Abstract
In the present Monte Carlo study, the empirical Type I error properties and power of several statistics for testing the homogeneity hypothesis in a one—way classification are examined in the case of small sample sizes. We compared these tests under several scenarios: normal populations under heterogeneous variances, nonnormal populations under homogeneous variances, nonnormal populations under heterogeneous variances, balanced and unbalanced sample sizes, and increasing number of populations. Overall, none of the tests considered is uniformly dominating the others. Under normality and variance heterogeneity, the Brown—Forsythe and the Welch test perform well over a wide range of parameter configurations, the modified Brown-Forsythe test by Mehrotra keeps generally the level, but other tests may also perform well, depending on the constellation of the parameters under study. The Welch test becomes liberal when the sample sizes are small and the number of populations is large. We propose a modified version of Welch’s test that keeps the nominal level in these cases. With the understanding that methods are unacceptable if they have Type I error rates that are too high, only the testing procedure associated with the modified Brown-Forsythe test can be recommended both for normal and nonnormal data. Under normality, the modified Welch test can also be recommended.
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References
Asiribo, O., Gurland, J. (1990). Coping with variance heterogeneity. Commun. Statist. Theory Meth., 19, 4029–4048.
Bockenhoff, A., Hartung, J. (1998). Some corrections of the significance level in meta-analysis. Biometrical Journal, 40, 937–947.
Box, G. E. P. (1954). Some theorems on quadratic forms applied in the study of analysis of variance problems, I. Effect of inequality of variance in the one-way classification. Annals of Mathematical Statistics, 25, 290–403.
Brown, M. B., Forsythe, A. B. (1974). The small sample behavior of some statistics which test the equality of several means. Technometrics, 16, 129–132.
Chalmers, T. C. (1991). Problems induced by meta-analyses. Statistics in Medicine, 10, 971–980.
Cochran, W. G. (1937). Problems arising in the analysis of a series of similar experiments. J. Roy. Stat. Soc. Supp., 4, 102–118.
Conover, W. J., Iman, R. L. (1981). Rank transformations as a bridge between parametric and nonparametric statistics. American Statistician, 35, 124–129.
De Beuckelaer, A. (1996). A closer examination on some parametric alternatives to the ANOVA F-test. Statistical Papers, 37, 291–305.
Fligner, M. A. (1981). Comment on rank transformations as a bridge between parametric and nonparametric statistics. American Statistician, 35, 131–133.
Hardy, R. J., Thompson, S. G. (1998). Detecting and describing heterogeneity in meta-analysis. Statistics in Medicine, 17, 841–856.
Hartung, J., Knapp, G. (2000). On tests of the overall treatment effect in the meta-analysis with normally distributed responses. Statistics in Medicine, to appear.
James, G. S. (1951). The comparison of several groups of observations when the ratios of population variances are unknown. Biometrika, 38, 324–329.
Keselman, H. J., Wilcox, R. R. (1999). The ’improved’ Brown and Forsythe test for mean equality: some things can’t be fixed. Commun. Statist. Simula., 28, 687–698.
Lehmann, E. L. (1975) Nonparametrics. Holden Day, San Francisco.
Lehmann, E. L. (1986) Testing Statistical Hypotheses. 2nd edn., Wiley, New York.
Li, Y., Shi, L., Roth, H. D. (1994). The bias of the commonly-used estimate of variance in meta-analysis. Commun. Statist.-Theory Meth., 23, 1063–1085.
Mehrotra, D. V. (1997). Improving the Brown-Forsythe solution to the generalized Behrens-Fisher problem. Commun. Statist.— Simula., 26, 1139–1145.
Noether, G. E. (1981). Comment on rank transformations as a bridge between parametric and nonparametric statistics. American Statistician, 35, 129–130.
Normand, S. T. (1999). Meta-analysis: Formulating, evaluating, combining, and reporting. Statistics in Medicine, 18, 321–359.
Patel, J. K., Kapadia, C. P., Owen, D. B. (1976) Handbook of Statistical Distributions. Marcel Dekker, New York.
Scheffe, H. (1959) The Analysis of Variance. Wiley, New York.
Welch, B. L. (1951). On the comparison of several mean values: An alternative approach. Biometrika, 38, 330–336.
Whitehead, A., Whitehead, J. (1991). A general parametric approach to the meta-analysis of randomized clinical trials. Statistics in Medicine, 10, 1665–1677.
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Hartung, J., Argaç, D. & Makambi, K.H. Small sample properties of tests on homogeneity in one—way Anova and Meta—analysis. Statistical Papers 43, 197–235 (2002). https://doi.org/10.1007/s00362-002-0097-8
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DOI: https://doi.org/10.1007/s00362-002-0097-8