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

Year 2017, Volume: 46 Issue: 3, 493 - 510, 01.06.2017

Abstract

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.

References

  • 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.
Year 2017, Volume: 46 Issue: 3, 493 - 510, 01.06.2017

Abstract

References

  • 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.
There are 45 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Statistics
Authors

Mustafa Cavus

Berna Yazici

Ahmet Sezer

Publication Date June 1, 2017
Published in Issue Year 2017 Volume: 46 Issue: 3

Cite

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.
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. June 2017;46(3):493-510.
Chicago Cavus, Mustafa, Berna Yazici, and Ahmet Sezer. “Modified Tests for Comparison of Group Means under Heteroskedasticity and Non-Normality Caused by outlier(s)”. Hacettepe Journal of Mathematics and Statistics 46, no. 3 (June 2017): 493-510.
EndNote Cavus M, Yazici B, Sezer A (June 1, 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.
IEEE M. Cavus, B. Yazici, and A. Sezer, “Modified tests for comparison of group means under heteroskedasticity and non-normality caused by outlier(s)”, Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 3, pp. 493–510, 2017.
ISNAD Cavus, Mustafa et al. “Modified Tests for Comparison of Group Means under Heteroskedasticity and Non-Normality Caused by outlier(s)”. Hacettepe Journal of Mathematics and Statistics 46/3 (June 2017), 493-510.
JAMA 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;46:493–510.
MLA Cavus, Mustafa et al. “Modified Tests for Comparison of Group Means under Heteroskedasticity and Non-Normality Caused by outlier(s)”. Hacettepe Journal of Mathematics and Statistics, vol. 46, no. 3, 2017, pp. 493-10.
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):493-510.