Winsorized Modified One Step M-Estimator As a Measure of the Central Tendency in the Alexander-Govern Test
DOI:
https://doi.org/10.21512/comtech.v7i3.2505Keywords:
Alexander-Govern (AG) test, MOM estimator, AGWMOM test, Type I error rates, Test StatisticAbstract
This research dealt with making comparison of the independent group tests with the use of parametric technique. This test used mean as its central tendency measure. It was a better alternative to the ANOVA, the Welch test and the James test, because it gave a good control of Type I error rates and high power with ease in its calculation, for variance heterogeneity under a normal data. But the test was found not to be robust to non-normal data. Trimmed mean was used on the test as its central tendency measure under non-normality for two group condition, but as the number of groups increased above two, the test failed to give a good control of Type I error rates. As a result of this, the MOM estimator was applied on the test as its central tendency measure and is not influenced by the number of groups. However, under extreme condition of skewness and kurtosis, the MOM estimator could no longer control the Type I error rates. In this study, the Winsorized MOM estimator was used in the AG test, as a measure of its central tendency under non-normality. 5,000 data sets were simulated and analysed for each of the test in the research design with the use of Statistical Analysis Software (SAS) package. The results of the analysis shows that the Winsorized modified one step M-estimator in the Alexander-Govern (AGWMOM) test, gave the best control of Type I error rates under non-normality compared to the AG test, the AGMOM test, and the ANOVA, with the highest number of conditions for both lenient and stringent criteria of robustness.
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References
Alexander, R. A., & Govern, D. M. (1994). A New and Simpler Approximation for ANOVA Under Variance Heterogeneity. Journal of Education Statistics, 19(2), 91-101.
Algina, J., Oshima, T. C., & Lin, W. Y. (1994). Type I Error Rates for Welch’s Test and James’s Second-Order Test Under Nonnormality and Inequality. When There Are Two Groups. Journal of Educational and Behavioural Statistics, 19(3), 275-291.
Bradley, J. V. (1978). Robustness?. British Journal of Mathematical and Statistical Psychology, (31), 144-152.
Brunner, e., Dette, H., & Munk, A. (1997). Box-Type Approximations in Nonparametric Factorial Designs. Journal of the American Statistical Association, 92(440), 1494-1502.
Efron, B., & Tibshirani. (1998). An introduction to the bootstrap. New York: Chapman & Hall.
Hill, G. W. (1970). Algorithm 395, Student’s t-distribution. Communication of the ACM, 13, 67-619.
James, G. S. (1951). The comparison of several groups of observations when the ratios of the population variances are unknown. Biometrika, 38, 324-329.
Keselman, H. J., Kowalchuk, R. K., Algina, J., Lix, L. M., & Wilcox, R. R. (2000). Testing treatment effects in repeated measure designs: Trimmed means and bootstrapping. British Journal of Mathematical and Statistical Psychology, 53, 175-191.
Kulinskaya, E., Staudte, R. G., & Gao, H. (2003). Power Approximations in Testing for Unequal Means in a One-Way ANOVA Weighted for Unequal Variances. Communication in Statistics
– Theory and Methods, 32(12), 2353-2371. Doi: 10.1081/STA-12002538.
Krishnamoorthy, K., Lu, F., & Matthew, T. (2007). A parametric bootstrap approach for ANOVA with unequal variances: Fixed and random models. Computational Statistics & Data Analysis, 51(12), 5731-5742.
Lix, L. M, Keselman, J. C., & Keselman, H. J. (1998). To trim or not to trim. Educational and Psychological Measurement, 58(3), 409-429.
Myers, L. (1998). Comparability of The James’ Second-Order Approximation Test and The Alexander and Govern A Statistic for Non-normal Heteroscedastic Data. Journal of Statistical Computation, Computational, 60, 207-222.
Ochuko, T. K., Abdullah, S., Zain, Z., & Yahaya, S. S. S. (2015). Winsorized Modified One Step M-estimator in Alexander-Govern test. Modern Applied Science, 9(11), 51-67.
Oshima, T. C., & Algina, J. (1992). Type I error rates for James’s second-order test and Wilcoxon’s Hm test under heteroscedaticity and non-normality. British Journal of Mathematical and Statistical Psychology, 45, 255-263.
Othman, A. R., Keselman, H. J., Padmanabhan, A. R., Wilcox, R. R., & Fradette, K. (2004). Comparing measures of the “typical” score across treatments groups. The British Journal of Mathematical and Statistical psychology, 57(Pt 2), 215-234.
Pardo, J. A., Pardo, M. C., Vincente, M. L., & Esteban, M. D. (1997). A statistical information theory approach to compare the homogeneity of several variances. Computational Statistics & Data Analysis, 24(4), 411-416.
Schneider, P. J., & Penfield, D. A. (1997). Alexander and Govern’s Approximation: Providing an alternative to ANOVA Under Variance Heterogeneity. Journal of Experimental Education, 65(3), 271-287.
Welch, B. L. (1951). On the comparison of several mean values: An alternative approach. Biometrika, 38, 330-336.
Wilcox, R. R. (1988). A new alternative to the ANOVA F and new results on James’s second-order method. British Journal of Mathematical and Statistical Psychology, 42, 203-213.
Wilcox, R. R. (1997). Introduction to robust estimation and hypothesis testing. San Diego, CA: Academic Press.
Wilcox, R. R., & Keselman, H. J. (2003). Modern Robust Data Analysis Methods: Measures of Central Tendency. Psychological Methods, 8(3), 254-274.
Wilcox, R. R. (2003). Multiple comparisons based on a modified one-step M-estimator. Journal of Applied Statistics, 30, 1231-1241.
Wilcox, R. R., & Keselman, H. J. (2000). Power analysis when comparing trimmed means. J. Modern Appl. Stat. Methods, 1(1), 24-31.
Yusof, Md. Z., Abdullah, S., & Yahaya, S. S. S. (2011). Type I Error Rates of Ft Statistic with Different Trimming Strategy for TWO Groups case. Modern Applied Science, 5(4), 236-242.
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