Simultaneous Test for Means: An Unblind Way to the F-test in One-way Analysis of Variance

  • Elsayed A. H Elamir University of Bahrian
Keywords: ANOVA, adjusted p-value, beta distribution, Bonferonni’s approximation, F-test, linear models.


After rejecting the null hypothesis in the analysis of variance, the next step is to make the pairwise comparisons to find out differences in means. The purpose of this paper is threefold. The foremost aim is to suggest expression for calculating decision limit that enables us to collect the test and pairwise comparisons in one step. This expression is proposed as the ratio of between square for each treatment and within sum of squares for all treatments. The second aim is to obtain the sampling distribution of the proposed ratio under the null hypothesis. This sampling distribution is derived exactly as the beta distribution of the second type. The third aim is to use beta distribution of second type and adjusted p-values to create adjusted points and decision limit. Therefore, reject the null hypothesis of equal means if any adjusted point falls outside the decision limit. Simulation study is conducted to compute type I error. The results show that the proposed method controls the type I error near the nominal values using Benjamini-Hochberg’adjusted p-values. Two applications are given to show the benefits of the proposed method.

Author Biography

Elsayed A. H Elamir, University of Bahrian
Associate professor Department of  Management & Marketing College of Business


H. Abdi, The Bonferonni and idk corrections for multiple comparisons, In: Neil Salkind. Encyclopedia of Measurement and Statistics: Thousand Oaks (CA), Sage, 2007.

Y. Benjamini, and T. Hochberg, Controlling the false discovery Rate: a practical and powerful approach to multiple testing, Journal of the Royal Statistical Society Series B, vol. 85,pp. 289–300, 1995.

Y. Benjamini, Simultaneous and selective inference: current successes and future challenges, Biometrical Journal, 52, 708–721, 2010.

F. Bretz, W. Maurer, and G. Hommel, Test and power considerations for multiple endpoint analyses using sequentially rejective graphical procedures, Statistics in Medicine, 30, 1489–1501, 2011.

W. Cochran, and G. Cox, Experimental Designs, 2nd edition. Wiley, New York, 1957.

C. Coelho, and J. Mexia, On the distribution of the product and ratio of independent generalized gamma-ratio random variables, Sankhya: The Indian Journal of Statistics, 69, 221–255, 2007.

O. Dunn, Multiple comparisons using rank sums, Technometrics, 6, 241–252, 1964.

R. Fisher, The correlation between relatives on the supposition of mendelian Inheritance , Royal Society of Edinburgh, 52, 399-433,1918.

R. Fisher, Statistical methods for research workers, Edinburgh: Oliver and Boyd, 1925.

J. Diaz-Garcia, and R. Jaimez, Bimatrix variate generalized beta distributions: theory and methods, South Africa Statistical Journal, 44, 193-208, 2010.

E. Elamir, On uses of mean absolute deviation: decomposition, skewness and correlation coefficients, Metron: International Journal of Statistics, 70, 145-164, 2012.

E. Elamir, Simultaneous test for means: An unblind way to the F-test in One-way analysis of variance, arXiv stat.ME, 2020.

N. Elssied, O. Ibrahim, A. Osman, A Novel Feature Selection Based on One-Way ANOVA F-Test for E-Mail Spam Classification, Research Journal of Applied Sciences, Engineering and Technology, 7, 625–638, 2014.

T. Grecki and T. Smaga, A comparison of tests for the one-way ANOVA problem for functional data, Comput Stat, 30, 987-1010, 2015.

G. Gven, . Grer, H. amkar, B. enolu, A fiducial-based approach to the one-way ANOVA in the presence of nonnormality and heterogeneous error variances, Journal of Statistical Computation and Simulation, 89, 1715–1729, 2019.

Y. Hochberg, A sharper Bonferroni procedure for multiple tests of significance, Biometrika, 75, 800802, 1988.

S. Holm, A simple sequential rejective multiple test procedure, Scandinavian Journal of Statistics, 6, 6570, 1979.

G. Hommel, A stagewise rejective multiple test procedure on a modified Bonferroni test, Biometrika, 75, 383 386, 1988.

G. Hommel, A comparison of two modified Bonferonii procedures, Biometrika, 76, 624-625, 1989.

T. Kim, Understanding one-way ANOVA using conceptual figures, Korean J Anesthesiolgy, 70, 22-26, 2017.

M. Kutner, C. Nachtsheim, J. Neter, and L. William, Applied linear statistical models, 5th Ed., McGraw-Hill/Irwin, 2004.

D. Mongomery, Design and analysis of experiments, 8th ed. Jhon Wiley and Sons. Inc., 2013.

A. Qamar and M. Alassaf, Improving sentiment analysis of Arabic tweets by One-Way ANOVA, Journal of King Saud University -Computer and Information Sciences, In press, 2020.

A. Ross and V.Willson, One-Way ANOVA, In: Basic and Advanced Statistical Tests, E-Book Publisher: Brill—Sense, 21–24, 2017.

Z. Sidak, Rectangular confidence regions for the means of multivariate normal distributions, Journal of the American Statistical Association, 62, 626–633, 1967.

R. Simes, An improved Bonferroni procedure for multiple tests of significance, Biometrika, 73, 751–754, 1986.

A. Vishwakarma, M. Choudhary, M. Chauhan, Applicability of SPI and RDI for forthcoming drought events: a non-parametric trend and one way ANOVA approach, Journal of Water and Climate Change, 11, 18-28, 2020.

P. Westfall, Combining p-values, in Encyclopedia of Biostatistics, eds. P. Armitage and T. Colton, Chichester: Wiley, 987–991, 2005.

S. Yigit and M. Mendes, Which effect size measure is appropriate for one-way and two-way ANOVA models? A Monte Carlo simulation study, REVSTAT Statistical Journal, 16, 295–313, 2018.

How to Cite
Elamir, E. A. H. (2021). Simultaneous Test for Means: An Unblind Way to the F-test in One-way Analysis of Variance. Statistics, Optimization & Information Computing, 11(2), 504-518.
Research Articles