Abstract

This study compares the effect of sample sizes on the empirical power of some homogeneity of variance tests that have been proposed to assess the homogeneity of within-group variances, prior to ANOVA. The Tests of Homogeneity of Variance (THV) compared are: Bartlett (1937), Levene (1960), Cochran (1941) and Hartley Fmax (1950) tests. Homogeneity of variances occurs when variances are equal across groups. Homogeneity of variance testing is a statistical method designed to provide evidence that groups are comparable by demonstrating that the variations found between groups are small enough that they are considered practically insignificant.Few recommendations exist regarding the appropriate use of these tests under varying data conditions. Monte Carlo simulation methods were used to generate data to examine and compare the power rates of the tests under conditions of equal and unequal sample sizes when the underlying distribution is normal 1,000 times through the use of R software. It was found that Hartley Fmax test performs best (highest power) when the sample sizes are equal, while, Cochran test has the worst performance. Generally, when the sample sizes are both equal and unequal, Levene test has the highest power followed by Bartlett test, hence, Bartlett and Levene tests will be recommended for both equal and unequal sample sizes since they give higher power (above 0.8).Thus it is important for researchers to conduct an initial analysis of the data in order to determine the distribution of the population and is also advised to pay attention to the amount of sample size required to obtain a powerful test.

Keywords

References

[1] Bartlett, M. S. (1937b) Properties of sufficiency and statistical tests.Proc. Roy. Statist. Soc. Ser. A 160: 268-282.
[2] Bishop, T. A. (1976). Heteroscedastic ANOVA, MANOVA, and multiple comparisons. Unpublished doctoral dissertation, Ohio State University, Columbus.
[3] Box, G. E. P. (1953) Non-normality and tests on variances.Biometrika40: 318- 335.
[4] Brown, M. B., Forsythe. A. B. (1974a). The small sample behavior of some statistics which test the equality of several means. Technometrics, 16, 129- 132.
[5] Brown, Morton B.; Forsythe, Alan B. (1974). "Robust tests for the equality of variances". Journal of the American Statistical Association.69: 364–367.
[6] Cochran, W. G. (1941). The distribution of the largest of a set of estimated variances as a fraction of their total. Ann. Eug., 11, 47–52.
[7] Cochran, W. G. and Cox, G. M. (1957).Experimental design. New York: John Willey and Sons Inc.
[8] Diep, T. Nguyen, ThanhV. Pham, Patricia Rodriguez de Gil, Tyler Hicks, Yan Wang, Isaac Li, Aarti Bellara, Jeanine L. Romano, EunSook Kim, Harold Holmes, Yi-Hsin Chen, Jefferey D. Kromry (2014). ANOVA-HOV: A SAS MACRO for Testing Homogeneity of Variances in one-Factor ANOVA models, University of South Florida.
[9] Gartside, P.S. (1972), "A Study of Methods for Comparing Several Variances”. J. Amer. Statist. Assoc., 67, 342 -346
[10] Hartley, H. O. (1950) The maximum F-ratio as a short-cut test for heterogeneity of variance. Biometrika 37:308-312.
[11] Levene, H. (1960). Robust tests for equality of variances. In I. Olkin (Ed.), Contributions to Probability and Statistics, pp. 278–292. Palo Alto, California: Stanford University Press.
[12] LiX, Qiu W, Morrow J DeMeo DL, Weiss ST, Fu Y.(2005). “A Comparative Study of Tests for Homogeneity of Variances with Application to DNA Methylation Data.PLOSONE10(12):eo145295,doi:0.1371/journal.pone.0145295.
[13] Mu, Zhiqiang. (2006)“ Comparing the Statistical Tests for Homogeneity of Variances”. Electronic Theses and Dissertations.Paper 2212. http://dc.etsu.edu/etd/2212.
[14] Neter, J, Kutner, M.H., Nachtsheim, C.J. and Wasserman, W. (1996). Applied linear regression model.3rd ed. Illinois: Irwin.
[15] Ostertagova , E. and Ostertag, O.(2013). “Methodology and Application of One Way ANOVA”.American Journal of Mechanical Engineering, 1(7), 256- 261.
[16] Overall, J.E. and J.A. Woodward.(1974). A simple test for homogeneity of variance in complex factorial design.Psychometrika, 39: 311- 218. DOI:10.1007/BF02291705.
[17] Overall, J.E. and J.A. Woodward.(1976). A robust and powerful test for heterogeneity of variance.University of Texas Medical Branch Psychometric Laboratory.
[18] Parra-Frutos,I. (2009). The behaviour of the modified Levene’s test when data are not normally distributed. Compute Stat, Springer, 671-693.
[19] Rogan, J. C., and Keselman, H. J. (1977). Is the ANOVA F -test robust to variance heterogeneity when sample sizes are equal? An investigation via a coefficient of variation. American Educational Research Journal, 14, 493–498.
[20] Underwood, A. J. (1997). Experiments in ecology – Their logical design and interpretation using analysis of variance.Cambridge: Cambridge University Press.
[21] Wilcox, R. R., Charlin, V. L. and Thomson, K. L. (1986). New Monte Carlo results on the robustness of the ANOVA F, W and F ∗ statistics. J. Statist. Comput. Simul.,15, 33–43.
[22] Zar, J. H. (1999). Biostatistical analysis (Fourth Edition). Upper Saddle River, N.J.:Prentice Hall.