A Comparison of Tests for Equality of Means under Heterogeneity of Variances in Completely Randomized Design
Abstract
The objective of the research is to compare the ability to control probability of type I error of testing the equality of means from a completely randomized design (CRD) by F test, Welch’s test and Generalized F test when the population variance is heterogeneous. The ability to control probability of type I error is measured by the probability of the number of “reject” when the null hypothesis is true and the percentage of accuracy is measured by the number of “reject” when the alternative hypothesis is true. The data are generated from the fixed effect CRD simulation of normal distributed error population with zero mean and six cases of variance. The sample sizes are and the replication in each case is 1,000. The research results can be summarized as follows. In the cases of homogeneous variance with small variance equal to 1, medium variance equal to 25 and large variance equal to 100, it is found that the F test yields the highest result. In the cases of heterogeneous variance with small variation in variance , medium variation in variance and large variation in variance , it is found that the generalized F test yields the highest result. However, the percentage of accuracy by the F test, Welch’s test and generalized F test are nearly the same and increases as the sample size increases in all cases. Keyword: Completely Randomized Design, Fixed Effect, F test, Welch’s test, Generalized F testReferences
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Welch, B. L. (1947). The generalization ofstudent's' problem when several different population variances are involved. Biometrika, 34(1/2), 28-35.
Welch, B. L. (1951). On the comparison of several mean values: an alternative approach. Biometrika, 38(3/4), 330-336.
Xu, L. W., & Wang, S. G. (2008a). A new generalized p-value for ANOVA under heteroscedasticity. Statistics & Probability Letters, 78(8), 963-969.
Xu, L., & Wang, S. (2008b). A new generalized p-value and its upper bound for ANOVA under unequal error variances. Communications in Statistics—Theory and Methods, 37(7), 1002-1010.
Brown, M. B., & Forsythe, A. B. (1974). The small sample behavior of some statistics which test the equality of several means. Technometrics, 16(1), 129-132.
Chen, S. Y. (2001). One-stage and two-stage statistical inference under heteroscedasticity. Communications in Statistics-Simulation and Computation, 30(4), 991-1009.
Chen, S. Y., & Chen, H. J. (1998). Single-stage analysis of variance under heteroscedasticity. Communications in Statistics-Simulation and Computation, 27(3), 641-666.
Cochran, W. G. (1954). The combination of estimates from different experiments. Biometrics, 10(1), 101-129.
Krishnamoorthy, K., Lu, F., & Mathew, T. (2007). A parametric bootstrap approach for ANOVA with unequal variances: Fixed and random models. Computational Statistics & Data Analysis, 51(12), 5731-5742.
Scott, A. J., & Smith, T. M. F. (1971). Interval estimates for linear combinations of means. Applied Statistics, 276-285.
Weerahandi, S. (1995). ANOVA under unequal error variances. Biometrics, 589-599.
Weerahandi, S. (2013). Exact statistical methods for data analysis. Springer Science & Business Media.
Weerahandi, S. (2004). Generalized inference in repeated measures: exact methods in MANOVA and mixed models (Vol. 500). John Wiley & Sons.
Welch, B. L. (1947). The generalization ofstudent's' problem when several different population variances are involved. Biometrika, 34(1/2), 28-35.
Welch, B. L. (1951). On the comparison of several mean values: an alternative approach. Biometrika, 38(3/4), 330-336.
Xu, L. W., & Wang, S. G. (2008a). A new generalized p-value for ANOVA under heteroscedasticity. Statistics & Probability Letters, 78(8), 963-969.
Xu, L., & Wang, S. (2008b). A new generalized p-value and its upper bound for ANOVA under unequal error variances. Communications in Statistics—Theory and Methods, 37(7), 1002-1010.
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2018-01-09
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