 1036 - 1054, 31.12.2021
https://doi.org/10.31801/cfsuasmas.800452

Abstract

The non-normality may occur in the data due to several reasons such as the presence of the outlier or skewness. It leads to lose the power and fail to control Type I error probability of the tests which are used to test the equality of the group means under heteroscedasticity. To overcome this problem, a revised generalized F-test (RGF) is proposed to test the equality of group means under heteroscedasticity in which non-normality caused by skewness in this study. An extensive Monte-Carlo simulation study is conducted to investigate the performance of the proposed test under several values of skewness for different number of groups. The proposed RGF is the best choice in the high level of skewness for k = 3, 4, 5. The Kruskal-Wallis test shows better performance than the others in small and moderate sample sizes for k = 6, and 7. It is shown that the proposed RGF test is superior than the non-parametric alternatives in the most of the conditions.

References

  • Bradley, J. V., Robustness?, British Journal of Mathematical and Statistical Psychology, 31 (1978), 144-152. https://doi.org/10.1111/j.2044-8317.1978.tb00581.x.
  • Brunner, E., Dette, H., Munk, A., Box-type approximations in nonparametric factorial designs, Journal of the American Statistical Association, 92 (1997), 1494-150. https://doi.org/10.1080/01621459.1997.10473671.
  • Cavus, M., Yazici, B., Sezer, A., Modified tests for comparison of group means under heteroskedasticity and non-normality caused by outlier(s), Hacettepe Journal of Mathematics and Statistics, 46 (2017), 493-510. https://doi.org/10.15672/HJMS.2017.417.
  • Cavus, M., Yazici, B., Sezer, A., Analyzing regional export data by the modified generalized F-test, International Journal of Economic and Administrative Studies, (2018), 541-552. https://doi.org/10.18092/ulikidince.348070.
  • Cavus, M., Yazici, B., Testing the equality of normal distributed and independent groups’ means under unequal variances by doex package, The R Journal, 12 (2020), 134-154. https://doi.org/10.32614/RJ-2021-008.
  • Cavus, M., Yazici, B., Sezer, A., Penalized power approach to compare the power of the tests when Type I error probabilities are different, Communications in Statistics - Simulation and Computation, 50 (7) (2021), 1912-1926. https://doi.org/10.1080/03610918.2019.1588310.
  • Cribbie, R. A., Fiksenbaum, L., Keselman, H. J., Wilcox, R. R., Effect of non-normality on test statistics for one-way independent group designs, British Journal of Mathematical and Statistical Psychology, 65 (2012), 56-73. https://doi.org/10.1111/j.2044-8317.2011.02014.x.
  • Cribbie, R. A., Wilcox, R. R., Bewell, C., and Keselman, H. J., Tests for treatment group equality when data are nonnormal and heteroscedastic, Journal of Modern Applied Statistical Methods, 6 (2007), 117-132. https://doi.org/10.22237/jmasm/1177992660.
  • Fagerland, M. W., Sandvik, L., The Wilcoxon-Mann-Whitney test under scrutiny, Statistics in Medicine, 28 (2009), 1487-1497. https://doi.org/10.1002/sim.3561.
  • Huber, P. J., Robust estimation of a location parameter, Annals of Mathematical Statistics, 35 (1964), 73-101. https://doi.org/10.1214/aoms/1177703732.
  • Huber, P. J. Robust Statistics. New York: John Wiley and Sons, 2013. ISBN: 978-0-470-12990-6.
  • Karagoz, D., Saracbasi, T., Robust Brown-Forsythe and robust modified Brown-Forsythe ANOVA tests under heteroscedasticity for contaminatedWeibull distribution, Revista Colombiana de Estadistica, 39 (2016), 17-32. https://doi.org/10.15446/rce.v39n1.55135.
  • Keselman, H.J., Wilcox, R. R., Othman, A. R., Fradette, K., Trimming, transforming statistics and bootstrapping: circumventing the biasing effects of heteroscedasticity and non-normality, Journal of Modern Applied Statistical Methods, 1 (2002), 288-309. https://doi.org/10.22237/jmasm/1036109820.
  • Kruskal, W. H., Wallis, W. A., Use of ranks in one-criterion variance analysis, Journal of the American Statistical Association, 47 (1952), 583-621. https://doi.org/10.2307/2280779.
  • Kulinskaya, E., Dollinger, M. B., Robust weighted one-way ANOVA: improved approximation and efficiency, Journal of Statistical Planning and Inference, 137 (2007), 462-472. https://doi.org/10.1016/j.jspi.2006.01.008.
  • Luh, W. M., Guo, J. H., A powerful transformation trimmed mean method for one-way fixed effects ANOVA model under non-normality and inequality of variances, British Journal of Mathematical and Statistical Psychology, 52 (1999), 303-320. https://doi.org/10.1348/000711099159125.
  • Luh, W. M., Guo, J. H., Heteroscedastic test statistics for one-way analysis of variance: the trimmed means and Hall’s transformation conjunction. The Journal of Experimental Education, 74 (2005), 75-100. https://doi.org/10.3200/JEXE.74.1.75-100.
  • Ochuko, T. K., Abdullah, S., Zain, Z. B., Yahaya, S. S. S., The modification and evaluation of the Alexander-Govern tests in terms of power, Modern Applied Science, 9 (2015), 1-21. https://doi.org/10.5539/mas.v9n13p1.
  • Oshima, T., Algina, J., Type I error rates for James’s second-order test and Wilcox’s HM test under heteroscedasticity and non-normality, British Journal of Mathematical and Statistical Psychology, 45 (1992), 255-263. https://doi.org/10.1111/j.2044-8317.1992.tb00991.x.
  • Ozdemir, A. F., Wilcox, R. R. Yildiztepe, E., Comparing j independent groups with a method based on trimmed mean, Communications in Statistics-Simulation and Computation, 47 (2018), 852-863. https://doi.org/10.1080/03610918.2017.1295152.
  • Weerahandi, S., ANOVA under unequal error variances. Biometric, 51 (1995), 589-599. https://doi.org/10.2307/2532947.
  • Yazici, B., Cavus, M., A comparative study of computation approaches of the generalized F-test. Journal of Applied Statistics, 48 (2021), 2906-2919. https://doi.org/10.1080/02664763.2021.1939660.
  • Yusof, Z., Abdullah, S., Yahaya, S. S. S., Comparing the performance of modified Ft statistic with ANOVA and Kruskal Wallis test. Applied Mathematics and Information Sciences, 7 (2013), 403-408. https://doi.org/10.12785/amis/072L04.
  • Zhu, D., Zinde-Walsh, V., Properties of estimation of asymmetric exponential power distribution. Journal of Econometrics, 148 (2009), 89-99. https://doi.org/10.1016/j.jeconom.2008.09.038.
There are 24 citations in total.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Research Articles
Authors

Mustafa Çavuş 0000-0002-6172-5449

Berna Yazıcı 0000-0001-9843-7355

Ahmet Sezer 0000-0002-5962-4999

Publication Date December 31, 2021
Submission Date October 5, 2020
Acceptance Date June 16, 2021
Published in Issue Year 2021 Volume: 70 Issue: 2

Cite

APA Çavuş, M., Yazıcı, B., & Sezer, A. (2021). A revised generalized F-test for testing the equality of group means under non-normality caused by skewness. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70(2), 1036-1054. https://doi.org/10.31801/cfsuasmas.800452
AMA Çavuş M, Yazıcı B, Sezer A. A revised generalized F-test for testing the equality of group means under non-normality caused by skewness. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. December 2021;70(2):1036-1054. doi:10.31801/cfsuasmas.800452
Chicago Çavuş, Mustafa, Berna Yazıcı, and Ahmet Sezer. “A Revised Generalized F-Test for Testing the Equality of Group Means under Non-Normality Caused by Skewness”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70, no. 2 (December 2021): 1036-54. https://doi.org/10.31801/cfsuasmas.800452.
EndNote Çavuş M, Yazıcı B, Sezer A (December 1, 2021) A revised generalized F-test for testing the equality of group means under non-normality caused by skewness. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70 2 1036–1054.
IEEE M. Çavuş, B. Yazıcı, and A. Sezer, “A revised generalized F-test for testing the equality of group means under non-normality caused by skewness”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 70, no. 2, pp. 1036–1054, 2021, doi: 10.31801/cfsuasmas.800452.
ISNAD Çavuş, Mustafa et al. “A Revised Generalized F-Test for Testing the Equality of Group Means under Non-Normality Caused by Skewness”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70/2 (December 2021), 1036-1054. https://doi.org/10.31801/cfsuasmas.800452.
JAMA Çavuş M, Yazıcı B, Sezer A. A revised generalized F-test for testing the equality of group means under non-normality caused by skewness. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2021;70:1036–1054.
MLA Çavuş, Mustafa et al. “A Revised Generalized F-Test for Testing the Equality of Group Means under Non-Normality Caused by Skewness”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 70, no. 2, 2021, pp. 1036-54, doi:10.31801/cfsuasmas.800452.
Vancouver Çavuş M, Yazıcı B, Sezer A. A revised generalized F-test for testing the equality of group means under non-normality caused by skewness. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2021;70(2):1036-54.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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