A Modification of ANOVA with Trimmed Mean
Keywords:
anova, modified anova, arithmetic mean, trimmed mean, type I error
Abstract
Analysis of Variance (ANOVA) is a well-known method to test the equality of mean for two or more groups. ANOVA is a robust test under the normality assumption. Arithmetic mean is used in the computation of the ANOVA test. Mean is known to be sensitive towards outlier and this problem will affect the robustness and power of ANOVA. In this study, modification of ANOVA was created using one type of mean to replace arithmetic mean namely trimmed mean. New approaches were be obtained for the computation of ANOVA. This study was conducted based on a simulation study and application on real data. The performance of the modified ANOVA is then compared with the classical ANOVA test in terms of Type I error rate. This innovation enhances the ability of modified ANOVA to provide good control of Type I error rates. The findings were in favor of the modified ANOVA or better known as ANOVATM.
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References
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Bieniek, M. (2016). Sharp bounds on the bias of trimean. 57, 365-379.
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Guo, J. H. and Luh, W. M. (2000). Statistics & Probability Letters, 49, 1-7.
Hasan, M. K., Othman, N. A., Karim, S. A. and Sulaiman, J. (2018). Semi Non-Standard Trimean Algorithm for Rosenzweig-MacArthur Interaction Model. International Journal on Advanced Science, Engineering and Information Technology, 8(4-2), 1520.
Heeren, T. and D'Agostino, R. B. (1987). Statistics in Medicine. 6, 79-90.
Huber, P. J. (2004). Robust statistics. New York: Wiley.
Hunter, M. A. and May, R. B. (1993). Some myths concerning parametric and nonparametric tests. Canadian Psychology/Psychologie canadienne, 34, 384-389.
Inc., S. I. (2006). SAS online doc. Cary, NC: SAS Institute Inc.
Keselman, H. J., Kowalchuk, R. K. Algina, J., Lix, L. M., and Wilcox, R. R. (2000). Testing treatment effects in repeated measures designs: Trimmed means and bootstrapping. British Journal of Mathematical and Statistical Psychology, 53, 175-191.
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Kundu, S. (2013). An Introduction to Business Statistics.
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Lix, L. and Keselman, H. (1998). To trim or not to trim: Tests of location equality under heteroscedasticity. Educ. Psychol. Meas, 58, 409-429.
Luh, W. M. and Guo, J. H. (2010). Approximate Sample Size Formulas for Testing Group Mean Differences When Variances Are Unequal in One-Way ANOVA. Educational and Psychological Measurement, 68(6), 959.
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Moder, K. (2016). How to keep the Type I Error Rate in ANOVA if Variances are Heteroscedastic. Austrian Journal of Statistics, 36(3), 179.
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Norris, N. (2000). The standard errors of the geometric and harmonic means and their application to index numbers. Ann. Math. Statistics, 11, 445-448.
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Rocke, D. M., Downs, G. W. and Rocke, A. J. (1982). Are Robust Estimators Really Necessary? Technometrics, 24(2), 95-101.
Rosenberger, J. L. and Gasko, M. (1983). Comparing location estimators: trimmed means, medians and trimean. Understanding Robust and Exploratory Data Analysis, 297-338.
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Santos, R. G., Giulianotti, M. A., Dooley, C. T., Pinilla, C., Appel, J. R. and Houghten, R. A. (2011). The Use and Implications of the Harmonic Mean Model on Mixtures for Basic Research and Drug Discovery. ACS combinatorial science, 13(3), 337-344.
Sarkar, J. and Rashid, M. (2016). A geometric view of the mean of a set of numbers. Teaching Statistics, 38(3), 77-82.
Shorack, G. R. (1974). Random means. Ann. Statist., 1, 661-675.
Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. New York: Wiley.
Stigler, S. M. (1973). The Asymptotic Distribution of the Trimmed Mean. Ann. Math. Statist., 1, 472-477.
Sullivan, I. M. and D`Agostino, R. B. (2003). Statistics in Medicine, 22, 1317-1334.
Sullivan, I. M. and D'Agostino, R. B. (1996). Statistics in Medicine. 15, 477-496.
Thelwall, M. (2015). The precision of the arithmetic mean, geometric mean and percentiles for citation data: An experimental simulation modelling approach. Journal of Informetrics.
Tiku, M. L., Tan, W. Y. and Balakrishnan, N. (1986). Robust inference. New York: Marcel Dekker.
Tukey, J. W. (1948). Approximate Weights. The Annals of Mathematical Statistics, 19(1), 91-92.
Tukey, J. W. (1977). Exploratory Data Analysis. Pearson.
Tukey, J. W. (1982). Teaching of Statistics and Statistical Consulting. Ft. Belvoir: Defense Technical Information Center.
Velina, M., Valeinis, J., Greco, L. and Luta, G. (2016). Empirical Likelihood-Based ANOVA for Trimmed Means. International journal of environmental research and public health, 13(10), 953.
Weisberg, H. F. (1992). Central Tendency and Variability. Sage University.
Weisstein, E. W. (2003). CRC Concise Encyclopedia of Mathematics. Chapman and Hall/CRC. Retrieved from Chapman and Hall/CRC.
Wilcox, R. R. (1990). Comparing the means of two independent groups. Biom. J., 32, 771-780.
Wilcox, R. R. (1995). ANOVA: A paradigm for low power and misleading measures of effect size? Review of Educational Research, 65(1), 51-77.
Wilcox, R. R. (2005). Trimming and Winsorization. Encyclopedia of Biostatistics.
Wilcox, R. R. (2012). Modern Statistics for the Social and Behavioral Sciences: A Practical Introduction. CRC Press.
Wilcox, R. R. and Keselman, H. J. (2003). Modern robust data analysis methods: Measures of central tendency. Psychological Methods, 8, 254-274.
Wilcox, R. R. and Keselman, H. J. (2010). Modern robust data analysis methods. Measures of central tendency, 1-43.
Xia, D., Xu, S. and Qi, F. (1999). A Proof of the Arithmetic Mean-Geometric Mean-Harmonic Mean Inequalities. RGMIA Research Report Collection, 2(1), 85-88.
Zitt, M. (2012). The journal impact factor: angel, devil, or scapegoat? A comment on JK. Scientometrics, 92(2), 485-503.
Bickel, P. J. (1965). On Some Robust Estimates of Location. Ann. Math. Stat., 36, 847-858.
Bieniek, M. (2016). Sharp bounds on the bias of trimean. 57, 365-379.
Bland, J. M. and Altman, D. G. (2007). Agreement between methods of measurement with multiple observations per individual. Journal of Biopharmaceutical Statistics, 17(4), 571-582.
Bradley, J. (1978). Robustness? British Journal of Mathematical and Statistical Psychology, 31, 144-152.
De, P. (2016). The arithmetic mean - Geometric mean - Harmonic mean: Inequalities and a spectrum of applications. Resonance, 21(12), 1119-1133.
Doyle, J. R. and Chen, C. H. (2009). On the efficiency of the Trimean and Q123. Journal of Statistics and Management Systems, 12(2), 319-323.
Evans. (1955). Appendix G. Inefficient statistics, 902-904. Retrieved from Inefficient statistics.
Ferger, W. F. (1931). The Nature and Use of the Harmonic Mean. Journal of the American Statistical Association, 26(173), 36-40.
Glass, G. V., Peckham, P. D. and Sanders, J. R. (1972). Consequences of failure to meet assumptions underlying the fixed effects analysis of variance and covariance. Review of Educational Research, 42, 237-288.
Gleason, J. H. (2013). Comparative Power of the ANOVA, Approximate Randomization ANOVA, and Kruskal-Wallis Test.Wayne State University.
Guo, J. H. and Luh, W. M. (2000). Statistics & Probability Letters, 49, 1-7.
Hasan, M. K., Othman, N. A., Karim, S. A. and Sulaiman, J. (2018). Semi Non-Standard Trimean Algorithm for Rosenzweig-MacArthur Interaction Model. International Journal on Advanced Science, Engineering and Information Technology, 8(4-2), 1520.
Heeren, T. and D'Agostino, R. B. (1987). Statistics in Medicine. 6, 79-90.
Huber, P. J. (2004). Robust statistics. New York: Wiley.
Hunter, M. A. and May, R. B. (1993). Some myths concerning parametric and nonparametric tests. Canadian Psychology/Psychologie canadienne, 34, 384-389.
Inc., S. I. (2006). SAS online doc. Cary, NC: SAS Institute Inc.
Keselman, H. J., Kowalchuk, R. K. Algina, J., Lix, L. M., and Wilcox, R. R. (2000). Testing treatment effects in repeated measures designs: Trimmed means and bootstrapping. British Journal of Mathematical and Statistical Psychology, 53, 175-191.
Komic, J. (2011). Harmonic Mean. International Encyclopedia of Statistical Science, 622-624.
Kundu, S. (2013). An Introduction to Business Statistics.
Kutner, M.H., Nachtsheim, C.J., Neter, J. and Li, W. (2005). Applied Linear Statistical Models. Singapore City: McGraw-Hill.
Li, N. (2015). Estimating Life Tables for Developing Countries. Technical Paper No. 2014/4, pp. 1-29.
Lix, L. and Keselman, H. (1998). To trim or not to trim: Tests of location equality under heteroscedasticity. Educ. Psychol. Meas, 58, 409-429.
Luh, W. M. and Guo, J. H. (2010). Approximate Sample Size Formulas for Testing Group Mean Differences When Variances Are Unequal in One-Way ANOVA. Educational and Psychological Measurement, 68(6), 959.
Mendes, M. and Yigit, S. (2012). Comparison of ANOVA-Fand ANOM tests with regard to type I error rate and test power. Journal of Statistical Computation and Simulation, 83(11), 2093-2104.
Moder, K. (2016). How to keep the Type I Error Rate in ANOVA if Variances are Heteroscedastic. Austrian Journal of Statistics, 36(3), 179.
Mosteller, F. (2006). On some useful “inefficient” statistics. Hoaglin DG (ed) Selected Papers of Frederick Mosteller, 69-100.
Mu, W. and Yuan, X. (2012). Statistical inference for ANOVA under heteroscedasticity: Statistical inference. 2012 2nd International Conference on Consumer Electronics, Communications and Networks (CECNet).
Norris, N. (2000). The standard errors of the geometric and harmonic means and their application to index numbers. Ann. Math. Statistics, 11, 445-448.
Othman, A. R., Keselman, H.j., Padmanabhan, A. R., Wilcox, R. R. and Fradette, K. (2004). Comparing measures of the ‘typical’ score across treatment groups. British Journal of Mathematical and Statistical Psychology, 57(2), 215-234.
Permutt, T. and Li, F. (2017). Trimmed means for symptom trials with dropouts. Pharm. Stat., 16(1), 21.
Rhoad, R., Milauskas, G. and Whipple, R. (1991). Geometry for enjoyment and challenge. Evanston, IL: McDougal Littell.
Rocke, D. M., Downs, G. W. and Rocke, A. J. (1982). Are Robust Estimators Really Necessary? Technometrics, 24(2), 95-101.
Rosenberger, J. L. and Gasko, M. (1983). Comparing location estimators: trimmed means, medians and trimean. Understanding Robust and Exploratory Data Analysis, 297-338.
Rousselet, G., Pernet, C. and Wilcox, R. R. (2017). Beyond differences in means: robust graphical methods to compare two groups in neuroscience. figshare.
Santos, R. G., Giulianotti, M. A., Dooley, C. T., Pinilla, C., Appel, J. R. and Houghten, R. A. (2011). The Use and Implications of the Harmonic Mean Model on Mixtures for Basic Research and Drug Discovery. ACS combinatorial science, 13(3), 337-344.
Sarkar, J. and Rashid, M. (2016). A geometric view of the mean of a set of numbers. Teaching Statistics, 38(3), 77-82.
Shorack, G. R. (1974). Random means. Ann. Statist., 1, 661-675.
Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. New York: Wiley.
Stigler, S. M. (1973). The Asymptotic Distribution of the Trimmed Mean. Ann. Math. Statist., 1, 472-477.
Sullivan, I. M. and D`Agostino, R. B. (2003). Statistics in Medicine, 22, 1317-1334.
Sullivan, I. M. and D'Agostino, R. B. (1996). Statistics in Medicine. 15, 477-496.
Thelwall, M. (2015). The precision of the arithmetic mean, geometric mean and percentiles for citation data: An experimental simulation modelling approach. Journal of Informetrics.
Tiku, M. L., Tan, W. Y. and Balakrishnan, N. (1986). Robust inference. New York: Marcel Dekker.
Tukey, J. W. (1948). Approximate Weights. The Annals of Mathematical Statistics, 19(1), 91-92.
Tukey, J. W. (1977). Exploratory Data Analysis. Pearson.
Tukey, J. W. (1982). Teaching of Statistics and Statistical Consulting. Ft. Belvoir: Defense Technical Information Center.
Velina, M., Valeinis, J., Greco, L. and Luta, G. (2016). Empirical Likelihood-Based ANOVA for Trimmed Means. International journal of environmental research and public health, 13(10), 953.
Weisberg, H. F. (1992). Central Tendency and Variability. Sage University.
Weisstein, E. W. (2003). CRC Concise Encyclopedia of Mathematics. Chapman and Hall/CRC. Retrieved from Chapman and Hall/CRC.
Wilcox, R. R. (1990). Comparing the means of two independent groups. Biom. J., 32, 771-780.
Wilcox, R. R. (1995). ANOVA: A paradigm for low power and misleading measures of effect size? Review of Educational Research, 65(1), 51-77.
Wilcox, R. R. (2005). Trimming and Winsorization. Encyclopedia of Biostatistics.
Wilcox, R. R. (2012). Modern Statistics for the Social and Behavioral Sciences: A Practical Introduction. CRC Press.
Wilcox, R. R. and Keselman, H. J. (2003). Modern robust data analysis methods: Measures of central tendency. Psychological Methods, 8, 254-274.
Wilcox, R. R. and Keselman, H. J. (2010). Modern robust data analysis methods. Measures of central tendency, 1-43.
Xia, D., Xu, S. and Qi, F. (1999). A Proof of the Arithmetic Mean-Geometric Mean-Harmonic Mean Inequalities. RGMIA Research Report Collection, 2(1), 85-88.
Zitt, M. (2012). The journal impact factor: angel, devil, or scapegoat? A comment on JK. Scientometrics, 92(2), 485-503.
Published
2019-08-21
How to Cite
Najdi, N. F. and Ahad, N. A. (2019) “A Modification of ANOVA with Trimmed Mean”, Malaysian Journal of Social Sciences and Humanities (MJSSH), 4(4), pp. 109 - 118. doi: 10.47405/mjssh.v4i4.247.
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