Modification of S₁ statistic with Hodges-Lehmann as the central tendency measure

Normality and variance homogeneity assumptions are usually the main concern of parametric procedures such as in testing the equality of central tendency measures. Violation of these assumptions can seriously inflate the Type I error rates, which will cause spurious rejection of null hypotheses. Para...

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Main Author:	Lee, Ping Yin
Format:	Thesis
Language:	eng eng eng
Published:	2018
Subjects:	QA273-280 Probabilities Mathematical statistics
Online Access:	https://etd.uum.edu.my/7349/1/Depositpermission_s813618.pdf https://etd.uum.edu.my/7349/2/s813618_01.pdf https://etd.uum.edu.my/7349/3/s813618_02.pdf
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id	my-uum-etd.7349
record_format	uketd_dc
institution	Universiti Utara Malaysia
collection	UUM ETD
language	eng eng eng
advisor	Syed Yahya, Sharipah Soaad Ahad, Aishah
topic	QA273-280 Probabilities Mathematical statistics
spellingShingle	QA273-280 Probabilities Mathematical statistics Lee, Ping Yin Modification of S₁ statistic with Hodges-Lehmann as the central tendency measure
description	Normality and variance homogeneity assumptions are usually the main concern of parametric procedures such as in testing the equality of central tendency measures. Violation of these assumptions can seriously inflate the Type I error rates, which will cause spurious rejection of null hypotheses. Parametric procedures such as ANOVA and t-test rely heavily on the assumptions which are hardly encountered in real data. Alternatively, nonparametric procedures do not rely on the distribution of the data, but the procedures are less powerful. In order to overcome the aforementioned issues, robust procedures are recommended. S₁ statistic is one of the robust procedures which uses median as the location parameter to test the equality of central tendency measures among groups, and it deals with the original data without having to trim or transform the data to attain normality. Previous works on S₁ showed lack of robustness in some of the conditions under balanced design. Hence, the objective of this study is to improve the original S₁ statistic by substituting median with Hodges-Lehmann estimator. The substitution was also done on the scale estimator using the variance of Hodges-Lehmann as well as several robust scale estimators. To examine the strengths and weaknesses of the proposed procedures, some variables like types of distributions, number of groups, balanced and unbalanced group sizes, equal and unequal variances, and the nature of pairings were manipulated. The findings show that all proposed procedures are robust across all conditions for every group case. Besides, three proposed procedures namely S₁(MADn), S₁(Tn) and S₁(Sn) show better performance than the original S₁ procedure under extremely skewed distribution. Overall, the proposed procedures illustrate the ability in controlling the inflation of Type I error. Hence, the objective of this study has been achieved as the three proposed procedures show improvement in robustness under skewed distributions.
format	Thesis
qualification_name	masters
qualification_level	Master's degree
author	Lee, Ping Yin
author_facet	Lee, Ping Yin
author_sort	Lee, Ping Yin
title	Modification of S₁ statistic with Hodges-Lehmann as the central tendency measure
title_short	Modification of S₁ statistic with Hodges-Lehmann as the central tendency measure
title_full	Modification of S₁ statistic with Hodges-Lehmann as the central tendency measure
title_fullStr	Modification of S₁ statistic with Hodges-Lehmann as the central tendency measure
title_full_unstemmed	Modification of S₁ statistic with Hodges-Lehmann as the central tendency measure
title_sort	modification of s₁ statistic with hodges-lehmann as the central tendency measure
granting_institution	Universiti Utara Malaysia
granting_department	Awang Had Salleh Graduate School of Arts & Sciences
publishDate	2018
url	https://etd.uum.edu.my/7349/1/Depositpermission_s813618.pdf https://etd.uum.edu.my/7349/2/s813618_01.pdf https://etd.uum.edu.my/7349/3/s813618_02.pdf
_version_	1747828201973874688
spelling	my-uum-etd.73492021-08-09T08:33:33Z Modification of S₁ statistic with Hodges-Lehmann as the central tendency measure 2018 Lee, Ping Yin Syed Yahya, Sharipah Soaad Ahad, Aishah Awang Had Salleh Graduate School of Arts & Sciences Awang Had Salleh Graduate School of Arts and Sciences QA273-280 Probabilities. Mathematical statistics Normality and variance homogeneity assumptions are usually the main concern of parametric procedures such as in testing the equality of central tendency measures. Violation of these assumptions can seriously inflate the Type I error rates, which will cause spurious rejection of null hypotheses. Parametric procedures such as ANOVA and t-test rely heavily on the assumptions which are hardly encountered in real data. Alternatively, nonparametric procedures do not rely on the distribution of the data, but the procedures are less powerful. In order to overcome the aforementioned issues, robust procedures are recommended. S₁ statistic is one of the robust procedures which uses median as the location parameter to test the equality of central tendency measures among groups, and it deals with the original data without having to trim or transform the data to attain normality. Previous works on S₁ showed lack of robustness in some of the conditions under balanced design. Hence, the objective of this study is to improve the original S₁ statistic by substituting median with Hodges-Lehmann estimator. The substitution was also done on the scale estimator using the variance of Hodges-Lehmann as well as several robust scale estimators. To examine the strengths and weaknesses of the proposed procedures, some variables like types of distributions, number of groups, balanced and unbalanced group sizes, equal and unequal variances, and the nature of pairings were manipulated. The findings show that all proposed procedures are robust across all conditions for every group case. Besides, three proposed procedures namely S₁(MADn), S₁(Tn) and S₁(Sn) show better performance than the original S₁ procedure under extremely skewed distribution. Overall, the proposed procedures illustrate the ability in controlling the inflation of Type I error. Hence, the objective of this study has been achieved as the three proposed procedures show improvement in robustness under skewed distributions. 2018 Thesis https://etd.uum.edu.my/7349/ https://etd.uum.edu.my/7349/1/Depositpermission_s813618.pdf text eng public https://etd.uum.edu.my/7349/2/s813618_01.pdf text eng public https://etd.uum.edu.my/7349/3/s813618_02.pdf text eng public masters masters Universiti Utara Malaysia Babu, G. J., Padmanabhan, A. R. and Puri, M. L. (1999). Robust one-way ANOVA under possibly non-regular conditions. Biometrical Journal, 41, 321-339. Bickel, P. J. (1965). On some robust estimates of location. The Annals of Mathematical Statistics, 36(3), 847-858. Bradley, J. V. (1978). Robustness? British Journal of Mathematical and Statistical Psychology, 31, 144-152. Boos, D. D. (1982). A test for asymmetry associated with the Hodges-Lehmann estimator. Journal of the American Statistical Association, 77(379), 647-651. Boos, D. D. and Monahan, J. F. (1986). Bootstrap methods using prior information. Biometrika, 73(1), 77-83. Box, G. E. P. (1953). Non-normality and tests on variance. Biometrika, 40, 318-355. Brownie, C and Boss, D. D. (1994). Type I error robustness of ANOVA and ANOVA on ranks when the number of treatments is large. Biometrics, 50(2), 542-549. Brunner, L. J. and Austin P. C. (2007). Inflation of Type I error in multiple regression when independent variables are measured with error. Retrieved from http://www.utstat.toronto.edu/~brunner/MeasurementError/MeasurementError3i.pdf Chernick, M. R. (1999). Bootstrap Methods: A Practitioner’s Guide. New Jersey, US: John Wiley & Sons, Inc. Cui, H., He, X. and Ng, K. W. (2003). Asymptotic distributions of principal components based on robust dispersions. Biometrika, 90(4), 953-966. Dixon, W. J. and Yuen, K. K. (1974). Trimming and winsorization: a review. Statistiche Hefte, 2, 157-170. Donoho, D. L. and Huber, P. J. (1983). The notion of breakdown point. In A Festscherift for Erich L. Lehmann (Brickel, P. J., Doksum, K. & Hidges, J. L., Jr., eds), 157-184. Fisher, R. A. (1935). The Design of Experiments. New York, US: Hafner. Geyer, C. J. (2006). Breakdown point theory notes. Retrieved from http://www.stat.umn.edu/geyer/5601/notes/break.pdf Gibbons, J. D. and Chakraborti, S. (2003). Nonparametric Statistical Inference, 4th ed. Florida, US: CRC Press. Guo, J. H. and Luh, W. M. (2000). An invertible transformation two-sample trimmed t-statistic under heterogeneity and nonnormality, Statistics and Probability Letters, 49(1), 1-7. Hall, P. and Sheather, S. (1988). On the distribution of a studentized quantile. Journal of the Royal Statistical Society, 381-391. Hampel, F. R. (1974). The influence curve and its role in robust estimation. Journal of the American Statistical Association, 69, 383-393. Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A. (1986). Robust statistics: The approach based on influence functions. New York, US: Wiley. Hodges, J. L. Jr. and Lehmann, E. L. (1963). Estimated of location based on rank tests. The Annals of Mathematical Statistics, 34(2), 598-611. Hogg, R. V. (1974). Adaptive robust procedures: a partial review and some suggestions for future applications and theory. Journal of the American Statistical Association, 69, 909-927. Huber, P. J. (1964). Robust estimation of a location parameter. The Annals of Mathematical Statistics, 35(1), 73-101. Huber, P. J. (1972). Robust statistics: a review. The Annals Mathematical Statistics, 43(4), 1041-1067. Huber, P. J. (1981). Robust Statistics. New York, US: Wiley. James, G. S. (1951). The comparison of several groups of observations when the ratios of the population variances are unknown. Biometrika, 38, 324-329. Kang, Y. and Harring, J. R. (2012). Investigating the impact of non-normality, effect size, and sample size on two-group comparison procedures: an empirical study. Presented at the Annual Meeting of the American Educational Research Association (AERA), SIG: Educational Statisticians, BC. Kazempour, K. (1995). Impact of stratification imbalance on probability of Type I error. The American Statistician, 49(2), 170-174. Kellermann, A. P., Bellara, A. P., Gil, P. R. D., Nguyen, D., Kim, E. S., Chen, Y. and Kromrey, J. D. (2013). Variance heterogeneity and non-normality: how SAS PROC TEST can keep us honest. Retrieved from http://support.sas.com/resources/papers/proceedings13/228-2013.pdf Keselman, H. J., Wilcox, R. R., Lix, L. M., Algina. J. and Fradette, K. (2007). Adaptive robust estimation and testing. British Journal of Mathematical and Statistical Psychology, 60, 267-293. Keselman, H. J., Wilcox, R. R., Othman, A. R. and Fradette, K. (2002). Trimming, transforming statistics, and bootstrapping: circumventing the biasing effects of heteroscedasticity and nonnormality. Journal of Modern Applied Statistical Methods, 1, 288-309. Khan, A. and Rayner, G. D. (2003). Robustness to non-normality of common tests for the many sample location problem. Journal of Applied Mathematics and Decision Sciences, 7(4), 187. Kruskal, W. H. and Wallis, W. A. (1952). Use of ranks in one-criterion variance analysis. Journal of the American Statistical Association, 47, 583-621. Kulinskaya, E., Staudte, R. G., and Gao, H. (2003). Power approximations in testing for unequal means in a one-way ANOVA weighted for unequal variances. Communications in Statistics – Theory and Methods, 32, 2353-2371. Lix, L. M. and Keselman, H. J. (1998). To trim or not to trim: tests of location equality under heteroscedasticity and nonnormality. Educational and Psychological Measurement, 58, 409-429. Lix, L. M., Keselman, J. C. and Keselman, H. J. (1996). Consequences of assumption violations revisited: a quantitative review of alternatives to the one-way analysis of variance “F” test. Review of Education Research, 66(4), 579-619. MacGillivray, H. L. and Cannon, W. H. (2002). Generalizations of the g-and-h distributions and their uses. Unpublished Thesis. Maxwell, S. E. and Delaney, H. D. (2004). Designing Experiments and Analyzing Data: A Model Comparison Perspective, 2nd ed. Mahwah, NJ, US: Lawrence Erlbaumn Associates Publishers. Md. Yusof, Z., Othman, A. R. and Syed Yahaya, S. S. (2010). Comparison of Type I error rates between T1 and Ft statistics for unequal population variance using variable trimming. Malaysian Journal of Mathematical Sciences, 4(2), 195-207. Morgenthaler, S. (1992). Least-absolute-deviations fits for generalized linear models. Biometrika, 79, 747-754. Nachar, N. (2008). The Mann-Whitney U: a test for assessing whether two independent samples come from the same distribution. Tutorials in Quantitative Methods for Psychology, 4(1), 13-20. 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. Oshima, T. C. and Algina, J. (1992). 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, 42, 255-263. Ott, R. L. and Longnecker, M. T. (2010). An Introduction to Statistical Methods and Data Analysis. Belmont, CA: Brooks/Cole Cengage Learning. Pitman, E. J. G. (1937a). Significance tests which may be applied to samples from any population. Royal Statistical Society Supplement, 4(1-2), 119-130. Pitman, E. J. G. (1937b). Significance tests which may be applied to samples from any population. Royal Statistical Society Supplement, 4(2), 225-232. Pitman, E. J. G. (1938). Significance tests which may be applied to samples from any population: the analysis of variance test. Biometrika, 29(3-4): 332-335. Quenouille, M. H. (1949). Approximate tests of correlation in time-series. Journal of the Royal Statistical Society, 11(1), 68-84. Ramsey, J. B. (2001). The Elements of Statistics: with Applications to Economics and the Social Sciences, 1st ed. Boston, US:Cengage Learning. Reed, J. F. (1998). Contributions to adaptive estimation. Journal of Applied Statistics, 25(5), 651-669. Ronchetti, E. M. (2006). The historical development of robust statistics. In Proceedings of the 7th International Conference on Teaching Statistics (ICOTS-7). Retrieved from https://iase-web.org/documents/papers/icots7/3B1 _RONC.pdf Rousseeuw, P. J. and Croux, C. (1993). Alternatives to the median absolute deviation. Journal of the American Statistical Association, 88, 1283-1283. SAS Institute. (1989). IML software: Usage and Reference, Version 6, 1st ed. Cary, NC: SAS Institute. Scheffe, H. (1959). The Analysis of Variance. New York, NY: Wiley. Sheskin, D. J. (2011). Handbook of Parametric and Nonparametric Statistical Procedures, 5th ed. Florida, US: CRC Press. Snedecor, G. W. and Cochran, W. G. (1980). Statistical Methods, 7th ed. U.S.A, Ames:Iowa State University Press. Staudte, R. G. and Sheather, S. J. (1990). Robust Estimation and Testing. New York, US: Wiley. Stigler, S. M. (1977). Do robust estimators work with real data? The Annals of Statistics, 5(6), 1055-1098. Stigler, S. M. (2010). The changing history of robustness. The American Statistician, 64(4), 277-281. Sullivan, M. III (2004). Statistics: Informed Decisions Using Data. Upper Saddle River, NJ: Pearson Education, Inc. Syed Yahaya, S. S. (2005). Robust statistical procedures for testing the equality of central tendency parameters under skewed distributions. Unpublished Thesis. Syed Yahaya, S. S., Md Yusof, Z. and Abdullah, S. (2011). A robust alternative to ANOVA. Proceedings of the Second International Soft Science Conference 2011 (ISSC 2011). Ho Chi Minh City, Vietnam. Syed Yahaya, S. S., Othman, A. R. and Keselman, H. J. (2004). Testing the equality of location parameters for skewed distributions using S1 with high breakdown robust scale estimators. Theory and Applications of Recent Robust Methods, Series: Statistics for industry and technology, 319 – 328. Syed Yahaya, S. S., Othman, A. R. and Keselman, H. J. (2006). Comparing the “typical score” across independent groups based on different criteria for trimming. Metodoloskizveki, 3(1), 49-62. Teh, S. Y., Md Yusof, Z., Yaacob, C. R. & Othman, A. R. (2010). Performance of the traditional pooled variance t-test against the bootstrap procedure of difference between sample means. Malaysian Journal of Mathematical Sciences, 4(1), 85-94. Tukey, J. W. (1960). A survey of sampling from contaminated distributions. In Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling (I. Olkin et al., eds), 448-485. Stanford University Press. Welch, B. L. (1951). On the comparison of several mean values: An alternative approach. Biometrika, 38, 330-336. Wilcox, R. R. (1994). A one-way random effects model for trimmed means. Psychometrika, 59(3), 289-306. Wilcox, R. R. (1997). ANCOVA based on comparing a robust measure of location at empirically determined design points. British Journal of Mathematical and Statistical Psychology, 50(1), 93-103. Wilcox, R. R. and Keselman, H. J. (2002). Power analysis when comparing trimmed means. Journal of Modern Applied Statistical Methods, 1(1), 24-31. 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. (2012). Introduction to Robust Estimation and Hypothesis Testing, 3rd ed. San Diego, CA: Academic Press. Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics Bulletin, 1, 80-83. Winter, J. C. F. D. and Dodou, D. (2010). Five-point likert items: t test versus mann-whitney-wilcoxon. Retrieved from http://pareonline.net/pdf/v15n11.pdf Yi, G. Y. and He, W. (2009). Median regression models for longitudinal data with dropouts. Biometrics, 65(2), 618-625. Yuen, K. K. (1974). The two-sample trimmed t for unequal population variances. Biometrika, 61, 165-170.

Modification of S₁ statistic with Hodges-Lehmann as the central tendency measure

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