Nonparametric Bootstrap Method for Location Testing between Two Populations under Combined Assumption Violations
Abstract
This research aims to compare the efficiency of four nonparametric bootstrap methods for location testing between two populations when the preliminary assumptions are violated. The four methods include Nonparametric Bootstrap t test (NBTT), Nonparametric Bootstrap Welch t test (NBWT), Nonparametric Bootstrap Welch test based on Rank (NBWR), and Nonparametric Bootstrap Yuen Test (NBYT). The data simulation designed to have log-normal, exponential, and gamma distribution. The test includes both equal and unequal of variances and sample size. The results show that when population has log-normal, exponential, and gamma distribution with equal variance, unequal variance and sample size , the NBWR method has the highest efficiency. When and unequal variance ratio of 1:4 and 1:9, the NBYT method has the highest efficiency. In case that the sample size and unequal variance, the NBYT method has the highest efficiency. Keywords : bootstrap method ; nonparametric test ; location testingReferences
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outcomes: comparative power of the t-test and Wilcoxon rank-sum test in small samples applied
research. Journal of Clinical Epidemiology, 52, 229-235.
George W. Divine, H. James Norton, Anna E. Barón & Elizabeth JuarezColunga
Dwivedi, A. K., Mallawaarachchi, I., & Alvarado, L. A. (2017). Analysis of small sample size studies using
nonparametric bootstrap test with pooled resampling method. Statistics in Medicine, 36, 2187-2205.
Efron, B., & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. New York: Chapman & Hall.
Fagerland, M. W., & Sandvik, L. (2009). Performance of five two-sample location tests for skewed distributions
with unequal variances. Contemporary Clinical Trial, 30, 490-496.
Harwell, M. R., Rubinstein, E. N., Hayes, W. S., & Olds, C. C. (1992). Summarizing Monte Carlo results in
methodological research: The one- and two-factor fixed effects ANOVA cases, Journal of Educational
Statistics, 17, 315-339.
Mickelson, W. T. (2013). A monte carlo simulation of the robust rank-order test under various population symmetry
conditions. Journal of Modern Applied Statistical Methods, 12(1), 21-33.
Neuhauser, M. (2012). Nonparametric statistical tests: A computational approach. Florida: CRC Press.
Nguyen, D. T., Kim, E. S., Gil, P. R., Kellermann, A., Chen, YH., Kromrey, J. D., & Bellara, A. (2016). Parametric
Tests for Two Population Means under Normal and Non-Normal Distribution. Journal of Modern Applied
Statistical Methods, 15(1), 141-159.
Reiczigel, J., Zakarias, I., & Rozsa, L. (2005). A bootstrap test of stochastic equality of two populations.
The American Statistician, 59(2), 1-6, DOI: 10.1198/000313005X23526.
Stonehouse J. M., & Forrester. G. J. (1998). Robustness of the t and U tests under combined assumption
violations. Journal of Applied Statistics, 25(1), 63-74.
Welch, B. L. (1938). The significance of the difference between two means when the population variances are
unequal, Biometrika, 29, 350-362.
Welz, A., Ruxton, G. D., & Neuhauser, M. (2018). A non-parametric maximum test for the Behrens-Fisher problem.
Journal of Statistical Computation and Simulation, DOI: 10.1080/00949655.2018.1431236.
Wilcox, R. R. (1990). Comparing the mean of two independent group, Biometrical Journal, 32, 771-780.
Wilcox, R. R. (1994). Some results on the Tukey-Mclaughlin and Yuen methods for trimmed means when
distribution are skewed. Biometrical Journal, 3, 259-273.
Wilcox, R. R. (2003). Applying contemporary statistical techniques. San Diego, CA: Academic Press.
Wilcox, R. R. (2005). Introduction to robust estimation and hypothesis testing. (2nd ed). San Diego, CA:
Academic Press.
Wilcox, R. R., & Keselman, H. J. (2003). Modern robust data analysis method: measures of central tendency.
Psychological Methods, 8, 254-274.
Yuen, K. K. (1974). The two-sample trimmed t for unequal population variances. Biometrika, 61, 165-170.
Zimmerman, D. W., & Zumbo, B. D. (1993a). Rank transformations and the power of the Student t test and
Welch t test non-normal populations with unequal variances. Canadian Journal of Experimental
Psychology, 47, 523-539.
Zimmerman, D. W., & Zumbo, B. D. (1993b). The relative power of parametric and nonparametric statistical
methods. In A handbook for data analysis in the behavioral sciences: Methodological issues, G. Keren
& C. Lewis (Eds.), 481-517. Hillsdale, NJ: Erlbaum.
Bridge, P. D., & Sawilowsky, S. S. (1999). Increasing physicians’ awareness of the impact of statistics on research
outcomes: comparative power of the t-test and Wilcoxon rank-sum test in small samples applied
research. Journal of Clinical Epidemiology, 52, 229-235.
George W. Divine, H. James Norton, Anna E. Barón & Elizabeth JuarezColunga
Dwivedi, A. K., Mallawaarachchi, I., & Alvarado, L. A. (2017). Analysis of small sample size studies using
nonparametric bootstrap test with pooled resampling method. Statistics in Medicine, 36, 2187-2205.
Efron, B., & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. New York: Chapman & Hall.
Fagerland, M. W., & Sandvik, L. (2009). Performance of five two-sample location tests for skewed distributions
with unequal variances. Contemporary Clinical Trial, 30, 490-496.
Harwell, M. R., Rubinstein, E. N., Hayes, W. S., & Olds, C. C. (1992). Summarizing Monte Carlo results in
methodological research: The one- and two-factor fixed effects ANOVA cases, Journal of Educational
Statistics, 17, 315-339.
Mickelson, W. T. (2013). A monte carlo simulation of the robust rank-order test under various population symmetry
conditions. Journal of Modern Applied Statistical Methods, 12(1), 21-33.
Neuhauser, M. (2012). Nonparametric statistical tests: A computational approach. Florida: CRC Press.
Nguyen, D. T., Kim, E. S., Gil, P. R., Kellermann, A., Chen, YH., Kromrey, J. D., & Bellara, A. (2016). Parametric
Tests for Two Population Means under Normal and Non-Normal Distribution. Journal of Modern Applied
Statistical Methods, 15(1), 141-159.
Reiczigel, J., Zakarias, I., & Rozsa, L. (2005). A bootstrap test of stochastic equality of two populations.
The American Statistician, 59(2), 1-6, DOI: 10.1198/000313005X23526.
Stonehouse J. M., & Forrester. G. J. (1998). Robustness of the t and U tests under combined assumption
violations. Journal of Applied Statistics, 25(1), 63-74.
Welch, B. L. (1938). The significance of the difference between two means when the population variances are
unequal, Biometrika, 29, 350-362.
Welz, A., Ruxton, G. D., & Neuhauser, M. (2018). A non-parametric maximum test for the Behrens-Fisher problem.
Journal of Statistical Computation and Simulation, DOI: 10.1080/00949655.2018.1431236.
Wilcox, R. R. (1990). Comparing the mean of two independent group, Biometrical Journal, 32, 771-780.
Wilcox, R. R. (1994). Some results on the Tukey-Mclaughlin and Yuen methods for trimmed means when
distribution are skewed. Biometrical Journal, 3, 259-273.
Wilcox, R. R. (2003). Applying contemporary statistical techniques. San Diego, CA: Academic Press.
Wilcox, R. R. (2005). Introduction to robust estimation and hypothesis testing. (2nd ed). San Diego, CA:
Academic Press.
Wilcox, R. R., & Keselman, H. J. (2003). Modern robust data analysis method: measures of central tendency.
Psychological Methods, 8, 254-274.
Yuen, K. K. (1974). The two-sample trimmed t for unequal population variances. Biometrika, 61, 165-170.
Zimmerman, D. W., & Zumbo, B. D. (1993a). Rank transformations and the power of the Student t test and
Welch t test non-normal populations with unequal variances. Canadian Journal of Experimental
Psychology, 47, 523-539.
Zimmerman, D. W., & Zumbo, B. D. (1993b). The relative power of parametric and nonparametric statistical
methods. In A handbook for data analysis in the behavioral sciences: Methodological issues, G. Keren
& C. Lewis (Eds.), 481-517. Hillsdale, NJ: Erlbaum.
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2020-09-01
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