Yıl 2017, Cilt 46 , Sayı 3, Sayfalar 493 - 510 2017-06-01

Modified tests for comparison of group means under heteroskedasticity and non-normality caused by outlier(s)

Mustafa Cavus [1] , Berna Yazici [2] , Ahmet Sezer [3]


There are several approximate tests proposed such as Welch's F-test (W), the Parametric Bootstrap Test (PB) and Generalized F-test (GF) for comparing several group means under heteroskedasticity. These tests are powerful and have nominal type 1 error rates but they are not performing satisfactorily under non-normality caused by outlier(s). To handle this problem, we investigate tests that are powerful and provide nominal type 1 error rates by using robust estimators both for location and scale parameters. The performance of the modified tests are examined with Monte-Carlo simulation studies. Results of simulations clearly indicate that Generalized F-test modied with Huber's M-estimators achieves the nominal type 1 error rate and provide higher power than alternative methods.
Outlier, Heteroskedasticity, Unbalanced, Huber's M-Estimators, Generalized F-test
  • Barnett, V. and Lewis, T. Outlier in Statistical Data, Wiley and Sons, 1994.
  • Ben, M.G. and Yohai, V.J. Robust analysis of variance for randomized block design, Com- munications in Statistics 216, 1779-1798, 1992.
  • Braun, H.I. and McNeil, D.R. Testing in robust ANOVA, Communications in Statistics 102, 149-165, 1981.
  • Chang, C.H., Pal, N., Lim, W.K., Lin, J.J. Comparing several population means: a para- metric bootstrap method and its comparison with usual ANOVA F-test as well as ANOM, Computational Statistics 25, 71-95, 2010.
  • Chang, C.H., Lin, J.J., Pal, N. Testing the equality of several gamma means: a parametric bootstrap method with applications, Computational Statistics 76, 26-55, 2011.
  • Cavus, M., Yazici, B., Sezer, A. Comparison of means under non-normality with outliers for unbalanced data, 8th International Conference of the ERCIM Working Group on Computing and Statistics Proceedings, London, ENGLAND, 2015.
  • Cavus, M., Yazici, B., Sezer, A. Robust methods to compare means under violation of nor- mality with outliers, 12th German Probability and Statistics Days Proceedings, Bochum, GERMANY, 2016.
  • Cavus, M., Yazici, B., Sezer, A. Modified tests for comparison of group means in the presence of outlier(s), 2th International Researchers-Statisticians and Young Statisticians Congress Proceedings, Ankara, TURKEY, 2016.
  • Fan, W. and Hancock, G. Robust mean modelling: an alternative for hypothesis testing of independent means under variance heterogeneity and nonnormality, Journal of Educational and Behavioral Statistics 37, 137-156, 2012.
  • Fisher, R.A. Statistical methods and scientific inference, Oliver and Boyd, 1925.
  • Gamage, J. and Weerahandi, S. Size performance of some tests in one-way ANOVA, Com- munications in Statistics 273, 625-640, 1998.
  • Gokpinar Yigit, E., Polat, E., Gokpinar, F., Gunay, S. A new computational approach for testing equality of inverse Gaussian means under heterogeneity, Hacettepe Journal of Mathematics and Statistics 42 5, 581-590, 2013.
  • Huber, P.J. Robust estimation of a location parameter, The Annals of Mathematical Statis- tics 69, 73-101, 1964.
  • Huber, P.J. and Ronchetti, E.M. Robust Statistics, John Wiley and Sons, New Jersey, 2009.
  • Jian-Hong, S. and Jiang-Long, L. A new generalized p-value for testing equality of inverse Gaussian means under heterogeneity, Statistics and Probability Letters 82, 96-102, 2012.
  • Karagoz, D. Modied Welch test statistic for ANOVA under Weibull distribution, Hacettepe Journal of Mathematics and Statistics 45 56, 2015.
  • Karagoz, D. and Saracbasi, T. Robust Brown-Forsythe and robust modified Brown-Forsythe ANOVA tests under heteroscedasticity for contaminated Weibull distribution, Revista Colombiana de Estadistica 39, 17-32, 2009.
  • Krishnamoorthy, K., Lu F., Mathew, T. A parametric bootstrap approach for ANOVA with unequal variances: Fixed and random models, Computational Statistics and Data Analysis 51, 5731-5742, 2007.
  • Lee, H. and Fung K.Y. Robust procedures for multi-sample location problems with unequal group variances, Journal of Statistical Computation and Simulation 18, 125-143, 1983.
  • Lee, S. and Ahn, C.H. Modified ANOVA for unequal variances, Communications in Statistics 32 4, 987-1004, 2003.
  • Li, X. A generalized p-value approach for comparing the means of several log-normal populations, Statistics and Probability Letters 79, 1404-1408, 2009.
  • Ozkip, E., Yazici, B., Sezer, A. Comparing the means for unbalanced several skewed data with unequal variances, 5th International Conference of the ERCIM Working Group on Computing and Statistics Proceedings , 2012a.
  • Ozkip, E., Yazici, B., Sezer, A. Comparison of population means for skewed data with unequal variances, Joint Statistical Meetings 2012 Proceedings , 2524-2529, 2012b.
  • Ozkip, E., Yazici, B., Sezer, A. Inference on the mean of unbalanced several lognormal population using generalized p-value and generalized condence intervals, 8th International Symposium of Statistics Proceedings , 2012c.
  • Ozkip, E., Yazici, B., Sezer, A. Performance of some tests compares for equality of means in skewed distribution data, Joint Statistical Meetings Proceedings, Montreal, CANADA, 2013.
  • Ozkip, E., Yazici, B., Sezer, A. A simulation study on tests for the Behrens-Fisher problem, Turkiye Klinikleri Biyoistatistik Dergisi,6 2, 59-66, 2014.
  • Ozkip, E., Yazici, B., Sezer, A. Comparison of tests for the ANOVA with unequal variance, Joint Statistical Meetings Proceedings, Boston, USA, 2014.
  • Reed, J.F. and Stark, D.B. Robust analysis of variance: a simulation study, Journal of Applied Statistics 22 1, 87-104, 1995.
  • Rousseeuw, P. and Croux, C. Alternatives to the median absolute deviation, Journal of American Statistical Association 88 424, 1273-1283, 1993.
  • Sadooghi-Alvandi, S.M., Jafari, A.A., Mardani-Ford, H.A. One-way ANOVA with unequal variances, Communications in Statistics 41, 4200-4221, 2012.
  • Schrader, R.M. Robust analysis of variance, Communications in Statistics 6 9, 879-894, 1977.
  • So, Y.C. and Sen, P.K. M-estimators based repeated signicance tests for one-way ANOVA with adaptation to multiple comparisons, Communications in Statistics 1 2, 101-119, 1982.
  • Tan, W.Y. and Tabatabai, M.A. Some robust ANOVA procedures under heteroskedasticity and non-normality, Communications in Statistics 14 4, 1007-1026, 1985.
  • Tiku, M. and Akkaya, A. Robust Estimation and Hypothesis Testing, New Age International, 2004.
  • Tsui, K. and Weerahandi, S. Generalized p-values in signicance testing of hypotheses in the presence of nuisance parameters, Journal of the American Statistical Association 84, 602-607, 1989.
  • Weerahandi, S. Testing variance components in mixed models with generalized p-values, Journal of the American Statistical Association 86, 151-153, 1991.
  • Weerahandi, S. ANOVA under unequal error variances, Biometrics 51, 589-599, 1994.
  • Weerahandi, S. Exact Statistical Methods for Data Analysis, Springer, 1995.
  • Weerahandi, S. Generalized Inference in Repeated Measures: Exact Methods in MANOVA and Mixed Models, Wiley, 2004.
  • Welch, B.L. On the comparison of several group means values, Biometrika 38, 330-336, 1951.
  • Wilcox, R.R. Simulation results on solutions to the multivariate Behrens-Fisher problem via trimmed means, The Statistician 44 2, 213-225, 1995.
  • Wilcox, R.R. Introduction to Robust Estimation and Hypothesis Testing, Elsevier, 2012.
  • Wu, L.L. Robust M-estimation of location and regression, Sociological Methodology 15, 316-388, 1985.
  • Yazici, B., Sezer, A., Ozkip, E. A simulation study to compare several population means under heteroskedasticity, Abstract Book of 5th International Conference of the ERCIM Working Group on Computing and Statistics, Oviedo, SPAIN, 2012.
  • Yazici, B., Sezer, A., Ozkip, E. Generalized p-value test to compare several population means for unequal variances, Joint Statistical Meetings Proceedings, San Diego, USA, 2012.
Birincil Dil en
Konular Matematik
Bölüm İstatistik
Yazarlar

Yazar: Mustafa Cavus (Sorumlu Yazar)

Yazar: Berna Yazici

Yazar: Ahmet Sezer

Tarihler

Yayımlanma Tarihi : 1 Haziran 2017

Bibtex @araştırma makalesi { hujms451019, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe Üniversitesi}, year = {2017}, volume = {46}, pages = {493 - 510}, doi = {}, title = {Modified tests for comparison of group means under heteroskedasticity and non-normality caused by outlier(s)}, key = {cite}, author = {Cavus, Mustafa and Yazici, Berna and Sezer, Ahmet} }
APA Cavus, M , Yazici, B , Sezer, A . (2017). Modified tests for comparison of group means under heteroskedasticity and non-normality caused by outlier(s). Hacettepe Journal of Mathematics and Statistics , 46 (3) , 493-510 . Retrieved from https://dergipark.org.tr/tr/pub/hujms/issue/38758/451019
MLA 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://dergipark.org.tr/tr/pub/hujms/issue/38758/451019>
Chicago 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
RIS TY - JOUR T1 - Modified tests for comparison of group means under heteroskedasticity and non-normality caused by outlier(s) AU - Mustafa Cavus , Berna Yazici , Ahmet Sezer Y1 - 2017 PY - 2017 N1 - DO - T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 493 EP - 510 VL - 46 IS - 3 SN - 2651-477X-2651-477X M3 - UR - Y2 - 2016 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics Modified tests for comparison of group means under heteroskedasticity and non-normality caused by outlier(s) %A Mustafa Cavus , Berna Yazici , Ahmet Sezer %T Modified tests for comparison of group means under heteroskedasticity and non-normality caused by outlier(s) %D 2017 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 46 %N 3 %R %U
ISNAD Cavus, Mustafa , Yazici, Berna , Sezer, Ahmet . "Modified tests for comparison of group means under heteroskedasticity and non-normality caused by outlier(s)". Hacettepe Journal of Mathematics and Statistics 46 / 3 (Haziran 2017): 493-510 .
AMA 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. 2017; 46(3): 493-510.
Vancouver 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. 2017; 46(3): 510-493.